Tensor Convolutional Dictionary Learning With CP Low-Rank Activations

• Published in: IEEE transactions on signal processing Vol. 70; pp. 785 - 796
• Main Authors: Humbert, Pierre, Oudre, Laurent, Vayatis, Nicolas, Audiffren, Julien
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; canonical polyadic decomposition; Constraints; Convergence; Convolution; Convolutional codes; Convolutional dictionary learning; convolutional sparse coding; Dictionaries; Engineering Sciences; Learning; Machine learning; Mathematical analysis; Mathematical models; Modularity; Representations; Signal and Image processing; Signal processing algorithms; tensor; Tensors; convolutional sparse coding; canonical polyadic decomposition; tensor; Convolutional dictionary learning
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2021.3135695

In this paper, we propose to extend the standard Convolutional Dictionary Learning problem to a tensor representation where the activations are constrained to be "low-rank" through a Canonical Polyadic decomposition. We show that this additional constraint increases the robustness of the CDL with respect to noise and improve the interpretability of the final results. In addition, we discuss in detail the advantages of this representation and introduce two algorithms, based on ADMM or FISTA, that efficiently solve this problem. We show that by exploiting the low rank property of activations, they achieve lower complexity than the main CDL algorithms. Finally, we evaluate our approach on a wide range of experiments, highlighting the modularity and the advantages of this tensorial low-rank formulation.