Vaguery + looking-to-see   85

[1209.5958] Diffusion-limited aggregation with polygon particles
Diffusion-limited aggregation (DLA) assumes that particles perform pure random walk at a finite temperature and aggregate when they come close enough and stick together. Although it is well known that DLA in two dimensions results in a ramified fractal structure, how the particle shape influences the formed morphology is still unclear. In this work, we perform the off-lattice two-dimensional DLA simulations with different particle shapes of triangle, quadrangle, pentagon, hexagon, and octagon, respectively, and compared with the results for circular particles. Our results indicate that different particle shapes only change the local structure, but have no effects on the global structure of the formed fractal cluster. The local compactness decreases as the number of polygon edges increases.
self-organization  DLA  fractals  looking-to-see  to-simulate  consider:locking-rules 
8 days ago by Vaguery
Are Pop Lyrics Getting More Repetitive?
In 1977, the great computer scientist Donald Knuth published a paper called The Complexity of Songs, which is basically one long joke about the repetitive lyrics of newfangled music (example quote: "the advent of modern drugs has led to demands for still less memory, and the ultimate improvement of Theorem 1 has consequently just been announced").

I'm going to try to test this hypothesis with data. I'll be analyzing the repetitiveness of a dataset of 15,000 songs that charted on the Billboard Hot 100 between 1958 and 2017.
visualization  graphic-design  data-analysis  essay  looking-to-see  javascript  rather-interesting  via:cdzombak 
24 days ago by Vaguery
[1803.11463] Arctic curves for paths with arbitrary starting points: a tangent method approach
We use the tangent method to investigate the arctic curve in a model of non-intersecting lattice paths with arbitrary fixed starting points aligned along some boundary and whose distribution is characterized by some arbitrary piecewise differentiable function. We find that the arctic curve has a simple explicit parametric representation depending of this function, providing us with a simple transform that maps the arbitrary boundary condition to the arctic curve location. We discuss generic starting point distributions as well as particular freezing ones which create additional frozen domains adjacent to the boundary, hence new portions for the arctic curve. A number of examples are presented, corresponding to both generic and freezing distributions.
combinatorics  arctic-curves  tiling  statistical-mechanics  simulation  looking-to-see  feature-construction  emergence  to-write-about  consider:prediction  consider:feature-discovery 
27 days ago by Vaguery
[1810.04909] Tangent method for the arctic curve arising from freezing boundaries
In the paper arXiv:1803.11463, the authors study the arctic curve arising in random tilings of some planar domains with an arbitrary distribution of defects on one edge. Using the tangent method they derive a parametric equation for portions of arctic curve in terms of an arbitrary piecewise differentiable function that describes the defect distribution. When this distribution presents "freezing" intervals, other portions of arctic curve appear and typically have a cusp. These freezing boundaries can be of two types, respectively with maximal or minimal density of defects. Our purpose here is to extend the tangent method derivation of arXiv:1803.11463 to include these portions, hence providing the proof of the conjectures made in arXiv:1803.11463.
combinatorics  tiling  phase-transitions  arctic-curve  statistical-mechanics  looking-to-see  rather-interesting  to-write-about  to-simulate 
27 days ago by Vaguery
How many landmarks are enough to characterize shape and size variation?
Accurate characterization of morphological variation is crucial for generating reliable results and conclusions concerning changes and differences in form. Despite the prevalence of landmark-based geometric morphometric (GM) data in the scientific literature, a formal treatment of whether sampled landmarks adequately capture shape variation has remained elusive. Here, I introduce LaSEC (Landmark Sampling Evaluation Curve), a computational tool to assess the fidelity of morphological characterization by landmarks. This task is achieved by calculating how subsampled data converge to the pattern of shape variation in the full dataset as landmark sampling is increased incrementally. While the number of landmarks needed for adequate shape variation is dependent on individual datasets, LaSEC helps the user (1) identify under- and oversampling of landmarks; (2) assess robustness of morphological characterization; and (3) determine the number of landmarks that can be removed without compromising shape information. In practice, this knowledge could reduce time and cost associated with data collection, maintain statistical power in certain analyses, and enable the incorporation of incomplete, but important, specimens to the dataset. Results based on simulated shape data also reveal general properties of landmark data, including statistical consistency where sampling additional landmarks has the tendency to asymptotically improve the accuracy of morphological characterization. As landmark-based GM data become more widely adopted, LaSEC provides a systematic approach to evaluate and refine the collection of shape data––a goal paramount for accumulation and analysis of accurate morphological information.
inference  data-analysis  looking-to-see  rather-interesting  training-data  data-balancing  to-write-about 
5 weeks ago by Vaguery
[cs/0610153] Most Programs Stop Quickly or Never Halt
Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k>0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{-k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that ``long'' runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density.
halting-problem  computer-science  rather-interesting  looking-to-see  to-write-about  ReQ  computational-complexity  probability-theory 
5 weeks ago by Vaguery
When Optimising Code Measure
This is a truism that lots of people quote, but it can be hard to remember, especially in the heat of battle (as it were). Rather fortunately it came to mind just when needed, as I found something completely unexpected.

I was writing a simple implementation of the Fermat difference of squares method of factoring. This involves writing the number to be factored as - you guessed it - the difference of two squares. If n=a2−b2

then n=(a−b)(a+b)

