Efficient Orthogonal Tensor Decomposition, with an Application to Latent Variable Model Learning

Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to be orthogonal across summands, by relating this orthogonal decomposition to the singular value decompositions of the flattenings. We show that it is a non-trivial assumption for a tensor to have such an orthogonal decomposition, and we show that it is unique (up to natural symmetries) in case it exists, in which case we also demonstrate how it can be efficiently and reliably obtained by a sequence of singular value decompositions. We demonstrate how the factoring algorithm can be applied for parameter identification in latent variable and mixture models

Blocking optimal kk-arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq A$ is called a \textbf{$k$-arborescence} if it is the disjoint union of $k$ spanning arborescences. The problem of finding a minimum cost $k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bern\'ath, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general $k$ that has polynomial running time if $k$ is fixed

Covering complete partite hypergraphs by monochromatic components

A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gy\'arf\'as: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Kir\'aly proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \rceil$ sets, each connected in some color. Here we investigate the analogue problem for complete r-uniform r-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of any color used in the coloring. We propose the following analogue of Ryser conjecture. In every spanning (r+t)-coloring of the edges of a complete r-uniform r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each connected in some color. Our main result is that the conjecture is true for 1 <= t <= r-1. We also prove a slightly weaker result for t >= r, namely that t+2 sets, each connected in some color, are enough to cover the vertex set. To build a bridge between complete r-uniform and complete r-uniform r-partite hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform (r,l)-partite if it has all r-sets that intersect each partite class in at most l vertices. Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l = 1+r-l, in every spanning k-coloring of the edges of a complete r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most 1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.Comment: 14 page

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mathrm{B}$? \textbf{Covering problem:} can we cover $E$ with an object of type $\mathrm{A}$, and an object of type $\mathrm{B}$? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an $s$-$t$ path $P$ and an $s'$-$t'$ path $P'$ that are edge-disjoint. However, many others were not, for example can we find an $s$-$t$ path $P\subseteq E$ and a spanning tree $T\subseteq E$ that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve a few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees