Shiqian Ma

An efficient algorithm for compressed MR imaging using total variation and wavelets

Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compressed sensing makes it possible to reconstruct the same MR image from a very limited set of measurements significantly reducing the MRI scan duration. In order to do that however, one has to solve the difficult problem of minimizing nonsmooth functions on large data sets. To handle this, we propose an efficient algorithm that jointly minimizes the lscr1 norm, total variation, and a least squares measure, one of the most powerful models for compressive MR imaging. Our algorithm is based upon an iterative operator-splitting framework. The calculations are accelerated by continuation and takes advantage of fast wavelet and Fourier transforms enabling our code to process MR images from actual real life applications. We show that faithful MR images can be reconstructed from a subset that represents a mere 20 percent of the complete set of measurements.

Convergence of Fixed-Point Continuation Algorithms for Matrix Rank Minimization

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem (Math. Program., doi:10.1007/s10107-009-0306-5, 2009). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.

Fast Multiple-Splitting Algorithms for Convex Optimization

We present in this paper two different classes of general multiple-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized can be written as the sum of

Positive-Definite ℓ1-Penalized Estimation of Large Covariance Matrices

K

On the Global Linear Convergence of the ADMM with MultiBlock Variables

convex functions, each of which has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an

Semi-supervised sparse metric learning using alternating linearization optimization

\epsilon

Alternating Proximal Gradient Method for Convex Minimization

-optimal solution is

Accelerated Linearized Bregman Method

O((K-1)L/\epsilon)

An alternating direction method for total variation denoising

, where

Tensor principal component analysis via convex optimization

L