Sub-exponential time Sum-of-Squares lower bounds for Principal Components Analysis
Authors: Aaron Potechin, Goutham Rajendran

Principal Components Analysis (PCA) is a dimension-reduction technique widely used in machine learning and statistics. However, due to the dependence of the principal components on all the dimensions, the components are notoriously hard to interpret. Therefore, a variant known as sparse PCA is often preferred. Sparse PCA learns principal components of the data but enforces that such components must be sparse. This has applications in diverse fields such as computational biology and image processing. To learn sparse principal components, it's well known that standard PCA will not work, especially in high dimensions, and therefore algorithms for sparse PCA are often studied as a separate endeavor. Various algorithms have been proposed for Sparse PCA over the years, but given how fundamental it is for applications in science, the limits of efficient algorithms are only partially understood. In this work, we study the limits of the powerful Sum of Squares (SoS) family of algorithms for Sparse PCA. SoS algorithms have recently revolutionized robust statistics, leading to breakthrough algorithms for long-standing open problems in machine learning, such as optimally learning mixtures of gaussians, robust clustering, robust regression, etc. Moreover, it is believed to be the optimal robust algorithm for many statistical problems. Therefore, for sparse PCA, it's plausible that it can beat simpler algorithms such as diagonal thresholding that have been traditionally used. In this work, we show that this is not the case, by exhibiting strong tradeoffs between the number of samples required, the sparsity and the ambient dimension, for which SoS algorithms, even if allowed sub-exponential time, will fail to optimally recover the component. Our results are complemented by known algorithms in literature, thereby painting an almost complete picture of the behavior of efficient algorithms for sparse PCA. Since SoS algorithms encapsulate many algorithmic techniques such as spectral or statistical query algorithms, this solidifies the message that  known algorithms are optimal for sparse PCA. Moreover, our techniques are strong enough to obtain similar tradeoffs for Tensor PCA, another important higher order variant of PCA with applications in topic modeling, video processing, etc.