Spectrum Estimation from Samples
This addresses a fundamental problem in statistics and machine learning for researchers and practitioners needing to infer distribution properties like structure and dimensionality from limited data, with incremental improvements in theoretical guarantees.
The paper tackles the problem of estimating the eigenvalues of a covariance matrix from samples, particularly when sample sizes are comparable to or smaller than dimensionality, by proposing an algorithm that recovers eigenvalue moments and uses Wasserstein distance to accurately estimate eigenvalues, achieving finite-sample error bounds and asymptotic consistency even in sublinear sample regimes.
We consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution (equivalently, the problem of approximating the "population spectrum"), given access to samples drawn from the distribution. The eigenvalues of the covariance of a distribution contain basic information about the distribution, including the presence or lack of structure in the distribution, the effective dimensionality of the distribution, and the applicability of higher-level machine learning and multivariate statistical tools. We consider this fundamental recovery problem in the regime where the number of samples is comparable, or even sublinear in the dimensionality of the distribution in question. First, we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix. We then leverage this accurate moment recovery, via a Wasserstein distance argument, to show that the vector of eigenvalues can be accurately recovered. We provide finite--sample bounds on the expected error of the recovered eigenvalues, which imply that our estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity, even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero. In addition to our theoretical results, we show that our approach performs well in practice for a broad range of distributions and sample sizes.