and we have a factorisation (provided we don't have a−b=1

the-mangle-in-practice  looking-to-see  learning-in-public  computer-science  computational-complexity  rather-interesting  to-write-about  contingency 
5 weeks ago by Vaguery
PsyArXiv Preprints | Multiple Perspectives on Inference for Two Simple Statistical Scenarios
When data analysts operate within different statistical frameworks (e.g., frequentist versus Bayesian, emphasis on estimation versus emphasis on testing), how does this impact the qualitative conclusions that are drawn for real data? To study this question empirically we selected from the literature two simple scenarios --involving a comparison of two proportions and a Pearson correlation-- and asked four teams of statisticians to provide a concise analysis and a qualitative interpretation of the outcome. The results showed considerable overall agreement; nevertheless, this agreement did not appear to diminish the intensity of the subsequent debate over which statistical framework is more appropriate to address the questions at hand.
statistics  looking-to-see  the-mangle-in-practice  science-studies  anthropology-of-data  rather-interesting  to-write-about 
5 weeks ago by Vaguery
[PDF] MEXICA: a computer model of a cognitive account of creative writing.
MEXICA is a computer model that produces frameworks for short stories based on the engagement-reflection cognitive account of writing. During engagement MEXICA generates material guided by content and rhetorical constraints, avoiding the use of explicit goals or story- structure information. During reflection the system breaks impasses, evaluates the novelty and interestingness of the story in progress and verifies that coherence requirements are satisfied. In this way, MEXICA complements and extends those models of computerised story-telling based on traditional problem-solving techniques where explicit goals drive the generation of stories. This paper describes the engagement-reflection account of writing, the general characteristics of MEXICA and reports an evaluation of the program
generative-models  cognition  simulation  looking-to-see  cultural-engineering  rather-interesting  representation  to-write-about 
9 weeks ago by Vaguery
Ecological theory provides insights about evolutionary computation [PeerJ Preprints]
Evolutionary algorithms often incorporate ecological concepts to help maintain diverse populations and drive continued innovation. However, while there is strong evidence for the value of ecological dynamics, a lack of overarching theoretical framework renders the precise mechanisms behind these results unclear. These gaps in our understanding make it challenging to predict which approaches will be most appropriate for a given problem. Biologists have been developing ecological theory for decades, but the resulting body of work has yet to be translated into an evolutionary computation context. This paper lays the groundwork for such a translation by applying ecological theory to three different selection mechanisms in evolutionary computation: fitness sharing, lexicase selection, and Eco-EA. First, we use ecological ideas to establish a framework that clarifies how these selection schemes are alike and how they differ. We then build upon this framework by using metrics from ecology to gather empirical data about the underlying differences in the population dynamics that these approaches produce. Specifically, we measure interaction networks and phylogenetic diversity within the population to explore long-term stable coexistence. Notably, we find that selection methods affect phylogenetic diversity differently than phenotypic diversity. These results can inform parameter selection, choice of selection scheme, and the development of new selection schemes.
evolutionary-algorithms  selection  looking-to-see  rather-interesting  hey-I-know-these-folks  artificial-life  feature-construction  community-formation 
10 weeks ago by Vaguery
Evolution of metazoan morphological disparity | PNAS
We attempt to quantify animal “bodyplans” and their variation within Metazoa. Our results challenge the view that maximum variation was achieved early in animal evolutionary history by nonuniformitarian mechanisms. Rather, they are compatible with the view that the capacity for fundamental innovation is not limited to the early evolutionary history of clades. We perform quantitative tests of the principal hypotheses of the molecular mechanisms underpinning the establishment of animal bodyplans and corroborate the hypothesis that animal evolution has been permitted or driven by gene regulatory evolution.
developmental-biology  biology  rather-interesting  clustering  looking-to-see  to-write-about  consider:feature-discovery 
12 weeks ago by Vaguery
[1804.10962] Stress anisotropy in shear-jammed packings of frictionless disks
We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain γ at fixed packing fraction ϕ. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the ϕ-γ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, dϕ/dγ>0 and dϕ/dγ<0. In contrast, MS packings from protocol 2 populate segments with dϕ/dγ<0 only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy dϕ/dγ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.
physics!  sandpiles  materials-science  simulation  rather-interesting  condensed-matter  phase-transitions  looking-to-see 
october 2018 by Vaguery
Coder-Physicists Are Simulating the Universe to Unlock Its Secrets | Quanta Magazine
These small, faint galaxies have always presented problems. The “missing satellite problem,” for instance, is the expectation, based on standard cold dark matter models, that hundreds of satellite galaxies should orbit every spiral galaxy. But the Milky Way has just dozens. This has caused some physicists to contemplate more complicated models of dark matter. However, when Hopkins and colleagues incorporated realistic superbubbles into their simulations, they saw many of those excess satellite galaxies go away. Hopkins has also found potential resolutions to two other problems, called “cusp-core” and “too-big-to-fail,” that have troubled the cold dark matter paradigm.
simulation  looking-to-see  astronomy  rather-interesting  to-write-about  the-mangle-in-practice  (totally) 
october 2018 by Vaguery
Orthogonal polygons | The Math Less Traveled
Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to draw all possible orthogonal polygons, that is, polygons with only right angles, organized by the total number of vertices. (So, for example, the picture above shows all orthogonal polygons with exactly ten vertices.) However, we have to be careful what we mean by the phrase “all possible”: there would be an infinite number of such polygons if we think about things like the precise lengths of edges. So we have to say when two polygons are considered the same, and when they are distinct. My rules are as follows:
mathematical-recreations  looking-to-see  geometry  polyominoes  to-write-about 
may 2018 by Vaguery
Cooking the books – Almost looks like work
Since Christmas, at my house we’ve been cooking with 5 ingredients or fewer thanks to the acquisition of Jamie Oliver’s new book, the contents of which are mostly available online here. The recipes are unanimously very tasty, but that’s besides the point. The real mark of culinary excellence (in my humble opinion) is how efficiently one can buy ingredients to make as many of the recipes as possible in one shopping trip. Let’s investigate while the lamb is on.

Each of the 135 recipes in the book consists of 5 ingredients, some of which overlap. It is therefore not necessary to purchase 675 ingredients, there are actually only 239 unique ones. (Yes, I did spend a Sunday morning typing 675 individual ingredients into a spreadsheet.)

The question is then this:

In which order should I buy my ingredients to maximise the number of possible recipes as a function of number of ingredients?

Let’s start simply, and look at the frequency of occurrence of the ingredients.
mathematical-recreations  looking-to-see  cooking  data-analysis  leading-questions  rather-interesting 
may 2018 by Vaguery
mathrecreation: Tigers and Treasure
The graph below shows all 196 possible puzzles. The puzzles that lead to contradictions are coloured as white squares, the puzzles that do not lead to contradictions are coloured in black. We can see a white strip across the bottom: these are all the puzzles where door 2 is labelled with statement 1. Can you find the statement that always leads to contradictions for door 1? There are 131 puzzles that are "good" in the sense that they do not lead to contradictions.
mathematical-recreations  logic  looking-to-see  to-write-about  consider:robustness  nudge-targets 
april 2018 by Vaguery
[1802.01396] To understand deep learning we need to understand kernel learning
Generalization performance of classifiers in deep learning has recently become a subject of intense study. Heavily over-parametrized deep models tend to fit training data exactly. Despite overfitting, they perform well on test data, a phenomenon not yet fully understood.
The first point of our paper is that strong performance of overfitted classifiers is not a unique feature of deep learning. Using real-world and synthetic datasets, we establish that kernel classifiers trained to have zero classification error (overfitting) or even zero regression error (interpolation) perform very well on test data.
We proceed to prove lower bounds on the norm of overfitted solutions for smooth kernels, showing that they increase nearly exponentially with the data size. Since most generalization bounds depend polynomially on the norm of the solution, this result implies that they diverge as data increases. Furthermore, the existing bounds do not apply to interpolated classifiers.
