Laplace Factor Models in High-Dimensional Data
Laplace factor models (LFMs) provide a heavy-tailed alternative to Gaussian factor models by representing high-dimensional observations through a low-rank common component and Laplace-distributed idiosyncratic errors. This paper develops an assumption-consistent finite-sample analysis of matrix concentration, covariance estimation, and Monte Carlo integration under this model. We first formulate the model with explicit dimensional, independence, covariance, and identifiability conditions. Standard matrix Laplace-transform and matrix Bernstein inequalities are then recalled with their precise applicability conditions. Because untruncated Laplace variables are neither almost surely bounded nor strongly log-concave, these standard results cannot be applied directly in the forms commonly used for bounded or Gaussian-like observations. To address this issue, we analyze a coordinatewise truncated covariance estimator and derive an operator-norm bound that separates the stochastic estimation error from the truncation bias. The resulting rate depends on the effective rank and the logarithm of the ambient dimension and is therefore not dimension-free. For Monte Carlo integration, we replace strong-log-concavity arguments by a sub-exponential concentration analysis that is compatible with independent Laplace errors and yields non-asymptotic absolute- and relative-error bounds. Simulation studies compare empirical tails with the classical matrix Bernstein bound, evaluate ordinary, truncated, winsorized, PCA, POET-type, and Huberized covariance estimators, and we compare Laplace-based and Studentized confidence intervals. The results show that the classical Bernstein bound can be conservative, and truncation involves a substantial bias–variance trade-off. In a Wine chemical-analysis application, three factors explain 66.53% of the standardized variance, and POET-type covariance estimation attains a cross-validated balanced accuracy of 0.9901. These findings clarify both the scope and the limitations of finite-sample analysis for LFMs.