Vaguery + consider:representation   364

[1711.03172] Curve Reconstruction via the Global Statistics of Natural Curves
Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes that seek visually pleasing or perceptually plausible completions. In this work we focus on reconstructing the underlying physically likely shape by utilizing the global statistics of natural curves. More specifically, we develop a reconstruction model that seeks the mean physical curve for a given inducer configuration. This simple model is both straightforward to compute and it is receptive to diverse additional information, but it requires enough samples for all curve configurations, a practical requirement that limits its effective utilization. To address this practical issue we explore and exploit statistical geometrical properties of natural curves, and in particular, we show that in many cases the mean curve is scale invariant and often times it is extensible. This, in turn, allows to boost the number of examples and thus the robustness of the statistics and its applicability. The reconstruction results are not only more physically plausible but they also lead to important insights on the reconstruction problem, including an elegant explanation why certain inducer configurations are more likely to yield consistent perceptual completions than others.
inference  computer-vision  rather-interesting  algorithms  nudge-targets  consider:looking-to-see  consider:representation  performance-measure  to-write-about 
9 weeks ago by Vaguery
[1705.01887] Streaming for Aibohphobes: Longest Palindrome with Mismatches
A palindrome is a string that reads the same as its reverse, such as "aibohphobia" (fear of palindromes). Given an integer d>0, a d-near-palindrome is a string of Hamming distance at most d from its reverse. We study the natural problem of identifying a longest d-near-palindrome in data streams. The problem is relevant to the analysis of DNA databases, and to the task of repairing recursive structures in documents such as XML and JSON. We present an algorithm that returns a d-near-palindrome whose length is within a multiplicative (1+ϵ)-factor of the longest d-near-palindrome. Our algorithm also returns the set of mismatched indices of the d-near-palindrome, using (dlog7nϵlog(1+ϵ)) bits of space, and (dlog6nϵlog(1+ϵ)) update time per arriving symbol. We show that Ω(dlogn) space is necessary for estimating the length of longest d-near-palindromes with high probability. We further obtain an additive-error approximation algorithm and a comparable lower bound, as well as an exact two-pass algorithm that solves the longest d-near-palindrome problem using (d2n‾√log6n) bits of space.
strings  computational-complexity  algorithms  probability-theory  inference  online-learning  rather-interesting  to-write-about  nudge-targets  consider:looking-to-see  consider:representation 
9 weeks ago by Vaguery
Proofs are not only often beautiful but also necessary | Dan McQuillan
I hope you get the idea. When mathematicians seem obsessed with rigor, it is at least in part due to our history of making mistakes, seemingly every time we tried to jump to a conclusion. But perhaps there’s something more. So for that, we end with a quote from William Thurston:

what we are doing is finding ways for people to understand and think about mathematics… what they really want is usually not some collection of of ‘answers’-what they want is understanding.
mathematics  proof  philosophy  to-write-about  consider:representation 
10 weeks ago by Vaguery
[1710.01410] Learning Registered Point Processes from Idiosyncratic Observations
A parametric point process model is developed, with modeling based on the assumption that sequential observations often share latent phenomena, while also possessing idiosyncratic effects. An alternating optimization method is proposed to learn a "registered" point process that accounts for shared structure, as well as "warping" functions that characterize idiosyncratic aspects of each observed sequence. Under reasonable constraints, in each iteration we update the sample-specific warping functions by solving a set of constrained nonlinear programming problems in parallel, and update the model by maximum likelihood estimation. The justifiability, complexity and robustness of the proposed method are investigated in detail. Experiments on both synthetic and real-world data demonstrate that the method yields explainable point process models, achieving encouraging results compared to state-of-the-art methods.
modeling-is-not-mathematics  rather-interesting  representation  machine-learning  algorithms  nudge  nudge-targets  consider:representation  consider:performance-measures 
10 weeks ago by Vaguery
[1212.4771] Necklaces, Convolutions, and X+Y
We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.
optimization  operations-research  combinatorics  distance  nudge-targets  consider:looking-to-see  consider:representation 
10 weeks ago by Vaguery
[1601.01298] Visibility Graphs, Dismantlability, and the Cops and Robbers Game
We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.
computational-geometry  game-theory  rather-interesting  to-write-about  nudge-targets  consider:representation  consider:splinegons 
10 weeks ago by Vaguery
[1702.02017] Pushing the Bounds for Matrix-Matrix Multiplication
A tight lower bound for required I/O when computing a matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of \textit{phases} needed to perform C:=AB, where each phase is a series of operations involving S reads and writes to and from fast memory, and S is the size of fast memory. A lower bound on the number of phases was then determined by obtaining an upper bound on the number of scalar multiplications performed per phase. This paper follows the same high level approach, but improves the lower bound by considering C:=AB+C instead of C:=AB, and obtains the maximum number of scalar fused multiply-adds (FMAs) per phase instead of scalar additions. Key to obtaining the new result is the decoupling of the per-phase I/O from the size of fast memory. The new lower bound is 2mnk/S‾√−2S. The constant for the leading term is an improvement of a factor 42‾√. A theoretical algorithm that attains the lower bound is given, and how the state-of-the-art Goto's algorithm also in some sense meets the lower bound is discussed.
computational-complexity  algorithms  matrices  rather-interesting  nudge-targets  consider:looking-to-see  consider:representation  consider:performance-measures 
10 weeks ago by Vaguery
[1604.08797] Ortho-polygon Visibility Representations of Embedded Graphs
An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O(n) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O(n) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.
graph-layout  computational-geometry  optimization  rather-interesting  to-write-about  nudge-targets  consider:representation  consider:feature-discovery  algorithms  computational-complexity 
10 weeks ago by Vaguery
[1710.01992] Fast and Accurate Image Super-Resolution with Deep Laplacian Pyramid Networks
Convolutional neural networks have recently demonstrated high-quality reconstruction for single image super-resolution. However, existing methods often require a large number of network parameters and entail heavy computational loads at runtime for generating high-accuracy super-resolution results. In this paper, we propose the deep Laplacian Pyramid Super-Resolution Network for fast and accurate image super-resolution. The proposed network progressively reconstructs the sub-band residuals of high-resolution images at multiple pyramid levels. In contrast to existing methods that involve the bicubic interpolation for pre-processing (which results in large feature maps), the proposed method directly extracts features from the low-resolution input space and thereby entails low computational loads. We train the proposed network with deep supervision using the robust Charbonnier loss functions and achieve high-quality image reconstruction. Furthermore, we utilize the recursive layers to share parameters across as well as within pyramid levels, and thus drastically reduce the number of parameters. Extensive quantitative and qualitative evaluations on benchmark datasets show that the proposed algorithm performs favorably against the state-of-the-art methods in terms of run-time and image quality.
superresolution  neural-networks  representation  algorithms  image-processing  generative-models  to-write-about  nudge-targets  consider:representation  consider:performance-measures 
11 weeks ago by Vaguery
[1707.05425] Fast and Accurate Image Super Resolution by Deep CNN with Skip Connection and Network in Network
We propose a highly efficient and faster Single Image Super-Resolution (SISR) model with Deep Convolutional neural networks (Deep CNN). Deep CNN have recently shown that they have a significant reconstruction performance on single-image super-resolution. Current trend is using deeper CNN layers to improve performance. However, deep models demand larger computation resources and is not suitable for network edge devices like mobile, tablet and IoT devices. Our model achieves state of the art reconstruction performance with at least 10 times lower calculation cost by Deep CNN with Residual Net, Skip Connection and Network in Network (DCSCN). A combination of Deep CNNs and Skip connection layers is used as a feature extractor for image features on both local and global area. Parallelized 1x1 CNNs, like the one called Network in Network, is also used for image reconstruction. That structure reduces the dimensions of the previous layer's output for faster computation with less information loss, and make it possible to process original images directly. Also we optimize the number of layers and filters of each CNN to significantly reduce the calculation cost. Thus, the proposed algorithm not only achieves the state of the art performance but also achieves faster and efficient computation. Code is available at this https URL
superresolution  image-processing  deep-learning  neural-networks  generative-models  nudge-targets  consider:representation 
11 weeks ago by Vaguery
[1710.00228] Physics-based Motion Planning: Evaluation Criteria and Benchmarking
Motion planning has evolved from coping with simply geometric problems to physics-based ones that incorporate the kinodynamic and the physical constraints imposed by the robot and the physical world. Therefore, the criteria for evaluating physics-based motion planners goes beyond the computational complexity (e.g. in terms of planning time) usually used as a measure for evaluating geometrical planners, in order to consider also the quality of the solution in terms of dynamical parameters. This study proposes an evaluation criteria and analyzes the performance of several kinodynamic planners, which are at the core of physics-based motion planning, using different scenarios with fixed and manipulatable objects. RRT, EST, KPIECE and SyCLoP are used for the benchmarking. The results show that KPIECE computes the time-optimal solution with heighest success rate, whereas, SyCLoP compute the most power-optimal solution among the planners used.
simulation  planning  algorithms  rather-interesting  feature-construction  approximation  nudge-targets  consider:representation 
12 weeks ago by Vaguery
[1706.10206] Sums of Palindromes: an Approach via Automata
Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome. However, the cases b = 2, 3, 4 were left unresolved.
