Tensor Decomposition With Generalized Lasso Penalties

ABSTRACT

We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multiway data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our approach presents many nontrivial challenges at the intersection of modeling and computation, which are studied in detail. An efficient coordinate-wise optimization algorithm for PTD is presented, and its convergence properties are characterized. The method is applied both to simulated data and real data on flu hospitalizations in Texas and motion-capture data from video cameras. These results show that our penalized tensor decomposition can offer major improvements on existing methods for analyzing multiway data that exhibit smooth spatial or temporal features.

KEYWORDS:

• Convex optimization
• Multiway data
• Penalized methods
• Tensors
• Trend filtering

Supplementary Materials

• R code: The supplemental files include R scripts to obtain our experimental results.

• Proofs: A pdf file titled Appendix to “Tensor decomposition with generalized lasso penalties” contains the proofs of the results presented in this article.

Acknowledgments

The authors thank two anonymous referees for their helpful feedback in improving the article. This research was partially supported by a CAREER grant (DMS-1255187) from the U.S. National Science Foundation. The motion capture data used in this project were obtained from mocap.cs.cmu.edu. The database was created with funding from NSF EIA-0196217.