**dynamical-systems**273

[1709.05601] Markov Brains: A Technical Introduction

9 hours ago by Vaguery

Markov Brains are a class of evolvable artificial neural networks (ANN). They differ from conventional ANNs in many aspects, but the key difference is that instead of a layered architecture, with each node performing the same function, Markov Brains are networks built from individual computational components. These computational components interact with each other, receive inputs from sensors, and control motor outputs. The function of the computational components, their connections to each other, as well as connections to sensors and motors are all subject to evolutionary optimization. Here we describe in detail how a Markov Brain works, what techniques can be used to study them, and how they can be evolved.

representation
evolutionary-algorithms
dynamical-systems
emergent-design
to-write-about
to-invite-to-speak
hey-I-know-these-folks
9 hours ago by Vaguery

Metrical properties for a class of continued fractions with increasing digits - ScienceDirect

25 days ago by Vaguery

In 2002, Hartono, Kraaikamp and Schweiger introduced the Engel continued fractions (ECF), whose partial quotients are increasing. Later, Schweiger generalized it into a class of continued fractions with increasing digits and a parameter ϵ, called generalized continued fractions (GCF). In this paper, we will give some arithmetic properties of such an expansion, and show that the GCF holds similar metric properties with ECF under the condition that

. But when it comes to the condition that

, it does not.

continued-fractions
number-theory
to-understand
algorithms
dynamical-systems
to-write-about
. But when it comes to the condition that

, it does not.

25 days ago by Vaguery

[1606.05099] Invariant measures for continued fraction algorithms with finitely many digits

25 days ago by Vaguery

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
25 days ago by Vaguery

[1710.05183] Inferring Mesoscale Models of Neural Computation

november 2017 by Vaguery

Recent years have seen dramatic progress in the development of techniques for measuring the activity and connectivity of large populations of neurons in the brain. However, as these techniques grow ever more powerful---allowing us to even contemplate measuring every neuron in entire brain---a new problem arises: how do we make sense of the mountains of data that these techniques produce? Here, we argue that the time is ripe for building an intermediate or "mesoscale" computational theory that can bridge between single-cell (microscale) accounts of neural function and behavioral (macroscale) accounts of animal cognition and environmental complexity. Just as digital accounts of computation in conventional computers abstract away the non-essential dynamics of the analog circuits that implementing gates and registers, so too a computational account of animal cognition can afford to abstract from the non-essential dynamics of neurons. We argue that the geometry of neural circuits is essential in explaining the computational limitations and technological innovations inherent in biological information processing. We propose a blueprint for how to employ tools from modern machine learning to automatically infer a satisfying mesoscale account of neural computation that combines functional and structural data, with an emphasis on learning and exploiting regularity and repeating motifs in neuronal circuits. Rather than suggest a specific theory, we present a new class of scientific instruments that can enable neuroscientists to design, propose, implement and test mesoscale theories of neural computation.

dynamical-systems
machine-learning
deep-learning
representation
temporal-models
rather-interesting
inference
to-write-about
november 2017 by Vaguery

Understanding LSTM Networks -- colah's blog

november 2017 by Vaguery

Humans don’t start their thinking from scratch every second. As you read this essay, you understand each word based on your understanding of previous words. You don’t throw everything away and start thinking from scratch again. Your thoughts have persistence.

Traditional neural networks can’t do this, and it seems like a major shortcoming. For example, imagine you want to classify what kind of event is happening at every point in a movie. It’s unclear how a traditional neural network could use its reasoning about previous events in the film to inform later ones.

recurrent-neural-nets
machine-learning
architecture
dynamical-systems
explainer
Traditional neural networks can’t do this, and it seems like a major shortcoming. For example, imagine you want to classify what kind of event is happening at every point in a movie. It’s unclear how a traditional neural network could use its reasoning about previous events in the film to inform later ones.

november 2017 by Vaguery

[1711.00963] The Computational Complexity of Finding Separators in Temporal Graphs

november 2017 by Vaguery

Vertex separators, that is, vertex sets whose deletion disconnects two distinguished vertices in a graph, play a pivotal role in algorithmic graph theory. For instance, the concept of tree decompositions of graphs is tightly connected to the separator concept. For many realistic models of the real world, however, it is necessary to consider graphs whose edge set changes with time. More specifically, the edges are labeled with time stamps. In the literature, these graphs are referred to as temporal graphs, temporal networks, time-varying networks, edge-scheduled networks, etc. While there is an extensive literature on separators in "static" graphs, much less is known for the temporal setting. Building on previous work (e.g., Kempe et al. [STOC '00]), for the first time we systematically investigate the (parameterized) complexity of finding separators in temporal graphs. Doing so, we discover a rich landscape of computationally (fixed-parameter) tractable and intractable cases. In particular, we shed light on the so far seemingly overlooked fact that two frequently used models of temporal separation may lead to quite significant differences in terms of computational complexity. More specifically, considering paths in temporal graphs one may distinguish between strict paths (the time stamps along a path are strictly increasing) and non-strict paths (the time stamps along a path are monotonically non-decreasing). We observe that the corresponding strict case of temporal separators leads to several computationally much easier to handle cases than the non-strict case does.

graph-theory
combinatorics
dynamical-systems
robustness
nudge-targets
heuristics
algorithms
to-write-about
november 2017 by Vaguery

[1310.1166] Flipping Edge-Labelled Triangulations

october 2017 by Vaguery

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips.

We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ(nlogn) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ(n) bounds in their respective unlabelled settings. The Ω(nlogn) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O(log2n), although we no longer have a matching lower bound.

Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of O(n2) on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter Ω(n3).

combinatorics
dynamical-systems
computational-complexity
rather-interesting
consider:looking-to-see
consider:simulation
to-write-about
to-draw
We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ(nlogn) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ(n) bounds in their respective unlabelled settings. The Ω(nlogn) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O(log2n), although we no longer have a matching lower bound.

Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of O(n2) on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter Ω(n3).

october 2017 by Vaguery

[1208.2762] Computing by Temporal Order: Asynchronous Cellular Automata

october 2017 by Vaguery

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity).

Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule.

We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.

cellular-automata
rather-interesting
asynchronous
collective-behavior
out-of-the-box
dynamical-systems
representation
to-write-about
Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule.

We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.

october 2017 by Vaguery

[1208.2762v1] Computing by Temporal Order: Asynchronous Cellular Automata

october 2017 by Vaguery

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity).

Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule.

We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.

cellular-automata
rather-interesting
out-of-the-box
asynchronous
collective-behavior
dynamical-systems
nudge-targets
consider:looking-to-see
to-write-about
Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule.

We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.

october 2017 by Vaguery

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