We prove, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem.
We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.
number-theory  rather-interesting  lovely  algorithms  proof  nudge-targets  consider:rediscovery  consider:representation  consider:looking-to-see 
12 weeks ago by Vaguery
[1305.6596] On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot colorability to this broader class of knot-like objects. We use these definitions to define the determinant of a pseudoknot (i.e. the pseudodeterminant) that agrees with the classical determinant for classical knots. Moreover, we extend Conway notation to pseudoknots to facilitate the investigation of families of pseudoknots and links. The general formulae for pseudodeterminants of pseudoknot families may then be used as a criterion for p-colorability of pseudoknots.
knot-theory  representation  topology  to-understand  nudge-targets  consider:representation  consider:looking-to-see  consider:algorithms 
october 2017 by Vaguery
[1603.00051] Algebraic Method in Tilings
In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.
polyominoes  algebra  representation  rather-interesting  nudge-targets  consider:representation 
october 2017 by Vaguery
[math/0204106] The Second Hull of a Knotted Curve
The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This provides a new proof of the Fary/Milnor theorem that every knotted curve has total curvature at least 4pi.
knot-theory  geometry  topology  rather-interesting  feature-construction  proof  nudge-targets  consider:representation  consider:looking-to-see 
october 2017 by Vaguery
[1505.00667] An Unusual Continued Fraction
We consider the real number σ with continued fraction expansion [a0,a1,a2,…]=[1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,…], where ai is the largest power of 2 dividing i+1. We compute the irrationality measure of σ2 and demonstrate that σ2 (and σ) are both transcendental numbers. We also show that certain partial quotients of σ2 grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.
continued-fractions  strings  number-theory  rather-interesting  to-understand  nudge-targets  consider:representation 
october 2017 by Vaguery
[1606.07163] Interpretable Machine Learning Models for the Digital Clock Drawing Test
The Clock Drawing Test (CDT) is a rapid, inexpensive, and popular neuropsychological screening tool for cognitive conditions. The Digital Clock Drawing Test (dCDT) uses novel software to analyze data from a digitizing ballpoint pen that reports its position with considerable spatial and temporal precision, making possible the analysis of both the drawing process and final product. We developed methodology to analyze pen stroke data from these drawings, and computed a large collection of features which were then analyzed with a variety of machine learning techniques. The resulting scoring systems were designed to be more accurate than the systems currently used by clinicians, but just as interpretable and easy to use. The systems also allow us to quantify the tradeoff between accuracy and interpretability. We created automated versions of the CDT scoring systems currently used by clinicians, allowing us to benchmark our models, which indicated that our machine learning models substantially outperformed the existing scoring systems.
machine-learning  benchmarking  rather-interesting  nudge-targets  consider:representation  consider:looking-to-see 
october 2017 by Vaguery
[1602.01241] Using separable non-negative matrix factorization techniques for the analysis of time-resolved Raman spectra
The key challenge of time-resolved Raman spectroscopy is the identification of the constituent species and the analysis of the kinetics of the underlying reaction network. In this work we present an integral approach that allows for determining both the component spectra and the rate constants simultaneously from a series of vibrational spectra. It is based on an algorithm for non-negative matrix factorization which is applied to the experimental data set following a few pre-processing steps. As a prerequisite for physically unambiguous solutions, each component spectrum must include one vibrational band that does not significantly interfere with vibrational bands of other species. The approach is applied to synthetic "experimental" spectra derived from model systems comprising a set of species with component spectra differing with respect to their degree of spectral interferences and signal-to-noise ratios. In each case, the species involved are connected via monomolecular reaction pathways. The potential and limitations of the approach for recovering the respective rate constants and component spectra are discussed.
spectroscopy  data-analysis  inference  numerical-methods  modeling  statistics  rather-interesting  nudge-targets  consider:representation 
october 2017 by Vaguery
[1708.08377] Two-Dimensional Indirect Binary Search for the Positive One-in-Three Satisfiability Problem
In this paper, we propose an algorithm for the positive one-in-three satisfiability problem (Pos1in3SAT). The proposed algorithm can efficiently decide the existence of a satisfying assignment in all assignments for a given formula by using a 2-dimensional binary search method without constructing an exponential number of assignments.
satisfiability  algorithms  computational-complexity  machine-learning  nudge-targets  consider:looking-to-see  consider:representation  matrices 
october 2017 by Vaguery
[1709.09022] On Integer Images of Max-plus Linear Mappings
Let us extend the pair of operations (max,+) over real numbers to matrices in the same way as in conventional linear algebra. We study integer images of max-plus linear mappings. The question whether Ax (in the max-plus algebra) is an integer vector for at least one x has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems. We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case.
representation  linear-algebra  out-of-the-box  mathematics  algebra  nudge-targets  consider:looking-to-see  consider:representation 
october 2017 by Vaguery
[1708.02891v1] Area difference bounds for dissections of a square into an odd number of triangles
Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur.
We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.
computational-geometry  limits  geometry  constraint-satisfaction  nudge-targets  consider:looking-to-see  consider:representation  to-write-about 
september 2017 by Vaguery
[1702.03615v2] Online Prediction with Selfish Experts
We consider the problem of binary prediction with expert advice in settings where experts have agency and seek to maximize their credibility. This paper makes three main contributions. First, it defines a model to reason formally about settings with selfish experts, and demonstrates that "incentive compatible" (IC) algorithms are closely related to the design of proper scoring rules. Designing a good IC algorithm is easy if the designer's loss function is quadratic, but for other loss functions, novel techniques are required. Second, we design IC algorithms with good performance guarantees for the absolute loss function. Third, we give a formal separation between the power of online prediction with selfish experts and online prediction with honest experts by proving lower bounds for both IC and non-IC algorithms. In particular, with selfish experts and the absolute loss function, there is no (randomized) algorithm for online prediction-IC or otherwise-with asymptotically vanishing regret.
mechanism-design  prediction  rather-interesting  collective-behavior  markets  game-theory  nudge-targets  consider:representation 
september 2017 by Vaguery
[1709.07241] Satisfiability Modulo Theory based Methodology for Floorplanning in VLSI Circuits
This paper proposes a Satisfiability Modulo Theory based formulation for floorplanning in VLSI circuits. The proposed approach allows a number of fixed blocks to be placed within a layout region without overlapping and at the same time minimizing the area of the layout region. The proposed approach is extended to allow a number of fixed blocks with ability to rotate and flexible blocks (with variable width and height) to be placed within a layout without overlap. Our target in all cases is reduction in area occupied on a chip which is of vital importance in obtaining a good circuit design. Satisfiability Modulo Theory combines the problem of Boolean satisfiability with domains such as convex optimization. Satisfiability Modulo Theory provides a richer modeling language than is possible with pure Boolean SAT formulas. We have conducted our experiments on MCNC and GSRC benchmark circuits to calculate the total area occupied, amount of deadspace and the total CPU time consumed while placing the blocks without overlapping. The results obtained shows clearly that the amount of dead space or wasted space is reduced if rotation is applied to the blocks.
