Convexity in source separation: Models, geometry, and algorithms

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems. Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work

Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\ast g$. This problem, known as {\em blind deconvolution}, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on $f$ and $g$. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework

DRUG TARGET DECONVOLUTION IN CANCER CELL LINES

The deconvolution problem to identify the critical protein targets behind drug sensitivity profiling is an important part of drug development. It helps us to understand the mechanism of action of anti-cancer drugs on the cell lines through protein targets in those cell lines. This problem can be formulated as a matrix deconvolution problem, with two matrices for the cell-based drug sensitivity profiling and drug target interaction data, respectively. The model needs to be solved to identify the vulnerability of the cell lines to inhibition of critical targets. We used drug sensitivity data for 265 anti-cancer compounds over 990 cell models taken from cancer patients and cultivated in the lab. Using the data on interaction of these drugs with the protein targets, we used a novel method called TDSBS (target deconvolution with semi-blind source separation) in order to determine the critical targets for each cell model. The critical protein targets determined using this method were found to be clinically relevant, as we could determine that the driver genes have higher TDSBS values compared to the non-driver genes in the cell models. In this thesis we demonstrate a general statistical model which can be used to identify the protein targets which are inhibited by anti-cancer drugs in drug/cell line sensitivity experiments