### Exploiting Numerical Sparsity for Efficient Learning : Faster Eigenvector Computation and Regression

In this paper, we obtain improved running times for regression and top eigenvector computation for numerically sparse matrices. Given a data matrix $\mat{A} \in \R^{n \times d}$ where every row $a \in \R^d$ has $\|a\|_2^2 \leq L$ and numerical sparsity $\leq s$, i.e. $\|a\|_1^2 / \|a\|_2^2 \leq s$, we provide faster algorithms for these problems for many parameter settings. For top eigenvector computation, when $\gap > 0$ is the relative gap between the top two eigenvectors of $\mat{A}^\top \mat{A}$ and $r$ is the stable rank of $\mat{A}$ we obtain a running time of $\otilde(nd + r(s + \sqrt{r s}) / \gap^2)$ improving upon the previous best unaccelerated running time of $O(nd + r d / \gap^2)$. As $r \leq d$ and $s \leq d$ our algorithm everywhere improves or matches the previous bounds for all parameter settings. For regression, when $\mu > 0$ is the smallest eigenvalue of $\mat{A}^\top \mat{A}$ we obtain a running time of $\otilde(nd + (nL / \mu) \sqrt{s nL / \mu})$ improving upon the previous best unaccelerated running time of $\otilde(nd + n L d / \mu)$. This result expands when regression can be solved in nearly linear time from when $L/\mu = \otilde(1)$ to when $L / \mu = \otilde(d^{2/3} / (sn)^{1/3})$. Furthermore, we obtain similar improvements even when row norms and numerical sparsities are non-uniform and we show how to achieve even faster running times by accelerating using approximate proximal point \cite{frostig2015regularizing} / catalyst \cite{lin2015universal}. Our running times depend only on the size of the input and natural numerical measures of the matrix, i.e. eigenvalues and $\ell_p$ norms, making progress on a key open problem regarding optimal running times for efficient large-scale learning.

### Efficient coordinate-wise leading eigenvector computation

We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods are based on combining a shift-and-invert approach with coordinate-wise algorithms for linear regression.

### The curse of non-unique eigenvectors

A SAS customer asked, "I computed the eigenvectors of a matrix in SAS and in another software package. How do I know which answer is correct?" I've been asked variations of this question dozens of times. The answer is usually "both answers are correct." The mathematical root of the problem is that eigenvectors are not unique.

### Gradient Descent Meets Shift-and-Invert Preconditioning for Eigenvector Computation

Shift-and-invert preconditioning, as a classic acceleration technique for the leading eigenvector computation, has received much attention again recently, owing to fast least-squares solvers for efficiently approximating matrix inversions in power iterations. In this work, we adopt an inexact Riemannian gradient descent perspective to investigate this technique on the effect of the step-size scheme. The shift-and-inverted power method is included as a special case with adaptive step-sizes. Particularly, two other step-size settings, i.e., constant step-sizes and Barzilai-Borwein (BB) step-sizes, are examined theoretically and/or empirically. We present a novel convergence analysis for the constant step-size setting that achieves a rate at $\tilde{O}(\sqrt{\frac{\lambda_{1}}{\lambda_{1}-\lambda_{p+1}}})$, where $\lambda_{i}$ represents the $i$-th largest eigenvalue of the given real symmetric matrix and $p$ is the multiplicity of $\lambda_{1}$. Our experimental studies show that the proposed algorithm can be significantly faster than the shift-and-inverted power method in practice.

### Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrt{\kappa}}{\rho} \log(1/\epsilon) \log \left(k\kappa/\rho\right))$ where $z$ is the total number of nonzero entries, $\kappa$ is the condition number and $\rho$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.