operations-research  optimization  representation  engineering-design  rather-interesting  nudge-targets  consider:representation  mathematical-programming  to-write-about 
september 2017 by Vaguery
[1709.06309] Aspect-Based Relational Sentiment Analysis Using a Stacked Neural Network Architecture
Sentiment analysis can be regarded as a relation extraction problem in which the sentiment of some opinion holder towards a certain aspect of a product, theme or event needs to be extracted. We present a novel neural architecture for sentiment analysis as a relation extraction problem that addresses this problem by dividing it into three subtasks: i) identification of aspect and opinion terms, ii) labeling of opinion terms with a sentiment, and iii) extraction of relations between opinion terms and aspect terms. For each subtask, we propose a neural network based component and combine all of them into a complete system for relational sentiment analysis. The component for aspect and opinion term extraction is a hybrid architecture consisting of a recurrent neural network stacked on top of a convolutional neural network. This approach outperforms a standard convolutional deep neural architecture as well as a recurrent network architecture and performs competitively compared to other methods on two datasets of annotated customer reviews. To extract sentiments for individual opinion terms, we propose a recurrent architecture in combination with word distance features and achieve promising results, outperforming a majority baseline by 18% accuracy and providing the first results for the USAGE dataset. Our relation extraction component outperforms the current state-of-the-art in aspect-opinion relation extraction by 15% F-Measure.
sentiment-analysis  natural-language-processing  deep-learning  neural-networks  machine-learning  architecture  nudge-targets  consider:representation 
september 2017 by Vaguery
[1709.06389] A General Framework for the Recognition of Online Handwritten Graphics
We propose a new framework for the recognition of online handwritten graphics. Three main features of the framework are its ability to treat symbol and structural level information in an integrated way, its flexibility with respect to different families of graphics, and means to control the tradeoff between recognition effectiveness and computational cost. We model a graphic as a labeled graph generated from a graph grammar. Non-terminal vertices represent subcomponents, terminal vertices represent symbols, and edges represent relations between subcomponents or symbols. We then model the recognition problem as a graph parsing problem: given an input stroke set, we search for a parse tree that represents the best interpretation of the input. Our graph parsing algorithm generates multiple interpretations (consistent with the grammar) and then we extract an optimal interpretation according to a cost function that takes into consideration the likelihood scores of symbols and structures. The parsing algorithm consists in recursively partitioning the stroke set according to structures defined in the grammar and it does not impose constraints present in some previous works (e.g. stroke ordering). By avoiding such constraints and thanks to the powerful representativeness of graphs, our approach can be adapted to the recognition of different graphic notations. We show applications to the recognition of mathematical expressions and flowcharts. Experimentation shows that our method obtains state-of-the-art accuracy in both applications.
OCR  handwriting-recognition  machine-learning  image-processing  nudge-targets  consider:looking-to-see  consider:representation 
september 2017 by Vaguery
[1709.05397] Zero-Shot Learning to Manage a Large Number of Place-Specific Compressive Change Classifiers
With recent progress in large-scale map maintenance and long-term map learning, the task of change detection on a large-scale map from a visual image captured by a mobile robot has become a problem of increasing criticality. Previous approaches for change detection are typically based on image differencing and require the memorization of a prohibitively large number of mapped images in the above context. In contrast, this study follows the recent, efficient paradigm of change-classifier-learning and specifically employs a collection of place-specific change classifiers. Our change-classifier-learning algorithm is based on zero-shot learning (ZSL) and represents a place-specific change classifier by its training examples mined from an external knowledge base (EKB). The proposed algorithm exhibits several advantages. First, we are required to memorize only training examples (rather than the classifier itself), which can be further compressed in the form of bag-of-words (BoW). Secondly, we can incorporate the most recent map into the classifiers by straightforwardly adding or deleting a few training examples that correspond to these classifiers. Thirdly, we can share the BoW vocabulary with other related task scenarios (e.g., BoW-based self-localization), wherein the vocabulary is generally designed as a rich, continuously growing, and domain-adaptive knowledge base. In our contribution, the proposed algorithm is applied and evaluated on a practical long-term cross-season change detection system that consists of a large number of place-specific object-level change classifiers.
machine-learning  data-fusion  image-processing  deep-learning  zero-shot-learning  algorithms  nudge-targets  consider:representation 
september 2017 by Vaguery
[1705.01991] Sharp Models on Dull Hardware: Fast and Accurate Neural Machine Translation Decoding on the CPU
Attentional sequence-to-sequence models have become the new standard for machine translation, but one challenge of such models is a significant increase in training and decoding cost compared to phrase-based systems. Here, we focus on efficient decoding, with a goal of achieving accuracy close the state-of-the-art in neural machine translation (NMT), while achieving CPU decoding speed/throughput close to that of a phrasal decoder.
We approach this problem from two angles: First, we describe several techniques for speeding up an NMT beam search decoder, which obtain a 4.4x speedup over a very efficient baseline decoder without changing the decoder output. Second, we propose a simple but powerful network architecture which uses an RNN (GRU/LSTM) layer at bottom, followed by a series of stacked fully-connected layers applied at every timestep. This architecture achieves similar accuracy to a deep recurrent model, at a small fraction of the training and decoding cost. By combining these techniques, our best system achieves a very competitive accuracy of 38.3 BLEU on WMT English-French NewsTest2014, while decoding at 100 words/sec on single-threaded CPU. We believe this is the best published accuracy/speed trade-off of an NMT system.
machine-learning  representation  algorithms  deep-learning  neural-networks  approximation  nudge-targets  consider:performance-measures  consider:representation  natural-language-processing 
september 2017 by Vaguery
[1608.08324] Topological Drawings of Complete Bipartite Graphs
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing.
We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodings of the drawings by enumerating the distinct drawings of subgraphs isomorphic to K2,2 and K3,2, and investigate the constraints they must satisfy. We prove that a drawing of Kk,n exists if and only if some simple local conditions are satisfied by the encodings. This directly yields a polynomial-time algorithm for deciding the existence of such a drawing given the encoding. We show the encoding is equivalent to specifying which pairs of edges cross, yielding a similar polynomial-time algorithm for the realizability of abstract topological graphs.
We also completely characterize and enumerate such drawings of Kk,n in which the order of the edges around each vertex is the same for vertices on the same side of the bipartition. Finally, we investigate drawings of Kk,n using straight lines and pseudolines, and consider the complexity of the corresponding realizability problems.
computational-geometry  graph-layout  rather-interesting  purdy-pitchers  nudge-targets  consider:representation  consider:performance-measures 
september 2017 by Vaguery
[1709.05769] Where to Focus: Deep Attention-based Spatially Recurrent Bilinear Networks for Fine-Grained Visual Recognition
Fine-grained visual recognition typically depends on modeling subtle difference from object parts. However, these parts often exhibit dramatic visual variations such as occlusions, viewpoints, and spatial transformations, making it hard to detect. In this paper, we present a novel attention-based model to automatically, selectively and accurately focus on critical object regions with higher importance against appearance variations. Given an image, two different Convolutional Neural Networks (CNNs) are constructed, where the outputs of two CNNs are correlated through bilinear pooling to simultaneously focus on discriminative regions and extract relevant features. To capture spatial distributions among the local regions with visual attention, soft attention based spatial Long-Short Term Memory units (LSTMs) are incorporated to realize spatially recurrent yet visually selective over local input patterns. All the above intuitions equip our network with the following novel model: two-stream CNN layers, bilinear pooling layer, spatial recurrent layer with location attention are jointly trained via an end-to-end fashion to serve as the part detector and feature extractor, whereby relevant features are localized and extracted attentively. We show the significance of our network against two well-known visual recognition tasks: fine-grained image classification and person re-identification.
image-processing  neural-networks  attention  feature-extraction  deep-learning  architecture  constraint-satisfaction  nudge-targets  consider:feature-discovery  consider:representation 
september 2017 by Vaguery
[1709.07153] Large Vocabulary Automatic Chord Estimation Using Deep Neural Nets: Design Framework, System Variations and Limitations
In this paper, we propose a new system design framework for large vocabulary automatic chord estimation. Our approach is based on an integration of traditional sequence segmentation processes and deep learning chord classification techniques. We systematically explore the design space of the proposed framework for a range of parameters, namely deep neural nets, network configurations, input feature representations, segment tiling schemes, and training data sizes. Experimental results show that among the three proposed deep neural nets and a baseline model, the recurrent neural network based system has the best average chord quality accuracy that significantly outperforms the other considered models. Furthermore, our bias-variance analysis has identified a glass ceiling as a potential hindrance to future improvements of large vocabulary automatic chord estimation systems.
