Sharp Analysis of Sketch-and-Project Methods via a Connection to Randomized Singular Value Decomposition

Sketch-and-project is a framework which unifies many known iterative methods for solving linear systems and their variants, as well as further extensions to non-linear optimization problems. It includes popular methods such as randomized Kaczmarz, coordinate descent, variants of the Newton method in convex optimization, and others. In this paper, we obtain sharp guarantees for the convergence rate of sketch-and-project methods via new tight spectral bounds for the expected sketched projection matrix. Our estimates reveal a connection between the sketch-and-project convergence rate and the approximation error of another well-known but seemingly unrelated family of algorithms, which use sketching to accelerate popular matrix factorizations such as QR and SVD. This connection brings us closer to precisely quantifying how the performance of sketch-and-project solvers depends on their sketch size. Our analysis covers not only Gaussian and sub-gaussian sketching matrices, but also a family of efficient sparse sketching methods known as LESS embeddings. Our experiments back up the theory and demonstrate that even extremely sparse sketches show the same convergence properties in practice.

Nonlinear Matrix Approximation with Radial Basis Function Components

We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the scalar multiplication between individual vector elements. Even though the RBF functions are positive definite, the summation across components is not restricted to convex combinations and allows us to compute the decomposition for any real matrix that is not necessarily symmetric or positive definite. We formulate the problem of seeking such a decomposition as an optimization problem with a nonlinear and non-convex loss function. Several modern versions of the gradient descent method, including their scalable stochastic counterparts, are used to solve this problem. We provide extensive empirical evidence of the effectiveness of the RBF decomposition and that of the gradient-based fitting algorithm. While being conceptually motivated by singular value decomposition (SVD), our proposed nonlinear counterpart outperforms SVD by drastically reducing the memory required to approximate a data matrix with the same L2 error for a wide range of matrix types. For example, it leads to 2 to 6 times memory save for Gaussian noise, graph adjacency matrices, and kernel matrices. Moreover, this proximity-based decomposition can offer additional interpretability in applications that involve, e.g., capturing the inner low-dimensional structure of the data, retaining graph connectivity structure, and preserving the acutance of images.

Analysis of Legal Documents via Non-negative Matrix Factorization Methods

The California Innocence Project (CIP), a clinical law school program aiming to free wrongfully convicted prisoners, evaluates thousands of mails containing new requests for assistance and corresponding case files. Processing and interpreting this large amount of information presents a significant challenge for CIP officials, which can be successfully aided by topic modeling techniques.In this paper, we apply Non-negative Matrix Factorization (NMF) method and implement various offshoots of it to the important and previously unstudied data set compiled by CIP. We identify underlying topics of existing case files and classify request files by crime type and case status (decision type). The results uncover the semantic structure of current case files and can provide CIP officials with a general understanding of newly received case files before further examinations. We also provide an exposition of popular variants of NMF with their experimental results and discuss the benefits and drawbacks of each variant through the real-world application.

On a Guided Nonnegative Matrix Factorization

Fully unsupervised topic models have found fantastic success in document clustering and classification. However, these models often suffer from the tendency to learn less-than-meaningful or even redundant topics when the data is biased towards a set of features. For this reason, we propose an approach based upon the nonnegative matrix factorization (NMF) model, deemed \textit{Guided NMF}, that incorporates user-designed seed word supervision. Our experimental results demonstrate the promise of this model and illustrate that it is competitive with other methods of this ilk with only very little supervision information.

COVID-19 Literature Topic-Based Search via Hierarchical NMF

A dataset of COVID-19-related scientific literature is compiled, combining the articles from several online libraries and selecting those with open access and full text available. Then, hierarchical nonnegative matrix factorization is used to organize literature related to the novel coronavirus into a tree structure that allows researchers to search for relevant literature based on detected topics. We discover eight major latent topics and 52 granular subtopics in the body of literature, related to vaccines, genetic structure and modeling of the disease and patient studies, as well as related diseases and virology. In order that our tool may help current researchers, an interactive website is created that organizes available literature using this hierarchical structure.

A Study of Clustering Techniques and Hierarchical Matrix Formats for Kernel Ridge Regression

We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the execution time are drastically reduced compared to standard dense linear algebra routines. We consider both the general $\mathcal{H}$ matrix hierarchical format as well as Hierarchically Semi-Separable (HSS) matrices. Furthermore, we investigate the impact of several preprocessing and clustering techniques on the hierarchical matrix compression. Effective clustering of the input leads to a ten-fold increase in efficiency of the compression. The algorithms are implemented using the STRUMPACK solver library. These results confirm that --- with correct tuning of the hyperparameters --- classification using kernel ridge regression with the compressed matrix does not lose prediction accuracy compared to the exact --- not compressed --- kernel matrix and that our approach can be extended to $\mathcal{O}(1M)$ datasets, for which computation with the full kernel matrix becomes prohibitively expensive. We present numerical experiments in a distributed memory environment up to 1,024 processors of the NERSC's Cori supercomputer using well-known datasets to the machine learning community that range from dimension 8 up to 784.