Related papers: The phase rank of a matrix
Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the…
Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…
Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations.…
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the…
The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the…
Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…
We consider multilevel low rank (MLR) matrices, defined as a row and column permutation of a sum of matrices, each one a block diagonal refinement of the previous one, with all blocks low rank given in factored form. MLR matrices extend low…
The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion…
Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, with every pencil one can associate a…
The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a…
We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix.…
This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and…
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…
We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks…
Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems,…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix with rank…
The minimum rank problem is to determine for a graph $G$ the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of $G$. Here $G$ is taken to be a circulant graph,…
This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is…
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system.…