sound-recognition  feature-extraction  algorithms  machine-learning  nudge-targets  consider:representation  consider:looking-to-see 
september 2017 by Vaguery
[1608.02025] Boundary-based MWE segmentation with text partitioning
This work presents a fine-grained, text-chunking algorithm designed for the task of multiword expressions (MWEs) segmentation. As a lexical class, MWEs include a wide variety of idioms, whose automatic identification are a necessity for the handling of colloquial language. This algorithm's core novelty is its use of non-word tokens, i.e., boundaries, in a bottom-up strategy. Leveraging boundaries refines token-level information, forging high-level performance from relatively basic data. The generality of this model's feature space allows for its application across languages and domains. Experiments spanning 19 different languages exhibit a broadly-applicable, state-of-the-art model. Evaluation against recent shared-task data places text partitioning as the overall, best performing MWE segmentation algorithm, covering all MWE classes and multiple English domains (including user-generated text). This performance, coupled with a non-combinatorial, fast-running design, produces an ideal combination for implementations at scale, which are facilitated through the release of open-source software.
natural-language-processing  algorithms  parsing  nudge-targets  consider:looking-to-see  consider:representation  consider:performance-measures 
september 2017 by Vaguery
[1606.06570] Metastability-Containing Circuits
In digital circuits, metastability can cause deteriorated signals that neither are logical 0 or logical 1, breaking the abstraction of Boolean logic. Unfortunately, any way of reading a signal from an unsynchronized clock domain or performing an analog-to-digital conversion incurs the risk of a metastable upset; no digital circuit can deterministically avoid, resolve, or detect metastability (Marino, 1981). Synchronizers, the only traditional countermeasure, exponentially decrease the odds of maintained metastability over time. Trading synchronization delay for an increased probability to resolve metastability to logical 0 or 1, they do not guarantee success.
We propose a fundamentally different approach: It is possible to contain metastability by fine-grained logical masking so that it cannot infect the entire circuit. This technique guarantees a limited degree of metastability in---and uncertainty about---the output.
At the heart of our approach lies a time- and value-discrete model for metastability in synchronous clocked digital circuits. Metastability is propagated in a worst-case fashion, allowing to derive deterministic guarantees, without and unlike synchronizers. The proposed model permits positive results and passes the test of reproducing Marino's impossibility results. We fully classify which functions can be computed by circuits with standard registers. Regarding masking registers, we show that they become computationally strictly more powerful with each clock cycle, resulting in a non-trivial hierarchy of computable functions.
robustness  engineering-design  performance-measure  rather-interesting  nudge-targets  consider:looking-to-see  simulation  consider:representation 
september 2017 by Vaguery
[1705.01842] A Deep Learning Perspective on the Origin of Facial Expressions
Facial expressions play a significant role in human communication and behavior. Psychologists have long studied the relationship between facial expressions and emotions. Paul Ekman et al., devised the Facial Action Coding System (FACS) to taxonomize human facial expressions and model their behavior. The ability to recognize facial expressions automatically, enables novel applications in fields like human-computer interaction, social gaming, and psychological research. There has been a tremendously active research in this field, with several recent papers utilizing convolutional neural networks (CNN) for feature extraction and inference. In this paper, we employ CNN understanding methods to study the relation between the features these computational networks are using, the FACS and Action Units (AU). We verify our findings on the Extended Cohn-Kanade (CK+), NovaEmotions and FER2013 datasets. We apply these models to various tasks and tests using transfer learning, including cross-dataset validation and cross-task performance. Finally, we exploit the nature of the FER based CNN models for the detection of micro-expressions and achieve state-of-the-art accuracy using a simple long-short-term-memory (LSTM) recurrent neural network (RNN).
computer-vision  deep-learning  face-recognition  rather-interesting  feature-extraction  feature-construction  nudge-targets  consider:looking-to-see  consider:representation 
september 2017 by Vaguery
[1704.08676] A quantitative assessment of the effect of different algorithmic schemes to the task of learning the structure of Bayesian Networks
One of the most challenging tasks when adopting Bayesian Networks (BNs) is the one of learning their structure from data. This task is complicated by the huge search space of possible solutions and turned out to be a well-known NP-hard problem and, hence, approximations are required. However, to the best of our knowledge, a quantitative analysis of the performance and characteristics of the different heuristics to solve this problem has never been done before.
For this reason, in this work, we provide a detailed study of the different state-of-the-arts methods for structural learning on simulated data considering both BNs with discrete and continuous variables, and with different rates of noise in the data. In particular, we investigate the characteristics of different widespread scores proposed for the inference and the statistical pitfalls within them.
learning-from-data  machine-learning  statistics  algorithms  rather-interesting  inference  nudge-targets  consider:looking-to-see  consider:representation 
august 2017 by Vaguery
[1410.3564v1] Randomized Triangle Algorithms for Convex Hull Membership
We present randomized versions of the {\it triangle algorithm} introduced in \cite{kal14}. The triangle algorithm tests membership of a distinguished point p∈ℝm in the convex hull of a given set S of n points in ℝm. Given any {\it iterate} p′∈conv(S), it searches for a {\it pivot}, a point v∈S so that d(p′,v)≥d(p,v). It replaces p′ with the point on the line segment p′v closest to p and repeats this process. If a pivot does not exist, p′ certifies that p∉conv(S). Here we propose two random variations of the triangle algorithm that allow relaxed steps so as to take more effective steps possible in subsequent iterations. One is inspired by the {\it chaos game} known to result in the Sierpinski triangle. The incentive is that randomized iterates together with a property of Sierpinski triangle would result in effective pivots. Bounds on their expected complexity coincides with those of the deterministic version derived in \cite{kal14}.
computational-geometry  algorithms  convex-hulls  rather-interesting  nudge-targets  consider:representation  consider:looking-to-see 
may 2017 by Vaguery
[1704.00899] Dynamic Rank Maximal Matchings
We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A U P,E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on.
We consider this problem in a dynamic setting, where vertices and edges can be added and deleted at any point. Let n and m be the number of vertices and edges in an instance G, and r be the maximum rank used by any rank-maximal matching in G. We give a simple O(r(m+n))-time algorithm to update an existing rank-maximal matching under each of these changes. When r = o(n), this is faster than recomputing a rank-maximal matching completely using a known algorithm like that of Irving et al., which takes time O(min((r + n, r*sqrt(n))m).
operations-research  optimization  rostering  scheduling  dynamic-optimization  to-write-about  consider:representation  consider:multiobjective-optimization 
may 2017 by Vaguery
[1703.08052] Dynamic Bernoulli Embeddings for Language Evolution
Word embeddings are a powerful approach for unsupervised analysis of language. Recently, Rudolph et al. (2016) developed exponential family embeddings, which cast word embeddings in a probabilistic framework. Here, we develop dynamic embeddings, building on exponential family embeddings to capture how the meanings of words change over time. We use dynamic embeddings to analyze three large collections of historical texts: the U.S. Senate speeches from 1858 to 2009, the history of computer science ACM abstracts from 1951 to 2014, and machine learning papers on the Arxiv from 2007 to 2015. We find dynamic embeddings provide better fits than classical embeddings and capture interesting patterns about how language changes.
natural-language-processing  representation  rather-interesting  to-write-about  nudge-targets  consider:representation 
may 2017 by Vaguery
[1704.00708] No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on optimization landscapes for matrix sensing and symmetric matrix completion. The framework naturally leads to new results for asymmetric matrix completion and robust PCA.
compressed-sensing  matrices  optimization  approximation  rather-interesting  machine-learning  nudge-targets  consider:representation 
may 2017 by Vaguery
[1704.08156] Locally optimal 2-periodic sphere packings
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi presented an algorithm to enumerate all local optima in a finite computation, which has been implemented in up to d = 8 dimensions. We generalize Voronoi's algorithm to m > 1 and use this new algorithm to enumerate all locally optimal 2-periodic sphere packings in d = 3, 4, and 5. In particular, we show that no 2-periodic packing surpasses the density of the optimal lattice in these dimensions. A partial enumeration is performed in d = 6.
