# A Practical Randomized CP Tensor Decomposition

## TL;DR

This paper introduces a randomized approach to CP tensor decomposition that significantly reduces computational workload and memory usage while maintaining quality, enabling faster and more robust multiway data analysis.

## Contribution

It extends randomized least squares methods to tensors, providing efficient preprocessing, sampling, and stopping techniques for CP decomposition, improving speed and robustness.

## Key findings

- Significant speed improvements in tensor decomposition
- Reductions in memory requirements
- Enhanced robustness to initialization

## Abstract

The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.

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Source: https://tomesphere.com/paper/1701.06600