When are the most informative components for inference also the principal components?
This addresses a foundational issue in statistical inference and machine learning for researchers and practitioners using PCA, revealing scenarios where standard approaches fail.
The paper tackles the problem of determining which singular value decomposition components are most informative for detecting or estimating a low-rank signal in a signal-plus-noise matrix, showing that principal components are justified for i.i.d. Gaussian noise but middle components can be more informative for heterogeneous noise models like mixtures of Gaussians, with results indicating phase transitions that can lead to suboptimal inference.
Which components of the singular value decomposition of a signal-plus-noise data matrix are most informative for the inferential task of detecting or estimating an embedded low-rank signal matrix? Principal component analysis ascribes greater importance to the components that capture the greatest variation, i.e., the singular vectors associated with the largest singular values. This choice is often justified by invoking the Eckart-Young theorem even though that work addresses the problem of how to best represent a signal-plus-noise matrix using a low-rank approximation and not how to best_infer_ the underlying low-rank signal component. Here we take a first-principles approach in which we start with a signal-plus-noise data matrix and show how the spectrum of the noise-only component governs whether the principal or the middle components of the singular value decomposition of the data matrix will be the informative components for inference. Simply put, if the noise spectrum is supported on a connected interval, in a sense we make precise, then the use of the principal components is justified. When the noise spectrum is supported on multiple intervals, then the middle components might be more informative than the principal components. The end result is a proper justification of the use of principal components in the setting where the noise matrix is i.i.d. Gaussian and the identification of scenarios, generically involving heterogeneous noise models such as mixtures of Gaussians, where the middle components might be more informative than the principal components so that they may be exploited to extract additional processing gain. Our results show how the blind use of principal components can lead to suboptimal or even faulty inference because of phase transitions that separate a regime where the principal components are informative from a regime where they are uninformative.