packing  generalization  geometry  optimization  nudge-targets  consider:rediscovery  consider:representation 
may 2017 by Vaguery
[1704.07378] Geometry-Based Optimization of One-Way Quantum Computation Measurement Patterns
In one-way quantum computation (1WQC) model, an initial highly entangled state called a graph state is used to perform universal quantum computations by a sequence of adaptive single-qubit measurements and post-measurement Pauli-X and Pauli-Z corrections. The needed computations are organized as measurement patterns, or simply patterns, in the 1WQC model. The entanglement operations in a pattern can be shown by a graph which together with the set of its input and output qubits is called the geometry of the pattern. Since a one-way quantum computation pattern is based on quantum measurements, which are fundamentally nondeterministic evolutions, there must be conditions over geometries to guarantee determinism. Causal flow is a sufficient and generalized flow (gflow) is a necessary and sufficient condition over geometries to identify a dependency structure for the measurement sequences in order to achieve determinism. Previously, three optimization methods have been proposed to simplify 1WQC patterns which are called standardization, signal shifting and Pauli simplification. These optimizations can be performed using measurement calculus formalism by rewriting rules. However, maintaining and searching these rules in the library can be complicated with respect to implementation. Moreover, serial execution of these rules is time consuming due to executing many ineffective commutation rules. To overcome this problem, in this paper, a new scheme is proposed to perform optimization techniques on patterns with flow or gflow only based on their geometries instead of using rewriting rules. Furthermore, the proposed scheme obtains the maximally delayed gflow order for geometries with flow. It is shown that the time complexity of the proposed approach is improved over the previous ones.
quantums  quantum-computing  algorithms  nudge-targets  consider:rediscovery  consider:representation  see-also:Lee's-book 
may 2017 by Vaguery
Balanced ternary - Wikipedia
Balanced ternary is a non-standard positional numeral system (a balanced form), useful for comparison logic. While it is a ternary (base 3) number system, in the standard (unbalanced) ternary system, digits have values 0, 1 and 2. The digits in the balanced ternary system have values −1, 0, and 1.

Different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T (which resembles a ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using Greek letter theta (Θ), which resembles a minus sign in a circle, to represent −1.
exotic-math  number-theory  nudge-targets  consider:representation  consider:rediscovery 
april 2017 by Vaguery
[1703.04381] On the Transformation Capability of Feasible Mechanisms for Programmable Matter
In this work, we study theoretical models of \emph{programmable matter} systems. The systems under consideration consist of spherical modules, kept together by magnetic forces and able to perform two minimal mechanical operations (or movements): \emph{rotate} around a neighbor and \emph{slide} over a line. In terms of modeling, there are n nodes arranged in a 2-dimensional grid and forming some initial \emph{shape}. The goal is for the initial shape A to \emph{transform} to some target shape B by a sequence of movements. Most of the paper focuses on \emph{transformability} questions, meaning whether it is in principle feasible to transform a given shape to another. We first consider the case in which only rotation is available to the nodes. Our main result is that deciding whether two given shapes A and B can be transformed to each other, is in P. We then insist on rotation only and impose the restriction that the nodes must maintain global connectivity throughout the transformation. We prove that the corresponding transformability question is in PSPACE and study the problem of determining the minimum \emph{seeds} that can make feasible, otherwise infeasible transformations. Next we allow both rotations and slidings and prove universality: any two connected shapes A,B of the same order, can be transformed to each other without breaking connectivity. The worst-case number of movements of the generic strategy is Ω(n2). We improve this to O(n) parallel time, by a pipelining strategy, and prove optimality of both by matching lower bounds. In the last part of the paper, we turn our attention to distributed transformations. The nodes are now distributed processes able to perform communicate-compute-move rounds. We provide distributed algorithms for a general type of transformations.
programmable-matter  engineering-design  self-organization  rather-interesting  to-write-about  nudge-targets  consider:planning  consider:representation 
april 2017 by Vaguery
[1602.05150v3] Approximation and Hardness for Token Swapping
Given a graph G=(V,E) with V={1,…,n}, we place on every vertex a token T1,…,Tn. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token Ti is on vertex i. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out any 2o(n) algorithm under the ETH. This is matched with a simple 2O(nlogn) algorithm based on a breadth-first search in an auxiliary graph. We show one general 4-approximation and show APX-hardness. Thus, there is a small constant δ>1 such that every polynomial time approximation algorithm has approximation factor at least δ. Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.
computational-complexity  graph-theory  computational-geometry  mathematical-recreations  rather-interesting  feature-construction  to-write-about  nudge-targets  consider:looking-to-see  consider:representation  consider:robustness 
april 2017 by Vaguery
[cs/0106019v2] Playing Games with Algorithms: Algorithmic Combinatorial Game Theory
Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
games  mathematical-recreations  rewriting-systems  rather-interesting  to-write-about  puzzles  nudge-targets  consider:looking-to-see  consider:representation 
april 2017 by Vaguery
[cond-mat/0203436] Entropy estimation of symbol sequences
We discuss algorithms for estimating the Shannon entropy h of finite symbol sequences with long range correlations. In particular, we consider algorithms which estimate h from the code lengths produced by some compression algorithm. Our interest is in describing their convergence with sequence length, assuming no limits for the space and time complexities of the compression algorithms. A scaling law is proposed for extrapolation from finite sample lengths. This is applied to sequences of dynamical systems in non-trivial chaotic regimes, a 1-D cellular automaton, and to written English texts.
statistics  information-theory  probability-theory  models  dynamical-systems  nudge-targets  consider:looking-to-see  consider:representation 
april 2017 by Vaguery
[1308.0419] Inverse Procedural Modeling of Facade Layouts
In this paper, we address the following research problem: How can we generate a meaningful split grammar that explains a given facade layout? To evaluate if a grammar is meaningful, we propose a cost function based on the description length and minimize this cost using an approximate dynamic programming framework. Our evaluation indicates that our framework extracts meaningful split grammars that are competitive with those of expert users, while some users and all competing automatic solutions are less successful.
grammar  L-systems  generative-models  image-processing  learning-from-data  machine-learning  inverse-problems  nudge-targets  consider:representation  consider:feature-discovery 
april 2017 by Vaguery
[1701.06790] Parallel Graph Rewriting with Overlapping Rules
We tackle the problem of simultaneous transformations of networks represented as graphs. Roughly speaking, one may distinguish two kinds of simultaneous or parallel rewrite relations over complex structures such as graphs: (i) those which transform disjoint subgraphs in parallel and hence can be simulated by successive mere sequential and local transformations and (ii) those which transform overlapping subgraphs simultaneously. In the latter situations, parallel transformations cannot be simulated in general by means of successive local rewrite steps. We investigate this last problem in the framework of overlapping graph transformation systems. As parallel transformation of a graph does not produce a graph in general, we propose first some sufficient conditions that ensure the closure of graphs by parallel rewrite relations. Then we mainly introduce and discuss two parallel rewrite relations over graphs. One relation is functional and thus deterministic, the other one is not functional for which we propose sufficient conditions which ensure its confluence.
rewriting-systems  graph-theory  generative-models  representation  dynamical-systems  to-write-about  nudge-targets  consider:looking-to-see  consider:representation  algorithms 
april 2017 by Vaguery
[1602.01401] Applications of AI for Magic Squares
In recreational mathematics, a normal magic square is an n×n square matrix whose entries are distinctly the integers 1…n2, such that each row, column, and major and minor traces sum to one constant μ. It has been proven that there are 7,040 fourth order normal magic squares and 2,202,441,792 fifth order normal magic squares, with higher orders unconfirmed. Previous work related to fourth order normal squares has shown that symmetries such as the dihedral group exist and that (under certain conditions) normal magic squares can be categorized into four distinct subsets. With the implementation of an efficient backtracking algorithm along with supervised machine learning techniques for classification, it will be shown that the entire set of fourth order normal magic squares can be generated by expanding the symmetry groups of 95 asymmetric parents. Discussion will suggest that methods employed in this project could similarly apply to higher orders.
