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Title: Are Latent Factor Regression and Sparse Regression Adequate?
Award ID(s):
2053832 2210833 2052926
PAR ID:
10415840
Author(s) / Creator(s):
; ;
Publisher / Repository:
American Statistical Association
Date Published:
Journal Name:
Journal of the American Statistical Association
Volume:
119
Issue:
546
ISSN:
0162-1459
Page Range / eLocation ID:
1076-1088
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  1. Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e., a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the nonconvex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design. 
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