Nonparametric Trace Regression in High Dimensions via Sign Series Representation

05/04/2021
by   Chanwoo Lee, et al.
0

Learning of matrix-valued data has recently surged in a range of scientific and business applications. Trace regression is a widely used method to model effects of matrix predictors and has shown great success in matrix learning. However, nearly all existing trace regression solutions rely on two assumptions: (i) a known functional form of the conditional mean, and (ii) a global low-rank structure in the entire range of the regression function, both of which may be violated in practice. In this article, we relax these assumptions by developing a general framework for nonparametric trace regression models via structured sign series representations of high dimensional functions. The new model embraces both linear and nonlinear trace effects, and enjoys rank invariance to order-preserving transformations of the response. In the context of matrix completion, our framework leads to a substantially richer model based on what we coin as the "sign rank" of a matrix. We show that the sign series can be statistically characterized by weighted classification tasks. Based on this connection, we propose a learning reduction approach to learn the regression model via a series of classifiers, and develop a parallelable computation algorithm to implement sign series aggregations. We establish the excess risk bounds, estimation error rates, and sample complexities. Our proposal provides a broad nonparametric paradigm to many important matrix learning problems, including matrix regression, matrix completion, multi-task learning, and compressed sensing. We demonstrate the advantages of our method through simulations and two applications, one on brain connectivity study and the other on high-rank image completion.

READ FULL TEXT
research
04/18/2019

On Low-rank Trace Regression under General Sampling Distribution

A growing number of modern statistical learning problems involve estimat...
research
07/02/2020

Partial Trace Regression and Low-Rank Kraus Decomposition

The trace regression model, a direct extension of the well-studied linea...
research
01/31/2021

Beyond the Signs: Nonparametric Tensor Completion via Sign Series

We consider the problem of tensor estimation from noisy observations wit...
research
11/29/2021

Rank-Constrained Least-Squares: Prediction and Inference

In this work, we focus on the high-dimensional trace regression model wi...
research
02/15/2021

Tight Risk Bound for High Dimensional Time Series Completion

Initially designed for independent datas, low-rank matrix completion was...
research
03/31/2019

Nonparametric Matrix Response Regression with Application to Brain Imaging Data Analysis

With the rapid growth of neuroimaging technologies, a great effort has b...
research
04/23/2015

Regularization-free estimation in trace regression with symmetric positive semidefinite matrices

Over the past few years, trace regression models have received considera...

Please sign up or login with your details

Forgot password? Click here to reset