combinatorics  magic-squares  constraint-satisfaction  rather-interesting  to-write-about  nudge-targets  consider:representation  consider:looking-to-see 
april 2017 by Vaguery
[1412.1545] On Magic Finite Projective Space
This paper studies a generalization of magic squares to finite projective space ℙn(q). We classify at all functions from ℙn(q) into a finite field where the sum along any r-flat is 0. In doing so we show connections to elementary number theory and the modular representation theory of GL(n,q).
mathematical-recreations  combinatorics  rather-interesting  to-write-about  generalization  nudge-targets  consider:looking-to-see  consider:representation 
april 2017 by Vaguery
[1107.4120] Generalized packing designs
Generalized t-designs, which form a common generalization of objects such as t-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of t-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with t=2 and block size k=3 or 4.
combinatorics  constraint-satisfaction  algorithms  representation  experimental-design  rather-interesting  to-write  nudge-targets  consider:representation 
april 2017 by Vaguery
[math/0611293] Descending Dungeons and Iterated Base-Changing
For real numbers a, b> 1, let as a_b denote the result of interpreting a in base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or n_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the logarithms are to the base 10. We also investigate the behavior as n increases of the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards. This converges either to a single number (e.g. to the golden ratio if a = 1.1), to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a = frac{100{99).
number-theory  mathematical-recreations  representation  rather-interesting  to-write-about  nudge-targets  consider:representation  consider:feature-discovery  sequences 
april 2017 by Vaguery
[1606.05172] Asynchronous simulation of Boolean networks by monotone Boolean networks
We prove that the fully asynchronous dynamics of a Boolean network f:{0,1}n→{0,1}n without negative loop can be simulated, in a very specific way, by a monotone Boolean network with 2n components. We then use this result to prove that, for every even n, there exists a monotone Boolean network f:{0,1}n→{0,1}n, an initial configuration x and a fixed point y of f such that: (i) y can be reached from x with a fully asynchronous updating strategy, and (ii) all such strategies contains at least 2n2 updates. This contrasts with the following known property: if f:{0,1}n→{0,1}n is monotone, then, for every initial configuration x, there exists a fixed point y such that y can be reached from x with a fully asynchronous strategy that contains at most n updates.
boolean-networks  Kauffmania  numerical-methods  rather-interesting  nudge-targets  distributed-processing  consider:looking-to-see  consider:representation  to-write-about 
april 2017 by Vaguery
[1511.02667] An Efficient Multilinear Optimization Framework for Hypergraph Matching
Hypergraph matching has recently become a popular approach for solving correspondence problems in computer vision as it allows to integrate higher-order geometric information. Hypergraph matching can be formulated as a third-order optimization problem subject to the assignment constraints which turns out to be NP-hard. In recent work, we have proposed an algorithm for hypergraph matching which first lifts the third-order problem to a fourth-order problem and then solves the fourth-order problem via optimization of the corresponding multilinear form. This leads to a tensor block coordinate ascent scheme which has the guarantee of providing monotonic ascent in the original matching score function and leads to state-of-the-art performance both in terms of achieved matching score and accuracy. In this paper we show that the lifting step to a fourth-order problem can be avoided yielding a third-order scheme with the same guarantees and performance but being two times faster. Moreover, we introduce a homotopy type method which further improves the performance.
algorithms  numerical-methods  graph-theory  hypergraphs  feature-construction  representation  computer-vision  nudge-targets  consider:looking-to-see  consider:representation  to-write-about 
april 2017 by Vaguery
[1703.04823] Classification in biological networks with hypergraphlet kernels
Biological and cellular systems are often modeled as graphs in which vertices represent objects of interest (genes, proteins, drugs) and edges represent relational ties among these objects (binds-to, interacts-with, regulates). This approach has been highly successful owing to the theory, methodology and software that support analysis and learning on graphs. Graphs, however, often suffer from information loss when modeling physical systems due to their inability to accurately represent multiobject relationships. Hypergraphs, a generalization of graphs, provide a framework to mitigate information loss and unify disparate graph-based methodologies. In this paper, we present a hypergraph-based approach for modeling physical systems and formulate vertex classification, edge classification and link prediction problems on (hyper)graphs as instances of vertex classification on (extended, dual) hypergraphs in a semi-supervised setting. We introduce a novel kernel method on vertex- and edge-labeled (colored) hypergraphs for analysis and learning. The method is based on exact and inexact (via hypergraph edit distances) enumeration of small simple hypergraphs, referred to as hypergraphlets, rooted at a vertex of interest. We extensively evaluate this method and show its potential use in a positive-unlabeled setting to estimate the number of missing and false positive links in protein-protein interaction networks.
machine-learning  representation  rather-interesting  hypergraphs  to-write-about  nudge-targets  consider:representation  consider:looking-to-see  bioinformatics  clustering  metrics 
april 2017 by Vaguery
[1305.1880] A Heuristic for Magic and Antimagic Graph Labellings
Graph labellings have been a very fruitful area of research in the last four decades. However, despite the staggering number of papers published in the field (over 1000), few general results are available, and most papers deal with particular classes of graphs and methods. Here we approach the problem from the computational viewpoint, and in a quite general way. We present the existence problem of a particular labelling as a combinatorial optimization problem, then we discuss the possible strategies to solve it, and finally we present a heuristic for finding different classes of labellings, like vertex-, edge-, or face-magic, and $(a, d)$-antimagic $(v, e, f)$-labellings. The algorithm has been implemented in C++ and MATLAB, and with its aid we have been able to derive new results for some classes of graphs, in particular, vertex-antimagic edge labellings for small graphs of the type $P_2^r \times P_3^s$, for which no general construction is known so far.
mathematical-recreations  integer-programming  constraint-satisfaction  construction  heuristics  nudge-targets  consider:looking-to-see  consider:representation 
april 2017 by Vaguery
[1509.07756] Unraveling the secret of Benjamin Franklin: Constructing Franklin squares of higher order
Benjamin Franklin constructed three squares which have amazing properties, and his method of construction has been a mystery to date. In this article, we divulge his secret and show how to construct such squares for any order.
mathematical-recreations  combinatorics  magic-squares  history  to-write-about  constraint-satisfaction  nudge-targets  consider:representation  consider:looking-to-see 
april 2017 by Vaguery
[math/0602300] Deterministic Random Walks on the Integers
Jim Propp's P-machine, also known as the "rotor router model" is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order.
We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c_1, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L), for the L_2 average of a contiguous set of intervals even to O(sqrt{log L}). All these bounds are tight.
graph-theory  rotor-router  feature-construction  combinatorics  rather-interesting  to-write-about  nudge-targets  consider:representation 
april 2017 by Vaguery
[1702.06135] Enabling Multi-Source Neural Machine Translation By Concatenating Source Sentences In Multiple Languages
In this paper, we propose a novel and elegant solution to "Multi-Source Neural Machine Translation" (MSNMT) which only relies on preprocessing a N-way multilingual corpus without modifying the Neural Machine Translation (NMT) architecture or training procedure. We simply concatenate the source sentences to form a single long multi-source input sentence while keeping the target side sentence as it is and train an NMT system using this preprocessed corpus. We evaluate our method in resource poor as well as resource rich settings and show its effectiveness (up to 4 BLEU using 2 source languages and up to 6 BLEU using 5 source languages) by comparing against existing methods for MSNMT. We also provide some insights on how the NMT system leverages multilingual information in such a scenario by visualizing attention.
