arXiv subjects

Andrea Montanari

Publications and source records attributed to Andrea Montanari.

At least 163 records · Page 9Linked to original sources

arXiv2011

Optimal coding for the deletion channel with small deletion probability

The deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Input bits are deleted independently with probability d, and when they are not deleted, they are not affected by the channel. Despite significant effort, little is known about the capacity of this channel, and even less about optimal coding schemes. In this paper we develop a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute three leading terms of this expansion, and find an input distribution that achieves capacity up to this order. This constitutes the first optimal coding result for the deletion channel. The key idea employed is the following: We understand perfectly the deletion channel with deletion probability d=0. It has capacity 1 and the optimal input distribution is i.i.d. Bernoulli(1/2). It is natural to expect that the channel with small deletion probabilities has a capacity that varies smoothly with d, and that the optimal input distribution is obtained by smoothly perturbing the i.i.d. Bernoulli(1/2) process. Our results show that this is indeed the case. We think that this general strategy can be useful in a number of capacity calculations.

cs.IT↗

arXiv2011

Subexponential convergence for information aggregation on regular trees

We consider the decentralized binary hypothesis testing problem on trees of bounded degree and increasing depth. For a regular tree of depth t and branching factor k>=2, we assume that the leaves have access to independent and identically distributed noisy observations of the 'state of the world' s. Starting with the leaves, each node makes a decision in a finite alphabet M, that it sends to its parent in the tree. Finally, the root decides between the two possible states of the world based on the information it receives. We prove that the error probability vanishes only subexponentially in the number of available observations, under quite general hypotheses. More precisely the case of binary messages, decay is subexponential for any decision rule. For general (finite) message alphabet M, decay is subexponential for 'node-oblivious' decision rules, that satisfy a mild irreducibility condition. In the latter case, we propose a family of decision rules with close-to-optimal asymptotic behavior.

cs.MA↗

arXiv2011

Compressed Sensing over $\ell_p$-balls: Minimax Mean Square Error

We consider the compressed sensing problem, where the object $x_0 \in \bR^N$ is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$. A popular method of sparsity-promoting reconstruction is $\ell^1$-penalized least-squares reconstruction (aka LASSO, Basis Pursuit). It is currently popular to consider the strict sparsity model, where the object $x_0$ is nonzero in only a small fraction of entries. In this paper, we instead consider the much more broadly applicable $\ell_p$-sparsity model, where $x_0$ is sparse in the sense of having $\ell_p$ norm bounded by $ξ\cdot N^{1/p}$ for some fixed $0 < p \leq 1$ and $ξ> 0$. We study an asymptotic regime in which $n$ and $N$ both tend to infinity with limiting ratio $n/N = δ\in (0,1)$, both in the noisy ($z \neq 0$) and noiseless ($z=0$) cases. Under weak assumptions on $x_0$, we are able to precisely evaluate the worst-case asymptotic minimax mean-squared reconstruction error (AMSE) for $\ell^1$ penalized least-squares: min over penalization parameters, max over $\ell_p$-sparse objects $x_0$. We exhibit the asymptotically least-favorable object (hardest sparse signal to recover) and the maximin penalization. Our explicit formulas unexpectedly involve quantities appearing classically in statistical decision theory. Occurring in the present setting, they reflect a deeper connection between penalized $\ell^1$ minimization and scalar soft thresholding. This connection, which follows from earlier work of the authors and collaborators on the AMP iterative thresholding algorithm, is carefully explained. Our approach also gives precise results under weak-$\ell_p$ ball coefficient constraints, as we show here.

cs.IT↗

arXiv2011

Gossip PCA

Eigenvectors of data matrices play an important role in many computational problems, ranging from signal processing to machine learning and control. For instance, algorithms that compute positions of the nodes of a wireless network on the basis of pairwise distance measurements require a few leading eigenvectors of the distances matrix. While eigenvector calculation is a standard topic in numerical linear algebra, it becomes challenging under severe communication or computation constraints, or in absence of central scheduling. In this paper we investigate the possibility of computing the leading eigenvectors of a large data matrix through gossip algorithms. The proposed algorithm amounts to iteratively multiplying a vector by independent random sparsification of the original matrix and averaging the resulting normalized vectors. This can be viewed as a generalization of gossip algorithms for consensus, but the resulting dynamics is significantly more intricate. Our analysis is based on controlling the convergence to stationarity of the associated Kesten-Furstenberg Markov chain.

