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Hypothesis testing of covariance matrices is an important problem in multivariate analysis. Given n data samples and a covariance matrix Σο, the goal is to determine whether or not the data is consistent with this matrix. In this paper we introduce a framework that we call sketched covariance testing, where the data is provided after being compressed by multiplying by a “sketching” matrix A chosen...
Signal demixing arises in many applications. Common among these are the separation of sparse and low rank components in image and video processing, sparse and group sparse models in multitask learning and spikes and sinusoids in source separation problems. For specific problems of interest, many methods exist to perform recovery, but an approach that generalizes to arbitrary notions of simplicity...
Line spectral estimation is a classical signal processing problem that finds numerous applications in array signal processing and speech analysis. We propose a robust approach for line spectral estimation based on atomic norm minimization that is able to recover the spectrum exactly even when the observations are corrupted by arbitrary but sparse outliers. The resulting optimization problem is reformulated...
This invited submission summarizes recent work by the authors on conic geometric programs (CGPs), which are convex optimization problems obtained by blending geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). GPs and SDPs are two prominent families of structured convex programs that each generalize linear programs (LPs) in different ways, and that are both...
In many applications in signal and image processing, communications, and system identification, one aims to recover a signal that has a simple representation in a given basis or frame. Key devices for obtaining such representations are objects called atoms, and functions called atomic norms. These concepts unify the idea of simple representations across several known applications, and motivate extensions...
This paper proposes a new algorithm for linear system identification from noisy measurements. The proposed algorithm balances a data fidelity term with a norm induced by the set of single pole filters. We pose a convex optimization problem that approximately solves the atomic norm minimization problem and identifies the unknown system from noisy linear measurements. This problem can be solved efficiently...
Learning covariance matrices from high-dimensional data is an important problem that has received a lot of attention recently. We are particularly interested in the high-dimensional setting, where the number of samples one has access to is fewer than the number of variates. Fortunately, in many applications of interest, the underlying covariance matrix is sparse and hence has limited degrees of freedom...
We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressive sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples,...
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the notion of a symmetric model via group invariance. We propose projection onto a group fixed point subspace as a fundamental way of regularizing covariance matrices in...
This paper considers the fully decentralized H2 model matching optimization for continuous-time LTI systems. The properties of vector operator and Khatri-Rao product are exploited to reduce the problem to a one-sided centralized setting of a higher dimension. The potential of this approach for the derivation of explicit state-space formulae is examined. First, a closed-form, though not minimal state-space...
We propose a novel and natural architecture for decentralized control, that is applicable whenever the underlying system has the structure of a partially ordered set (poset). This controller architecture is based on the Möbius transform of the poset, and enjoys simple and appealing separation properties, since the closed-loop dynamics can be analyzed in terms of decoupled subsystems. The controller...
In this paper, we propose a novel exemplar based technique for classification problems where for every new test sample the classification model is re-estimated from a subset of relevant samples of the training data.We formulate the exemplar-based classification paradigm as a sparse representation (SR) problem, and explore the use of convex hull constraints to enforce both regularization and sparsity...
We introduce an algorithm known as Manifold Iterative Projection to solve the problem of recovering an unknown high-dimensional signal contained in a low-dimensional sub-manifold from a few linear measurements. The algorithm provably and robustly recovers any unknown signal on the manifold, provided the measurement operator is benign with respect to the manifold. A variant of the algorithm provably...
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