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 pr...

Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. This dissertation extends the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix compl...

In multivariate linear regression, it is often assumed that the response matrix is intrinsically of lower rank. This could be because of the correlation structure among the prediction variables or the coefficient matrix being lower rank. To accommodate both, we propose a reduced rank ridge regression for multivariate linear regression. Specifically, we combine the ridge penalty with the reduced...

The low-rank matrix recovery is an important machine learning research topic with various scientific applications. Most existing low-rank matrix recovery methods relax the rank minimization problem via the trace norm minimization. However, such a relaxation makes the solution seriously deviate from the original one. Meanwhile, most matrix recovery methods minimize the squared prediction errors ...

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient, but have off-diagonal blocks that are; specifically, the class of so called Hierarchically Semi-Separable (HSS) matric...

A new method of using the numerical range a matrix to bound optimal value certain optimization problems over real tensor product vectors is presented. This stronger than trivial bounds based on eigenvalues and can be computed significantly faster provided by semidefinite programming relaxations. Numerous applications other hard linear algebra are discussed, such as showing that subspace matrice...

a new three-dimensional unit cell model has been developed for modeling three constituent phases including inclusion, interphase and matrix. the total elastic modulus of nano-composite is evaluated.  numerical results are in good agreement with the previous proposed theoretical modeling. higher matrix and inclusion elastic modulus both can dramatically influence the total elastic modulus.

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume m entries of an n× n rank r matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient d...

Differential Riccati equations are at the heart of many applications in control theory. They time-dependent, matrix-valued, and particular nonlinear that require special methods for their solution. Low-rank have been used heavily computing a low-rank solution every step time-discretization. We propose use an all-at-once space-time leading to large problem which we Newton–Kleinman iteration. App...

In this paper we consider the low-rank matrix completion problem with specific application to forecasting in time series analysis. Briefly, the low-rank matrix completion problem is the problem of imputing missing values of a matrix under a rank constraint. We consider a matrix completion problem for Hankel matrices and a convex relaxation based on the nuclear norm. Based on new theoretical res...