arXiv subjects

Yury Polyanskiy

Publications and source records attributed to Yury Polyanskiy.

At least 91 records · Page 5Linked to original sources

arXiv2020

Information Storage in the Stochastic Ising Model

Most information storage devices write data by modifying the local state of matter, in the hope that sub-atomic local interactions stabilize the state for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution of physical states in magnetic storage media affects their capacity, this work initiates the study of information retention in locally-interacting particle systems. The system dynamics follow the stochastic Ising model (SIM) over a 2-dimensional $\sqrt{n}\times\sqrt{n}$ grid. The initial spin configuration $X_0$ serves as the user-controlled input. The output configuration $X_t$ is produced by running $t$ steps of Glauber dynamics. Our main goal is to evaluate the information capacity $I_n(t):=\max_{p_{X_0}}I(X_0;X_t)$ when time $t$ scales with the system's size $n$. While the positive (but low) temperature regime is our main interest, we start by exploring the simpler zero-temperature dynamics. We first show that at zero temperature, order of $\sqrt{n}$ bits can be stored in the system indefinitely by coding over stable, striped configurations. While $\sqrt{n}$ is order optimal for infinite time, backing off to $t<\infty$, higher orders of $I_n(t)$ are achievable. First, linear coding arguments imply that $I_n(t) = Θ(n)$ for $t=O(n)$. To go beyond the linear scale, we develop a droplet-based achievability scheme that reliably stores $Ω\left(n/\log n\right)$ for $t=O(n\log n)$ time ($\log n$ can be replaced with any $o(n)$ function). Moving to the positive but low temperature regime, two main results are provided. First, we show that an initial configuration drawn from the Gibbs measure cannot retain more than a single bit for $t\geq \exp(Cβn^{1/4+ε})$ time. On the other hand, when scaling time with the inverse temperature $β$, the stripe-based coding scheme is shown to retain its bits for $e^{cβ}$.

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arXiv2020

Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models

Introduced by Kiefer and Wolfowitz \cite{KW56}, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture odels and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme form of overparameterization. In this paper we discover a surprising property of the NPMLE solution. Consider, for example, a Gaussian mixture model on the real line with a subgaussian mixing distribution. Leveraging complex-analytic techniques, we show that with high probability the NPMLE based on a sample of size $n$ has $O(\log n)$ atoms (mass points), significantly improving the deterministic upper bound of $n$ due to Lindsay \cite{lindsay1983geometry1}. Notably, any such Gaussian mixture is statistically indistinguishable from a finite one with $O(\log n)$ components (and this is tight for certain mixtures). Thus, absent any explicit form of model selection, NPMLE automatically chooses the right model complexity, a property we term \emph{self-regularization}. Extensions to other exponential families are given. As a statistical application, we show that this structural property can be harnessed to bootstrap existing Hellinger risk bound of the (parametric) MLE for finite Gaussian mixtures to the NPMLE for general Gaussian mixtures, recovering a result of Zhang \cite{zhang2009generalized}.

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arXiv2020

Note on approximating the Laplace transform of a Gaussian on a complex disk

In this short note we study how well a Gaussian distribution can be approximated by distributions supported on $[-a,a]$. Perhaps, the natural conjecture is that for large $a$ the almost optimal choice is given by truncating the Gaussian to $[-a,a]$. Indeed, such approximation achieves the optimal rate of $e^{-Θ(a^2)}$ in terms of the $L_\infty$-distance between characteristic functions. However, if we consider the $L_\infty$-distance between Laplace transforms on a complex disk, the optimal rate is $e^{-Θ(a^2 \log a)}$, while truncation still only attains $e^{-Θ(a^2)}$. The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of $Θ(a^2)$ components with means in $[-a,a]$ and whose density has all derivatives bounded by $e^{-Ω(a^2 \log(a))}$ in the $O(1)$-neighborhood of the origin.