We also show experimentally that (non-smooth) Laplacian kernels easily fit random labels using a version of SGD, a finding that parallels results reported for ReLU neural networks. In contrast, fitting noisy data requires many more epochs for smooth Gaussian kernels. The observation that the performance of overfitted Laplacian and Gaussian classifiers on the test is quite similar, suggests that generalization is tied to the properties of the kernel function rather than the optimization process.
We see that some key phenomena of deep learning are manifested similarly in kernel methods in the overfitted regime. We argue that progress on understanding deep learning will be difficult, until more analytically tractable "shallow" kernel methods are better understood. The experimental and theoretical results presented in this paper indicate a need for new theoretical ideas for understanding classical kernel methods.
machine-learning  philosophy-of-engineering  looking-to-see  statistics  algorithms  explanation  to-write-about  explanatory-power-and-narrative-discovery 
april 2018 by Vaguery
[1709.04594] Revisiting Spectral Graph Clustering with Generative Community Models
The methodology of community detection can be divided into two principles: imposing a network model on a given graph, or optimizing a designed objective function. The former provides guarantees on theoretical detectability but falls short when the graph is inconsistent with the underlying model. The latter is model-free but fails to provide quality assurance for the detected communities. In this paper, we propose a novel unified framework to combine the advantages of these two principles. The presented method, SGC-GEN, not only considers the detection error caused by the corresponding model mismatch to a given graph, but also yields a theoretical guarantee on community detectability by analyzing Spectral Graph Clustering (SGC) under GENerative community models (GCMs). SGC-GEN incorporates the predictability on correct community detection with a measure of community fitness to GCMs. It resembles the formulation of supervised learning problems by enabling various community detection loss functions and model mismatch metrics. We further establish a theoretical condition for correct community detection using the normalized graph Laplacian matrix under a GCM, which provides a novel data-driven loss function for SGC-GEN. In addition, we present an effective algorithm to implement SGC-GEN, and show that the computational complexity of SGC-GEN is comparable to the baseline methods. Our experiments on 18 real-world datasets demonstrate that SGC-GEN possesses superior and robust performance compared to 6 baseline methods under 7 representative clustering metrics.
community-detection  network-theory  algorithms  looking-to-see  clustering  to-write-about  the-pragmatics-of-the-thing 
march 2018 by Vaguery
[1801.00548] A Machine Learning Approach to Adaptive Covariance Localization
Data assimilation plays a key role in large-scale atmospheric weather forecasting, where the state of the physical system is estimated from model outputs and observations, and is then used as initial condition to produce accurate future forecasts. The Ensemble Kalman Filter (EnKF) provides a practical implementation of the statistical solution of the data assimilation problem and has gained wide popularity as. This success can be attributed to its simple formulation and ease of implementation. EnKF is a Monte-Carlo algorithm that solves the data assimilation problem by sampling the probability distributions involved in Bayes theorem. Because of this, all flavors of EnKF are fundamentally prone to sampling errors when the ensemble size is small. In typical weather forecasting applications, the model state space has dimension 109−1012, while the ensemble size typically ranges between 30−100 members. Sampling errors manifest themselves as long-range spurious correlations and have been shown to cause filter divergence. To alleviate this effect covariance localization dampens spurious correlations between state variables located at a large distance in the physical space, via an empirical distance-dependent function. The quality of the resulting analysis and forecast is greatly influenced by the choice of the localization function parameters, e.g., the radius of influence. The localization radius is generally tuned empirically to yield desirable results.This work, proposes two adaptive algorithms for covariance localization in the EnKF framework, both based on a machine learning approach. The first algorithm adapts the localization radius in time, while the second algorithm tunes the localization radius in both time and space. Numerical experiments carried out with the Lorenz-96 model, and a quasi-geostrophic model, reveal the potential of the proposed machine learning approaches.
modeling  machine-learning  prediction  rather-interesting  looking-to-see  approximation  algorithms  to-write-about 
march 2018 by Vaguery
[1704.03636] Energy Propagation in Deep Convolutional Neural Networks
Many practical machine learning tasks employ very deep convolutional neural networks. Such large depths pose formidable computational challenges in training and operating the network. It is therefore important to understand how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across layers. In addition, it is desirable that the feature extractor generated by the network be informative in the sense of the only signal mapping to the all-zeros feature vector being the zero input signal. This "trivial null-set" property can be accomplished by asking for "energy conservation" in the sense of the energy in the feature vector being proportional to that of the corresponding input signal. This paper establishes conditions for energy conservation (and thus for a trivial null-set) for a wide class of deep convolutional neural network-based feature extractors and characterizes corresponding feature map energy decay rates. Specifically, we consider general scattering networks employing the modulus non-linearity and we find that under mild analyticity and high-pass conditions on the filters (which encompass, inter alia, various constructions of Weyl-Heisenberg filters, wavelets, ridgelets, (α)-curvelets, and shearlets) the feature map energy decays at least polynomially fast. For broad families of wavelets and Weyl-Heisenberg filters, the guaranteed decay rate is shown to be exponential. Moreover, we provide handy estimates of the number of layers needed to have at least ((1−ε)⋅100)% of the input signal energy be contained in the feature vector.
deep-learning  signal-processing  rather-interesting  information-theory  to-understand  looking-to-see  machine-learning 
march 2018 by Vaguery
[1802.09979] The Emergence of Spectral Universality in Deep Networks
Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network's input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spectra of Jacobians at initialization. To this end, we leverage powerful tools from free probability theory to provide a detailed analytic understanding of how a deep network's Jacobian spectrum depends on various hyperparameters including the nonlinearity, the weight and bias distributions, and the depth. For a variety of nonlinearities, our work reveals the emergence of new universal limiting spectral distributions that remain concentrated around one even as the depth goes to infinity.
deep-learning  looking-to-see  machine-learning  neural-networks  to-understand 
march 2018 by Vaguery
[1606.05099] Invariant measures for continued fraction algorithms with finitely many digits
In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss-Kuzmin-L\'evy based approximation method is used. Finally, a subfamily of the N-expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N=2 and N=36. Interesting behavior can be observed from numerical results.
continued-fractions  number-theory  looking-to-see  dynamical-systems  to-write-about 
january 2018 by Vaguery
[1711.01711] Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as 'miraculous' because of its surprisingly powerful properties and its application as the optimal theoretical solution to induction/inference, (approximations to) Algorithmic Probability (AP) and the Universal Distribution are of the greatest importance in computer science and science in general. Here we investigate the emergence, the rates of emergence and convergence, and the Coding-theorem like behaviour of AP in subuniversal models of computation. We investigate empirical distributions of computer programs of weaker computational power according to the Chomsky hierarchy. We introduce measures of algorithmic probability and algorithmic complexity based upon resource-bounded computation, in contrast to previously thoroughly investigated distributions produced from the output distribution of Turing machines. This approach allows for numerical approximations to algorithmic (Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a computational hierarchy. We demonstrate that all these estimations are correlated in rank and that they converge both in rank and values as a function of computational power, despite the fundamental differences of each computational model. In the context of natural processes that may operate below the Turing universal level due to the constraint of resources and physical degradation, the investigation of natural biases coming from algorithmic laws is highly relevant. We show that the simplicity/complexity bias in distributions produced even by the weakest of the computational models can be accounted up to 60% by Algorithmic Probability in its approximation to the Universal Distribution.
information-theory  computational-complexity  to-understand  to-write-about  looking-to-see 
january 2018 by Vaguery
CTK Insights » Blog Archive A Discovery of Hirotaka Ebisui And Thanos Kalogerakis - CTK Insights
Hirotaka Ebisui has found that in the case of the right triangle,






