171 research outputs found
Approximating the moments of marginals of high-dimensional distributions
For probability distributions on , we study the optimal sample
size N = N(n,p) that suffices to uniformly approximate the pth moments of all
one-dimensional marginals. Under the assumption that the marginals have bounded
4p moments, we obtain the optimal bound for p > 2. This bound
goes in the direction of bridging the two recent results: a theorem of Guedon
and Rudelson [Adv. Math. 208 (2007) 798-823] which has an extra logarithmic
factor in the sample size, and a result of Adamczak et al. [J. Amer. Math. Soc.
23 (2010) 535-561] which requires stronger subexponential moment assumptions.Comment: Published in at http://dx.doi.org/10.1214/10-AOP589 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Concentration inequalities for random tensors
We show how to extend several basic concentration inequalities for simple
random tensors where all are
independent random vectors in with independent coefficients. The
new results have optimal dependence on the dimension and the degree . As
an application, we show that random tensors are well conditioned: independent copies of the simple random tensor
are far from being linearly dependent with high probability. We prove this fact
for any degree and conjecture that it is true for any
.Comment: A few more typos were correcte
Four lectures on probabilistic methods for data science
Methods of high-dimensional probability play a central role in applications
for statistics, signal processing theoretical computer science and related
fields. These lectures present a sample of particularly useful tools of
high-dimensional probability, focusing on the classical and matrix Bernstein's
inequality and the uniform matrix deviation inequality. We illustrate these
tools with applications for dimension reduction, network analysis, covariance
estimation, matrix completion and sparse signal recovery. The lectures are
geared towards beginning graduate students who have taken a rigorous course in
probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of
Data. Some typos, inaccuracies fixe
Frame expansions with erasures: an approach through the non-commutative operator theory
In modern communication systems such as the Internet, random losses of
information can be mitigated by oversampling the source. This is equivalent to
expanding the source using overcomplete systems of vectors (frames), as opposed
to the traditional basis expansions. Dependencies among the coefficients in
frame expansions often allow for better performance comparing to bases under
random losses of coefficients. We show that for any n-dimensional frame, any
source can be linearly reconstructed from only (n log n) randomly chosen frame
coefficients, with a small error and with high probability. Thus every frame
expansion withstands random losses better (for worst case sources) than the
orthogonal basis expansion, for which the (n log n) bound is attained. The
proof reduces to M.Rudelson's selection theorem on random vectors in the
isotropic position, which is based on the non-commutative Khinchine's
inequality.Comment: 12 page
Integer cells in convex sets
Every convex body K in R^n has a coordinate projection PK that contains at
least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at
least one. Our proof of this counterpart of Minkowski's theorem is based on an
extension of the combinatorial density theorem of Sauer, Shelah and
Vapnik-Chervonenkis to Z^n. This leads to a new approach to sections of convex
bodies. In particular, fundamental results of the asymptotic convex geometry
such as the Volume Ratio Theorem and Milman's duality of the diameters admit
natural versions for coordinate sections.Comment: Historical remarks on the notion of the combinatorial dimension are
added. This is a published version in Advances in Mathematic
Isoperimetry of waists and local versus global asymptotic convex geometries
Existence of nicely bounded sections of two symmetric convex bodies K and L
implies that the intersection of random rotations of K and L is nicely bounded.
For L = subspace, this main result immediately yields the unexpected
phenomenon: "If K has one nicely bounded section, then most sections of K are
nicely bounded". This 'existence implies randomness' consequence was proved
independently in [Giannopoulos, Milman and Tsolomitis]. The main result
represents a new connection between the local asymptotic convex geometry (study
of sections of convex bodies) and the global asymptotic convex geometry (study
of convex bodies as a whole). The method relies on the new 'isoperimetry of
waists' on the sphere due to Gromov
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