6,463 research outputs found
Universal Approximation Depth and Errors of Narrow Belief Networks with Discrete Units
We generalize recent theoretical work on the minimal number of layers of
narrow deep belief networks that can approximate any probability distribution
on the states of their visible units arbitrarily well. We relax the setting of
binary units (Sutskever and Hinton, 2008; Le Roux and Bengio, 2008, 2010;
Mont\'ufar and Ay, 2011) to units with arbitrary finite state spaces, and the
vanishing approximation error to an arbitrary approximation error tolerance.
For example, we show that a -ary deep belief network with layers of width for some can approximate any probability
distribution on without exceeding a Kullback-Leibler
divergence of . Our analysis covers discrete restricted Boltzmann
machines and na\"ive Bayes models as special cases.Comment: 19 pages, 5 figures, 1 tabl
Evidence for a Lattice Weak Gravity Conjecture
The Weak Gravity Conjecture postulates the existence of superextremal charged
particles, i.e. those with mass smaller than or equal to their charge in Planck
units. We present further evidence for our recent observation that in known
examples a much stronger statement is true: an infinite tower of superextremal
particles of different charges exists. We show that effective Kaluza-Klein
field theories and perturbative string vacua respect the Sublattice Weak
Gravity Conjecture, namely that a finite index sublattice of the full charge
lattice exists with a superextremal particle at each site. In perturbative
string theory we show that this follows from modular invariance. However, we
present counterexamples to the stronger possibility that a superextremal
particle exists at every lattice site, including an example in which the
lightest charged particle is subextremal. The Sublattice Weak Gravity
Conjecture has many implications both for abstract theories of quantum gravity
and for real-world physics. For instance, it implies that if a gauge group with
very small coupling exists, then the fundamental gravitational cutoff
energy of the theory is no higher than .Comment: v2: 41 pages, typos fixed, references added, substantial revisions
and clarifications (conclusions unchanged
Robust regression with imprecise data
We consider the problem of regression analysis with imprecise data. By imprecise data we mean imprecise observations of precise quantities in the form of sets of values. In this paper, we explore a recently introduced likelihood-based approach to regression with such data. The approach is very general, since it covers all kinds of imprecise data (i.e. not only intervals) and it is not restricted to linear regression. Its result consists of a set of functions, reflecting the entire uncertainty of the regression problem. Here we study in particular a robust special case of the likelihood-based imprecise regression, which can be interpreted as a generalization of the method of least median of squares. Moreover, we apply it to data from a social survey, and compare it with other approaches to regression with imprecise data. It turns out that the likelihood-based approach is the most generally applicable one and is the only approach accounting for multiple sources of uncertainty at the same time
Hurewicz Theorem for Assouad-Nagata dimension
Given a function of metric spaces, its {\it asymptotic
dimension} \asdim(f) is the supremum of \asdim(A) such that
and \asdim(f(A))=0. Our main result is \begin{Thm} \label{ThmAInAbstract}
\asdim(X)\leq \asdim(f)+\asdim(Y) for any large scale uniform function
. \end{Thm}
\ref{ThmAInAbstract} generalizes a result of Bell and Dranishnikov in which
is Lipschitz and is geodesic. We provide analogs of
\ref{ThmAInAbstract} for Assouad-Nagata dimension and asymptotic
Assouad-Nagata dimension \ANasdim. In case of linearly controlled asymptotic
dimension \Lasdim we provide counterexamples to three questions in a list of
problems of Dranishnikov.
As an application of analogs of \ref{ThmAInAbstract} we prove \begin{Thm}
\label{ThmBInAbstract} If is an exact sequence of
groups and is finitely generated, then \ANasdim (G,d_G)\leq \ANasdim
(K,d_G|K)+\ANasdim (H,d_H) for any word metrics metrics on and
on . \end{Thm}
\ref{ThmBInAbstract} extends a result of Bell and Dranishnikov for asymptotic
dimension
Power-law distributions in binned empirical data
Many man-made and natural phenomena, including the intensity of earthquakes,
population of cities and size of international wars, are believed to follow
power-law distributions. The accurate identification of power-law patterns has
significant consequences for correctly understanding and modeling complex
systems. However, statistical evidence for or against the power-law hypothesis
is complicated by large fluctuations in the empirical distribution's tail, and
these are worsened when information is lost from binning the data. We adapt the
statistically principled framework for testing the power-law hypothesis,
developed by Clauset, Shalizi and Newman, to the case of binned data. This
approach includes maximum-likelihood fitting, a hypothesis test based on the
Kolmogorov--Smirnov goodness-of-fit statistic and likelihood ratio tests for
comparing against alternative explanations. We evaluate the effectiveness of
these methods on synthetic binned data with known structure, quantify the loss
of statistical power due to binning, and apply the methods to twelve real-world
binned data sets with heavy-tailed patterns.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS710 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Evaluating geometric queries using few arithmetic operations
Let \cp:=(P_1,...,P_s) be a given family of -variate polynomials with
integer coefficients and suppose that the degrees and logarithmic heights of
these polynomials are bounded by and , respectively. Suppose furthermore
that for each the polynomial can be evaluated using
arithmetic operations (additions, subtractions, multiplications and the
constants 0 and 1). Assume that the family \cp is in a suitable sense
\emph{generic}. We construct a database , supported by an algebraic
computation tree, such that for each the query for the signs of
can be answered using h d^{\cO(n^2)} comparisons and
arithmetic operations between real numbers. The arithmetic-geometric tools
developed for the construction of are then employed to exhibit example
classes of systems of polynomial equations in unknowns whose
consistency may be checked using only few arithmetic operations, admitting
however an exponential number of comparisons
On the asymptotic geometry of abelian-by-cyclic groups
A finitely presented, torsion free, abelian-by-cyclic group can always be
written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n
integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only
if |det(M)|=1. We give a complete classification of the nonpolycyclic groups
Gamma_M up to quasi-isometry: given n x n integer matrices M,N with |det(M)|,
|det(N)| > 1, the groups Gamma_M, Gamma_N are quasi-isometric if and only if
there exist positive integers r,s such that M^r, N^s have the same absolute
Jordan form. We also prove quasi-isometric rigidity: if Gamma_M is an
abelian-by-cyclic group determined by an n x n integer matrix M with |det(M)| >
1, and if G is any finitely generated group quasi-isometric to Gamma_M, then
there is a finite normal subgroup K of G such that G/K is abstractly
commensurable to Gamma_N, for some n x n integer matrix N with |det(N)| > 1.Comment: 65 pages, 2 figures. To appear in Acta Mathematic
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