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arXiv2020

A Note on the Probability of Rectangles for Correlated Binary Strings

Consider two sequences of $n$ independent and identically distributed fair coin tosses, $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$, which are $ρ$-correlated for each $j$, i.e. $\mathbb{P}[X_j=Y_j] = {1+ρ\over 2}$. We study the question of how large (small) the probability $\mathbb{P}[X \in A, Y\in B]$ can be among all sets $A,B\subset\{0,1\}^n$ of a given cardinality. For sets $|A|,|B| = Θ(2^n)$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|A|,|B| = 2^{Θ(n)}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb{P}[X \in A, Y\in B]$ in the regime of $ρ\to 1$. We also prove a similar tight lower bound, i.e. show that for $ρ\to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb{P}[X \in A, Y\in B]$.

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arXiv2020

Attracting Random Walks

This paper introduces the Attracting Random Walks model, which describes the dynamics of a system of particles on a graph with $n$ vertices. At each step, a single particle moves to an adjacent vertex (or stays at the current one) with probability proportional to the exponent of the number of other particles at a vertex. From an applied standpoint, the model captures the rich get richer phenomenon. We show that the Markov chain exhibits a phase transition in mixing time, as the parameter governing the attraction is varied. Namely, mixing time is $O(n\log n)$ when the temperature is sufficiently high and $\exp(Ω(n))$ when temperature is sufficiently low. When $\mathcal{G}$ is the complete graph, the model is a projection of the Potts model, whose mixing properties and the critical temperature have been known previously. However, for any other graph our model is non-reversible and does not seem to admit a simple Gibbsian description of a stationary distribution. Notably, we demonstrate existence of the dynamic phase transition without decomposing the stationary distribution into phases.

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arXiv2020

Application of information-percolation method to reconstruction problems on graphs

In this paper we propose a method of proving impossibility results based on applying strong data-processing inequalities to estimate mutual information between sets of variables forming certain Markov random fields. The end result is that mutual information between two "far away" (as measured by the graph distance) variables is bounded by the probability of the existence of an open path in a bond-percolation problem on the same graph. Furthermore, stronger bounds can be obtained by establishing mutual information comparison results with an erasure model on the same graph, with erasure probabilities given by the contraction coefficients. As applications, we show that our method gives sharp threshold for partially recovering a rank-one perturbation of a random Gaussian matrix (spiked Wigner model), yields the best known upper bound on the noise level for group synchronization (obtained concurrently by Abbe and Boix), and establishes new impossibility result for community detection on the stochastic block model with $k$ communities.

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arXiv2020

Extrapolating the profile of a finite population

We study a prototypical problem in empirical Bayes. Namely, consider a population consisting of $k$ individuals each belonging to one of $k$ types (some types can be empty). Without any structural restrictions, it is impossible to learn the composition of the full population having observed only a small (random) subsample of size $m = o(k)$. Nevertheless, we show that in the sublinear regime of $m =ω(k/\log k)$, it is possible to consistently estimate in total variation the \emph{profile} of the population, defined as the empirical distribution of the sizes of each type, which determines many symmetric properties of the population. We also prove that in the linear regime of $m=c k$ for any constant $c$ the optimal rate is $Θ(1/\log k)$. Our estimator is based on Wolfowitz's minimum distance method, which entails solving a linear program (LP) of size $k$. We show that there is a single infinite-dimensional LP whose value simultaneously characterizes the risk of the minimum distance estimator and certifies its minimax optimality. The sharp convergence rate is obtained by evaluating this LP using complex-analytic techniques.

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arXiv2020

Broadcasting on trees near criticality

We revisit the problem of broadcasting on $d$-ary trees: starting from a Bernoulli$(1/2)$ random variable $X_0$ at a root vertex, each vertex forwards its value across binary symmetric channels $\mathrm{BSC}_δ$ to $d$ descendants. The goal is to reconstruct $X_0$ given the vector $X_{L_h}$ of values of all variables at depth $h$. It is well known that reconstruction (better than a random guess) is possible as $h\to \infty$ if and only if $δ< δ_c(d)$. In this paper, we study the behavior of the mutual information and the probability of error when $δ$ is slightly subcritical. The innovation of our work is application of the recently introduced "less-noisy" channel comparison techniques. For example, we are able to derive the positive part of the phase transition (reconstructability when $δ<δ_c$) using purely information-theoretic ideas. This is in contrast with previous derivations, which explicitly analyze distribution of the Hamming weight of $X_{L_h}$ (a so-called Kesten-Stigum bound).