. On informing me of that result, Thanos added an identity for the next layer




That was exciting enough for me to investigate. I can happily and proudly report that, for the next layer,



plane-geometry  rather-interesting  mathematical-recreations  looking-to-see  to-write-about 
november 2017 by Vaguery
[1711.00867] The (Un)reliability of saliency methods
Saliency methods aim to explain the predictions of deep neural networks. These methods lack reliability when the explanation is sensitive to factors that do not contribute to the model prediction. We use a simple and common pre-processing step ---adding a constant shift to the input data--- to show that a transformation with no effect on the model can cause numerous methods to incorrectly attribute. In order to guarantee reliability, we posit that methods should fulfill input invariance, the requirement that a saliency method mirror the sensitivity of the model with respect to transformations of the input. We show, through several examples, that saliency methods that do not satisfy input invariance result in misleading attribution.
via:?  machine-learning  looking-to-see  saliency  define-your-terms  algorithms  criticism  to-write-about 
november 2017 by Vaguery
Knots, Links, & Learning | arbitrarilyclose
If any of this is compelling to you, I would love it if you wanted to join me in this investigation. I don’t want someone to show up and say, “here’s the final solution, dummy, it’s already all been solved,” because I’m sure that this is already a chapter in a textbook somewhere. I don’t care about that. I care about investigating for myself and with friends what’s happening. Feel free to share ideas below, and I would love it if you did some of your own drawings and joined in on the party.
learning-in-public  mathematical-recreations  looking-to-see  lovely 
november 2017 by Vaguery
[1710.00479] Factor selection by permutation
Researchers often have data measuring features xij of samples, such as test scores of students. In factor analysis and PCA, these features are thought to be influenced by unobserved factors, such as skills. Can we determine how many factors affect the data? Many approaches have been developed for this factor selection problem. The popular Parallel Analysis method randomly permutes each feature of the data. It selects factors if their singular values are larger than those of the permuted data. It is used by leading applied statisticians, including T Hastie, M Stephens, J Storey, R Tibshirani and WH Wong. Despite empirical evidence for its accuracy, there is currently no theoretical justification. This prevents us from knowing when it will work in the future.
In this paper, we show that parallel analysis consistently selects the significant factors in certain high-dimensional factor models. The intuition is that permutations keep the noise invariant, while "destroying" the low-rank signal. This provides justification for permutation methods in PCA and factor models under some conditions. A key requirement is that the factors must load on several variables. Our work points to improvements of permutation methods.
modeling  statistics  rather-interesting  information-theory  signal-processing  feature-extraction  looking-to-see  to-write-about  consider:lexicase 
october 2017 by Vaguery
[1512.03421] The extended 1-perfect trades in small hypercubes
An extended 1-perfect trade is a pair (T0,T1) of two disjoint binary distance-4 even-weight codes such that the set of words at distance 1 from T0 coincides with the set of words at distance 1 from T1. Such trade is called primary if any pair of proper subsets of T0 and T1 is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended 1-perfect trades of length 10, constant-weight extended 1-perfect trades of length 12, and Steiner trades derived from them. In particular, all Steiner trades with parameters (5,6,12) are classified.
combinatorics  to-understand  graph-theory  strings  optimization  constraint-satisfaction  looking-to-see  nudge-targets  consider:looking-to-see  consider:classification  consider:feature-discovery 
october 2017 by Vaguery
[1502.04644] Beyond the Runs Theorem
Recently, a short and elegant proof was presented showing that a binary word of length n contains at most n−3 runs. Here we show, using the same technique and a computer search, that the number of runs in a binary word of length n is at most 2223n<0.957n.
strings  proof  permutations  looking-to-see  to-write-about  nudge-targets 
october 2017 by Vaguery
[1605.04550] Finite-size effects and percolation properties of Poisson geometries
Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering and life sciences. In this work, we investigate the statistical properties of d-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case d=3. We first analyse the behaviour of the key features of these stochastic geometries as a function of the dimension d and the linear size L of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two `labels' with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster and the average cluster size.
probability-theory  computational-geometry  statistical-mechanics  looking-to-see  simulation  rather-interesting  feature-extraction  to-write-about 
october 2017 by Vaguery
[1608.08225] Why does deep and cheap learning work so well?
We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through "cheap learning" with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various "no-flattening theorems" showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss, for example, we show that n variables cannot be multiplied using fewer than 2^n neurons in a single hidden layer.
deep-learning  machine-learning  approximation  looking-to-see  algorithms  representation  rather-interesting  learning-algorithms  information-theory  to-write-about 
october 2017 by Vaguery
[1608.08087] Equisectional equivalence of triangles
We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of triangles introduced by Nakamura and Oguiso, we give characterization of equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of circles) and properties of special equivalent triangles. We also study rationality problem and constructibility problem.
plane-geometry  compass-and-straightedge  looking-to-see  rather-interesting  algebra  nudge-targets  consider:feature-discovery 
october 2017 by Vaguery
[1412.1913] A Portfolio Approach to Algorithm Selection for Discrete Time-Cost Trade-off Problem
It is a known fact that the performance of optimization algorithms for NP-Hard problems vary from instance to instance. We observed the same trend when we comprehensively studied multi-objective evolutionary algorithms (MOEAs) on a six benchmark instances of discrete time-cost trade-off problem (DTCTP) in a construction project. In this paper, instead of using a single algorithm to solve DTCTP, we use a portfolio approach that takes multiple algorithms as its constituent. We proposed portfolio comprising of four MOEAs, Non-dominated Sorting Genetic Algorithm II (NSGA-II), the strength Pareto Evolutionary Algorithm II (SPEA-II), Pareto archive evolutionary strategy (PAES) and Niched Pareto Genetic Algorithm II (NPGA-II) to solve DTCTP. The result shows that the portfolio approach is computationally fast and qualitatively superior to its constituent algorithms for all benchmark instances. Moreover, portfolio approach provides an insight in selecting the best algorithm for all benchmark instances of DTCTP.
multiobjective-optimization  benchmarking  trade-offs  looking-to-see  computational-complexity  define-your-terms  rather-interesting  to-write-about 
september 2017 by Vaguery
the power of doing things just for fun – Navigating Complexity
I recently watched a documentary called The Pleasure of Finding Things Out (1981) which focussed on the life of Richard Feynman, a physicist who has widely been called a scientific genius. I was delighted to hear him describe how valuable ‘doing things just for fun’ was to his work. He told of how, at a certain moment, he let go of trying to make his work ‘useful’ and started doing things ‘just for the fun of it’. It was then that the insights started to flow which eventually led him to winning the Nobel Prize.
via:?  ludic-impulse  looking-to-see  cultural-norms  philosophy-of-science  philosophy  psychology  social-norms 
september 2017 by Vaguery
[1708.03046] When Does the First Spurious Variable Get Selected by Sequential Regression Procedures?
Applied statisticians use sequential regression procedures to produce a ranking of explanatory variables and, in settings of low correlations between variables and strong true effect sizes, expect that variables at the very top of this ranking are true. In a regime of certain sparsity levels, however, three examples of sequential procedures---forward stepwise, the lasso, and least angle regression---are shown to include the first spurious variable unexpectedly early. We derive a rigorous, sharp prediction of the rank of the first spurious variable for the three procedures, demonstrating that the first spurious variable occurs earlier and earlier as the regression coefficients get denser. This counterintuitive phenomenon persists for independent Gaussian random designs and an arbitrarily large magnitude of the true effects. We further gain a better understanding of the phenomenon by identifying the underlying cause and then leverage the insights to introduce a simple visualization tool termed the "double-ranking diagram" to improve on sequential methods.
As a byproduct of these findings, we obtain the first provable result certifying the exact equivalence between the lasso and least angle regression in the early stages of solution paths beyond orthogonal designs. This equivalence can seamlessly carry over many important model selection results concerning the lasso to least angle regression.
statistics  variable-selection  rather-interesting  looking-to-see  algorithms  to-write-about  performance-measure  feature-construction 
august 2017 by Vaguery
Heesch Numbers, Part 2: Polyforms – Isohedral
In the first post in this series, I introduced the concept of a shape’s Heesch number. In brief, if a shape doesn’t tile the plane, its Heesch number is a measure of the maximum number of times you can surround the shape with layers of copies of itself. (Shapes that do tile are defined to have a Heesch number of infinity.) Shapes with positive, finite Heesch numbers are entertaining mathematical curiosities. Far more mysterious—and infuriating—is the fact that we know of examples of Heesch numbers only up to five, and nothing higher. Learning more about shapes with high Heesch numbers could offer insights into deep unsolved problems in tiling theory.
tiling  combinatorics  mathematical-recreations  rather-interesting  number-theory  looking-to-see  nudge-targets  consider:feature-discovery  to-write-about 
august 2017 by Vaguery
[1704.01565] Charging changes contact composition in binary sphere packings
Equal volume mixtures of small and large polytetrafluorethylene (PTFE) spheres are shaken in an atmosphere of controlled humidity which allows to also control their tribo-charging. We find that the contact numbers are charge-dependent: as the charge density of the beads increases, the number of same-type contacts decreases and the number of opposite-type contacts increases. This change is not caused by a global segregation of the sample. Hence, tribo-charging can be a way to tune the local composition of a granular material.
packing  condensed-matter  looking-to-see  experiment  rather-interesting  granular-materials  to-write-about  it's-more-complicated-than-you-think 
august 2017 by Vaguery
[1607.00363] Using smartphone pressure sensors to measure vertical velocities of elevators, stairways, and drones
We measure the vertical velocities of elevators, pedestrians climbing stairs, and drones (flying unmanned aerial vehicles), by means of smartphone pressure sensors. The barometric pressure obtained with the smartphone is related to the altitude of the device via the hydrostatic approximation. From the altitude values, vertical velocities are derived. The approximation considered is valid in the first hundred meters of the inner layers of the atmosphere. In addition to pressure, acceleration values were also recorded using the built-in accelerometer. Numerical integration was performed, obtaining both vertical velocity and altitude. We show that data obtained using the pressure sensor is significantly less noisy than that obtained using the accelerometer. Error accumulation is also evident in the numerical integration of the acceleration values. In the proposed experiments, the pressure sensor also outperforms GPS, because this sensor does not receive satellite signals indoors and, in general, the operating frequency is considerably lower than that of the pressure sensor. In the cases in which it is possible, comparison with reference values taken from the architectural plans of buildings validates the results obtained using the pressure sensor. This proposal is ideally performed as an external or outreach activity with students to gain insight about fundamental questions in mechanics, fluids, and thermodynamics.
physics  looking-to-see  experiment  rather-interesting  want  to-write-about  to-do 
august 2017 by Vaguery
[1706.04791] Obstacle-shape effect in a two-dimensional granular silo flow field
We conducted simple experiment and numerical simulation of two-dimensional granular discharge flow driven by gravity under the influence of an obstacle. According to the previous work (Zuriguel {\it et al.,\,Phys.\,Rev.\,Lett.}\,{\bf 107}: 278001, 2011), the clogging of granular discharge flow can be suppressed by putting a circular obstacle at a proper position. In order to investigate the details of obstacle effect in granular flow, we focused on particle dynamics in this study. From the experimental and numerical data, we found that the obstacle remarkably affects the horizontal-velocity distribution and packing fraction at the vicinity of the exit. In addition to the circular obstacle, we utilized triangular, inverted-triangular, and horizontal-bar obstacles to discuss the obstacle-shape effect in granular discharge flow. Based on the investigation of dynamical quantities such as velocity distributions, granular temperature, and volume fraction, we found that the triangular obstacle or horizontal bar could be very effective to prevent the clogging. From the obtained result, we consider that the detouring of particles around the obstacle and resultant low packing fraction at the exit region effectively prevent the clogging in a certain class of granular discharge flow.
granular-materials  looking-to-see  experiment  physics  rather-interesting  to-write-about 
august 2017 by Vaguery
Extractor attractor – Almost looks like work
Recently the extractor fan in my bathroom has started malfunctioning, occasionally grinding and stalling. The infuriating thing is that the grinding noise isn’t perfectly periodic – it is approximately so, but there are occasionally long gaps and the short gaps vary slightly. This lack of predictability makes the noise incredibly annoying, and hard to tune out. Before getting it fixed, I decided to investigate it a bit further.

