arXiv subjects

Maxim Raginsky

Publications and source records attributed to Maxim Raginsky.

At least 73 records · Page 4Linked to original sources

arXiv2015

On MMSE estimation from quantized observations in the nonasymptotic regime

This paper studies MMSE estimation on the basis of quantized noisy observations. It presents nonasymptotic bounds on MMSE regret due to quantization for two settings: (1) estimation of a scalar random variable given a quantized vector of $n$ conditionally independent observations, and (2) estimation of a $p$-dimensional random vector given a quantized vector of $n$ observations (not necessarily independent) when the full MMSE estimator has a subgaussian concentration property.

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arXiv2015

Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)

During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory, information theory, theoretical computer science, and learning theory. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors. The first part of the monograph introduces classical concentration inequalities for martingales, as well as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication. The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities, which underlie the so-called functional approach to concentration of measure, and then from a complementary information-theoretic viewpoint based on transportation-cost inequalities and probability in metric spaces. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, we discuss several applications of the entropy method to problems in communications and coding, including strong converses, empirical distributions of good channel codes, and an information-theoretic converse for concentration of measure.

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arXiv2015

Online discrete optimization in social networks in the presence of Knightian uncertainty

We study a model of collective real-time decision-making (or learning) in a social network operating in an uncertain environment, for which no a priori probabilistic model is available. Instead, the environment's impact on the agents in the network is seen through a sequence of cost functions, revealed to the agents in a causal manner only after all the relevant actions are taken. There are two kinds of costs: individual costs incurred by each agent and local-interaction costs incurred by each agent and its neighbors in the social network. Moreover, agents have inertia: each agent has a default mixed strategy that stays fixed regardless of the state of the environment, and must expend effort to deviate from this strategy in order to respond to cost signals coming from the environment. We construct a decentralized strategy, wherein each agent selects its action based only on the costs directly affecting it and on the decisions made by its neighbors in the network. In this setting, we quantify social learning in terms of regret, which is given by the difference between the realized network performance over a given time horizon and the best performance that could have been achieved in hindsight by a fictitious centralized entity with full knowledge of the environment's evolution. We show that our strategy achieves the regret that scales polylogarithmically with the time horizon and polynomially with the number of agents and the maximum number of neighbors of any agent in the social network.

math.OC↗

arXiv2014

Poisson's equation in nonlinear filtering

The aim of this paper is to provide a variational interpretation of the nonlinear filter in continuous time. A time-stepping procedure is introduced, consisting of successive minimization problems in the space of probability densities. The weak form of the nonlinear filter is derived via analysis of the first-order optimality conditions for these problems. The derivation shows the nonlinear filter dynamics may be regarded as a gradient flow, or a steepest descent, for a certain energy functional with respect to the Kullback-Leibler divergence. The second part of the paper is concerned with derivation of the feedback particle filter algorithm, based again on the analysis of the first variation. The algorithm is shown to be exact. That is, the posterior distribution of the particle matches exactly the true posterior, provided the filter is initialized with the true prior.

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arXiv2014

Online Markov decision processes with Kullback-Leibler control cost

This paper considers an online (real-time) control problem that involves an agent performing a discrete-time random walk over a finite state space. The agent's action at each time step is to specify the probability distribution for the next state given the current state. Following the set-up of Todorov, the state-action cost at each time step is a sum of a state cost and a control cost given by the Kullback-Leibler (KL) divergence between the agent's next-state distribution and that determined by some fixed passive dynamics. The online aspect of the problem is due to the fact that the state cost functions are generated by a dynamic environment, and the agent learns the current state cost only after selecting an action. An explicit construction of a computationally efficient strategy with small regret (i.e., expected difference between its actual total cost and the smallest cost attainable using noncausal knowledge of the state costs) under mild regularity conditions is presented, along with a demonstration of the performance of the proposed strategy on a simulated target tracking problem. A number of new results on Markov decision processes with KL control cost are also obtained.

