10,978 research outputs found
A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces
A Wasserstein spaces is a metric space of sufficiently concentrated
probability measures over a general metric space. The main goal of this paper
is to estimate the largeness of Wasserstein spaces, in a sense to be precised.
In a first part, we generalize the Hausdorff dimension by defining a family of
bi-Lipschitz invariants, called critical parameters, that measure largeness for
infinite-dimensional metric spaces. Basic properties of these invariants are
given, and they are estimated for a naturel set of spaces generalizing the
usual Hilbert cube. In a second part, we estimate the value of these new
invariants in the case of some Wasserstein spaces, as well as the dynamical
complexity of push-forward maps. The lower bounds rely on several embedding
results; for example we provide bi-Lipschitz embeddings of all powers of any
space inside its Wasserstein space, with uniform bound and we prove that the
Wasserstein space of a d-manifold has "power-exponential" critical parameter
equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all
part I on generalized Hausdorff dimension is new, as well as the embedding of
Hilbert cubes into Wasserstein spaces. v3 modified according to the referee
final remarks ; to appear in Journal of Topology and Analysi
Indexability, concentration, and VC theory
Degrading performance of indexing schemes for exact similarity search in high
dimensions has long since been linked to histograms of distributions of
distances and other 1-Lipschitz functions getting concentrated. We discuss this
observation in the framework of the phenomenon of concentration of measure on
the structures of high dimension and the Vapnik-Chervonenkis theory of
statistical learning.Comment: 17 pages, final submission to J. Discrete Algorithms (an expanded,
improved and corrected version of the SISAP'2010 invited paper, this e-print,
v3
Poincar\'e profiles of groups and spaces
We introduce a spectrum of monotone coarse invariants for metric measure
spaces called Poincar\'{e} profiles. The two extremes of this spectrum
determine the growth of the space, and the separation profile as defined by
Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the
Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic
spaces, where we deduce a connection between these profiles and conformal
dimension. As applications, we use these invariants to show the non-existence
of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican
Impossibility of dimension reduction in the nuclear norm
Let (the Schatten--von Neumann trace class) denote the Banach
space of all compact linear operators whose nuclear norm
is finite, where
are the singular values of . We prove that
for arbitrarily large there exists a subset
with that cannot be
embedded with bi-Lipschitz distortion into any -dimensional
linear subspace of . is not even a -Lipschitz
quotient of any subset of any -dimensional linear subspace of
. Thus, does not admit a dimension reduction
result \'a la Johnson and Lindenstrauss (1984), which complements the work of
Harrow, Montanaro and Short (2011) on the limitations of quantum dimension
reduction under the assumption that the embedding into low dimensions is a
quantum channel. Such a statement was previously known with
replaced by the Banach space of absolutely summable sequences via the
work of Brinkman and Charikar (2003). In fact, the above set can
be taken to be the same set as the one that Brinkman and Charikar considered,
viewed as a collection of diagonal matrices in . The challenge is
to demonstrate that cannot be faithfully realized in an arbitrary
low-dimensional subspace of , while Brinkman and Charikar
obtained such an assertion only for subspaces of that consist of
diagonal operators (i.e., subspaces of ). We establish this by proving
that the Markov 2-convexity constant of any finite dimensional linear subspace
of is at most a universal constant multiple of
Euclidean quotients of finite metric spaces
This paper is devoted to the study of quotients of finite metric spaces. The
basic type of question we ask is: Given a finite metric space M, what is the
largest quotient of (a subset of) M which well embeds into Hilbert space. We
obtain asymptotically tight bounds for these questions, and prove that they
exhibit phase transitions. We also study the analogous problem for embedings
into l_p, and the particular case of the hypercube.Comment: 36 pages, 0 figures. To appear in Advances in Mathematic
Shattering Thresholds for Random Systems of Sets, Words, and Permutations
This paper considers a problem that relates to the theories of covering
arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability
thresholds. Specifically, we want to find the number of subsets of
[n]:={1,2,....,n} we need to randomly select, in a certain probability space,
so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to
words, we ask for the number of n-letter words on a q-letter alphabet that are
needed to shatter all t-subwords of the q^n words of length n. Finally, we
explore the number of random permutations of [n] needed to shatter
(specializing to t=3), all length 3 permutation patterns in specified
positions. We uncover a very sharp zero-one probability threshold for the
emergence of such shattering; Talagrand's isoperimetric inequality in product
spaces is used as a key tool.Comment: 25 page
Topological Conformal Dimension
We investigate a quasisymmetrically invariant counterpart of the topological
Hausdorff dimension of a metric space. This invariant, called the topological
conformal dimension, gives a lower bound on the topological Hausdorff dimension
of quasisymmetric images of the space. We obtain results concerning the
behavior of this quantity under products and unions, and compute it for some
classical fractals. The range of possible values of the topological conformal
dimension is also considered, and we show that this quantity can be fractional.Comment: 16 pages, revised after referee's reports. To appear in Conformal
Geometry and Dynamic
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