natural-language-processing  representation  neural-networks  rather-interesting  nudge-targets  consider:representation  to-write-about  consider:other-applications 
april 2017 by Vaguery
[1704.00224] A Time-Frequency Domain Approach of Heart Rate Estimation From Photoplethysmographic (PPG) Signal
Objective- Heart rate monitoring using wrist type Photoplethysmographic (PPG) signals is getting popularity because of construction simplicity and low cost of wearable devices. The task becomes very difficult due to the presence of various motion artifacts. The objective is to develop algorithms to reduce the effect of motion artifacts and thus obtain accurate heart rate estimation. Methods- Proposed heart rate estimation scheme utilizes both time and frequency domain analyses. Unlike conventional single stage adaptive filter, multi-stage cascaded adaptive filtering is introduced by using three channel accelerometer data to reduce the effect of motion artifacts. Both recursive least squares (RLS) and least mean squares (LMS) adaptive filters are tested. Moreover, singular spectrum analysis (SSA) is employed to obtain improved spectral peak tracking. The outputs from the filter block and SSA operation are logically combined and used for spectral domain heart rate estimation. Finally, a tracking algorithm is incorporated considering neighbouring estimates. Results- The proposed method provides an average absolute error of 1.16 beat per minute (BPM) with a standard deviation of 1.74 BPM while tested on publicly available database consisting of recordings from 12 subjects during physical activities. Conclusion- It is found that the proposed method provides consistently better heart rate estimation performance in comparison to that recently reported by TROIKA, JOSS and SPECTRAP methods. Significance- The proposed method offers very low estimation error and a smooth heart rate tracking with simple algorithmic approach and thus feasible for implementing in wearable devices to monitor heart rate for fitness and clinical purpose.
signal-processing  time-series  engineering-design  rather-interesting  inference  nudge-targets  consider:looking-to-see  consider:representation 
april 2017 by Vaguery
[1701.01649] Existence of Some Signed Magic Arrays
We consider the notion of a signed magic array, which is an $m \times n$ rectangular array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We attempt to make progress toward a characterization of for which $(m, n, s, t)$ there exists such an array. This characterization is complete in the case where $n = s$ and in the case where $n = m$; we also characterize three-fourths of the cases where $n = 2m$.
number-theory  constraint-satisfaction  mathematical-recreations  nudge-targets  consider:representation  consider:looking-to-see 
april 2017 by Vaguery
[1704.00794] Time Series Cluster Kernel for Learning Similarities between Multivariate Time Series with Missing Data
Similarity-based approaches represent a promising direction for time series analysis. However, many such methods rely on parameter tuning and have shortcomings if the time series are multivariate (MTS) and contain missing data. In this paper, we address these challenges within the powerful context of kernel methods by proposing the robust \emph{time series cluster kernel} (TCK). The approach taken is to leverage the missing data handling properties of Gaussian mixture models (GMM) augmented with informative prior distributions. An ensemble learning approach is exploited to ensure robustness to parameters by combining the clustering results of many GMM to form the final kernel.
We evaluate the TCK on synthetic and real data and compare to other state-of-the-art techniques. The experimental results demonstrate that the TCK is robust to parameter choices, provides competitive results for MTS without missing data and outstanding results for missing data.
time-series  clustering  feature-extraction  statistics  algorithms  nudge-targets  consider:representation 
april 2017 by Vaguery
[1703.09097] An explicit formula for the pressure of box-like affine iterated function systems
In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser. Combining previous results of V. Yu. Protasov, A. K\"aenm\"aki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J.M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.
geometry  algebra  algorithms  fractals  nudge-targets  consider:rediscovery  consider:representation 
april 2017 by Vaguery
[1701.06794] Computations with p-adic numbers
This document contains the notes of a lecture I gave at the "Journ\'ees Nationales du Calcul Formel" (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of p-adic numbers and compare them and second, we introduce a general framework for studying precision issues and apply it in several concrete situations.
number-theory  rather-interesting  algorithms  group-theory  nudge-targets  consider:representation  consider:looking-to-see 
april 2017 by Vaguery
[1606.06329] Recognizing Surgical Activities with Recurrent Neural Networks
We apply recurrent neural networks to the task of recognizing surgical activities from robot kinematics. Prior work in this area focuses on recognizing short, low-level activities, or gestures, and has been based on variants of hidden Markov models and conditional random fields. In contrast, we work on recognizing both gestures and longer, higher-level activites, or maneuvers, and we model the mapping from kinematics to gestures/maneuvers with recurrent neural networks. To our knowledge, we are the first to apply recurrent neural networks to this task. Using a single model and a single set of hyperparameters, we match state-of-the-art performance for gesture recognition and advance state-of-the-art performance for maneuver recognition, in terms of both accuracy and edit distance. Code is available at this https URL .
neural-networks  recurrent-networks  control-theory  robotics  machine-learning  rather-interesting  representation  nudge-targets  consider:representation 
march 2017 by Vaguery
[1612.03471] Reinforcement Learning With Temporal Logic Rewards
Reinforcement learning (RL) depends critically on the choice of reward functions used to capture the de- sired behavior and constraints of a robot. Usually, these are handcrafted by a expert designer and represent heuristics for relatively simple tasks. Real world applications typically involve more complex tasks with rich temporal and logical structure. In this paper we take advantage of the expressive power of temporal logic (TL) to specify complex rules the robot should follow, and incorporate domain knowledge into learning. We propose Truncated Linear Temporal Logic (TLTL) as specifications language, that is arguably well suited for the robotics applications, together with quantitative semantics, i.e., robustness degree. We propose a RL approach to learn tasks expressed as TLTL formulae that uses their associated robustness degree as reward functions, instead of the manually crafted heuristics trying to capture the same specifications. We show in simulated trials that learning is faster and policies obtained using the proposed approach outperform the ones learned using heuristic rewards in terms of the robustness degree, i.e., how well the tasks are satisfied. Furthermore, we demonstrate the proposed RL approach in a toast-placing task learned by a Baxter robot.
machine-learning  performance-measure  planning  rather-interesting  what-gets-measured-gets-fudged  "reward-hacking"  nudge-targets  consider:representation 
march 2017 by Vaguery
[1409.3588] Density Classification Quality of the Traffic-majority Rules
The density classification task is a famous problem in the theory of cellular automata. It is unsolvable for deterministic automata, but recently solutions for stochastic cellular automata have been found. One of them is a set of stochastic transition rules depending on a parameter η, the traffic-majority rules.
Here I derive a simplified model for these cellular automata. It is valid for a subset of the initial configurations and uses random walks and generating functions. I compare its prediction with computer simulations and show that it expresses recognition quality and time correctly for a large range of η values.
cellular-automata  hey-I-know-this-guy  emergent-design  to-write-about  nudge-targets  consider:representation  consider:higher-order-problems  consider:classification-of-rules 
march 2017 by Vaguery
[1703.02671] Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces
We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the problem is strongly NP-complete even if the pieces are all polyominos. On the positive side, we show that the problem can be solved in polynomial time if the number of pieces is a fixed constant.
puzzles  computational-complexity  rather-interesting  nudge-targets  consider:looking-to-see  consider:representation  tangrams  to-write-about 
march 2017 by Vaguery
[0908.2442] Detecting all regular polygons in a point set
In this paper, we analyze the time complexity of finding regular polygons in a set of n points. We combine two different approaches to find regular polygons, depending on their number of edges. Our result depends on the parameter alpha, which has been used to bound the maximum number of isosceles triangles that can be formed by n points. This bound has been expressed as O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068.
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through the set of points, while larger polygons are found by random sampling. We can find all regular polygons with high probability in O(n^{2+alpha+epsilon}) expected time for every positive epsilon. This compares well to the O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.
computational-geometry  algorithms  nudge-targets  consider:looking-to-see  consider:classification  consider:representation 
march 2017 by Vaguery
[1306.5527] Computing the Fr'echet Distance with a Retractable Leash
All known algorithms for the Fr\'echet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fr\'echet distance between polygonal curves in Rd under polyhedral distance functions (e.g., L1 and L∞). We also get a (1+ε)-approximation of the Fr\'echet distance under the Euclidean metric, in quadratic time for any fixed ε>0. For the exact Euclidean case, our framework currently yields an algorithm with running time O(n2log2n). However, we conjecture that it may eventually lead to a faster exact algorithm.
metrics  algorithms  numerical-methods  rather-interesting  nudge-targets  approximation  consider:looking-to-see  consider:representation  to-write-about 
march 2017 by Vaguery
[1310.6900] Unsplittable coverings in the plane
A system of sets forms an {\em m-fold covering} of a set X if every point of X belongs to at least m of its members. A 1-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body C, there exists a constant m=m(C) such that every m-fold covering of the plane with translates of C splits into 2 coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every m, an unsplittable m-fold covering of the plane with translates of any open convex body C which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets C do not misbehave, they satisfy the conjecture: every 3-fold covering of any region of the plane by translates of such a set C splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant c>0 such that, for any positive integer m, every m-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} c2m/2 sets.
