The cut-rank of a set X in graph G is the rank X×(V(G)?X) submatrix adjacency matrix over binary field. A split partition vertex into two sets (X,Y) such that less than 2 and both Y have at least vertices. prime (with respect to decomposition) if it connected has no splits. k+?-rank-connected for every vertices with k, |X| or |V(G)?X| k+?. We prove 3+2-rank-connected 10 3+3-rank-connected pivot...

This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 dv/6e which in turn shows that the psd ran...

Let G be a directed graph with n vertices, subject to dynamic updates, and such that each edge weight can assume at most S diierent arbitrary real values throughout the sequence of updates. We present a new algorithm for maintaining all pairs shortest n) amortized time per update and in O(1) worst-case time per distance query. This improves over previous bounds. We also show how to obtain query...

MP1 : min y,η cy + η s.t. Ay ≥ d η ≥ (h−Ey −Mul ) π∗ l , ∀l ≤ k y ∈ Sy, η ∈ R. Derive an optimal solution (y∗ k+1, η ∗ k+1) and update LB = c Ty∗ k+1 + η ∗ k+1. 3. Call the oracle to solve SP1 in (2) in Section 2, i.e. Q(y∗ k+1), and derive an optimal solution (uk+1, π ∗ k+1). Update UB = min{UB, cTy∗ k+1 +Q(y∗ k+1)}. 4. If UB − LB ≤ 2, return y∗ k+1 and terminate. Otherwise, update k = k + 1 a...

In many real-world applications, data are represented by matrices or high-order tensors. Despite the promising performance, the existing 2-D discriminant analysis algorithms employ a single projection model to exploit the discriminant information for projection, making the model less flexible. In this paper, we propose a novel compound rank- k projection (CRP) algorithm for bilinear analysis. T...

In this work, we propose modification for PSB update with a new extended Quasi – Newton condition unconstrained optimization problem, so it's called (α PSB) method. kind of rank two update, which solve the but can't guarantee positive definite property Hessian matrix. matrix be confirmed by updating vector sk, represent difference between next gradient and current objective function assume to c...

We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-k graphs gives rise to pullbacks of the associated C∗algebras. We describe a combinat...

We prove that any real matrix A contains a subset of at most 4k/ + 2k log(k + 1) rows whose span “contains” a matrix of rank at most k with error only (1 + ) times the error of the best rank-k approximation of A. We complement it with an almost matching lower bound by constructing matrices where the span of any k/2 rows does not “contain” a relative (1 + )-approximation of rank k. Our existence...

Aiming at convex optimization under structural constraints, this work introduces and analyzes a variant of the Frank Wolfe (FW) algorithm termed ExtraFW. The distinct feature ExtraFW is pair gradients leveraged per iteration, thanks to which decision variable updated in prediction-correction (PC) format. Relying on no problem dependent parameters step sizes, convergence rate for general problem...