The terminally curious may listen to the sound here:

This was recorded from my phone, you can also hear me puttering around in the background.

After dumping the audio data, I looked at the waveform and realised it was quite difficult to extract the temporal locations of the grinding noises from the volume alone. As a good physicist I therefore had another look in the frequency domain, making a spectrogram.
mathematical-recreations  looking-to-see  data-analysis  visualization  physics  nonlinear-dynamics  amusing 
august 2017 by Vaguery
Beyond subjective and objective in statistics [PDF]
Decisions in statistical data analysis are often justified, criticized, or avoided using concepts of objectivity and subjectivity. We argue that the words “objective” and “subjective” in statis- tics discourse are used in a mostly unhelpful way, and we propose to replace each of them with broader collections of attributes, with objectivity replaced by transparency, consensus, im- partiality, and correspondence to observable reality, and subjectivity replaced by awareness of multiple perspectives and context dependence. Together with stability, these make up a collection of virtues that we think is helpful in discussions of statistical foundations and practice. The advantage of these reformulations is that the replacement terms do not oppose each other and that they give more specific guidance about what statistical science strives to achieve. Instead of debating over whether a given statistical method is subjective or objective (or normatively debating the relative merits of subjectivity and objectivity in statistical practice), we can rec- ognize desirable attributes such as transparency and acknowledgment of multiple perspectives as complementary goals. We demonstrate the implications of our proposal with recent applied examples from pharmacology, election polling, and socioeconomic stratification. The aim of this paper is to push users and developers of statistical methods toward more effective use of diverse sources of information and more open acknowledgement of assumptions and goals.
statistics  philosophy-of-science  data-analysis  looking-to-see  hypothesis-testing  learning  to-read 
july 2017 by Vaguery
[1706.08224] Do GANs actually learn the distribution? An empirical study
Do GANS (Generative Adversarial Nets) actually learn the target distribution? The foundational paper of (Goodfellow et al 2014) suggested they do, if they were given sufficiently large deep nets, sample size, and computation time. A recent theoretical analysis in Arora et al (to appear at ICML 2017) raised doubts whether the same holds when discriminator has finite size. It showed that the training objective can approach its optimum value even if the generated distribution has very low support ---in other words, the training objective is unable to prevent mode collapse. The current note reports experiments suggesting that such problems are not merely theoretical. It presents empirical evidence that well-known GANs approaches do learn distributions of fairly low support, and thus presumably are not learning the target distribution. The main technical contribution is a new proposed test, based upon the famous birthday paradox, for estimating the support size of the generated distribution.
deep-learning  neural-networks  looking-to-see  probability-theory  algorithms  experiment  rather-interesting  to-write-about 
july 2017 by Vaguery
The zombie robot argument lurches on: There is no evidence that automation leads to joblessness or inequality | Economic Policy Institute
What is remarkable about this media narrative is that there is a strong desire to believe it despite so little evidence to support these claims. There clearly are serious problems in the labor market that have suppressed job and wage growth for far too long; but these problems have their roots in intentional policy decisions regarding globalization, collective bargaining, labor standards, and unemployment levels, not technology.

This report highlights the paucity of the evidence behind the alleged robot apocalypse, particularly as mischaracterized in the media coverage of the 2017 Acemoglu and Restrepo (A&R) report. Yes, automation has led to job displacements in particular occupations and industries in the past, but there is no basis for claiming that automation has led—or will lead—to increased joblessness, unemployment, or wage stagnation overall. We argue that the current excessive media attention to robots and automation destroying the jobs of the past and leaving us jobless in the future is a distraction from the main issues that need to be addressed: the poor wage growth and inequality caused by policies that have shifted economic power away from low- and moderate-wage workers. It is also the case that, as Atkinson and Wu (2017) argue, our productivity growth is too low, not too high.