cs.DS↗

arXiv2011

Graphical Models Concepts in Compressed Sensing

This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ell_1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on `Compressed Sensing' edited by Yonina Eldar and Gitta Kutyniok.

cs.IT↗

arXiv2011

Information Theoretic Limits on Learning Stochastic Differential Equations

Consider the problem of learning the drift coefficient of a stochastic differential equation from a sample path. In this paper, we assume that the drift is parametrized by a high dimensional vector. We address the question of how long the system needs to be observed in order to learn this vector of parameters. We prove a general lower bound on this time complexity by using a characterization of mutual information as time integral of conditional variance, due to Kadota, Zakai, and Ziv. This general lower bound is applied to specific classes of linear and non-linear stochastic differential equations. In the linear case, the problem under consideration is the one of learning a matrix of interaction coefficients. We evaluate our lower bound for ensembles of sparse and dense random matrices. The resulting estimates match the qualitative behavior of upper bounds achieved by computationally efficient procedures.

cs.IT↗

arXiv2011

The dynamics of message passing on dense graphs, with applications to compressed sensing

Approximate message passing algorithms proved to be extremely effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper we provide the first rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with large number of short loops in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory.

cs.IT↗

arXiv2010

Tight Thresholds for Cuckoo Hashing via XORSAT

We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k >= 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds alpha_k such that, as n goes to infinity, if n/m <= alpha for some alpha < alpha_k then a hash table can be constructed successfully with high probability, and if n/m >= alpha for some alpha > alpha_k a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of k > 2 by showing that they are in fact the same as the previously known thresholds for the random k-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.

cs.DS↗

arXiv2010

Lossy compression of discrete sources via Viterbi algorithm

We present a new lossy compressor for discrete-valued sources. For coding a sequence $x^n$, the encoder starts by assigning a certain cost to each possible reconstruction sequence. It then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor. The cost of each sequence is a linear combination of its distance from the sequence $x^n$ and a linear function of its $k^{\rm th}$ order empirical distribution. The structure of the cost function allows the encoder to employ the Viterbi algorithm to recover the minimizer of the cost. We identify a choice of the coefficients comprising the linear function of the empirical distribution used in the cost function which ensures that the algorithm universally achieves the optimum rate-distortion performance of any stationary ergodic source in the limit of large $n$, provided that $k$ diverges as $o(\log n)$. Iterative techniques for approximating the coefficients, which alleviate the computational burden of finding the optimal coefficients, are proposed and studied.

cs.IT↗

arXiv2010

Learning Networks of Stochastic Differential Equations

We consider linear models for stochastic dynamics. To any such model can be associated a network (namely a directed graph) describing which degrees of freedom interact under the dynamics. We tackle the problem of learning such a network from observation of the system trajectory over a time interval $T$. We analyze the $\ell_1$-regularized least squares algorithm and, in the setting in which the underlying network is sparse, we prove performance guarantees that are \emph{uniform in the sampling rate} as long as this is sufficiently high. This result substantiates the notion of a well defined `time complexity' for the network inference problem.

math.ST↗

arXiv2010

Ising models on locally tree-like graphs

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.

math.PR↗

arXiv2010

On the concentration of the number of solutions of random satisfiability formulas

Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $α$. Assume that the probability that $F$ is unsatisfiable is $O(1/\log(n)^{1+\e})$ for $\e>0$. We show that (possibly excluding a countable set of `exceptional' $α$'s) the number of solutions concentrate in the logarithmic scale, i.e., there exists a non-random function $ϕ(α)$ such that, for any $δ>0$, $(1/n)\log Z(F)\in [ϕ-δ,ϕ+δ]$ with high probability. In particular, the assumption holds for all $α<1$, which proves the above concentration claim in the whole satisfiability regime of random $2$-SAT. We also extend these results to a broad class of constraint satisfaction problems. The proof is based on an interpolation technique from spin-glass theory, and on an application of Friedgut's theorem on sharp thresholds for graph properties.

cs.DM↗

arXiv2010

On the deletion channel with small deletion probability

The deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Despite significant effort, little is known about its capacity, and even less about optimal coding schemes. In this paper we intiate a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute two leading terms of this expansion, and show that capacity is achieved, up to this order, by i.i.d. uniform random distribution of the input. We think that this strategy can be useful in a number of capacity calculations.

