2,603 research outputs found
A concise proof of the Multiplicative Ergodic Theorem on Banach spaces
We give a streamlined proof of the multiplicative ergodic theorem for
quasi-compact operators on Banach spaces with a separable dual.Comment: 18 page
Classification and Geometry of General Perceptual Manifolds
Perceptual manifolds arise when a neural population responds to an ensemble
of sensory signals associated with different physical features (e.g.,
orientation, pose, scale, location, and intensity) of the same perceptual
object. Object recognition and discrimination requires classifying the
manifolds in a manner that is insensitive to variability within a manifold. How
neuronal systems give rise to invariant object classification and recognition
is a fundamental problem in brain theory as well as in machine learning. Here
we study the ability of a readout network to classify objects from their
perceptual manifold representations. We develop a statistical mechanical theory
for the linear classification of manifolds with arbitrary geometry revealing a
remarkable relation to the mathematics of conic decomposition. Novel
geometrical measures of manifold radius and manifold dimension are introduced
which can explain the classification capacity for manifolds of various
geometries. The general theory is demonstrated on a number of representative
manifolds, including L2 ellipsoids prototypical of strictly convex manifolds,
L1 balls representing polytopes consisting of finite sample points, and
orientation manifolds which arise from neurons tuned to respond to a continuous
angle variable, such as object orientation. The effects of label sparsity on
the classification capacity of manifolds are elucidated, revealing a scaling
relation between label sparsity and manifold radius. Theoretical predictions
are corroborated by numerical simulations using recently developed algorithms
to compute maximum margin solutions for manifold dichotomies. Our theory and
its extensions provide a powerful and rich framework for applying statistical
mechanics of linear classification to data arising from neuronal responses to
object stimuli, as well as to artificial deep networks trained for object
recognition tasks.Comment: 24 pages, 12 figures, Supplementary Material
Convex bodies of states and maps
We give a general solution to the question when the convex hulls of orbits of
quantum states on a finite-dimensional Hilbert space under unitary actions of a
compact group have a non-empty interior in the surrounding space of all density
states. The same approach can be applied to study convex combinations of
quantum channels. The importance of both problems stems from the fact that,
usually, only sets with non-vanishing volumes in the embedding spaces of all
states or channels are of practical importance. For the group of local
transformations on a bipartite system we characterize maximally entangled
states by properties of a convex hull of orbits through them. We also compare
two partial characteristics of convex bodies in terms of largest balls and
maximum volume ellipsoids contained in them and show that, in general, they do
not coincide. Separable states, mixed-unitary channels and k-entangled states
are also considered as examples of our techniques.Comment: 18 pages, 1 figur
Classifying homogeneous ultrametric spaces up to coarse equivalence
For every metric space we introduce two cardinal characteristics
and describing the capacity of
balls in . We prove that these cardinal characteristics are invariant under
coarse equivalence and prove that two ultrametric spaces are coarsely
equivalent if
.
This result implies that an ultrametric space is coarsely equivalent to an
isometrically homogeneous ultrametric space if and only if
. Moreover, two isometrically
homogeneous ultrametric spaces are coarsely equivalent if and only if
if and only if each of these
spaces coarsely embeds into the other space. This means that the coarse
structure of an isometrically homogeneous ultrametric space is completely
determined by the value of the cardinal
.Comment: arXiv admin note: text overlap with arXiv:1103.5118, arXiv:0908.368
Algebraic genericity of strict-order integrability
We provide sharp conditions on a measure µ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue
space Lp (µ, X) (p ≥ 1) which are not integrable with order q for any q > p (or any q < p) contains, except for zero, large subspaces of
Lp (µ, X). This improves recent results due to Aron, GarcĂa, Muñoz, Palmberg, PĂ©rez, Puglisi and Seoane. It is also shown that many nonintegrable functions of order q can be obtained even on any nonempty open subset of X, assuming that X is a topological space and µ is a Borel measure on X satisfying appropriate properties.Plan Andaluz de InvestigaciĂłn (Junta de AndalucĂa)Ministerio de Ciencia e InnovaciĂłnMinisterio de Ciencia y TecnologĂa (MCYT). Españ
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