Compressed Sensing with Adversarial Sparse Noise via L1 Regression

We present a simple and effective algorithm for the problem of sparse robust linear regression. In this problem, one would like to estimate a sparse vector w^* in R^n from linear measurements corrupted by sparse noise that can arbitrarily change an adversarially chosen eta fraction of measured responses y, as well as introduce bounded norm noise to the responses. For Gaussian measurements, we show that a simple algorithm based on L1 regression can successfully estimate w^* for any eta < eta_0 ~~ 0.239, and that this threshold is tight for the algorithm. The number of measurements required by the algorithm is O(k log n/k) for k-sparse estimation, which is within constant factors of the number needed without any sparse noise. Of the three properties we show - the ability to estimate sparse, as well as dense, w^*; the tolerance of a large constant fraction of outliers; and tolerance of adversarial rather than distributional (e.g., Gaussian) dense noise - to the best of our knowledge, no previous polynomial time algorithm was known to achieve more than two

Compressed sensing recovery with unlearned neural networks

This report investigates methods for solving the problem of compressed sensing, in which the goal is to recover a signal from noisy, linear measurements. Compressed sensing techniques enable signal recovery with far fewer measurements than required by traditional methods such as Nyquist sampling. Signal recovery is an incredibly important area in application domains such as consumer electronics, medical imaging, and many others. While classical methods for compressed sensing recovery are well established, recent developments in machine learning have created wide opportunity for improvement. In this report I first discuss pre-existing approaches, both classical and modern. I then present my own contribution to this field: creating a method using untrained machine learning models. This approach has several advantages which enable its use in complex domains such as medical imagingElectrical and Computer Engineerin