50 research outputs found
On the mean width of log-concave functions
In this work we present a new, natural, definition for the mean width of
log-concave functions. We show that the new definition coincide with a previous
one by B. Klartag and V. Milman, and deduce some properties of the mean width,
including an Urysohn type inequality. Finally, we prove a functional version of
the finite volume ratio estimate and the low-M* estimate.Comment: 15 page
A characterization of the support map
AbstractIn this short note we give a characterization of the support map from classical convexity. We show it is the unique additive transformation from the class of closed convex sets in Rn which include 0 to the class of positive 1-homogeneous functions on Rn. This will be a consequence of a theorem about transforms from the class of convex sets to itself which preserve Minkowski addition
Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits
It is well known that R^N has subspaces of dimension proportional to N on
which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit
constructions are known. Extending earlier work by Artstein--Avidan and Milman,
we prove that such a subspace can be generated using O(N) random bits.Comment: 16 pages; minor changes in the introduction to make it more
accessible to both Math and CS reader
Almost-Euclidean subspaces of via tensor products: a simple approach to randomness reduction
It has been known since 1970's that the N-dimensional -space contains
nearly Euclidean subspaces whose dimension is . However, proofs of
existence of such subspaces were probabilistic, hence non-constructive, which
made the results not-quite-suitable for subsequently discovered applications to
high-dimensional nearest neighbor search, error-correcting codes over the
reals, compressive sensing and other computational problems. In this paper we
present a "low-tech" scheme which, for any , allows to exhibit nearly
Euclidean -dimensional subspaces of while using only
random bits. Our results extend and complement (particularly) recent work
by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1)
simplicity (we use only tensor products) and (2) yielding "almost Euclidean"
subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor
change
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change