Rather than debating possible problems that are more than a decade way, policymakers need to focus on addressing the decades-long crisis of wage stagnation by creating good jobs and supporting wage growth. And as it turns out, policies to expand good jobs and increase wages are the same measures needed to ensure that workers potentially displaced by automation have good jobs to transition to. For these workers, the education and training touted as solutions in the mainstream robot narrative will be inadequate, just as they were not adequate to help displaced manufacturing workers over the last few decades.
public-policy  looking-to-see  economics  automation 
june 2017 by Vaguery
[1606.03686] Does Having More Options Mean Harder to Reach Consensus?
We generalize a binary majority-vote model on adaptive networks to a plurality-vote counterpart. When opinions are uniformly distributed in the population of voters in the initial state, it is found that having more available opinions in the initial state actually accelerate the time to consensus. In particular, we investigate the three-state plurality-vote model. While time to consensus in two state model scales exponentially with population size N, for finite-size system, there is a non-zero probability that either the population reaches the consensus state in a time that is very short and independent of N (in the heterophily regime), or in a time that scales exponentially with N but is still much faster than two-state model.
voting-models  agent-based  evolutionary-economics  diversity  rather-interesting  to-write-about  looking-to-see 
may 2017 by Vaguery
[1409.4178] The frustrated brain: From dynamics on motifs to communities and networks
Cognitive function depends on an adaptive balance between flexible dynamics and integrative processes in distributed cortical networks. Patterns of zero-lag synchrony likely underpin numerous perceptual and cognitive functions. Synchronization fulfils integration by reducing entropy, whilst adaptive function mandates that a broad variety of stable states be readily accessible. Here, we elucidate two complementary influences on patterns of zero-lag synchrony that derive from basic properties of brain networks. First, mutually coupled pairs of neuronal subsystems -- resonance pairs -- promote stable zero-lag synchrony amongst the small motifs in which they are embedded, and whose effects can propagate along connected chains. Second, frustrated closed-loop motifs disrupt synchronous dynamics, enabling metastable configurations of zero-lag synchrony to coexist. We document these two complementary influences in small motifs and illustrate how these effects underpin stable versus metastable phase-synchronization patterns in prototypical modular networks and in large-scale cortical networks of the macaque (CoCoMac). We find that the variability of synchronization patterns depends on the inter-node time delay, increases with the network size, and is maximized for intermediate coupling strengths. We hypothesize that the dialectic influences of resonance versus frustration may form a dynamic substrate for flexible neuronal integration, an essential platform across diverse cognitive processes.
dynamical-systems  network-theory  coupled-oscillators  emergent-design  looking-to-see  systems-biology  nudge-targets  consider:robustness  consider:looking-to-see 
may 2017 by Vaguery
[1304.5008] Mechanisms of Zero-Lag Synchronization in Cortical Motifs
Zero-lag synchronization between distant cortical areas has been observed in a diversity of experimental data sets and between many different regions of the brain. Several computational mechanisms have been proposed to account for such isochronous synchronization in the presence of long conduction delays: Of these, the phenomenon of "dynamical relaying" - a mechanism that relies on a specific network motif - has proven to be the most robust with respect to parameter mismatch and system noise. Surprisingly, despite a contrary belief in the community, the common driving motif is an unreliable means of establishing zero-lag synchrony. Although dynamical relaying has been validated in empirical and computational studies, the deeper dynamical mechanisms and comparison to dynamics on other motifs is lacking. By systematically comparing synchronization on a variety of small motifs, we establish that the presence of a single reciprocally connected pair - a "resonance pair" - plays a crucial role in disambiguating those motifs that foster zero-lag synchrony in the presence of conduction delays (such as dynamical relaying) from those that do not (such as the common driving triad). Remarkably, minor structural changes to the common driving motif that incorporate a reciprocal pair recover robust zero-lag synchrony. The findings are observed in computational models of spiking neurons, populations of spiking neurons and neural mass models, and arise whether the oscillatory systems are periodic, chaotic, noise-free or driven by stochastic inputs. The influence of the resonance pair is also robust to parameter mismatch and asymmetrical time delays amongst the elements of the motif. We call this manner of facilitating zero-lag synchrony resonance-induced synchronization, outline the conditions for its occurrence, and propose that it may be a general mechanism to promote zero-lag synchrony in the brain.
systems-biology  dynamical-systems  coupled-oscillators  engineering-design  emergent-design  looking-to-see  nudge-targets  consider:looking-to-see  consider:robustness  consider:reQ-like-systems 
may 2017 by Vaguery
[1704.06304] k-Majority Digraphs and the Hardness of Voting with a Constant Number of Voters
Many hardness results in computational social choice make use of the fact that every directed graph may be induced as the pairwise majority relation of some preference profile. However, this fact requires a number of voters that is almost linear in the number of alternatives. It is therefore unclear whether these results remain intact when the number of voters is bounded, as is, for example, typically the case in search engine aggregation settings. In this paper, we provide a systematic study of majority digraphs for a constant number of voters resulting in analytical, experimental, and complexity-theoretic insights. First, we characterize the set of digraphs that can be induced by two and three voters, respectively, and give sufficient conditions for larger numbers of voters. Second, we present a surprisingly efficient implementation via SAT solving for computing the minimal number of voters that is required to induce a given digraph and experimentally evaluate how many voters are required to induce the majority digraphs of real-world and generated preference profiles. Finally, we leverage our sufficient conditions to show that the winner determination problem of various well-known voting rules remains hard even when there is only a small constant number of voters. In particular, we show that Kemeny's rule is hard to evaluate for 7 voters, while previous methods could only establish such a result for constant even numbers of voters.
game-theory  yeah-but-what-if-papers  existence-is-not-expectation  rather-interesting  looking-to-see  probability-theory  nudge-targets  consider:stress-testing  graph-theory  Arrow's-theorem-too  to-write-about 
may 2017 by Vaguery
[1704.08798] Word Affect Intensities
Words often convey affect -- emotions, feelings, and attitudes. Lexicons of word-affect association have applications in automatic emotion analysis and natural language generation. However, existing lexicons indicate only coarse categories of affect association. Here, for the first time, we create an affect intensity lexicon with real-valued scores of association. We use a technique called best-worst scaling that improves annotation consistency and obtains reliable fine-grained scores. The lexicon includes terms common from both general English and terms specific to social media communications. It has close to 6,000 entries for four basic emotions. We will be adding entries for other affect dimensions shortly.
natural-language-processing  looking-to-see  affect  rather-interesting  to-write-about  consider:feature-discovery 
may 2017 by Vaguery
[1703.01143] Why is it hard to beat $O(n^2)$ for Longest Common Weakly Increasing Subsequence?
The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant of the classic Longest Common Subsequence problem (LCS). Both problems can be solved with simple quadratic time algorithms. A recent line of research led to a number of matching conditional lower bounds for LCS and other related problems. However, the status of LCWIS remained open.
In this paper we show that LCWIS cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis (SETH) is false.
The ideas which we developed can also be used to obtain a lower bound based on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC circuits instead of less expressive CNF formulas.
computational-complexity  robustness  algorithms  looking-to-see  rather-interesting  to-write-about  extreme-values  outliers  hard-problems  feature-construction  nudge-targets  consider:feature-discovery 
may 2017 by Vaguery
[1307.0587] Self-Assembly on a Cylinder: A Model System for Understanding the Constraint of Commensurability
