DeepTensor: Low-Rank Tensor Decomposition with Deep Network Priors

Abstract

DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors (e.g., a matrix as the outer product of two vectors), where each low-rank tensor is generated by a deep network (DN) that is trained in a self-supervised manner to minimize the mean-square approximation error. Our key observation is that the implicit regularization inherent in DNs enables them to capture nonlinear signal structures (e.g., manifolds) that are out of the reach of classical linear methods like the singular value decomposition (SVD) and principal components analysis (PCA). Furthermore, in contrast to the SVD and PCA, whose performance deteriorates when the tensor's entries deviate from additive white Gaussian noise, we demonstrate that the performance of DeepTensor is robust to a wide range of distributions. We validate that DeepTensor is a robust and computationally efficient drop-in replacement for the SVD, PCA, nonnegative matrix factorization (NMF), and similar decompositions by exploring a range of real-world applications, including hyperspectral image denoising, 3D MRI tomography, and image classification. In particular, DeepTensor offers a 6dB signal-to-noise ratio improvement over standard denoising methods for signal corrupted by Poisson noise and learns to decompose 3D tensors 60 times faster than a single DN equipped with 3D convolutions.

Tensor Decomposition-Inspired Convolutional Autoencoders for Hyperspectral Anomaly Detection

“…Besides that, the most similar works to TDNet are [47], [48], [49], but these works are mainly designed for hyperspectral image recovery tasks. In [47], the authors used deep generative networks for low-rank decomposition, and the method is robust to a wide range of distributions.…”

“…Besides that, the most similar works to TDNet are [47], [48], [49], but these works are mainly designed for hyperspectral image recovery tasks. In [47], the authors used deep generative networks for low-rank decomposition, and the method is robust to a wide range of distributions. Luo et al [48] utilized a multilayer neural network to learn the nonlinear nature of multidimensional images, which is more effective than the linear transforms.…”

WIRE: Wavelet Implicit Neural Representations

“…However, DIPs exhibit good performance only when over-parameterized and are tied to a grid-like discretized representation of the signal, implying DIPs do not scale to high dimensional signals such as point clouds with a large number of points. The issue of computational cost has been addressed to a certain extent by the deep decoder [21] and the DeepTensor [39], but they still need the signal to be defined as a regular data grid such as a 2D matrix or 3D tensor.…”