44 research outputs found
May We Have Your Attention: Analysis of a Selective Attention Task
In this paper we present a deeper analysis than has previously been carried out of a selective attention problem, and the evolution of continuous-time recurrent neural networks to solve it. We show that the task has a rich structure, and agents must solve a variety of subproblems to perform well. We consider the relationship between the complexity of an agent and the ease with which it can evolve behavior that generalizes well across subproblems, and demonstrate a shaping protocol that improves generalization
An efficiency upper bound for inverse covariance estimation
We derive an upper bound for the efficiency of estimating entries in the
inverse covariance matrix of a high dimensional distribution. We show that in
order to approximate an off-diagonal entry of the density matrix of a
-dimensional Gaussian random vector, one needs at least a number of samples
proportional to . Furthermore, we show that with samples, the
hypothesis that two given coordinates are fully correlated, when all other
coordinates are conditioned to be zero, cannot be told apart from the
hypothesis that the two are uncorrelated.Comment: 7 Page
Bounding the norm of a log-concave vector via thin-shell estimates
Chaining techniques show that if X is an isotropic log-concave random vector
in R^n and Gamma is a standard Gaussian vector then E |X| < C n^{1/4} E |Gamma|
for any norm |*|, where C is a universal constant. Using a completely different
argument we establish a similar inequality relying on the thin-shell constant
sigma_n = sup ((var|X|^){1/2} ; X isotropic and log-concave on R^n).
In particular, we show that if the thin-shell conjecture sigma_n = O(1)
holds, then n^{1/4} can be replaced by log (n) in the inequality.
As a consequence, we obtain certain bounds for the mean-width, the dual
mean-width and the isotropic constant of an isotropic convex body.
In particular, we give an alternative proof of the fact that a positive
answer to the thin-shell conjecture implies a positive answer to the slicing
problem, up to a logarithmic factor.Comment: preliminary version, 13 page
On the equivalence of modes of convergence for log-concave measures
An important theme in recent work in asymptotic geometric analysis is that
many classical implications between different types of geometric or functional
inequalities can be reversed in the presence of convexity assumptions. In this
note, we explore the extent to which different notions of distance between
probability measures are comparable for log-concave distributions. Our results
imply that weak convergence of isotropic log-concave distributions is
equivalent to convergence in total variation, and is further equivalent to
convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note
Remarks on the KLS conjecture and Hardy-type inequalities
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary
functions on a convex body , not necessarily
vanishing on the boundary . This reduces the study of the
Neumann Poincar\'e constant on to that of the cone and Lebesgue
measures on ; these may be bounded via the curvature of
. A second reduction is obtained to the class of harmonic
functions on . We also study the relation between the Poincar\'e
constant of a log-concave measure and its associated K. Ball body
. In particular, we obtain a simple proof of a conjecture of
Kannan--Lov\'asz--Simonovits for unit-balls of , originally due to
Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in
final form in GAFA seminar note
Optimal Concentration of Information Content For Log-Concave Densities
An elementary proof is provided of sharp bounds for the varentropy of random
vectors with log-concave densities, as well as for deviations of the
information content from its mean. These bounds significantly improve on the
bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543,
2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with
more general sufficient conditions for equality in Theorem 2.3. Also some
minor corrections and added reference
Convex hulls of random walks, hyperplane arrangements, and Weyl chambers
We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments.
This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement.
We obtain analogous distribution-free results for Weyl chambers of type An−1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1 ×···× B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin.
We also give an asymptotic analysis of the obtained absorption probabilities as n → ∞, in both cases of fixed and increasing dimension d
Evaluation of a Rapid Immunochromatographic ODK-0901 Test for Detection of Pneumococcal Antigen in Middle Ear Fluids and Nasopharyngeal Secretions
Since the incidence of penicillin-resistant Streptococcus pneumoniae has been increasing at an astonishing rate throughout the world, the need for accurate and rapid identification of pneumococci has become increasingly important to determine the appropriate antimicrobial treatment. We have evaluated an immunochromatographic test (ODK-0901) that detects pneumococcal antigens using 264 middle ear fluids (MEFs) and 268 nasopharyngeal secretions (NPSs). A sample was defined to contain S. pneumoniae when optochin and bile sensitive alpha hemolytic streptococcal colonies were isolated by culture. The sensitivity and specificity of the ODK-0901 test were 81.4% and 80.5%, respectively, for MEFs from patients with acute otitis media (AOM). In addition, the sensitivity and specificity were 75.2% and 88.8%, respectively, for NPSs from patients with acute rhinosinusitis. The ODK-0901 test may provide a rapid and highly sensitive evaluation of the presence of S. pneumoniae and thus may be a promising method of identifying pneumococci in MEFs and NPSs