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Stephen Becker

Publications and source records attributed to Stephen Becker.

At least 73 records · Page 4Linked to original sources
arXiv2019

Tensor Robust Principal Component Analysis: Better recovery with atomic norm regularization

This paper studies tensor-based Robust Principal Component Analysis (RPCA) using atomic-norm regularization. Given the superposition of a sparse and a low-rank tensor, we present conditions under which it is possible to exactly recover the sparse and low-rank components. Our results improve on existing performance guarantees for tensor-RPCA, including those for matrix RPCA. Our guarantees also show that atomic-norm regularization provides better recovery for tensor-structured data sets than other approaches based on matricization. In addition to these performance guarantees, we study a nonconvex formulation of the tensor atomic-norm and identify a class of local minima of this nonconvex program that are globally optimal. We demonstrate the strong performance of our approach in numerical experiments, where we show that our nonconvex model reliably recovers tensors with ranks larger than all of their side lengths, significantly outperforming other algorithms that require matricization.

math.OC↗
arXiv2018

On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank-$r$ modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few.

math.OC↗
arXiv2017

Adapting Regularized Low Rank Models for Parallel Architectures

We introduce a reformulation of regularized low-rank recovery models to take advantage of GPU, multiple CPU, and hybridized architectures. Low-rank recovery often involves nuclear-norm minimization through iterative thresholding of singular values. These models are slow to fit and difficult to parallelize because of their dependence on computing a singular value decomposition at each iteration. Regularized low-rank recovery models also incorporate non-smooth terms to separate structured components (e.g. sparse outliers) from the low-rank component, making these problems more difficult. Using Burer-Monteiro splitting and marginalization, we develop a smooth, non-convex formulation of regularized low-rank recovery models that can be fit with first-order solvers. We develop a computable certificate of convergence for this non-convex program, and use it to establish bounds on the suboptimality of any point. Using robust principal component analysis (RPCA) as an example, we include numerical experiments showing that this approach is an order-of-magnitude faster than existing RPCA solvers on the GPU. We also show that this acceleration allows new applications for RPCA, including real-time background subtraction and MR image analysis.

math.OC↗
arXiv2017

Efficient Adjoint Computation for Wavelet and Convolution Operators

First-order optimization algorithms, often preferred for large problems, require the gradient of the differentiable terms in the objective function. These gradients often involve linear operators and their adjoints, which must be applied rapidly. We consider two example problems and derive methods for quickly evaluating the required adjoint operator. The first example is an image deblurring problem, where we must compute efficiently the adjoint of multi-stage wavelet reconstruction. Our formulation of the adjoint works for a variety of boundary conditions, which allows the formulation to generalize to a larger class of problems. The second example is a blind channel estimation problem taken from the optimization literature where we must compute the adjoint of the convolution of two signals. In each example, we show how the adjoint operator can be applied efficiently while leveraging existing software.

math.OC↗
arXiv2017

URV Factorization with Random Orthogonal System Mixing

The unpivoted and pivoted Householder QR factorizations are ubiquitous in numerical linear algebra. A difficulty with pivoted Householder QR is the communication bottleneck introduced by pivoting. In this paper we propose using random orthogonal systems to quickly mix together the columns of a matrix before computing an unpivoted QR factorization. This method computes a URV factorization which forgoes expensive pivoted QR steps in exchange for mixing in advance, followed by a cheaper, unpivoted QR factorization. The mixing step typically reduces the variability of the column norms, and in certain experiments, allows us to compute an accurate factorization where a plain, unpivoted QR performs poorly. We experiment with linear least-squares, rank-revealing factorizations, and the QLP approximation, and conclude that our randomized URV factorization behaves comparably to a similar randomized rank-revealing URV factorization, but at a fraction of the computational cost. Our experiments provide evidence that our proposed factorization might be rank-revealing with high probability.

math.NA↗
arXiv2016

Randomized Clustered Nystrom for Large-Scale Kernel Machines

The Nystrom method has been popular for generating the low-rank approximation of kernel matrices that arise in many machine learning problems. The approximation quality of the Nystrom method depends crucially on the number of selected landmark points and the selection procedure. In this paper, we present a novel algorithm to compute the optimal Nystrom low-approximation when the number of landmark points exceed the target rank. Moreover, we introduce a randomized algorithm for generating landmark points that is scalable to large-scale data sets. The proposed method performs K-means clustering on low-dimensional random projections of a data set and, thus, leads to significant savings for high-dimensional data sets. Our theoretical results characterize the tradeoffs between the accuracy and efficiency of our proposed method. Extensive experiments demonstrate the competitive performance as well as the efficiency of our proposed method.

stat.ML↗
arXiv2016

A Randomized Approach to Efficient Kernel Clustering

Kernel-based K-means clustering has gained popularity due to its simplicity and the power of its implicit non-linear representation of the data. A dominant concern is the memory requirement since memory scales as the square of the number of data points. We provide a new analysis of a class of approximate kernel methods that have more modest memory requirements, and propose a specific one-pass randomized kernel approximation followed by standard K-means on the transformed data. The analysis and experiments suggest the method is accurate, while requiring drastically less memory than standard kernel K-means and significantly less memory than Nystrom based approximations.

stat.ML↗
arXiv2016

Preconditioned Data Sparsification for Big Data with Applications to PCA and K-means

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.