A crystal lattice, when confined to the surface of a cylinder, must have a periodic structure that is commensurate with the cylinder circumference. This constraint can frustrate the system, leading to oblique crystal lattices or to structures with a chiral seam known as a "line slip" phase, neither of which are stable for isotropic particles in equilibrium on flat surfaces. In this study, we use molecular dynamics simulations to find the steady-state structure of spherical particles with short-range repulsion and long-range attraction far below the melting temperature. We vary the range of attraction using the Lennard-Jones and Morse potentials and find that a shorter-range attraction favors the line-slip. We develop a simple model based only on geometry and bond energy to predict when the crystal or line-slip phases should appear, and find reasonable agreement with the simulations. The simplicity of this model allows us to understand the influence of the commensurability constraint, an understanding that might be extended into the more general problem of self-assembling particles in strongly confined spaces.
packing  tiling  approximation  looking-to-see  out-of-the-box  physics!  simulation  frustrated-systems  self-organization 
april 2017 by Vaguery
[1008.4633] Carryless Arithmetic Mod 10
We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .
number-theory  mathematical-recreations  looking-to-see  Martin-Gardner  nudge-targets  consider:looking-to-see  rather-interesting  to-write-about 
april 2017 by Vaguery
[0805.2128] Eight Hateful Sequences
In his July 1974 Scientific American column, Martin Gardner mentioned the Handbook of Integer Sequences, which then contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (the OEIS) contains 140000 sequences. This paper discusses eight of them, suggested by the theme of the Eighth Gathering For Gardner: they are all infinite, and all 'ateful in one way or another. Each one is connected with an unsolved problem. The sequences are related to: hateful numbers, Angelini's 1995 puzzle, the persistence of a number, Alekseyev's 123 sequence, the curling number conjecture, Quet's prime-generating recurrence, the traveling salesman's problem, and the Riemann Hypothesis.
number-theory  Martin-Gardner  mathematical-recreations  sequences  looking-to-see  nudge-targets  consider:looking-to-see 
april 2017 by Vaguery
[math/0305308] Numerical Analogues of Aronson's Sequence
Aronson's sequence 1, 4, 11, 16, ... is defined by the English sentence ``t is the first, fourth, eleventh, sixteenth, ... letter of this sentence.'' This paper introduces some numerical analogues, such as: a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition ``n is a member of the sequence if and only if a(n) is odd.'' This sequence can also be characterized by its ``square'', the sequence a^(2)(n) = a(a(n)), which equals 2n+3 for n >= 1. There are many generalizations of this sequence, some of which are new, while others throw new light on previously known sequences.
number-theory  sequences  looking-to-see  define-your-terms  nudge-targets  consider:feature-discovery  consider:performance-measures 
april 2017 by Vaguery
[math/0505295] Sloping Binary Numbers: A New Sequence Related to the Binary Numbers
If the list of binary numbers is read by upward-sloping diagonals, the resulting ``sloping binary numbers'' 0, 11, 110, 101, 100, 1111, 1010, ... (or 0, 3, 6, 5, 4, 15, 10, ...) have some surprising properties. We give formulae for the n-th term and the n-th missing term, and discuss a number of related sequences.
number-theory  looking-to-see  to-write-about  nudge-targets  consider:feature-discovery 
april 2017 by Vaguery
[1106.5531] Balls in Boxes: Variations on a Theme of Warren Ewens and Herbert Wilf
We comment on, elaborate, and extend the work of Warren Ewens and Herbert Wilf, described in their this http URL about the maximum in balls-and-boxes problem. In particular we meta-apply their ingenious method to show that it is not really needed, and that one is better off using the so-called Poisson Approximation, at least in applications to the real world, because extremely unlikely events mever happen in real life. This article is accompanied by the Maple package this http URL">BallsInBoxes.
probability-theory  statistics  hey-I-know-this-guy  combinatorics  looking-to-see  define-your-terms  to-write-about  mathematical-recreations  nudge-targets  consider:rediscovery 
april 2017 by Vaguery
[1304.1226] On Euler's "Misleading Induction", Andrews' "Fix", and How to Fully Automate them
One of the greatest experimental mathematicians of all time was also one of the greatest mathematicians of all time, the great Leonhard Euler. Usually he had an uncanny intuition on how many "special cases" one needs before one can formulate a plausible conjecture, but one time he was "almost fooled", only to find out that his conjecture was premature.
In 1990, George Andrews found a way to "correct" Euler. Here we show how to generate, AUTOMATICALLY, rigorously-proved Euler-Andrews Style formulas, that enables one to generate Euler-style "cautionary tales" about the "danger" of using naive empirical induction. Ironically, the way we prove the Andrews-style corrections is empirical! But in order to turn the empirical proof into a full-fledged rigorous proof, we must make sure that we check sufficiently many (but still not that many!) special cases.
combinatorics  looking-to-see  induction  philosophy-of-science  pattern-discovery  nudge-targets  consider:looking-to-see  to-write-about  mathematical-recreations 
april 2017 by Vaguery
[1502.04377] The Method(!) of "Guess and Check"
The problems of enumerating lattice walks, with an arbitrary finite set of allowed steps, both in one and two dimensions, where one must always stay in the non-negative half-line and quarter-plane respectively, are used, as case studies, to illustrate the `naive' methodology of guess-and-check, where rigorous proofs are possible, but not worth the trouble. We argue that this is a metaphor for future math.
philosophy-of-science  mathematical-recreations  rather-interesting  to-write-about  looking-to-see 
april 2017 by Vaguery
[1703.04792] Frustration and thermalisation in an artificial magnetic quasicrystal
We have created and studied artificial magnetic quasicrystals based on Penrose tiling patterns of interacting nanomagnets that lack the translational symmetry of spatially periodic artificial spin ices. Vertex-level degeneracy and frustration induced by the network topology of the Penrose pattern leads to a low energy configuration that we propose as a ground state. It has two parts, a quasi-one-dimensional rigid "skeleton" that spans the entire pattern and is capable of long-range order, and clusters of macrospins within it that are degenerate in a nearest neighbour model, and so are "flippable". These lead to macroscopic degeneracy for the array as a whole. Magnetic force microscopy imaging of Penrose tiling arrays revealed superdomains that are larger for more strongly coupled arrays. The superdomain size is larger after AC-demagnetisation and especially after annealing the array above its blocking temperature.
experiment  frustrated-networks  complexology  quasicrystals  looking-to-see  rather-interesting 
april 2017 by Vaguery
[1704.00828] A Probabilistic Linear Genetic Programming with Stochastic Context-Free Grammar for solving Symbolic Regression problems
Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a fitter individual. Probabilistic Model Building Genetic Programming (PMB-GP) methods were proposed to overcome this issue through a probability model that captures the structure of the fit individuals and use it to sample new individuals. This work proposes the use of LGP with a Stochastic Context-Free Grammar (SCFG), that has a probability distribution that is updated according to selected individuals. We proposed a method for adapting the grammar into the linear representation of LGP. Tests performed with the proposed probabilistic method, and with two hybrid approaches, on several symbolic regression benchmark problems show that the results are statistically better than the obtained by the traditional LGP.
genetic-programming  search-operators  generative-models  looking-to-see  rather-interesting  algorithms  metaheuristics 
april 2017 by Vaguery
The GRIM test — a method for evaluating published research. – Medium
Ever had one of those ideas where you thought: “No, this is too simple, someone must have already thought of it.”
And then found that no-one had?
And that it was a good idea after all?
Well, that’s what happened to us.
(Who is us? I’m going to use pronouns messily through the following almost-3000 words, but let the record show: ‘us’ and ‘we’ is Nick Brown and myself.)
The pre-print of this paper is HERE — “The GRIM test: A simple technique detects numerous anomalies in the reporting of results in psychology”.
via:arthegall  statistics  looking-to-see  inverse-problems  rather-interesting  academic-culture  publishing  science  peer-review 
march 2017 by Vaguery
[1310.5971] Jammed lattice sphere packings
We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.
statistical-mechanics  physics!  packing  simulation  looking-to-see 
march 2017 by Vaguery
You Can’t Break Math – dy/dan