cs.IT↗

arXiv2010

A Natural Dynamics for Bargaining on Exchange Networks

Bargaining networks model the behavior of a set of players that need to reach pairwise agreements for making profits. Nash bargaining solutions are special outcomes of such games that are both stable and balanced. Kleinberg and Tardos proved a sharp algorithmic characterization of such outcomes, but left open the problem of how the actual bargaining process converges to them. A partial answer was provided by Azar et al. who proposed a distributed algorithm for constructing Nash bargaining solutions, but without polynomial bounds on its convergence rate. In this paper, we introduce a simple and natural model for this process, and study its convergence rate to Nash bargaining solutions. At each time step, each player proposes a deal to each of her neighbors. The proposal consists of a share of the potential profit in case of agreement. The share is chosen to be balanced in Nash's sense as far as this is feasible (with respect to the current best alternatives for both players). We prove that, whenever the Nash bargaining solution is unique (and satisfies a positive gap condition) this dynamics converges to it in polynomial time. Our analysis is based on an approximate decoupling phenomenon between the dynamics on different substructures of the network. This approach may be of general interest for the analysis of local algorithms on networks.

cs.GT↗

arXiv2010

Applications of Lindeberg Principle in Communications and Statistical Learning

We use a generalization of the Lindeberg principle developed by Sourav Chatterjee to prove universality properties for various problems in communications, statistical learning and random matrix theory. We also show that these systems can be viewed as the limiting case of a properly defined sparse system. The latter result is useful when the sparse systems are easier to analyze than their dense counterparts. The list of problems we consider is by no means exhaustive. We believe that the ideas can be used in many other problems relevant for information theory.

cs.IT↗

arXiv2010

The Noise-Sensitivity Phase Transition in Compressed Sensing

Consider the noisy underdetermined system of linear equations: y=Ax0 + z0, with n x N measurement matrix A, n < N, and Gaussian white noise z0 ~ N(0,σ^2 I). Both y and A are known, both x0 and z0 are unknown, and we seek an approximation to x0. When x0 has few nonzeros, useful approximations are obtained by l1-penalized l2 minimization, in which the reconstruction \hxl solves min || y - Ax||^2/2 + λ||x||_1. Evaluate performance by mean-squared error (MSE = E ||\hxl - x0||_2^2/N). Consider matrices A with iid Gaussian entries and a large-system limit in which n,N\to\infty with n/N \to δand k/n \to ρ. Call the ratio MSE/σ^2 the noise sensitivity. We develop formal expressions for the MSE of \hxl, and evaluate its worst-case formal noise sensitivity over all types of k-sparse signals. The phase space 0 < δ, ρ< 1 is partitioned by curve ρ= \rhoMSE(δ) into two regions. Formal noise sensitivity is bounded throughout the region ρ< \rhoMSE(δ) and is unbounded throughout the region ρ> \rhoMSE(δ). The phase boundary ρ= \rhoMSE(δ) is identical to the previously-known phase transition curve for equivalence of l1 - l0 minimization in the k-sparse noiseless case. Hence a single phase boundary describes the fundamental phase transitions both for the noiseless and noisy cases. Extensive computational experiments validate the predictions of this formalism, including the existence of game theoretical structures underlying it. Underlying our formalism is the AMP algorithm introduced earlier by the authors. Other papers by the authors detail expressions for the formal MSE of AMP and its close connection to l1-penalized reconstruction. Here we derive the minimax formal MSE of AMP and then read out results for l1-penalized reconstruction.

math.ST↗

arXiv2010

Regularization for Matrix Completion

We consider the problem of reconstructing a low rank matrix from noisy observations of a subset of its entries. This task has applications in statistical learning, computer vision, and signal processing. In these contexts, "noise" generically refers to any contribution to the data that is not captured by the low-rank model. In most applications, the noise level is large compared to the underlying signal and it is important to avoid overfitting. In order to tackle this problem, we define a regularized cost function well suited for spectral reconstruction methods. Within a random noise model, and in the large system limit, we prove that the resulting accuracy undergoes a phase transition depending on the noise level and on the fraction of observed entries. The cost function can be minimized using OPTSPACE (a manifold gradient descent algorithm). Numerical simulations show that this approach is competitive with state-of-the-art alternatives.

stat.ML↗

arXiv2009

The weak limit of Ising models on locally tree-like graphs

We consider the Ising model with inverse temperature beta and without external field on sequences of graphs G_n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weak converges to the symmetric mixture of the Ising model with + boundary conditions and the - boundary conditions on the k-regular tree with inverse temperature β. In the case where the graphs G_n are expanders we derive a more detailed understanding by showing convergence of the Ising measure condition on positive magnetization (sum of spins) to the + measure on the tree.

math.PR↗