optimization  rather-interesting  computational-geometry  to-understand  nudge-targets  consider:representation 
march 2017 by Vaguery
[1609.07766] Separating Overlapped Intervals on a Line
Given n intervals on a line ℓ, we consider the problem of moving these intervals on ℓ such that no two intervals overlap and the maximum moving distance of the intervals is minimized. The difficulty for solving the problem lies in determining the order of the intervals in an optimal solution. By interesting observations, we show that it is sufficient to consider at most n "candidate" lists of ordered intervals. Further, although explicitly maintaining these lists takes Ω(n2) time and space, by more observations and a pruning technique, we present an algorithm that can compute an optimal solution in O(nlogn) time and O(n) space. We also prove an Ω(nlogn) time lower bound for solving the problem, which implies the optimality of our algorithm.
mathematical-recreations  optimization  to-write-about  lovely-little-problems  computational-complexity  nudge-targets  consider:looking-to-see  consider:representation  planning  computational-geometry 
march 2017 by Vaguery
[1612.03540] Sweeping costs of planar domains
Let D be a Jordan domain in the plane. We consider a pursuit-evasion, contamination clearing, or sensor sweep problem in which the pursuer at each point in time is modeled by a continuous curve, called the sensor curve. Both time and space are continuous, and the intruders are invisible to the pursuer. Given D, what is the shortest length of a sensor curve necessary to provide a sweep of domain D, so that no continuously-moving intruder in D can avoid being hit by the curve? We define this length to be the sweeping cost of D. We provide an analytic formula for the sweeping cost of any Jordan domain in terms of the geodesic Frechet distance between two curves on the boundary of D with non-equal winding numbers. As a consequence, we show that the sweeping cost of any convex domain is equal to its width, and that a convex domain of unit area with maximal sweeping cost is the equilateral triangle.
topology  feature-construction  rather-interesting  to-write-about  plane-geometry  nudge-targets  consider:approximation  consider:representation 
march 2017 by Vaguery
[1606.07312] Unsupervised preprocessing for Tactile Data
Tactile information is important for gripping, stable grasp, and in-hand manipulation, yet the complexity of tactile data prevents widespread use of such sensors. We make use of an unsupervised learning algorithm that transforms the complex tactile data into a compact, latent representation without the need to record ground truth reference data. These compact representations can either be used directly in a reinforcement learning based controller or can be used to calibrate the tactile sensor to physical quantities with only a few datapoints. We show the quality of our latent representation by predicting important features and with a simple control task.
rather-interesting  sensors  robotics  clustering  machine-learning  nudge-targets  consider:looking-to-see  consider:representation 
march 2017 by Vaguery
[1701.07540] Sample Complexity of the Boolean Multireference Alignment Problem
The Boolean multireference alignment problem consists in recovering a Boolean signal from multiple shifted and noisy observations. In this paper we obtain an expression for the error exponent of the maximum A posteriori decoder. This expression is used to characterize the number of measurements needed for signal recovery in the low SNR regime, in terms of higher order autocorrelations of the signal. The characterization is explicit for various signal dimensions, such as prime and even dimensions.
signal-processing  time-series  information-theory  benchmarks  nudge-targets  consider:looking-to-see  consider:representation  to-understand 
march 2017 by Vaguery
[1611.01988] Differentiable Functional Program Interpreters
Programming by Example (PBE) is the task of inducing computer programs from input-output examples. It can be seen as a type of machine learning where the hypothesis space is the set of legal programs in some programming language. Recent work on differentiable interpreters relaxes the discrete space of programs into a continuous space so that search over programs can be performed using gradient-based optimization. While conceptually powerful, so far differentiable interpreter-based program synthesis has only been capable of solving very simple problems. In this work, we study modeling choices that arise when constructing a differentiable programming language and their impact on the success of synthesis. The main motivation for the modeling choices comes from functional programming: we study the effect of memory allocation schemes, immutable data, type systems, and built-in control-flow structures. Empirically we show that incorporating functional programming ideas into differentiable programming languages allows us to learn much more complex programs than is possible with existing differentiable languages.
program-synthesis  system-of-professions  to-write-about  amusing-because-no-refs  rather-interesting  probabilistic-languages  consider:representation  consider:algorithmic-approach 
march 2017 by Vaguery
[1703.00998] randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization
This manuscript describes the randomized algorithm randUTV for computing a so called UTV factorization efficiently. Given a matrix A, the algorithm computes a factorization A=UTV∗, where U and V have orthonormal columns, and T is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, determining bases for various subspaces associated with the matrix, etc. Moreover, randUTV produces highly accurate approximations to the singular values of A. Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and non-iterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. These experiments demonstrate that in comparison to column pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy.
matrices  algorithms  numerical-methods  to-understand  nudge-targets  consider:representation  consider:libraries  consider:looking-to-see 
march 2017 by Vaguery
[1703.01054] When Hashes Met Wedges: A Distributed Algorithm for Finding High Similarity Vectors
Finding similar user pairs is a fundamental task in social networks, with numerous applications in ranking and personalization tasks such as link prediction and tie strength detection. A common manifestation of user similarity is based upon network structure: each user is represented by a vector that represents the user's network connections, where pairwise cosine similarity among these vectors defines user similarity. The predominant task for user similarity applications is to discover all similar pairs that have a pairwise cosine similarity value larger than a given threshold τ. In contrast to previous work where τ is assumed to be quite close to 1, we focus on recommendation applications where τ is small, but still meaningful. The all pairs cosine similarity problem is computationally challenging on networks with billions of edges, and especially so for settings with small τ. To the best of our knowledge, there is no practical solution for computing all user pairs with, say τ=0.2 on large social networks, even using the power of distributed algorithms.
Our work directly addresses this challenge by introducing a new algorithm --- WHIMP --- that solves this problem efficiently in the MapReduce model. The key insight in WHIMP is to combine the "wedge-sampling" approach of Cohen-Lewis for approximate matrix multiplication with the SimHash random projection techniques of Charikar. We provide a theoretical analysis of WHIMP, proving that it has near optimal communication costs while maintaining computation cost comparable with the state of the art. We also empirically demonstrate WHIMP's scalability by computing all highly similar pairs on four massive data sets, and show that it accurately finds high similarity pairs. In particular, we note that WHIMP successfully processes the entire Twitter network, which has tens of billions of edges.
algorithms  numerical-methods  linear-algebra  metrics  approximation  nudge-targets  consider:representation 
march 2017 by Vaguery
Surface Evolver
My Surface Evolver is an interactive program for the modelling of liquid surfaces shaped by various forces and constraints. The program is available free of charge.
geometry  software  simulation  rather-interesting  to-write-about  to-explore  nudge-targets  consider:prediction  consider:representation 
march 2017 by Vaguery
[1003.3301] A method for dense packing discovery
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density ϕ=128/219≈0.5845 and with a similar structure to the densest known tetrahedron packing.
packing  numerical-methods  simulation  rather-interesting  algorithms  to-write-about  nudge-targets  consider:looking-to-see  consider:representation 
march 2017 by Vaguery
[1103.4518] On the Reinhardt Conjecture
In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density. He conjectured that the unique solution up to an affine transformation is the smoothed octagon (an octagon rounded at corners by arcs of hyperbolas). This article offers a detailed strategy of proof. In particular, we show that the problem is an instance of the classical problem of Bolza in the calculus of variations. A minimizing solution is known to exist. The boundary of every minimizer is a differentiable curve with Lipschitz continuous derivative. If a minimizer is piecewise analytic, then it is a smoothed polygon (a polygon rounded at corners by arcs of hyperbolas). To complete the proof of the Reinhardt conjecture, the assumption of piecewise analyticity must be removed, and the conclusion of smoothed polygon must be strengthened to smoothed octagon.
plane-geometry  mathematical-recreations  nudge-targets  consider:representation  consider:looking-to-see  consider:simulation 
march 2017 by Vaguery
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