I haven’t found a way to generate these kinds of insights about math without surrounding myself with people learning math for the first time.
One of my most enduring shortcomings as a teacher is how much I plan and revise those plans, even if the lesson I have on file will suffice. I’ll chase a scintilla of an improvement for hours, which was never sustainable. I spent most of the previous day prepping this Desmos activity. We used 10% of it.
Language from the day that I’m still pondering: “We cancel the 2x’s because we want to get x by itself.” I’m trying to decide if those italicized expressions contribute to a student’s understanding of large ideas about mathematics or of small ideas about solving a particular kind of equation.
pedagogy  the-mangle-in-practice  learning-by-doing  looking-to-see  rather-interesting  mathematics  to-write-about 
february 2017 by Vaguery
[1604.04722] Where is the global corporate elite? A large-scale network study of local and nonlocal interlocking directorates
Business elites reconfigure their locus of organization over time, from the city level, to the national level, and beyond. We ask what the current level of elite organization is and propose a novel theoretical and empirical approach to answer this question. Building on the universal distinction between local and nonlocal ties we use network analysis and community detection to dissect the global network of interlocking directorates among over five million firms. We find that elite orientation is indeed changing from the national to the transnational plane, but we register a considerable heterogeneity across different regions in the world. In some regions the business communities are organized along national borders, whereas in other areas the locus of organization is at the city level or international level. London dominates the global corporate elite network. Our findings underscore that the study of corporate elites requires an approach that is sensitive to levels of organization that go beyond the confines of nation states.
social-networks  rather-interesting  corporatism  globalism  looking-to-see  to-write-about 
february 2017 by Vaguery
[1701.01329] Generating Focussed Molecule Libraries for Drug Discovery with Recurrent Neural Networks
In de novo drug design, computational strategies are used to generate novel molecules with good affinity to the desired biological target. In this work, we show that recurrent neural networks can be trained as generative models for molecular structures, similar to statistical language models in natural language processing. We demonstrate that the properties of the generated molecules correlate very well with the properties of the molecules used to train the model. In order to enrich libraries with molecules active towards a given biological target, we propose to fine-tune the model with small sets of molecules, which are known to be active against that target.
Against Staphylococcus aureus, the model reproduced 14% of 6051 hold-out test molecules that medicinal chemists designed, whereas against Plasmodium falciparum (Malaria) it reproduced 28% of 1240 test molecules. When coupled with a scoring function, our model can perform the complete de novo drug design cycle to generate large sets of novel molecules for drug discovery.
pharmaceutical  engineering-design  machine-learning  neural-networks  feature-construction  nudge-targets  consider:representation  consider:looking-to-see  looking-to-see  cheminformatics  party-like-its-1999 
february 2017 by Vaguery
[1606.03225] In a search for a shape maximizing packing fraction for two-dimensional random sequential adsorption
Random sequential adsorption (RSA) of various two dimensional objects is studied in order to find a shape which maximizes the saturated packing fraction. This investigation was begun in our previous paper [Cie\'sla et al., Phys. Chem. Chem. Phys. 17, 24376 (2015)], where the densest packing was studied for smoothed dimers. Here this shape is compared with a smoothed n-mers, spherocylinders and ellipses. It is found that the highest packing fraction out of the studied shapes is 0.58405±0.0001 and is obtained for ellipses having long-to-short axis ratio of 1.85, which is also the largest anisotropy among the investigated shapes.
packing  simulation  looking-to-see  granular-materials  rather-interesting  phase-transitions  nudge-targets  consider:looking-to-see  consider:symbolic-regression  consider:robustness 
december 2016 by Vaguery
[1507.03323] Boolean Gossiping Networks
This paper proposes and investigates a Boolean gossiping model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossiping process defines absorbing Markov chains. These results illustrate possibilities of relating detailed dynamical properties of Boolean networks to graphical properties of the underlying interactions.
boolean-networks  Kauffmania  network-theory  systems-biology  looking-to-see  rather-interesting  dynamical-systems  nudge-targets  consider:feature-discovery  consider:approximation 
november 2016 by Vaguery
Image Synthesis from Yahoo's open_nsfw
I remember proposing almost this exact conjecture to Cosma Shalizi and D. Eric Smith, sitting in the garden at SFI in about 2000. It makes me happy to see it coming into existence. I certainly never got around to it.
image-processing  generative-art  generative-models  deep-learning  aesthetics  looking-to-see  the-posthuman-aesthetic 
october 2016 by Vaguery
[1605.06396] Soft Covering with High Probability
Wyner's soft-covering lemma is the central analysis step for achievability proofs of information theoretic security, resolvability, and channel synthesis. It can also be used for simple achievability proofs in lossy source coding. This work sharpens the claim of soft-covering by moving away from an expected value analysis. Instead, a random codebook is shown to achieve the soft-covering phenomenon with high probability. The probability of failure is super-exponentially small in the block-length, enabling many applications through the union bound. This work gives bounds for both the exponential decay rate of total variation and the second-order codebook rate for soft covering.
information-theory  looking-to-see  rather-interesting  heuristics  approximation  nudge-targets  consider:looking-to-see  consider:feature-discovery 
july 2016 by Vaguery
[1606.07262] On the Theoretical Capacity of Evolution Strategies to Statistically Learn the Landscape Hessian
We study the theoretical capacity to statistically learn local landscape information by Evolution Strategies (ESs). Specifically, we investigate the covariance matrix when constructed by ESs operating with the selection operator alone. We model continuous generation of candidate solutions about quadratic basins of attraction, with deterministic selection of the decision vectors that minimize the objective function values. Our goal is to rigorously show that accumulation of winning individuals carries the potential to reveal valuable information about the search landscape, e.g., as already practically utilized by derandomized ES variants. We first show that the statistically-constructed covariance matrix over such winning decision vectors shares the same eigenvectors with the Hessian matrix about the optimum. We then provide an analytic approximation of this covariance matrix for a non-elitist multi-child (1,λ)-strategy, which holds for a large population size λ. Finally, we also numerically corroborate our results.
evolutionary-algorithms  looking-to-see  machine-learning  optimization  formalization 
july 2016 by Vaguery
[1604.08658] Dependence between External Path-Length and Size in Random Tries
We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results that the internal path length is totally positively correlated to the size and that both tend to the same normal limit law. These two examples provide concrete instances of bivariate normal distributions (as limit laws) whose correlation is 0, 1 and periodically oscillating.
probability-theory  data-structures  looking-to-see  computer-science  pattern-formation  rather-odd  nudge-targets  consider:looking-to-see 
june 2016 by Vaguery
Could a neuroscientist understand a microprocessor? | bioRxiv
There is a popular belief in neuroscience that we are primarily data limited, that producing large, multimodal, and complex datasets will, enabled by data analysis algorithms, lead to fundamental insights into the way the brain processes information. Microprocessors are among those artificial information processing systems that are both complex and that we understand at all levels, from the overall logical flow, via logical gates, to the dynamics of transistors. Here we take a simulated classical microprocessor as a model organism, and use our ability to perform arbitrary experiments on it to see if popular data analysis methods from neuroscience can elucidate the way it processes information. We show that the approaches reveal interesting structure in the data but do not meaningfully describe the hierarchy of information processing in the processor. This suggests that current approaches in neuroscience may fall short of producing meaningful models of the brain.
via:numerous  neural-networks  inference  modeling  experiment  rather-interesting  big-data  bioinformatics  looking-to-see 
may 2016 by Vaguery
[1605.03390] Variance of the Internal Profile in Suffix Trees
The precise analysis of the variance of the profile of a suffix tree has been a longstanding open problem. We analyze three regimes of the asymptotic growth of the variance of the profile of a suffix tree built from a randomly generated binary string, in the nonuniform case. We utilize combinatorics on words, singularity analysis, and the Mellin transform.
data-structures  algorithms  computer-science  combinatorics  looking-to-see  nudge-targets  consider:feature-discovery 
may 2016 by Vaguery
[1201.6035] How Accurate is inv(A)*b?
Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse inv(A) is inaccurate. Virtually all other textbooks on numerical analysis and numerical linear algebra advise against using computed inverses without stating whether this is accurate or not. In fact, under reasonable assumptions on how the inverse is computed, x = inv(A)*b is as accurate as the solution computed by the best backward-stable solvers. This fact is not new, but obviously obscure. We review the literature on the accuracy of this computation and present a self-contained numerical analysis of it.
numerical-methods  looking-to-see  approximation  rather-interesting  error  algorithms  nudge-targets  consider:looking-to-see  consider:performance-measures 
may 2016 by Vaguery
[1601.04273] Statistical-mechanical Analysis of Linear Programming Relaxation for Combinatorial Optimization Problems
Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for α=2 that the LP optimal solution is typically equal to that given by the IP below the critical average degree c=e in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result c=1, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with α≥3, minimum vertex covers on α-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree c=e/(α−1) where the replica symmetry is broken.
linear-programming  relaxation  approximation  looking-to-see  rather-interesting  consider:Nk-models  nudge-targets  consider:feature-discovery 
april 2016 by Vaguery
[1509.03327] Optimal Strategy in "Guess Who?": Beyond Binary Search
"Guess Who?" is a popular two player game where players ask "Yes"/"No" questions to search for their opponent's secret identity from a pool of possible candidates. This is modeled as a simple stochastic game. Using this model, the optimal strategy is explicitly found. Contrary to popular belief, performing a binary search is \emph{not} always optimal. Instead, the optimal strategy for the player who trails is to make certain bold plays in an attempt catch up. This is discovered by first analyzing a continuous version of the game where players play indefinitely and the winner is never decided after finitely many rounds.
game-theory  probability-theory  planning  looking-to-see  nudge-targets  consider:strategies 
february 2016 by Vaguery
[1601.03420] Critical fluctuations in proteins native states
We study a large data set of protein structure ensembles of very diverse sizes determined by nuclear magnetic resonance. By examining the distance-dependent correlations in the displacement of residues pairs and conducting finite size scaling analysis it was found that the correlations and susceptibility behave as in systems near a critical point implying that, at the native state, the motion of each amino acid residue is felt by every other residue up to the size of the protein molecule. Furthermore certain protein's shapes corresponding to maximum susceptibility were found to be more probable than others. Overall the results suggest that the protein's native state is critical, implying that despite being posed near the minimum of the energy landscape, they still preserve their dynamic flexibility.
biophysics  protein-folding  physics  phase-transitions  self-organization  looking-to-see  nonlinear-dynamics 
february 2016 by Vaguery
[1507.08245] Epistasis and the structure of fitness landscapes: are experimental fitness landscapes compatible with Fisher's model?
The fitness landscape defines the relationship between genotypes and fitness in a given environment, and underlies fundamental quantities such as the distribution of selection coefficient, or the magnitude and type of epistasis. A better understanding of variation of landscape structure across species and environments is thus necessary to understand and predict how populations adapt. An increasing number of experiments access the properties of fitness landscapes by identifying mutations, constructing genotypes with combinations of these mutations, and measuring fitness of these genotypes. Yet these empirical landscapes represent a very small sample of the vast space of all possible genotypes, and this sample is biased by the protocol used to identify mutations. Here we develop a rigorous and flexible statistical framework based on Approximate Bayesian Computation to address these concerns, and use this framework to fit a broad class of phenotypic fitness models (including Fisher's model) to 24 empirical landscapes representing 9 diverse biological systems. In spite of uncertainty due to the small size of most published empirical landscapes, the inferred landscapes have similar structure in similar biological systems. Surprisingly, goodness of fit tests reveal that this class of phenotypic models, which has been successful so far in interpreting experimental data, is a plausible model in only 3 out of 9 biological systems. In most cases, including notably the landscapes of drug resistance, Fisher's model is not able to explain the structure of empirical landscapes and patterns of epistasis.
fitness-landscapes  looking-to-see  theory-and-practice-sitting-in-a-tree  theoretical-biology  population-biology  modeling-is-not-mathematics 
february 2016 by Vaguery
[1509.04316] Sums of seven octahedral numbers
We show that for a large class of cubic polynomials f, every sufficiently large number can be written as a sum of seven positive values of f. As a special case, we show that every number greater than e107 is a sum of seven positive octahedral numbers, where an octahedral number is a number of the form 2x3+x3, reducing an open problem due to Pollock to a finite computation.
number-theory  existence-proof  looking-to-see  nudge-targets  consider:looking-to-see 
december 2015 by Vaguery
[1512.00908] Lattice Surfaces and smallest triangles
We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than .05 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects.
tiling  algebra  integer-lattices  computational-geometry  looking-to-see  rather-interesting  nudge-targets  consider:feature-discovery 
december 2015 by Vaguery
[1507.04381] Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks
Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted in order to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based upon the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. The understanding of how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe here a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically-equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an electro-optic experiment on a 5 node network that confirms the synchronization patterns predicted by the theory.
complexology  network-theory  coupled-oscillators  nonlinear-dynamics  structure-and-function  looking-to-see  nudge-targets  consider:looking-to-see  consider:feature-discovery 
december 2015 by Vaguery
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