Spectral Properties of Elementwise-Transformed Spiked Matrices

Source abstract

Abstract. This work concerns elementwise transformations of spiked matrices: [Formula: see text]. Here, [Formula: see text] is a function applied elementwise, [Formula: see text] is a low-rank signal matrix, and [Formula: see text] is white noise. We find that principal component analysis (PCA) is capable of recovering signal under highly nonlinear or discontinuous transformations. Specifically, in the high-dimensional setting where [Formula: see text] is of size [Formula: see text] with [Formula: see text] and [Formula: see text], we uncover a phase transition: for signal-to-noise ratios above a precise threshold—depending on [Formula: see text], the distribution of elements of [Formula: see text], and the limiting aspect ratio [Formula: see text]—the principal components of [Formula: see text] (partially) recover those of [Formula: see text]. Below this threshold, the principal components of [Formula: see text] are asymptotically orthogonal to the signal. In contrast, in the standard setting where PCA is applied to [Formula: see text] directly, the analogous phase transition depends only on [Formula: see text]. Similar phenomena occur with [Formula: see text] square and symmetric and [Formula: see text] a generalized Wigner matrix. This model accommodates diverse data types not covered by prior spiked-matrix theory, including forms of discrete data, preprocessed data, and data with missing values. Our results provide theoretical justification for applying PCA to such data, helping to elucidate PCA’s empirical success.