Low-rank approximation is a classic tool in data analysis, where the goal is to approximate a matrix $A$ with a low-rank matrix $L$ so as to minimize the error $\norm{A - L}_F^2$. However in many applications, approximating some entries is more important than others, which leads to the weighted low rank approximation problem. However, the addition of weights makes the low-rank approximation problem intractable. Thus many works have obtained efficient algorithms under additional structural assumptions on the weight matrix (such as low rank, and appropriate block structure). We study a natural greedy algorithm for weighted low rank approximation and develop a simple condition under which it yields bi-criteria approximation up to a small additive factor in the error. The algorithm involves iteratively computing the top singular vector of an appropriately varying matrix, and is thus easy to implement at scale. Our methods also allow us to study the problem of low rank approximation under $\ell_p$ norm error.

We study the problem of identifying anomalies in a low-rank matrix observed with sub-exponential noise, motivated by applications in retail and inventory management. State of the art approaches to anomaly detection in low-rank matrices apparently fall short, since they require that non-anomalous entries be observed with vanishingly small noise (which is not the case in our problem, and indeed in many applications). So motivated, we propose a conceptually simple entrywise approach to anomaly detection in low-rank matrices. Our approach accommodates a general class of probabilistic anomaly models. We extend recent work on entrywise error guarantees for matrix completion, establishing such guarantees for sub-exponential matrices, where in addition to missing entries, a fraction of entries are corrupted by (an also unknown) anomaly model. Viewing the anomaly detection as a classification task, to the best of our knowledge, we are the first to achieve the min-max optimal detection rate (up to log factors). Using data from a massive consumer goods retailer, we show that our approach provides significant improvements over incumbent approaches to anomaly detection.

Kernel methods are fundamental in machine learning, and faster algorithms for kernel approximation provide direct speedups for many core tasks in machine learning. The polynomial kernel is especially important as other kernels can often be approximated by the polynomial kernel via a Taylor series expansion. Recent techniques in oblivious sketching reduce the dependence in the running time on the degree $q$ of the polynomial kernel from exponential to polynomial, which is useful for the Gaussian kernel, for which $q$ can be chosen to be polylogarithmic. However, for more slowly growing kernels, such as the neural tangent and arc cosine kernels, $q$ needs to be polynomial, and previous work incurs a polynomial factor slowdown in the running time. We give a new oblivious sketch which greatly improves upon this running time, by removing the dependence on $q$ in the leading order term. Combined with a novel sampling scheme, we give the fastest algorithms for approximating a large family of slow-growing kernels.

The recently introduced Latent $k-$ Polytope($\LkP$) encompasses several stochastic Mixed Membership models including Topic Models. The problem of finding $k$, the number of extreme points of $\LkP$, is a fundamental challenge and includes several important open problems such as determination of number of components in Ad-mixtures. This paper addresses this challenge by introducing Interpolative Convex Rank(\INR) of a matrix defined as the minimum number of its columns whose convex hull is within Hausdorff distance $\varepsilon$ of the convex hull of all columns. The first important contribution of this paper is to show that under \emph{standard assumptions} $k$ equals the \INR of a \emph{subset smoothed data matrix} defined from Data generated from an $\LkP$. The second important contribution of the paper is a polynomial time algorithm for finding $k$ under standard assumptions. An immediate corollary is the first polynomial time algorithm for finding the \emph{inner dimension} in Non-negative matrix factorisation(NMF) with assumptions which are qualitatively different than existing ones such as \emph{Separability}. %An immediate corollary is the first polynomial time algorithm for finding the \emph{inner dimension} in Non-negative matrix factorisation(NMF) with assumptions considerably weaker than \emph{Separability}.

This paper studies Principal Component Analysis (PCA) for data lying in hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of subspaces spanned by these directions, (2) a method of projection onto subspaces that preserves information in these directions, and (3) an objective to optimize, namely the variance explained by projections. We generalize each of these concepts to the hyperbolic space and propose HoroPCA, a method for hyperbolic dimensionality reduction. By focusing on the core problem of extracting principal directions, HoroPCA theoretically better preserves information in the original data such as distances, compared to previous generalizations of PCA. Empirically, we validate that HoroPCA outperforms existing dimensionality reduction methods, significantly reducing error in distance preservation. As a data whitening method, it improves downstream classification by up to 3.9% compared to methods that don’t use whitening. Finally, we show that HoroPCA can be used to visualize hyperbolic data in two dimensions.

We give the first single-pass streaming algorithm for Column Subset Selection with respect to the entrywise $\ell_p$-norm with $1 \leq p < 2$. We study the $\ell_p$ norm loss since it is often considered more robust to noise than the standard Frobenius norm. Given an input matrix $A \in \mathbb{R}^{d \times n}$ ($n \gg d$), our algorithm achieves a multiplicative $k^{\frac{1}{p} - \frac{1}{2}}\poly(\log nd)$-approximation to the error with respect to the \textit{best possible column subset} of size $k$. Furthermore, the space complexity of the streaming algorithm is optimal up to a logarithmic factor. Our streaming algorithm also extends naturally to a 1-round distributed protocol with nearly optimal communication cost. A key ingredient in our algorithms is a reduction to column subset selection in the $\ell_{p,2}$-norm, which corresponds to the $p$-norm of the vector of Euclidean norms of each of the columns of $A$. This enables us to leverage strong coreset constructions for the Euclidean norm, which previously had not been applied in this context. We also give the first provable guarantees for greedy column subset selection in the $\ell_{1, 2}$ norm, which can be used as an alternative, practical subroutine in our algorithms. Finally, we show that our algorithms give significant practical advantages on real-world data analysis tasks.