Abstract Low-rank matrix recovery problem is difficult due to its non-convex properties and it is usually solved using convex relaxation approaches. In this paper, we formulate the non-convex low-rank matrix recovery problem exactly using novel Ky Fan 2-k-norm-based models. A general difference of convex functions algorithm (DCA) is developed to solve these models. A proximal point algorithm (PPA) framework is proposed to solve sub-problems within the DCA, which allows us to handle large instances. Numerical results show that the proposed models achieve high recoverability rates as compared to the truncated nuclear norm method and the alternating bilinear optimization approach. The results also demonstrate that the proposed DCA with the PPA framework is efficient in handling larger instances.

Community detection is a widely-studied unsupervised learning problem in which the task is to group similar entities together based on observed pairwise entity interactions. This problem has applications in diverse domains such as social network analysis and computational biology. There is a significant amount of literature studying this problem under the assumption that the communities do not overlap. When the communities are allowed to overlap, often a \textit{pure nodes} assumption is made, i.e. each community has a node that belongs exclusively to that community. This assumption, however, may not always be satisfied in practice. In this paper, we provide a provable method to detect overlapping communities in weighted graphs without explicitly making the pure nodes assumption. Moreover, contrary to most existing algorithms, our approach is based on convex optimization, for which many useful theoretical properties are already known. We demonstrate the success of our algorithm on artificial and real-world datasets.

In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We present a family of fast recursive algorithms and prove they are robust under any small perturbations of the input data matrix. This family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance.

Nonnegative matrix factorization (NMF) was popularized as a tool for data mining by Lee and Seung in 1999. NMF attempts to approximate a matrix with nonnegative entries by a product of two low-rank matrices, also with nonnegative entries. We propose an algorithm called rank-one downdate (R1D) for computing an NMF that is partly motivated by the singular value decomposition. This algorithm computes the dominant singular values and vectors of adaptively determined sub-matrices of a matrix. On each iteration, R1D extracts a rank-one submatrix from the original matrix according to an objective function. We establish a theoretical result that maximizing this objective function corresponds to correctly classifying articles in a nearly separable corpus. We also provide computational experiments showing the success of this method in identifying features in realistic datasets. The method is also much faster than other NMF routines.

We consider the problem of triangulating a d-dimensional region. Our mesh generation algorithm, called QMG, is a quadtree-based algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to overrefine the domain, in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method.

A collection of n balls in d dimensions forms a k -ply system if no point in the space is covered by more than k balls. We show that for every k -ply system Γ, there is a sphere S that intersects at most O ( k 1/ d n 1−1/ d ) balls of Γ and divides the remainder of Γ into two parts: those in the interior and those in the exterior of the sphere S , respectively, so that the larger part contains at most (1−1/( d +2)) n balls. This bound of ( O ( k 1/ d n 1−1/ d ) is the best possible in both n and k . We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k -nearest neighbor graphs of n points in d dimensions has a separator of size O ( k 1/ d n 1−1/ d ). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

We consider the problem of triangulating a ddimensional uniform grid of d-cubes that is cut by a k-dimensional affine subspace.The goal is to obtain a triangulation with bounded aspect ratio.To achieve this goal, we allow some of the box faces near the affine subspace to be displaced.This problem has applications to finite element mesh generation.For general d and k, the bound on aspect ratio that we attain is double-exponential in d.For the import ant special case of d = 3, the aspect ratio bound is small enough that the technique is useful in practice.

We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds.

A class of graphs called k-overlap graphs is proposed. Special cases of k-overlap graphs include planar graphs, k-nearest neighbor graphs, and earlier classes of graphs associated with finite element methods. A separator bound is proved for k-overlap graphs embedded in d dimensions. The result unifies several earlier separator results. All the arguments are based on geometric properties of embedding. The separator bounds come with randomized linear-time and randomized NC algorithms. Moreover, the bounds are the best possible up to the leading term.>

Gaussian elimination with partial pivoting is the standard numerical algorithm for solving unstructured linear systems. Here it is shown that Gaussian elimination with partial pivoting or complete pivoting is log-space complete for P. This provides theoretical evidence that these algorithms cannot be efficiently implemented on a highly parallel computer with a large number of processors. Since other algorithms for linear systems that are efficient on parallel computers are already known, this suggests that elimination-based approaches should not be pursued in a parallel environment with many processors.

The Brouwer fixed point theorem has become a major tool for modeling economic systems during the 20th century. It was intractable to use the theorem in a computational manner until 1965 when Scarf provided the first practical algorithm for finding a fixed point of a Brouwer map. Scarf's work left open the question of worstcase complexity, although he hypothesized that his algorithm had "typical" behavior of polynomial time in the number of variables of the problem. Here we show that any algorithm for fixed points based on function evaluation (which includes all general purpose fixed-point algorithrna) must in the worst case take a number of steps which is exponential both in the number of digits of accuracy and in the number of variables.