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arXiv2020

Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_σ$, for $\mathcal{N}_σ\triangleq\mathcal{N}(0,σ^2 \mathrm{I}_d)$, by $\hat{P}_n\ast\mathcal{N}_σ$, where $\hat{P}_n$ is the empirical measure, under different statistical distances. The convergence is examined in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $χ^2$-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf{W}_1$) converges at rate $e^{O(d)}n^{-\frac{1}{2}}$ in remarkable contrast to a typical $n^{-\frac{1}{d}}$ rate for unsmoothed $\mathsf{W}_1$ (and $d\ge 3$). For the KL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and $χ^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$ achieves finite input-output $χ^2$ mutual information across the additive white Gaussian noise channel. If the latter condition is not met, the rate changes to $ω(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, while the $χ^2$-divergence becomes infinite - a curious dichotomy. As a main application we consider estimating the differential entropy $h(P\ast\mathcal{N}_σ)$ in the high-dimensional regime. The distribution $P$ is unknown but $n$ i.i.d samples from it are available. We first show that any good estimator of $h(P\ast\mathcal{N}_σ)$ must have sample complexity that is exponential in $d$. Using the empirical approximation results we then show that the absolute-error risk of the plug-in estimator converges at the parametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimax rate-optimality of the plug-in. Numerical results that demonstrate a significant empirical superiority of the plug-in approach to general-purpose differential entropy estimators are provided.

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arXiv2020

The Information Bottleneck Problem and Its Applications in Machine Learning

Inference capabilities of machine learning (ML) systems skyrocketed in recent years, now playing a pivotal role in various aspect of society. The goal in statistical learning is to use data to obtain simple algorithms for predicting a random variable $Y$ from a correlated observation $X$. Since the dimension of $X$ is typically huge, computationally feasible solutions should summarize it into a lower-dimensional feature vector $T$, from which $Y$ is predicted. The algorithm will successfully make the prediction if $T$ is a good proxy of $Y$, despite the said dimensionality-reduction. A myriad of ML algorithms (mostly employing deep learning (DL)) for finding such representations $T$ based on real-world data are now available. While these methods are often effective in practice, their success is hindered by the lack of a comprehensive theory to explain it. The information bottleneck (IB) theory recently emerged as a bold information-theoretic paradigm for analyzing DL systems. Adopting mutual information as the figure of merit, it suggests that the best representation $T$ should be maximally informative about $Y$ while minimizing the mutual information with $X$. In this tutorial we survey the information-theoretic origins of this abstract principle, and its recent impact on DL. For the latter, we cover implications of the IB problem on DL theory, as well as practical algorithms inspired by it. Our goal is to provide a unified and cohesive description. A clear view of current knowledge is particularly important for further leveraging IB and other information-theoretic ideas to study DL models.

