Final Report Summary - GPHDPD (Geometric Phenomena in High-Dimensional Probability Distributions) The project's objective is to develop new mathematical methods for the study of geometric distributions on spaces of a very high dimension, tending to infinity. Probability distributions on high-dimensional spaces appear in quite a few branches of mathematics and mathematical physics. From probability theory to quantum physics, from analysis and combinatorics to statistical mechanics, it is not uncommon to study a distribution, or a family of distributions, on a space of many 'equally important' parameters. These high-dimensional measures are usually, but not always, quite concrete. The naive intuition (based on two- or three-dimensional experience) is that a general study of probability distributions in high dimension is hopeless, since the high dimension entails huge diversity which is almost impossible to grasp. The underlying theme of this project is that the opposite is sometimes true, and that high dimensionality, when viewed correctly, may create order and simplicity rather than complication.In this project, my students and I were able to make progress in several research directions that are related to this project. We present below a more detailed account of some of these research directions.1. Approximately Gaussian marginals and the hyperplane conjectureTogether with our Ph.D. student, Ronen Eldan, we were able to connect two seemingly unrelated conjectures in high-dimensional convex geometry. These two problems were described in detail in the grant proposal, so we will mention them only briefly.The first conjecture is the 'slicing problem', suggesting that for any n-dimensional convex body of volume one, there exists a hyperplane section, whose (n-1)-dimensional volume is at least c, for a universal constant c > 0. The second conjecture is pertaining to the rate of convergence in the central limit theorem for convex sets.We showed that the second conjecture implies the first, with good quantitative dependence. The proof is quite unexpected, and is very much different from all of the techniques employed in this field so far. It uses a certain Riemannian structure.2. Thin shell implies spectral gap up to polylog via a stochastic localisation schemeIn this remarkable work, my Ph.D. student Ronen Eldan was able to make substantial progress towards an almost-untouchable conjecture by Kannan, Lovasz and Simonovits (KLS in short). The KLS conjecture suggests that up to a universal constant, the most efficient way to bisect an n-dimensional convex body to two pieces of equal volumes is by using a hyperplane cut. Here, by 'most efficient' we mean that the surface area of the interface between these two regions should be minimal. By using the remarkable novel idea of an infinitesimal stochastic localisation, Eldan was able to prove that up to log(n)-factors, for the purpose of the KLS conjecture it suffices to consider cuts by an ellipsoid. This has two important consequences. First, this way Eldan improved the best known estimate regarding the KLS conjecture in n-dimensions, from n^{5/12} to n^{1/3} log n. Second, up to log(n)-factors Eldan reduced the study of the KLS conjecture from general hypersurfaces to quadratic hypersurfaces, boundaries of ellipsoids.Ronen's paper was accepted for publication in the prestigious journal Geometric and Functional Analysis (GAFA). This result is the cornerstone of Ronen's Ph.D. dissertation, which received its final approval in the last month.3. On nearly radial marginals of high-dimensional probability measuresOne of the phenomena we observe in high dimensions is the surprising emergence of symmetries, or at least approximate symmetries, in very general situations. Answering a question of Gromov related to Dvoretzky's theorem, we proved the following theorem: Suppose X is a random vector in n-dimensional Euclidean space, which is non-degenerate (say, X has a density with respect to the Lebesgue measure). Then there exists a d-dimensional subspace E in R^n, for which the projection of X onto E is approximately a spherically-symmetric random vector.The approximation is good only when n is large enough with respect to d, specifically n^{1 / d} >> 1. One of the interesting consequences of this theorem, is that for any absolutely-continuous distribution in high dimensions, there exists a one-dimensional marginal which is 'super-Gaussian'. That is, its tail distribution is at least exp(-c t^2) for t in a wide range.For exact statements and details, see our paper in the Journal of the European Mathematical Society, Vol. 12, pages 723 - 754.4. Dimensionality and the stability of the Brunn-Minkowski inequalityThe equality case in the classical Brunn-Minkowski inequality is well-known: Equality holds when the involved sets are convex and are equivalent up to a translation and a homothety. The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is almost-equality, then K and T are almost-translates of each other. These estimates, due to Diskant, Groemer, and Figalli, Maggi and Pratelli, share a common feature: Their estimates deteriorate quickly as the dimension increases. Only when the Brunn-Minkowski inequality is tight up to a factor of (1 + 1/n) or so, when n is the dimension, then these estimates can yield meaningful information.In this project, which is joint with my Ph.D. student Ronen Eldan, we raise the possibility that the stability of the Brunn-Minkowski inequality actually improves as the dimension increases. In particular, we conclude that the convex bodies are almost-translates of each other in a certain sense, already when the Brunn-Minkowski inequality is off by a factor of 10, or even log(n) or a power of the dimension. For instance, for many reasonable functions in n variables, the averages on the two convex bodies are very close to each other. Other than this example of 'reversing the curse of dimensionality'', we were able to connect the stability of the Brunn-Minkowski inequality to other standard problems in high-dimensional convex geometry, such as the 'thin shell problem'.This paper was accepted for publication in Ann. Sc. Norm. Super. Pisa. 5. Quantum one-way communication is exponentially stronger than classical communicationThis is a joint work with O. Regev, in which we settle an open problem in theoretical computer science. In 1999, Raz presented a (partial) function for which there is a quantum protocol communicating only O(log n) qubits, but for which any classical (randomised, bounded-error) protocol requires poly(n) bits of communication. That quantum protocol requires two rounds of communication. Ever since Raz's paper, it was open whether the same exponential separation can be achieved with a quantum protocol that uses only one round of communication. Here, we settle this question in the affirmative.The mathematical component of the argument is very much in spirit of 'geometric phenomena in high-dimensional probability distributions'. For details see STOC 2011 (Proceedings of a Symposium on Theory of Computing), Assoc. Comput. Mach., (2011), 31 - 40.6. Poincare inequalities and moment mapsHere, Poincare-type inequalities refer to inequalities in which the variance of a function is bounded in terms of an integral of a quadratic form involving the gradient of the function. These inequalities are indispensible in high dimensions, and for instance they are used to prove thin shell bounds in convexity. In our earlier work, we used the Bochner formula in order to study optimal thin shell bounds and Poincare-type inequalities for the uniform measure on high-dimensional convex bodies. Our technique, which was exploited also in the related work by Barthe and Cordero-Erausquin, relied very much on symmetries of the probability distribution under consideration. The method seemed quite irrelevant for arbitrary convex bodies, possessing no symmetries. The following twist is proposed: Introduce additional symmetries by considering a certain transportation of measure from a space of twice or thrice the dimension. The plan is to apply Bochner's formula in this higher dimensional space, and deduce a Poincare-type inequality for the original measure. This work provides an interesting connection to complex geometry: For instance, one may obtain Poincare-type inequalities on the simplex from corresponding inequalities on the complex projective space. Among the conclusions of this work is a central limit theorem for a family of non-convex bodies which includes the unit balls of L_p for 0 < p < 1.
