New perturbation bounds for low rank approximation of matrices: Beyond Eckart-Young-Mirsky

Abstract

Let $A$ be an $m \times n$ matrix with rank $r$ and spectral decomposition $A = \sum_{i=1}^r \sigma_i u_i v_i^\top,$ where $\sigma_i$ are its singular values, ordered decreasingly, and $u_i, v_i$ are the corresponding left and right singular vectors. For a parameter $1 \le p \le r$, $A_p := \sum_{i=1}^p \sigma_i u_i v_i^\top$ is the best rank $p$ approximation of $A$. In practice, one often chooses $p$ to be small, leading to the commonly used phrase "low-rank approximation". Low-rank approximation plays a central role in data science because it can substantially reduce the dimensionality of the original data, the matrix $A$. For a large data matrix $A$, one typically computes a rank-$p$ approximation $A_p$ for a suitably chosen small $p$, stores $A_p$, and uses it as input for further computations. The reduced dimension of $A_p$ enables faster computations and significant data compression. In practice, noise is inevitable. We often have access only to noisy data $\tilde A = A + E$, where $E$ represents the noise. Consequently, the low-rank approximation used as input in many downstream tasks is $\tilde A_p$, the best rank $p$ approximation of $\tilde A$, rather than $A_p$. Therefore, it is natural and important to estimate the error $ \| \tilde A_p - A_p \|$. This error plays a critical role in estimating the accuracy of the output of any process involving a low-rank approximation of noisy input. In this paper, we develop a new method (based on contour analysis) to bound $\| \tilde A_p - A_p \|$. With this method, we can exploit new parameters that measure the skewness between the noise matrix $E$ and the singular vectors of $A$, avoiding the worst-case analysis used in traditional approaches. In many settings, we obtain notable quantitative improvements compared to classical approaches (using the Eckart-Young-Mirsky theorem or the Davis-Kahan theorem).