2,237 research outputs found
Path coverings with prescribed ends in faulty hypercubes
We discuss the existence of vertex disjoint path coverings with prescribed
ends for the -dimensional hypercube with or without deleted vertices.
Depending on the type of the set of deleted vertices and desired properties of
the path coverings we establish the minimal integer such that for every such path coverings exist. Using some of these results, for ,
we prove Locke's conjecture that a hypercube with deleted vertices of each
parity is Hamiltonian if Some of our lemmas substantially
generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the
paper we formulate some conjectures supported by our results.Comment: 26 page
When Does a Mixture of Products Contain a Product of Mixtures?
We derive relations between theoretical properties of restricted Boltzmann
machines (RBMs), popular machine learning models which form the building blocks
of deep learning models, and several natural notions from discrete mathematics
and convex geometry. We give implications and equivalences relating
RBM-representable probability distributions, perfectly reconstructible inputs,
Hamming modes, zonotopes and zonosets, point configurations in hyperplane
arrangements, linear threshold codes, and multi-covering numbers of hypercubes.
As a motivating application, we prove results on the relative representational
power of mixtures of product distributions and products of mixtures of pairs of
product distributions (RBMs) that formally justify widely held intuitions about
distributed representations. In particular, we show that a mixture of products
requiring an exponentially larger number of parameters is needed to represent
the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table
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