265 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
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
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
On almost randomizing channels with a short Kraus decomposition
For large d, we study quantum channels on C^d obtained by selecting randomly
N independent Kraus operators according to a probability measure mu on the
unitary group U(d). When mu is the Haar measure, we show that for
N>d/epsilon^2. For d=2^k (k qubits), this includes Kraus operators
obtained by tensoring k random Pauli matrices. The proof uses recent results on
empirical processes in Banach spaces.Comment: We added some background on geometry of Banach space
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
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