Saturating Constructions for Normed Spaces II

We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log (dim X) = O(log (dim V)) and (2) every subspace of X, whose dimension is not "too small," contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps.Comment: 35 p., LATEX; the paper is a follow up on math.FA/040723

On the nontrivial projection problem

The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension greater than one) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true "up to a logarithmic factor."Comment: 17 page

Learning Arbitrary Statistical Mixtures of Discrete Distributions

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, $\vartheta$, is a probability distribution over probability distributions $p$, where each such $p$ is a probability distribution over $[n] = \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$ directly, but only indirectly and in very noisy fashion, by sampling from $[n]$ repeatedly, independently $K$ times from the distribution $p$. The problem is to infer $\vartheta$ to high accuracy in transportation (earthmover) distance. We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution $\vartheta$. We bound the quality of the solution as a function of the size of the samples $K$ and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.Comment: 23 pages. Preliminary version in the Proceeding of the 47th ACM Symposium on the Theory of Computing (STOC15

Almost-Euclidean subspaces of ℓ1N\ell_1^N via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. 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 $a > 0$, allows to exhibit nearly Euclidean $\Omega(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ 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

Survey on nonlocal games and operator space theory

This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states

Evidence of Band Bending Induced by Hole Trapping at MAPbI3 Perovskite / Metal Interface

International audienceElectron injection by tunneling from a gold electrode and hole transport properties in polycrystalline MAPbI3 has been investigated using variable temperature experiments and numerical simulations. The presence of a large and unexpected band bending at the Au/MAPbI3 interface is revealed and attributed to the trapping of holes, which enhances the injection of electrons via tunneling. These results elucidate the role of volume and interface defects in state-of-the-art hybrid perovskite semiconductors

Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. {}.Comment: 18 p., princeton/ecel/7-12-9

Chemical and electronic characterization of methyl-terminated Si(111) surfaces by high-resolution synchrotron photoelectron spectroscopy

The chemical state, electronic properties, and geometric structure of methyl-terminated Si(111) surfaces prepared using a two-step chlorination/alkylation process were investigated using high-resolution synchrotron photoelectron spectroscopy and low-energy electron diffraction methods. The electron diffraction data indicated that the methylated Si surfaces maintained a (1×1) structure, where the dangling bonds of the silicon surface atoms were terminated by methyl groups. The surfaces were stable to annealing at 720 K. The high degree of ordering was reflected in a well-resolved vibrational fine structure of the carbon 1s photoelectron emission, with the fine structure arising from the excitation of C-H stretching vibrations having hnu=0.38±0.01 eV. The carbon-bonded surface Si atoms exhibited a well-defined x-ray photoelectron signal having a core level shift of 0.30±0.01 eV relative to bulk Si. Electronically, the Si surface was close to the flat-band condition. The methyl termination produced a surface dipole of –0.4 eV. Surface states related to piCH3 and sigmaSi-C bonding orbitals were identified at binding energies of 7.7 and 5.4 eV, respectively. Nearly ideal passivation of Si(111) surfaces can thus be achieved by methyl termination using the two-step chlorination/alkylation process