The problem of recovering a structured signal x ∈ C p from a set of dimensionality-reduced linear measurements b = Ax arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix A ∈ C n×p is often of the form A = PΩΨ for some orthonormal basis matrix Ψ ∈ C p×p and subsampling operator PΩ : C p → C n that selects the rows indexed by Ω. This raises the fundamental question of how best to choose the index set Ω in order to optimize the recovery performance. Previous approaches to addressing this question rely on non-uniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals x1, . . . , xm. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets.
We study a phase retrieval problem in the Poisson noise model. Motivated by the PhaseLift approach, we approximate the maximumlikelihood estimator by solving a convex program with a nuclear norm constraint. While the Frank-Wolfe algorithm, together with the Lanczos method, can efficiently deal with nuclear norm constraints, our objective function does not have a Lipschitz continuous gradient, and hence existing convergence guarantees for the Frank-Wolfe algorithm do not apply. In this paper, we show that the Frank-Wolfe algorithm works for the Poisson phase retrieval problem, and has a global convergence rate of O(1/t), where t is the iteration counter. We provide rigorous theoretical guarantee and illustrating numerical results.
In the area of magnetic resonance imaging (MRI), an extensive range of non-linear reconstruction algorithms has been proposed which can be used with general Fourier subsampling patterns. However, the design of these subsampling patterns has typically been considered in isolation from the reconstruction rule and the anatomy under consideration. In this paper, we propose a learning-based framework for optimizing MRI subsampling patterns for a specific reconstruction rule and anatomy, considering both the noiseless and noisy settings. Our learning algorithm has access to a representative set of training signals, and searches for a sampling pattern that performs well on average for the signals in this set. We present a novel parameter-free greedy mask selection method and show it to be effective for a variety of reconstruction rules and performance metrics. Moreover, we also support our numerical findings by providing a rigorous justification of our framework via statistical learning theory.
It is proved that in a non-Bayesian parametric estimation problem, if the Fisher information matrix (FIM) is singular, unbiased estimators for the unknown parameter will not exist. Cramér-Rao bound (CRB), a popular tool to lower bound the variances of unbiased estimators, seems inapplicable in such situations. In this paper, we show that the Moore-Penrose generalized inverse of a singular FIM can be interpreted as the CRB corresponding to the minimum variance among all choices of minimum constraint functions. This result ensures the logical validity of applying the Moore-Penrose generalized inverse of an FIM as the covariance lower bound when the FIM is singular.Furthermore, the result can be applied as a performance bound on the joint design of constraint functions and unbiased estimators. Index TermsConstrained parameters, Cramér-Rao bound (CRB), singular Fisher information matrix (FIM).
The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the selfconcordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we develop a variable metric framework of minimizing the sum of a "simple" convex function and a self-concordant-like function. We introduce a new analytic step-size selection procedure and prove that the basic gradient algorithm has improved convergence guarantees as compared to "fast" algorithms that rely on the Lipschitz gradient property. Our numerical tests with real-data sets show that the practice indeed follows the theory.
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