Kernel Approximation Methods for Speech Recognition

Over the past five years or so, deep learning methods have dramatically improved the state of the art performance in a variety of domains, including speech recognition, computer vision, and natural language processing. Importantly, however, they suffer from a number of drawbacks: 1. Training these models is a non-convex optimization problem, and thus it is difficult to guarantee that a trained model minimizes the desired loss function. 2. These models are difficult to interpret. In particular, it is difficult to explain, for a given model, why the computations it performs make accurate predictions. In contrast, kernel methods are straightforward to interpret, and training them is a convex optimization problem. Unfortunately, solving these optimization problems exactly is typically prohibitively expensive, though one can use approximation methods to circumvent this problem. In this thesis, we explore to what extent kernel approximation methods can compete with deep learning, in the context of large-scale prediction tasks. Our contributions are as follows: 1. We perform the most extensive set of experiments to date using kernel approximation methods in the context of large-scale speech recognition tasks, and compare performance with deep neural networks. 2. We propose a feature selection algorithm which significantly improves the performance of the kernel models, making their performance competitive with fully-connected feedforward neural networks. 3. We perform an in-depth comparison between two leading kernel approximation strategies — random Fourier features [Rahimi and Recht, 2007] and the Nyström method [Williams and Seeger, 2001] — showing that although the Nyström method is better at approximating the kernel, it performs worse than random Fourier features when used for learning. We believe this work opens the door for future research to continue to push the boundary of what is possible with kernel methods. This research direction will also shed light on the question of when, if ever, deep models are needed for attaining strong performance

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

NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks

The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a hard task. A typical application is the computation of a number of its eigenvalues and eigenvectors. Standard methods become infeasible as the number of nodes in the graph is too large. We propose the use of the fast summation based on the nonequispaced fast Fourier transform (NFFT) to perform the dense matrix-vector product with the graph Laplacian fast without ever forming the whole matrix. The enormous flexibility of the NFFT algorithm allows us to embed the accelerated multiplication into Lanczos-based eigenvalues routines or iterative linear system solvers and even consider other than the standard Gaussian kernels. We illustrate the feasibility of our approach on a number of test problems from image segmentation to semi-supervised learning based on graph-based PDEs. In particular, we compare our approach with the Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of the Nystr\"om method, which internally uses the NFFT.Comment: 28 pages, 9 figure

Learning with SGD and Random Features

Sketching and stochastic gradient methods are arguably the most common techniques to derive efficient large scale learning algorithms. In this paper, we investigate their application in the context of nonparametric statistical learning. More precisely, we study the estimator defined by stochastic gradient with mini batches and random features. The latter can be seen as form of nonlinear sketching and used to define approximate kernel methods. The considered estimator is not explicitly penalized/constrained and regularization is implicit. Indeed, our study highlights how different parameters, such as number of features, iterations, step-size and mini-batch size control the learning properties of the solutions. We do this by deriving optimal finite sample bounds, under standard assumptions. The obtained results are corroborated and illustrated by numerical experiments