Practical sketching algorithms for low-rank matrix approximation

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data

Revisiting the Nystrom Method for Improved Large-Scale Machine Learning

We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds---e.g. improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error---and they point to future directions to make these algorithms useful in even larger-scale machine learning applications.Comment: 60 pages, 15 color figures; updated proof of Frobenius norm bounds, added comparison to projection-based low-rank approximations, and an analysis of the power method applied to SPSD sketche

Network Sketching: Exploiting Binary Structure in Deep CNNs

Convolutional neural networks (CNNs) with deep architectures have substantially advanced the state-of-the-art in computer vision tasks. However, deep networks are typically resource-intensive and thus difficult to be deployed on mobile devices. Recently, CNNs with binary weights have shown compelling efficiency to the community, whereas the accuracy of such models is usually unsatisfactory in practice. In this paper, we introduce network sketching as a novel technique of pursuing binary-weight CNNs, targeting at more faithful inference and better trade-off for practical applications. Our basic idea is to exploit binary structure directly in pre-trained filter banks and produce binary-weight models via tensor expansion. The whole process can be treated as a coarse-to-fine model approximation, akin to the pencil drawing steps of outlining and shading. To further speedup the generated models, namely the sketches, we also propose an associative implementation of binary tensor convolutions. Experimental results demonstrate that a proper sketch of AlexNet (or ResNet) outperforms the existing binary-weight models by large margins on the ImageNet large scale classification task, while the committed memory for network parameters only exceeds a little.Comment: To appear in CVPR201

Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data

Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nystrom approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques for fixed-rank psd matrix approximation across a wide range of examples