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Joel A. Tropp

Publications and source records attributed to Joel A. Tropp.

At least 73 records · Page 4Linked to original sources
arXiv2013

Deriving Matrix Concentration Inequalities from Kernel Couplings

This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for concentration of measure, as introduced by Chatterjee. Recent work of Mackey et al. uses these techniques to analyze random matrices with additive structure, while the enhancements in this paper cover a wider class of matrix-valued random elements. In particular, these ideas lead to a bounded differences inequality that applies to random matrices constructed from weakly dependent random variables. The proofs require novel trace inequalities that may be of independent interest.

math.PR↗
arXiv2013

Factoring nonnegative matrices with linear programs

This paper describes a new approach, based on linear programming, for computing nonnegative matrix factorizations (NMFs). The key idea is a data-driven model for the factorization where the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C such that X approximately equals CX and some linear constraints. The constraints are chosen to ensure that the matrix C selects features; these features can then be used to find a low-rank NMF of X. A theoretical analysis demonstrates that this approach has guarantees similar to those of the recent NMF algorithm of Arora et al. (2012). In contrast with this earlier work, the proposed method extends to more general noise models and leads to efficient, scalable algorithms. Experiments with synthetic and real datasets provide evidence that the new approach is also superior in practice. An optimized C++ implementation can factor a multigigabyte matrix in a matter of minutes.

math.OC↗
arXiv2012

Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method

The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into well-conditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined least-squares problem.

math.NA↗
arXiv2012

The Masked Sample Covariance Estimator: An Analysis via Matrix Concentration Inequalities

Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyze this approach, Levina and Vershynin (2011) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighted to reflect an a priori assessment of its importance. This paper provides a short analysis of the masked sample covariance estimator by means of a matrix concentration inequality. The main result applies to general distributions with at least four moments. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n = O(B log^2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n = O(B log^5 p) obtained by Levina and Vershynin.

math.PR↗
arXiv2011

Tail bounds for all eigenvalues of a sum of random matrices

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is used to derive eigenvalue analogues of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonormal rows. Here, the behavior of the singular values can be described in terms of coherence-like quantities. The second example addresses the question of relative accuracy in the estimation of eigenvalues of the covariance matrix of a random process. Standard results on the convergence of sample covariance matrices provide bounds on the number of samples needed to obtain relative accuracy in the spectral norm, but these results only guarantee relative accuracy in the estimate of the maximum eigenvalue. The minimax Laplace transform argument establishes that if the lowest eigenvalues decay sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the condition number of an optimal rank-r approximation to C, are sufficient to ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C) random vector are estimated to within a factor of 1+-eps with high probability.

math.PR↗
arXiv2011

Improved analysis of the subsampled randomized Hadamard transform

This paper presents an improved analysis of a structured dimension-reduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous approaches, and it offers---for the first time---optimal constants in the estimate on the number of dimensions required for the embedding.

math.NA↗
arXiv2011

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.

math.PR↗
arXiv2011

A comparison principle for functions of a uniformly random subspace

This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid.

math.PR↗
arXiv2011

Freedman's inequality for matrix martingales

Freedman's inequality is a martingale counterpart to Bernstein's inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman's inequality that provides tail bounds for the maximum singular value of a matrix-valued martingale. This note describes a different proof of the matrix Freedman inequality that depends on a deep theorem of Lieb from matrix analysis. This argument delivers sharp constants in the matrix Freedman inequality, and it also yields tail bounds for other types of matrix martingales. The new techniques are adapted from recent work by the present author.

math.PR↗
arXiv2010

Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed---either explicitly or implicitly---to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.

math.NA↗
arXiv2010

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.

cs.IT↗
arXiv2010

The Sparsity Gap: Uncertainty Principles Proportional to Dimension

In an incoherent dictionary, most signals that admit a sparse representation admit a unique sparse representation. In other words, there is no way to express the signal without using strictly more atoms. This work demonstrates that sparse signals typically enjoy a higher privilege: each nonoptimal representation of the signal requires far more atoms than the sparsest representation-unless it contains many of the same atoms as the sparsest representation. One impact of this finding is to confer a certain degree of legitimacy on the particular atoms that appear in a sparse representation. This result can also be viewed as an uncertainty principle for random sparse signals over an incoherent dictionary.

cs.IT↗
arXiv2009

Error Bounds for Random Matrix Approximation Schemes

Randomized matrix sparsification has proven to be a fruitful technique for producing faster algorithms in applications ranging from graph partitioning to semidefinite programming. In the decade or so of research into this technique, the focus has been--with few exceptions--on ensuring the quality of approximation in the spectral and Frobenius norms. For certain graph algorithms, however, the $(\infty,1)$ norm may be a more natural measure of performance. This paper addresses the problem of approximating a real matrix A by a sparse random matrix X with respect to several norms. It provides the first results on approximation error in the $(\infty, 1)$ and $(\infty, 2)$ norms, and it uses a result of Latala to study approximation error in the spectral norm. These bounds hold for random sparsification schemes which ensure that the entries of X are independent and average to the corresponding entries of A. Optimality of the $(\infty, 1)$ and $(\infty,2)$ error estimates is established. Concentration results for the three norms hold when the entries of X are uniformly bounded. The spectral error bound is used to predict the performance of several sparsification and quantization schemes that have appeared in the literature; the results are competitive with the performance guarantees given by earlier scheme-specific analyses.

math.NA↗
arXiv2009

Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals

Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations.

cs.IT↗
arXiv2009

Efficient Sampling of Sparse Wideband Analog Signals

Periodic nonuniform sampling is a known method to sample spectrally sparse signals below the Nyquist rate. This strategy relies on the implicit assumption that the individual samplers are exposed to the entire frequency range. This assumption becomes impractical for wideband sparse signals. The current paper proposes an alternative sampling stage that does not require a full-band front end. Instead, signals are captured with an analog front end that consists of a bank of multipliers and lowpass filters whose cutoff is much lower than the Nyquist rate. The problem of recovering the original signal from the low-rate samples can be studied within the framework of compressive sampling. An appropriate parameter selection ensures that the samples uniquely determine the analog input. Moreover, the analog input can be stably reconstructed with digital algorithms. Numerical experiments support the theoretical analysis.

cs.IT↗
arXiv2008

Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization

Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing {\sf QR}, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the $(\infty, 1)$ norm of a matrix, which is generally {\sf NP}-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous {\sc maxcut} semidefinite program.

math.NA↗
arXiv2008

On the linear independence of spikes and sines

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves depends on an extrapolation argument of Bourgain and Tzafriri.

math.FA↗