arXiv subjects

David L. Donoho

Publications and source records attributed to David L. Donoho.

33 records · Page 2Linked to original sources

arXiv2013

Information-Theoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing

We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. \cite{KrzakalaEtAl}, message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate $δ$ exceeds the (upper) Rényi information dimension of the signal, $\uRenyi(p_X)$. More precisely, for a sequence of signals of diverging dimension $n$ whose empirical distribution converges to $p_X$, reconstruction is with high probability successful from $\uRenyi(p_X)\, n+o(n)$ measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension $n$ and $k(n)$ non-zero entries, this implies reconstruction from $k(n)+o(n)$ measurements. For `discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from $o(n)$ measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal $p_X$.

cs.IT↗

arXiv2010

Microlocal Analysis of the Geometric Separation Problem

Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically `pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations. We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the $\ell_1$ norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by $\ell_1$ minimization; microlocal phase space helps organize the calculations that cluster coherence requires.

math.FA↗

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↗

arXiv2009

Message Passing Algorithms for Compressed Sensing: I. Motivation and Construction

In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.

cs.IT↗

arXiv2009

Message Passing Algorithms for Compressed Sensing: II. Analysis and Validation

In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the second of two conference papers describing the derivation of these algorithms, connection with related literature, extensions of original framework, and new empirical evidence. This paper describes the state evolution formalism for analyzing these algorithms, and some of the conclusions that can be drawn from this formalism. We carried out extensive numerical simulations to confirm these predictions. We present here a few representative results.

cs.IT↗

arXiv2009

Optimally Tuned Iterative Reconstruction Algorithms for Compressed Sensing

We conducted an extensive computational experiment, lasting multiple CPU-years, to optimally select parameters for two important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations available at {\tt sparselab.stanford.edu}; they run `out of the box' with no user tuning: it is not necessary to select thresholds or know the likely degree of sparsity. Our class of algorithms includes iterative hard and soft thresholding with or without relaxation, as well as CoSaMP, subspace pursuit and some natural extensions. As a result, our optimally tuned algorithms dominate such proposals. Our notion of optimality is defined in terms of phase transitions, i.e. we maximize the number of nonzeros at which the algorithm can successfully operate. We show that the phase transition is a well-defined quantity with our suite of random underdetermined linear systems. Our tuning gives the highest transition possible within each class of algorithms.

math.NA↗

arXiv2009

Message Passing Algorithms for Compressed Sensing

Compressed sensing aims to undersample certain high-dimensional signals, yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization -- which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity-undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity-undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity-undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity-undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this new, apparently very different theoretical formalism.

cs.IT↗

arXiv2009

Observed Universality of Phase Transitions in High-Dimensional Geometry, with Implications for Modern Data Analysis and Signal Processing

We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now-ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity tradeoff in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume the underlying matrices have independent and identically distributed (iid) Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are {\it universal} across a range of underlying matrix ensembles. The experimental results are consistent with an asymptotic large-$n$ universality across matrix ensembles; finite-sample universality can be rejected.

math.ST↗

arXiv2009

Does median filtering truly preserve edges better than linear filtering?

Image processing researchers commonly assert that "median filtering is better than linear filtering for removing noise in the presence of edges." Using a straightforward large-$n$ decision-theory framework, this folk-theorem is seen to be false in general. We show that median filtering and linear filtering have similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel noise level in a digital signal. To see dramatic benefits of median smoothing in an asymptotic setting, the per-pixel noise level should tend to zero (i.e., SNR should grow very large). We show that a two-stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges. In this two-stage procedure, the first pass, at a fine scale, aims at increasing the SNR. The second pass, at a coarser scale, correctly exploits the nonlinearity of the median. Image processing methods based on nonlinear partial differential equations (PDEs) are often said to improve on linear filtering in the presence of edges. Such methods seem difficult to analyze rigorously in a decision-theoretic framework. A popular example is mean curvature motion (MCM), which is formally a kind of iterated median filtering. Our results on iterated median filtering suggest that some PDE-based methods are candidates to rigorously outperform linear filtering in an asymptotic framework.

