  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-squared 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 component
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 signals
corrupted by Poisson noise and learns to decompose 3D tensors 60 times faster
than a single DN equipped with 3D convolutions.