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arXiv2012

A recursive procedure for density estimation on the binary hypercube

This paper describes a recursive estimation procedure for multivariate binary densities (probability distributions of vectors of Bernoulli random variables) using orthogonal expansions. For $d$ covariates, there are $2^d$ basis coefficients to estimate, which renders conventional approaches computationally prohibitive when $d$ is large. However, for a wide class of densities that satisfy a certain sparsity condition, our estimator runs in probabilistic polynomial time and adapts to the unknown sparsity of the underlying density in two key ways: (1) it attains near-minimax mean-squared error for moderate sample sizes, and (2) the computational complexity is lower for sparser densities. Our method also allows for flexible control of the trade-off between mean-squared error and computational complexity.

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arXiv2012

Empirical processes, typical sequences and coordinated actions in standard Borel spaces

This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable "test functions." In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers) and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios, in which the relevant operational criteria pertain to reproducing empirical averages of a general-alphabet stationary memoryless source with respect to a suitable function class.

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arXiv2012

Target Detection Performance Bounds in Compressive Imaging

This paper describes computationally efficient approaches and associated theoretical performance guarantees for the detection of known targets and anomalies from few projection measurements of the underlying signals. The proposed approaches accommodate signals of different strengths contaminated by a colored Gaussian background, and perform detection without reconstructing the underlying signals from the observations. The theoretical performance bounds of the target detector highlight fundamental tradeoffs among the number of measurements collected, amount of background signal present, signal-to-noise ratio, and similarity among potential targets coming from a known dictionary. The anomaly detector is designed to control the number of false discoveries. The proposed approach does not depend on a known sparse representation of targets; rather, the theoretical performance bounds exploit the structure of a known dictionary of targets and the distance preservation property of the measurement matrix. Simulation experiments illustrate the practicality and effectiveness of the proposed approaches.

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arXiv2012

Sequential anomaly detection in the presence of noise and limited feedback

This paper describes a methodology for detecting anomalies from sequentially observed and potentially noisy data. The proposed approach consists of two main elements: (1) {\em filtering}, or assigning a belief or likelihood to each successive measurement based upon our ability to predict it from previous noisy observations, and (2) {\em hedging}, or flagging potential anomalies by comparing the current belief against a time-varying and data-adaptive threshold. The threshold is adjusted based on the available feedback from an end user. Our algorithms, which combine universal prediction with recent work on online convex programming, do not require computing posterior distributions given all current observations and involve simple primal-dual parameter updates. At the heart of the proposed approach lie exponential-family models which can be used in a wide variety of contexts and applications, and which yield methods that achieve sublinear per-round regret against both static and slowly varying product distributions with marginals drawn from the same exponential family. Moreover, the regret against static distributions coincides with the minimax value of the corresponding online strongly convex game. We also prove bounds on the number of mistakes made during the hedging step relative to the best offline choice of the threshold with access to all estimated beliefs and feedback signals. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality, as well as on the Enron email dataset.

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arXiv2011

Directed information and Pearl's causal calculus

Probabilistic graphical models are a fundamental tool in statistics, machine learning, signal processing, and control. When such a model is defined on a directed acyclic graph (DAG), one can assign a partial ordering to the events occurring in the corresponding stochastic system. Based on the work of Judea Pearl and others, these DAG-based "causal factorizations" of joint probability measures have been used for characterization and inference of functional dependencies (causal links). This mostly expository paper focuses on several connections between Pearl's formalism (and in particular his notion of "intervention") and information-theoretic notions of causality and feedback (such as causal conditioning, directed stochastic kernels, and directed information). As an application, we show how conditional directed information can be used to develop an information-theoretic version of Pearl's "back-door" criterion for identifiability of causal effects from passive observations. This suggests that the back-door criterion can be thought of as a causal analog of statistical sufficiency.

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arXiv2011

Information-based complexity, feedback and dynamics in convex programming

We study the intrinsic limitations of sequential convex optimization through the lens of feedback information theory. In the oracle model of optimization, an algorithm queries an {\em oracle} for noisy information about the unknown objective function, and the goal is to (approximately) minimize every function in a given class using as few queries as possible. We show that, in order for a function to be optimized, the algorithm must be able to accumulate enough information about the objective. This, in turn, puts limits on the speed of optimization under specific assumptions on the oracle and the type of feedback. Our techniques are akin to the ones used in statistical literature to obtain minimax lower bounds on the risks of estimation procedures; the notable difference is that, unlike in the case of i.i.d. data, a sequential optimization algorithm can gather observations in a {\em controlled} manner, so that the amount of information at each step is allowed to change in time. In particular, we show that optimization algorithms often obey the law of diminishing returns: the signal-to-noise ratio drops as the optimization algorithm approaches the optimum. To underscore the generality of the tools, we use our approach to derive fundamental lower bounds for a certain active learning problem. Overall, the present work connects the intuitive notions of information in optimization, experimental design, estimation, and active learning to the quantitative notion of Shannon information.

