10 research outputs found
Some upper and lower bounds on PSD-rank
Positive semidefinite rank (PSD-rank) is a relatively new quantity with
applications to combinatorial optimization and communication complexity. We
first study several basic properties of PSD-rank, and then develop new
techniques for showing lower bounds on the PSD-rank. All of these bounds are
based on viewing a positive semidefinite factorization of a matrix as a
quantum communication protocol. These lower bounds depend on the entries of the
matrix and not only on its support (the zero/nonzero pattern), overcoming a
limitation of some previous techniques. We compare these new lower bounds with
known bounds, and give examples where the new ones are better. As an
application we determine the PSD-rank of (approximations of) some common
matrices.Comment: 21 page
Asymptotic nonnegative rank of matrices
The nonnegative rank of nonnegative matrices is an important quantity that
appears in many fields, such as combinatorial optimization, communication
complexity, and information theory. In this paper, we study the asymptotic
growth of the nonnegative rank of a fixed nonnegative matrix under Kronecker
product. This quantity is called the asymptotic nonnegative rank, which is
already studied in information theory. By applying the theory of asymptotic
spectra of V. Strassen (J. Reine Angew. Math. 1988), we introduce the
asymptotic spectrum of nonnegative matrices and give a dual characterization of
the asymptotic nonnegative rank. As the opposite of nonnegative rank, we
introduce the notion of the subrank of a nonnegative matrix and show that it is
exactly equal to the size of the maximum induced matching of the bipartite
graph defined on the support of the matrix (therefore, independent of the value
of entries). Finally, we show that two matrix parameters, namely rank and
fractional cover number, belong to the asymptotic spectrum of nonnegative
matrices
Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082)
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices"
Extension complexity of low-dimensional polytopes
Sometimes, it is possible to represent a complicated polytope as a projection
of a much simpler polytope. To quantify this phenomenon, the extension
complexity of a polytope is defined to be the minimum number of facets in a
(possibly higher-dimensional) polytope from which can be obtained as a
(linear) projection. This notion has been studied for several decades,
motivated by its relevance for combinatorial optimisation problems. It is an
important question to understand the extent to which the extension complexity
of a polytope is controlled by its dimension, and in this paper we prove three
different results along these lines. First, we prove that for a fixed dimension
, the extension complexity of a random -dimensional polytope (obtained as
the convex hull of random points in a ball or on a sphere) is typically on the
order of the square root of its number of vertices. Second, we prove that any
cyclic -vertex polygon (whose vertices lie on a circle) has extension
complexity at most . This bound is tight up to the constant factor
. Finally, we show that there exists an -dimensional polytope
with at most facets and extension complexity .Comment: We fixed an issue with Lemma 6.9 (the exponential Efron-Stein
inequality was previously used incorrectly
Heuristics for Exact Nonnegative Matrix Factorization
The exact nonnegative matrix factorization (exact NMF) problem is the
following: given an -by- nonnegative matrix and a factorization rank
, find, if possible, an -by- nonnegative matrix and an -by-
nonnegative matrix such that . In this paper, we propose two
heuristics for exact NMF, one inspired from simulated annealing and the other
from the greedy randomized adaptive search procedure. We show that these two
heuristics are able to compute exact nonnegative factorizations for several
classes of nonnegative matrices (namely, linear Euclidean distance matrices,
slack matrices, unique-disjointness matrices, and randomly generated matrices)
and as such demonstrate their superiority over standard multi-start strategies.