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arXiv2020

Sample complexity of population recovery

The problem of population recovery refers to estimating a distribution based on incomplete or corrupted samples. Consider a random poll of sample size $n$ conducted on a population of individuals, where each pollee is asked to answer $d$ binary questions. We consider one of the two polling impediments: (a) in lossy population recovery, a pollee may skip each question with probability $ε$, (b) in noisy population recovery, a pollee may lie on each question with probability $ε$. Given $n$ lossy or noisy samples, the goal is to estimate the probabilities of all $2^d$ binary vectors simultaneously within accuracy $δ$ with high probability. This paper settles the sample complexity of population recovery. For lossy model, the optimal sample complexity is $\tildeΘ(δ^{-2\max\{\fracε{1-ε},1\}})$, improving the state of the art by Moitra and Saks in several ways: a lower bound is established, the upper bound is improved and the result depends at most on the logarithm of the dimension. Surprisingly, the sample complexity undergoes a phase transition from parametric to nonparametric rate when $ε$ exceeds $1/2$. For noisy population recovery, the sharp sample complexity turns out to be more sensitive to dimension and scales as $\exp(Θ(d^{1/3} \log^{2/3}(1/δ)))$ except for the trivial cases of $ε=0,1/2$ or $1$. For both models, our estimators simply compute the empirical mean of a certain function, which is found by pre-solving a linear program (LP). Curiously, the dual LP can be understood as Le Cam's method for lower-bounding the minimax risk, thus establishing the statistical optimality of the proposed estimators. The value of the LP is determined by complex-analytic methods.

math.ST↗

arXiv2020

Broadcasting on Random Directed Acyclic Graphs

We study a generalization of the well-known model of broadcasting on trees. Consider a directed acyclic graph (DAG) with a unique source vertex $X$, and suppose all other vertices have indegree $d\geq 2$. Let the vertices at distance $k$ from $X$ be called layer $k$. At layer $0$, $X$ is given a random bit. At layer $k\geq 1$, each vertex receives $d$ bits from its parents in layer $k-1$, which are transmitted along independent binary symmetric channel edges, and combines them using a $d$-ary Boolean processing function. The goal is to reconstruct $X$ with probability of error bounded away from $1/2$ using the values of all vertices at an arbitrarily deep layer. This question is closely related to models of reliable computation and storage, and information flow in biological networks. In this paper, we analyze randomly constructed DAGs, for which we show that broadcasting is only possible if the noise level is below a certain degree and function dependent critical threshold. For $d\geq 3$, and random DAGs with layer sizes $Ω(\log k)$ and majority processing functions, we identify the critical threshold. For $d=2$, we establish a similar result for NAND processing functions. We also prove a partial converse for odd $d\geq 3$ illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex. Finally, for any noise level, we construct explicit DAGs (using expander graphs) with bounded degree and layer sizes $Θ(\log k)$ admitting reconstruction. In particular, we show that such DAGs can be generated in deterministic quasi-polynomial time or randomized polylogarithmic time in the depth. These results portray a doubly-exponential advantage for storing a bit in DAGs compared to trees, where $d=1$ but layer sizes must grow exponentially with depth in order to enable broadcasting.

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arXiv2020

Massive Access for Future Wireless Communication Systems

Multiple access technology played an important role in wireless communication in the last decades: it increases the capacity of the channel and allows different users to access the system simultaneously. However, the conventional multiple access technology, as originally designed for current human-centric wireless networks, is not scalable for future machine-centric wireless networks. Massive access (studied in the literature under such names as massive-device multiple access, unsourced massive random access, massive connectivity, massive machine-type communication, and many-access channels) exhibits a clean break with current networks by potentially supporting millions of devices in each cellular network. The tremendous growth in the number of connected devices requires a fundamental rethinking of the conventional multiple access technologies in favor of new schemes suited for massive random access. Among the many new challenges arising in this setting, the most relevant are: the fundamental limits of communication from a massive number of bursty devices transmitting simultaneously with short packets, the design of low complexity and energy-efficient massive access coding and communication schemes, efficient methods for the detection of a relatively small number of active users among a large number of potential user devices with sporadic transmission pattern, and the integration of massive access with massive MIMO and other important wireless communication technologies. This paper presents an overview of the concept of massive access wireless communication and of the contemporary research on this important topic.

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arXiv2019

On Low Density Majority Codes

We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful") input-output BER curves (as opposed to threshold-like curves typical for long error-correcting codes). To that end we introduce the notion of Low Density Majority Codes (LDMCs). These codes are non-linear sparse-graph codes, which output majority function evaluated on randomly chosen small subsets of the data bits. This is similar to Low Density Generator Matrix codes (LDGMs), except that the XOR function is replaced with the majority. We show that even with a few iterations of belief propagation (BP) the attained input-output curves provably improve upon performance of any linear systematic code. The effect of non-linearity bootstraping the initial iterations of BP, suggests that LDMCs should improve performance in various applications, where LDGMs have been used traditionally (e.g., pre-coding for optics, tornado raptor codes, protograph constructions). As a side result of separate interest we establish a lower (impossibility) bound for the achievable BER of a systematic linear code at one value of erasure noise given its BER at another value. We show that this new bound is superior to the results inferred from the area theorem for EXIT functions.

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arXiv2019

Low density majority codes and the problem of graceful degradation

We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful'') input-output BER curves (as opposed to threshold-like curves typical for long error-correcting codes). This paper restricts attention to binary erasure channels (BEC) and contains three contributions. First, we introduce the notion of Low Density Majority Codes (LDMCs). These codes are non-linear sparse-graph codes, which output majority function evaluated on randomly chosen small subsets of the data bits. This is similar to Low Density Generator Matrix codes (LDGMs), except that the XOR function is replaced with the majority. We show that even with a few iterations of belief propagation (BP) the attained input-output curves provably improve upon performance of any linear systematic code. The effect of non-linearity bootstraping the initial iterations of BP, suggests that LDMCs should improve performance in various applications, where LDGMs have been used traditionally. Second, we establish several \textit{two-point converse bounds} that lower bound the BER achievable at one erasure probability as a function of BER achieved at another one. The novel nature of our bounds is that they are specific to subclasses of codes (linear systematic and non-linear systematic) and outperform similar bounds implied by the area theorem for the EXIT function.

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arXiv2019

Information-Distilling Quantizers

Let $X$ and $Y$ be dependent random variables. This paper considers the problem of designing a scalar quantizer for $Y$ to maximize the mutual information between the quantizer's output and $X$, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low $I(X;Y)$, where it is shown that, if $X$ is binary, a constant fraction of the mutual information can always be preserved using $\mathcal{O}(\log(1/I(X;Y)))$ quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets $2 < |\mathcal{X}| < \infty$, it is established that an $η$-fraction of the mutual information can be preserved using roughly $(\log(| \mathcal{X} | /I(X;Y)))^{η\cdot(|\mathcal{X}| - 1)}$ quantization levels.

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arXiv2019

A Lower Bound on the Expected Distortion of Joint Source-Channel Coding

We consider the classic joint source-channel coding problem of transmitting a memoryless source over a memoryless channel. The focus of this work is on the long-standing open problem of finding the rate of convergence of the smallest attainable expected distortion to its asymptotic value, as a function of blocklength $n$. Our main result is that in general the convergence rate is not faster than $n^{-1/2}$. In particular, we show that for the problem of transmitting i.i.d uniform bits over a binary symmetric channels with Hamming distortion, the smallest attainable distortion (bit error rate) is at least $Ω(n^{-1/2})$ above the asymptotic value, if the ``bandwidth expansion ratio'' is above $1$.

cs.IT↗

arXiv2019

Sampling of the Wiener Process for Remote Estimation over a Channel with Random Delay

In this paper, we consider a problem of sampling a Wiener process, with samples forwarded to a remote estimator over a channel that is modeled as a queue. The estimator reconstructs an estimate of the real-time signal value from causally received samples. We study the optimal online sampling strategy that minimizes the mean square estimation error subject to a sampling rate constraint. We prove that the optimal sampling strategy is a threshold policy, and find the optimal threshold. This threshold is determined by how much the Wiener process varies during the random service time and the maximum allowed sampling rate. Further, if the sampling times are independent of the observed Wiener process, the above sampling problem for minimizing the estimation error is equivalent to a sampling problem for minimizing the age of information. This reveals an interesting connection between the age of information and remote estimation error. Our comparisons show that the estimation error achieved by the optimal sampling policy can be much smaller than those of age-optimal sampling, zero-wait sampling, and periodic sampling.

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