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arXiv2011

Performance bounds for expander-based compressed sensing in Poisson noise

This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. We support our results with experimental demonstrations of reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process.

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arXiv2010

Divergence-based characterization of fundamental limitations of adaptive dynamical systems

Adaptive dynamical systems arise in a multitude of contexts, e.g., optimization, control, communications, signal processing, and machine learning. A precise characterization of their fundamental limitations is therefore of paramount importance. In this paper, we consider the general problem of adaptively controlling and/or identifying a stochastic dynamical system, where our {\em a priori} knowledge allows us to place the system in a subset of a metric space (the uncertainty set). We present an information-theoretic meta-theorem that captures the trade-off between the metric complexity (or richness) of the uncertainty set, the amount of information acquired online in the process of controlling and observing the system, and the residual uncertainty remaining after the observations have been collected. Following the approach of Zames, we quantify {\em a priori} information by the Kolmogorov (metric) entropy of the uncertainty set, while the information acquired online is expressed as a sum of information divergences. The general theory is used to derive new minimax lower bounds on the metric identification error, as well as to give a simple derivation of the minimum time needed to stabilize an uncertain stochastic linear system.

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arXiv2010

Compressed sensing performance bounds under Poisson noise

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical $\ell_2$--$\ell_1$ minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity- and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the signal-dependent part of the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.

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arXiv2010

Fishing in Poisson streams: focusing on the whales, ignoring the minnows

This paper describes a low-complexity approach for reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process. Assuming that the rate vector of this Poisson process is sparse or approximately sparse, the goal is to maintain a compressed summary of the process sample paths using a small number of counters, such that at any time it is possible to reconstruct both the total number of packets in each flow and the underlying rate vector. We show that these tasks can be accomplished efficiently and accurately using compressed sensing with expander graphs. In particular, the compressive counts are a linear transformation of the underlying counting process by the adjacency matrix of an unbalanced expander. Such a matrix is binary and sparse, which allows for efficient incrementing when new packets arrive. We describe, analyze, and compare two methods that can be used to estimate both the current vector of total packet counts and the underlying vector of arrival rates.

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arXiv2009

Performance Bounds for Expander-based Compressed Sensing in the presence of Poisson Noise

This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. The geometry of the expander graphs makes them provably superior to random dense sensing matrices, such as Gaussian or partial Fourier ensembles, for the Poisson noise model. We support our results with experimental demonstrations.

cs.IT↗

arXiv2009

Performance bounds on compressed sensing with Poisson noise

This paper describes performance bounds for compressed sensing in the presence of Poisson noise when the underlying signal, a vector of Poisson intensities, is sparse or compressible (admits a sparse approximation). The signal-independent and bounded noise models used in the literature to analyze the performance of compressed sensing do not accurately model the effects of Poisson noise. However, Poisson noise is an appropriate noise model for a variety of applications, including low-light imaging, where sensing hardware is large or expensive, and limiting the number of measurements collected is important. In this paper, we describe how a feasible positivity-preserving sensing matrix can be constructed, and then analyze the performance of a compressed sensing reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which could be used as a measure of signal sparsity.

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arXiv2009

Achievability results for statistical learning under communication constraints

The problem of statistical learning is to construct an accurate predictor of a random variable as a function of a correlated random variable on the basis of an i.i.d. training sample from their joint distribution. Allowable predictors are constrained to lie in some specified class, and the goal is to approach asymptotically the performance of the best predictor in the class. We consider two settings in which the learning agent only has access to rate-limited descriptions of the training data, and present information-theoretic bounds on the predictor performance achievable in the presence of these communication constraints. Our proofs do not assume any separation structure between compression and learning and rely on a new class of operational criteria specifically tailored to joint design of encoders and learning algorithms in rate-constrained settings.

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