# Learning the Sparse and Low Rank PARAFAC Decomposition via the Elastic Net

## TL;DR

This paper introduces a Bayesian model with elastic net regularization for sparse, low-rank PARAFAC tensor decomposition, capable of handling missing data, noise, and large-scale problems, with an efficient solution path for rank and sparsity selection.

## Contribution

It proposes a novel Bayesian framework and algorithms for sparse, low-rank tensor decomposition that automatically determine the true rank and sparsity, improving robustness and scalability.

## Key findings

- Effective on simulation data
- Successful application to real data
- Robust to noise and missing values

## Abstract

In this article, we derive a Bayesian model to learning the sparse and low rank PARAFAC decomposition for the observed tensor with missing values via the elastic net, with property to find the true rank and sparse factor matrix which is robust to the noise. We formulate efficient block coordinate descent algorithm and admax stochastic block coordinate descent algorithm to solve it, which can be used to solve the large scale problem. To choose the appropriate rank and sparsity in PARAFAC decomposition, we will give a solution path by gradually increasing the regularization to increase the sparsity and decrease the rank. When we find the sparse structure of the factor matrix, we can fixed the sparse structure, using a small to regularization to decreasing the recovery error, and one can choose the proper decomposition from the solution path with sufficient sparse factor matrix with low recovery error. We test the power of our algorithm on the simulation data and real data, which show it is powerful.