Statistics Papers

Document Type

Technical Report

Date of this Version

2016

Publication Source

Journal of the American Statistical Association

Volume

111

Issue

516

Start Page

1608

Last Page

1622

DOI

10.1080/01621459.2015.1100620

Abstract

Rotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis. Regularization methods also serve to achieve this goal by prioritizing sparse loading matrices. In this work, we bridge these two paradigms with a unifying Bayesian framework. Our approach deploys intermediate factor rotations throughout the learning process, greatly enhancing the effectiveness of sparsity inducing priors. These automatic rotations to sparsity are embedded within a PXL-EM algorithm, a Bayesian variant of parameter-expanded EM for posterior mode detection. By iterating between soft-thresholding of small factor loadings and transformations of the factor basis, we obtain (a) dramatic accelerations, (b) robustness against poor initializations, and (c) better oriented sparse solutions. To avoid the prespecification of the factor cardinality, we extend the loading matrix to have infinitely many columns with the Indian buffet process (IBP) prior. The factor dimensionality is learned from the posterior, which is shown to concentrate on sparse matrices. Our deployment of PXL-EM performs a dynamic posterior exploration, outputting a solution path indexed by a sequence of spike-and-slab priors. For accurate recovery of the factor loadings, we deploy the spike-and-slab LASSO prior, a two-component refinement of the Laplace prior. A companion criterion, motivated as an integral lower bound, is provided to effectively select the best recovery. The potential of the proposed procedure is demonstrated on both simulated and real high-dimensional data, which would render posterior simulation impractical. Supplementary materials for this article are available online.

Copyright/Permission Statement

This is an Accepted Manuscript of an article published by Taylor & Francis in the Journal of the American Statistical Association on 5 Jan 2017, available online: http://dx.doi.org/10.1080/01621459.2015.1100620

Keywords

EM algorithm, factor rotations, parameter expansion, sparsity, spike-and-slab LASSO

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Date Posted: 25 October 2018

This document has been peer reviewed.