2,486 research outputs found
Computing a Nonnegative Matrix Factorization -- Provably
In the Nonnegative Matrix Factorization (NMF) problem we are given an nonnegative matrix and an integer . Our goal is to express
as where and are nonnegative matrices of size
and respectively. In some applications, it makes sense to ask
instead for the product to approximate -- i.e. (approximately)
minimize \norm{M - AW}_F where \norm{}_F denotes the Frobenius norm; we
refer to this as Approximate NMF. This problem has a rich history spanning
quantum mechanics, probability theory, data analysis, polyhedral combinatorics,
communication complexity, demography, chemometrics, etc. In the past decade NMF
has become enormously popular in machine learning, where and are
computed using a variety of local search heuristics. Vavasis proved that this
problem is NP-complete. We initiate a study of when this problem is solvable in
polynomial time:
1. We give a polynomial-time algorithm for exact and approximate NMF for
every constant . Indeed NMF is most interesting in applications precisely
when is small.
2. We complement this with a hardness result, that if exact NMF can be solved
in time , 3-SAT has a sub-exponential time algorithm. This rules
out substantial improvements to the above algorithm.
3. We give an algorithm that runs in time polynomial in , and
under the separablity condition identified by Donoho and Stodden in 2003. The
algorithm may be practical since it is simple and noise tolerant (under benign
assumptions). Separability is believed to hold in many practical settings.
To the best of our knowledge, this last result is the first example of a
polynomial-time algorithm that provably works under a non-trivial condition on
the input and we believe that this will be an interesting and important
direction for future work.Comment: 29 pages, 3 figure
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model
A dual formulation of group field theories, obtained by a Fourier transform
mapping functions on a group to functions on its Lie algebra, has been proposed
recently. In the case of the Ooguri model for SO(4) BF theory, the variables of
the dual field variables are thus so(4) bivectors, which have a direct
interpretation as the discrete B variables. Here we study a modification of the
model by means of a constraint operator implementing the simplicity of the
bivectors, in such a way that projected fields describe metric tetrahedra. This
involves a extension of the usual GFT framework, where boundary operators are
labelled by projected spin network states. By construction, the Feynman
amplitudes are simplicial path integrals for constrained BF theory. We show
that the spin foam formulation of these amplitudes corresponds to a variant of
the Barrett-Crane model for quantum gravity. We then re-examin the arguments
against the Barrett-Crane model(s), in light of our construction.Comment: revtex, 24 page
Extended matter coupled to BF theory
Recently, a topological field theory of membrane-matter coupled to BF theory
in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss
various aspects of the four-dimensional theory. Firstly, we study classical
solutions leading to an interpretation of the theory in terms of strings
propagating on a flat spacetime. We also show that the general classical
solutions of the theory are in one-to-one correspondence with solutions of
Einstein's equations in the presence of distributional matter (cosmic strings).
Secondly, we quantize the theory and present, in particular, a prescription to
regularize the physical inner product of the canonical theory. We show how the
resulting transition amplitudes are dual to evaluations of Feynman diagrams
coupled to three-dimensional quantum gravity. Finally, we remove the regulator
by proving the topological invariance of the transition amplitudes.Comment: 27 pages, 7 figure
Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects
We consider three a priori totally different setups for Hopf algebras from
number theory, mathematical physics and algebraic topology. These are the Hopf
algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for
renormalization, and a Hopf algebra constructed by Baues to study double loop
spaces. We show that these examples can be successively unified by considering
simplicial objects, co-operads with multiplication and Feynman categories at
the ultimate level. These considerations open the door to new constructions and
reinterpretations of known constructions in a large common framework, which is
presented step-by-step with examples throughout. In this first part of two
papers, we concentrate on the simplicial and operadic aspects.Comment: This replacement is part I of the final version of the paper, which
has been split into two parts. The second part is available from the arXiv
under the title "Three Hopf algebras from number theory, physics & topology,
and their common background II: general categorical formulation"
arXiv:2001.0872
Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
We use convex polyhedral cones to study a large class of multivariate
meromorphic germs, namely those with linear poles, which naturally arise in
various contexts in mathematics and physics. We express such a germ as a sum of
a holomorphic germ and a linear combination of special non-holomorphic germs
called polar germs. In analyzing the supporting cones -- cones that reflect the
pole structure of the polar germs -- we obtain a geometric criterion for the
non-holomorphicity of linear combinations of polar germs. This yields the
uniqueness of the above sum when required to be supported on a suitable family
of cones and assigns a Laurent expansion to the germ. Laurent expansions
provide various decompositions of such germs and thereby a uniformized proof of
known results on decompositions of rational fractions. These Laurent expansions
also yield new concepts on the space of such germs, all of which are
independent of the choice of the specific Laurent expansion. These include a
generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct
in the space of such germs. When applied to exponential sums on rational convex
polyhedral cones, the filtered residue yields back exponential integrals.Comment: 30 page
A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity
Group Field Theories, a generalization of matrix models for 2d gravity,
represent a 2nd quantization of both loop quantum gravity and simplicial
quantum gravity. In this paper, we construct a new class of Group Field Theory
models, for any choice of spacetime dimension and signature, whose Feynman
amplitudes are given by path integrals for clearly identified discrete gravity
actions, in 1st order variables. In the 3-dimensional case, the corresponding
discrete action is that of 1st order Regge calculus for gravity (generalized to
include higher order corrections), while in higher dimensions, they correspond
to a discrete BF theory (again, generalized to higher order) with an imposed
orientation restriction on hinge volumes, similar to that characterizing
discrete gravity. The new models shed also light on the large distance or
semi-classical approximation of spin foam models. This new class of group field
theories may represent a concrete unifying framework for loop quantum gravity
and simplicial quantum gravity approaches.Comment: 48 pages, 4 figures, RevTeX, one reference adde
- …