stat.ML↗
arXiv2016

Dual Smoothing and Level Set Techniques for Variational Matrix Decomposition

We focus on the robust principal component analysis (RPCA) problem, and review a range of old and new convex formulations for the problem and its variants. We then review dual smoothing and level set techniques in convex optimization, present several novel theoretical results, and apply the techniques on the RPCA problem. In the final sections, we show a range of numerical experiments for simulated and real-world problems.

stat.ML↗
arXiv2015

Robust Partially-Compressed Least-Squares

Randomized matrix compression techniques, such as the Johnson-Lindenstrauss transform, have emerged as an effective and practical way for solving large-scale problems efficiently. With a focus on computational efficiency, however, forsaking solutions quality and accuracy becomes the trade-off. In this paper, we investigate compressed least-squares problems and propose new models and algorithms that address the issue of error and noise introduced by compression. While maintaining computational efficiency, our models provide robust solutions that are more accurate--relative to solutions of uncompressed least-squares--than those of classical compressed variants. We introduce tools from robust optimization together with a form of partial compression to improve the error-time trade-offs of compressed least-squares solvers. We develop an efficient solution algorithm for our Robust Partially-Compressed (RPC) model based on a reduction to a one-dimensional search. We also derive the first approximation error bounds for Partially-Compressed least-squares solutions. Empirical results comparing numerous alternatives suggest that robust and partially compressed solutions are effectively insulated against aggressive randomized transforms.

stat.ML↗
arXiv2015

General Optimization Framework for Robust and Regularized 3D Full Waveform Inversion

Scarcity of hydrocarbon resources and high exploration risks motivate the development of high fidelity algorithms and computationally viable approaches to exploratory geophysics. Whereas early approaches considered least-squares minimization, recent developments have emphasized the importance of robust formulations, as well as formulations that allow disciplined encoding of prior information into the inverse problem formulation. The cost of a more flexible optimization framework is a greater computational complexity, as least-squares optimization can be performed using straightforward methods (e.g., steepest descent, Gauss-Newton, L-BFGS), whilst incorporation of robust (non-smooth) penalties requires custom changes that may be difficult to implement in the context of a general seismic inversion workflow. In this study, we propose a generic, flexible optimization framework capable of incorporating a broad range of noise models, forward models, regularizers, and reparametrization transforms. This framework covers seamlessly robust noise models (such as Huber and Student's $t$), as well as sparse regularizers, projected constraints, and Total Variation regularization. The proposed framework is also expandable --- we explain the adjustments that are required for any new formulation to be included. Lastly, we conclude with few numerical examples demonstrating the versatility of the formulation.

math.OC↗
arXiv2015

Efficient Dictionary Learning via Very Sparse Random Projections

Performing signal processing tasks on compressive measurements of data has received great attention in recent years. In this paper, we extend previous work on compressive dictionary learning by showing that more general random projections may be used, including sparse ones. More precisely, we examine compressive K-means clustering as a special case of compressive dictionary learning and give theoretical guarantees for its performance for a very general class of random projections. We then propose a memory and computation efficient dictionary learning algorithm, specifically designed for analyzing large volumes of high-dimensional data, which learns the dictionary from very sparse random projections. Experimental results demonstrate that our approach allows for reduction of computational complexity and memory/data access, with controllable loss in accuracy.

stat.ML↗
arXiv2014

Convex Optimization for Big Data

This article reviews recent advances in convex optimization algorithms for Big Data, which aim to reduce the computational, storage, and communications bottlenecks. We provide an overview of this emerging field, describe contemporary approximation techniques like first-order methods and randomization for scalability, and survey the important role of parallel and distributed computation. The new Big Data algorithms are based on surprisingly simple principles and attain staggering accelerations even on classical problems.

math.OC↗
arXiv2014

A variational approach to stable principal component pursuit

We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthetic and real data experiments, that our approach offers advantages over the classical SPCP formulations in scalability and practical parameter selection.

math.OC↗
arXiv2013

Randomized Low-Memory Singular Value Projection

Affine rank minimization algorithms typically rely on calculating the gradient of a data error followed by a singular value decomposition at every iteration. Because these two steps are expensive, heuristic approximations are often used to reduce computational burden. To this end, we propose a recovery scheme that merges the two steps with randomized approximations, and as a result, operates on space proportional to the degrees of freedom in the problem. We theoretically establish the estimation guarantees of the algorithm as a function of approximation tolerance. While the theoretical approximation requirements are overly pessimistic, we demonstrate that in practice the algorithm performs well on the quantum tomography recovery problem.

math.OC↗
arXiv2013

An Algorithm for Splitting Parallel Sums of Linearly Composed Monotone Operators, with Applications to Signal Recovery

We present a new primal-dual splitting algorithm for structured monotone inclusions in Hilbert spaces and analyze its asymptotic behavior. A novelty of our framework, which is motivated by image recovery applications, is to consider inclusions that combine a variety of monotonicity-preserving operations such as sums, linear compositions, parallel sums, and a new notion of parallel composition. The special case of minimization problems is studied in detail, and applications to signal recovery are discussed. Numerical simulations are provided to illustrate the implementation of the algorithm.

math.OC↗
arXiv2013

Sparse projections onto the simplex

Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the $\ell_1$-norm. However, several important learning applications cannot benefit from this approach as they feature these convex norms as constraints in addition to the non-convex rank and sparsity constraints. In this setting, we derive efficient sparse projections onto the simplex and its extension, and illustrate how to use them to solve high-dimensional learning problems in quantum tomography, sparse density estimation and portfolio selection with non-convex constraints.

cs.LG↗