### 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.

### 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.

### 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.

### 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.

### Mapping the Similarities of Spectra: Global and Locally-biased Approaches to SDSS Galaxy Data

We apply a novel spectral graph technique, that of locally-biased semi-supervised eigenvectors, to study the diversity of galaxies. This technique permits us to characterize empirically the natural variations in observed spectra data, and we illustrate how this approach can be used in an exploratory manner to highlight both large-scale global as well as small-scale local structure in Sloan Digital Sky Survey (SDSS) data. We use this method in a way that simultaneously takes into account the measurements of spectral lines as well as the continuum shape. Unlike Principal Component Analysis, this method does not assume that the Euclidean distance between galaxy spectra is a good global measure of similarity between all spectra, but instead it only assumes that local difference information between similar spectra is reliable. Moreover, unlike other nonlinear dimensionality methods, this method can be used to characterize very finely both small-scale local as well as large-scale global properties of realistic noisy data. The power of the method is demonstrated on the SDSS Main Galaxy Sample by illustrating that the derived embeddings of spectra carry an unprecedented amount of information. By using a straightforward global or unsupervised variant, we observe that the main features correlate strongly with star formation rate and that they clearly separate active galactic nuclei. Computed parameters of the method can be used to describe line strengths and their interdependencies. By using a locally-biased or semi-supervised variant, we are able to focus on typical variations around specific objects of astronomical interest. We present several examples illustrating that this approach can enable new discoveries in the data as well as a detailed understanding of very fine local structure that would otherwise be overwhelmed by large-scale noise and global trends in the data.