math.ST↗

arXiv2008

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Let $A$ be an $n$ by $N$ real valued random matrix, and $\h$ denote the $N$-dimensional hypercube. For numerous random matrix ensembles, the expected number of $k$-dimensional faces of the random $n$-dimensional zonotope $A\h$ obeys the formula $E f_k(A\h) /f_k(\h) = 1-P_{N-n,N-k}$, where $P_{N-n,N-k}$ is a fair-coin-tossing probability. The formula applies, for example, where the columns of $A$ are drawn i.i.d. from an absolutely continuous symmetric distribution. The formula exploits Wendel's Theorem\cite{We62}. Let $\po$ denote the positive orthant; the expected number of $k$-faces of the random cone$A \po$ obeys $ {\cal E} f_k(A\po) /f_k(\po) = 1 - P_{N-n,N-k}$. The formula applies to numerous matrix ensembles, including those with iid random columns from an absolutely continuous, centrally symmetric distribution. There is an asymptotically sharp threshold in the behavior of face counts of the projected hypercube; thresholds known for projecting the simplex and the cross-polytope, occur at very different locations. We briefly consider face counts of the projected orthant when $A$ does not have mean zero; these do behave similarly to those for the projected simplex. We consider non-random projectors of the orthant; the 'best possible' $A$ is the one associated with the first $n$ rows of the Fourier matrix. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Most of these flow in some way from the fact that face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations.

math.MG↗

arXiv2006

Multi-scale morphology of the galaxy distribution

Many statistical methods have been proposed in the last years for analyzing the spatial distribution of galaxies. Very few of them, however, can handle properly the border effects of complex observational sample volumes. In this paper, we first show how to calculate the Minkowski Functionals (MF) taking into account these border effects. Then we present a multiscale extension of the MF which gives us more information about how the galaxies are spatially distributed. A range of examples using Gaussian random fields illustrate the results. Finally we have applied the Multiscale Minkowski Functionals (MMF) to the 2dF Galaxy Redshift Survey data. The MMF clearly indicates an evolution of morphology with scale. We also compare the 2dF real catalog with mock catalogs and found that Lambda-CDM simulations roughly fit the data, except at the finest scale.

astro-ph↗

arXiv2006

Counting faces of randomly-projected polytopes when the projection radically lowers dimension

This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Three such implications concern: convex hulls of Gaussian point clouds, signal recovery from random projections, and how many gross errors can be efficiently corrected from Gaussian error correcting codes.

math.MG↗

arXiv2006

Adaptive multiscale detection of filamentary structures in a background of uniform random points

We are given a set of $n$ points that might be uniformly distributed in the unit square $[0,1]^2$. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with $C^α$-norm bounded by $β$. An asymptotic detection threshold exists in this problem; for a constant $T_-(α,β)>0$, if the number of points sampled from the curve is smaller than $T_-(α,β)n^{1/(1+α)}$, reliable detection is not possible for large $n$. We describe a multiscale significant-runs algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information $α$ or $β$ in advance, provided that the number of points on the curve exceeds $T_*(α,β)n^{1/(1+α)}$. This algorithm therefore has an optimal detection threshold, up to a factor $T_*/T_-$. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will have area $2/n$ and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices all are strips of the same anisotropy and whose edges link such strips to their ``good continuations.'' The point-cloud data are reduced to counts that measure membership in strips. Each anisotropy graph is reduced to a subgraph that consist of strips with significant counts. The algorithm rejects $\mathbf{H}_0$ whenever some such subgraph contains a path that connects many consecutive significant counts.

math.ST↗

arXiv2005

Adapting to Unknown Sparsity by controlling the False Discovery Rate

We attempt to recover an $n$-dimensional vector observed in white noise, where $n$ is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the $\ell_p$ norm for $p$ small. We obtain a procedure which is asymptotically minimax for $\ell^r$ loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the {\it False Discovery Rate} (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter $q_n$ also plays a determining role in asymptotic minimaxity. If $q = \lim q_n \in [0,1/2]$ and also $q_n > γ/\log(n)$ we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, $ q = \lim q_n \in (1/2,1]$, forces the risk to exceed the minimax risk by a factor growing with $q$. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form $2 \cdot \log({potential model size} / {actual model size})$. We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.

math.ST↗

arXiv2002

Emerging applications of geometric multiscale analysis

Classical multiscale analysis based on wavelets has a number of successful applications, e.g. in data compression, fast algorithms, and noise removal. Wavelets, however, are adapted to point singularities, and many phenomena in several variables exhibit intermediate-dimensional singularities, such as edges, filaments, and sheets. This suggests that in higher dimensions, wavelets ought to be replaced in certain applications by multiscale analysis adapted to intermediate-dimensional singularities. My lecture described various initial attempts in this direction. In particular, I discussed two approaches to geometric multiscale analysis originally arising in the work of Harmonic Analysts Hart Smith and Peter Jones (and others): (a) a directional wavelet transform based on parabolic dilations; and (b) analysis via anistropic strips. Perhaps surprisingly, these tools have potential applications in data compression, inverse problems, noise removal, and signal detection; applied mathematicians, statisticians, and engineers are eagerly pursuing these leads.

math.ST↗