# Multivariate Convolutional Sparse Coding with Low Rank Tensor

## TL;DR

This paper proposes a novel multivariate convolutional sparse coding model using tensor algebra and low-rank constraints, improving efficiency and performance in high-dimensional signal encoding.

## Contribution

It introduces a tensor-based convolutional sparse coding model with low-rank and sparsity constraints, along with an efficient optimization algorithm and theoretical guarantees.

## Key findings

- Enhanced encoding efficiency in high-dimensional settings
- Improved performance over existing methods
- Theoretical links to Kruskal tensor regression

## Abstract

This paper introduces a new multivariate convolutional sparse coding based on tensor algebra with a general model enforcing both element-wise sparsity and low-rankness of the activations tensors. By using the CP decomposition, this model achieves a significantly more efficient encoding of the multivariate signal-particularly in the high order/ dimension setting-resulting in better performance. We prove that our model is closely related to the Kruskal tensor regression problem, offering interesting theoretical guarantees to our setting. Furthermore, we provide an efficient optimization algorithm based on alternating optimization to solve this model. Finally, we evaluate our algorithm with a large range of experiments, highlighting its advantages and limitations.