Self-Calibration and Biconvex Compressive Sensing

The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations y = DAx, where both x and the diagonal matrix D (which models the calibration error) are unknown. By "lifting" this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both x and D can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis

Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing

We study the question of extracting a sequence of functions $\{\boldsymbol{f}_i, \boldsymbol{g}_i\}_{i=1}^s$ from observing only the sum of their convolutions, i.e., from $\boldsymbol{y} = \sum_{i=1}^s \boldsymbol{f}_i\ast \boldsymbol{g}_i$. While convex optimization techniques are able to solve this joint blind deconvolution-demixing problem provably and robustly under certain conditions, for medium-size or large-size problems we need computationally faster methods without sacrificing the benefits of mathematical rigor that come with convex methods. In this paper, we present a non-convex algorithm which guarantees exact recovery under conditions that are competitive with convex optimization methods, with the additional advantage of being computationally much more efficient. Our two-step algorithm converges to the global minimum linearly and is also robust in the presence of additive noise. While the derived performance bounds are suboptimal in terms of the information-theoretic limit, numerical simulations show remarkable performance even if the number of measurements is close to the number of degrees of freedom. We discuss an application of the proposed framework in wireless communications in connection with the Internet-of-Things.Comment: Accepted to Information and Inference: a Journal of the IM

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

Neural Collapse for Unconstrained Feature Model under Cross-entropy Loss with Imbalanced Data

Recent years have witnessed the huge success of deep neural networks (DNNs) in various tasks of computer vision and text processing. Interestingly, these DNNs with massive number of parameters share similar structural properties on their feature representation and last-layer classifier at terminal phase of training (TPT). Specifically, if the training data are balanced (each class shares the same number of samples), it is observed that the feature vectors of samples from the same class converge to their corresponding in-class mean features and their pairwise angles are the same. This fascinating phenomenon is known as Neural Collapse (N C), first termed by Papyan, Han, and Donoho in 2019. Many recent works manage to theoretically explain this phenomenon by adopting so-called unconstrained feature model (UFM). In this paper, we study the extension of N C phenomenon to the imbalanced data under cross-entropy loss function in the context of unconstrained feature model. Our contribution is multi-fold compared with the state-of-the-art results: (a) we show that the feature vectors exhibit collapse phenomenon, i.e., the features within the same class collapse to the same mean vector; (b) the mean feature vectors no longer form an equiangular tight frame. Instead, their pairwise angles depend on the sample size; (c) we also precisely characterize the sharp threshold on which the minority collapse (the feature vectors of the minority groups collapse to one single vector) will take place; (d) finally, we argue that the effect of the imbalance in datasize diminishes as the sample size grows. Our results provide a complete picture of the N C under the cross-entropy loss for the imbalanced data. Numerical experiments confirm our theoretical analysis.Comment: 38 pages, 10 figure