On the Power of Compressed Sensing with Generative Models
Akshay Kamath, Eric Price, Sushrut Karmalkar
Abstract
The goal of compressed sensing is to learn a structured signal $x$ from a limited number of noisy linear measurements $y \approx Ax$. In traditional compressed sensing, “structure” is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with Bora-Jalal-Price-Dimakis’17 has instead considered structure to come from a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. We present two results establishing the difficulty and strength of this latter task, showing that existing bounds are tight: First, we provide a lower bound matching the Bora et.al upper bound for compressed sensing with $L$-Lipschitz generative models $G$ which holds even for the more relaxed goal of \emph{non-uniform} recovery. Second, we show that generative models generalize sparsity as a representation of structure by constructing a ReLU-based neural network with $2$ hidden layers and $O(n)$ activations per layer whose range is precisely the set of all $k$-sparse vectors.
BibTeX
@InProceedings{pmlr-v119-kamath20a,
title = {On the Power of Compressed Sensing with Generative Models},
author = {Kamath, Akshay and Price, Eric and Karmalkar, Sushrut},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {5101--5109},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/kamath20a/kamath20a.pdf},
url = {https://proceedings.mlr.press/v119/kamath20a.html},
abstract = {The goal of compressed sensing is to learn a structured signal $x$ from a limited number of noisy linear measurements $y \approx Ax$. In traditional compressed sensing, “structure” is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with Bora-Jalal-Price-Dimakis’17 has instead considered structure to come from a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. We present two results establishing the difficulty and strength of this latter task, showing that existing bounds are tight: First, we provide a lower bound matching the Bora et.al upper bound for compressed sensing with $L$-Lipschitz generative models $G$ which holds even for the more relaxed goal of \emph{non-uniform} recovery. Second, we show that generative models generalize sparsity as a representation of structure by constructing a ReLU-based neural network with $2$ hidden layers and $O(n)$ activations per layer whose range is precisely the set of all $k$-sparse vectors.}
}