We also consider a hybridization between these two heuristics that allows us to
combine the advantages of both methods. Finally, we discuss the use of these
heuristics to gain insight on the behavior of the nonnegative rank, i.e., the
minimum factorization rank such that an exact NMF exists. In particular, we
disprove a conjecture on the nonnegative rank of a Kronecker product, propose a
new upper bound on the extension complexity of generic -gons and conjecture
the exact value of (i) the extension complexity of regular -gons and (ii)
the nonnegative rank of a submatrix of the slack matrix of the correlation
polytope.Comment: 32 pages, 2 figures, 16 table
Polütoopide laienditega seotud ülesanded
Väitekirja elektrooniline versioon ei sisalda publikatsiooneLineaarplaneerimine on optimeerimine matemaatilise mudeliga, mille sihi¬funktsioon ja kitsendused on esitatud lineaarsete seostega. Paljusid igapäeva elu väljakutseid võime vaadelda lineaarplaneerimise vormis, näiteks miinimumhinna või maksimaalse tulu leidmist. Sisepunkti meetod saavutab häid tulemusi nii teoorias kui ka praktikas ning lahendite leidmise tööaeg ja lineaarsete seoste arv on polünomiaalses seoses. Sellest tulenevalt eksponentsiaalne arv lineaarseid seoseid väljendub ka ekponentsiaalses tööajas.
Iga vajalik lineaarne seos vastab ühele polütoobi P tahule, mis omakorda tähistab lahendite hulka. Üks võimalus tööaja vähendamiseks on suurendada dimensiooni, mille tulemusel väheneks ka polütoobi tahkude arv. Saadud polütoopi Q nimeta¬takse polütoobi P laiendiks kõrgemas dimensioonis ning polütoobi Q minimaalset tahkude arvu nimetakakse polütoobi P laiendi keerukuseks, sellisel juhul optimaalsete lahendite hulk ei muutu. Tekib küsimus, millisel juhul on võimalik leida laiend Q, mille korral tahkude arv on polünomiaalne.
Mittedeterministlik suhtluskeerukus mängib olulist rolli tõestamaks polütoopide laiendite keerukuse alampiiri. Polütoobile P vastava suhtluskeerukuse leidmine ning alamtõkke tõestamine väistavad võimalused leida laiend Q, mis ei oleks eksponentsiaalne.
Käesolevas töös keskendume me juhuslikele Boole'i funktsioonidele f, mille tihedusfunktsioon on p = p(n). Me pakume välja vähima ülemtõkke ning suurima alamtõkke mittedeterministliku suhtluskeerukuse jaoks. Lisaks uurime me ka pedigree polütoobi graafi. Pedigree polütoop on rändkaupmehe ülesande polütoobi laiend, millel on kombinatoorne struktuur. Polütoobi graafi võib vaadelda kui abstraktset graafi ning see annab informatsiooni polütoobi omaduste kohta.The linear programming (LP for short) is a method for finding an optimal solution, such as minimum cost or maximum profit for a linear function subject to linear constraints. But having an exponential number of inequalities gives the exponential running time in solving linear program. A polytope, let's say P, represents the space of the feasible solution. One idea for decreasing the running time of the problem, is lifting the polytope P tho the higher dimensions with the goal of decresing the number of inequalities. The polytope in higher dimension, let's say Q, is the extension of the original polytope P and the minimum number of facets that Q can have is the extension complexity of P. Then the optimal solution of the problem over Q, gives the optimal solution over P. The natural question may raise is when is it possible to have an extension with a polynomial number of inequalities?
Nondeterministic communication complexity is a powerful tool for proving lower bound on the extension complexity of a polytopes. Finding a suitable communication complexity problem corresponded to a polytope P and proving a linear lower bound for the nondeterministic communication complexity of it, will rule out all the attempts for finding sub-exponential size extension Q of P.
In this thesis, we focus on the random Boolean functions f, with density p = p(n). We give tight upper and lower bounds for the nondeterministic communication complexity and parameters related to it. Also, we study the rank of fooling set matrix which is an important lower bound for nondeterministic communication complexity.
Finally, we investigate the graph of the pedigree polytope. Pedigree polytope is an extension of TSP (traveling salesman problem; the most extensively studied problem in combinatorial optimization) polytopes with a nice combinatorial structure. The graph of a polytope can be regarded as an abstract graph and it reveals meaningful information about the properties of the polytope