Asymptotic Geometric Analysis Research Articles

Random sections of [formula omitted]-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

We study the circumradius of a random section of an ℓp-ellipsoid, 0<p≤∞, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if 1≤p≤∞ and compressed sensing if 0<p≤1. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an ℓp-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and 1≤p≤∞, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value 1−1p or not. If 1≤p≤2, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021] for the case p=2.

Improved bounds for Hadwiger’s covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number N(n) is such that every convex body in \mathbb{R}^{n} can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of {2n \choose n}n\ln n . In this note, we improve this bound by a subexponential factor. That is, we prove a bound of the order of {2n \choose n}e^{-c\sqrt{n}} for some universal constant c&gt;0 . Our approach combines ideas from [3] by Artstein-Avidan and the second named author with tools from asymptotic geometric analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body K with a translate of -K ; in fact, we get the same lower bound for the volume of the intersection of K and -K when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for N(n) when restricting our attention to convex bodies that are \psi_{2} . By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity.

New Perspectives and Computational Challenges in High Dimensions

High-dimensional systems are frequent in mathematics and applied sciences, and the understanding of high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most strongly related to such phenomena are functional analysis, convex geometry, and probability theory. In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and theoretical computer science. The last two decades have seen a tremendous growth in this area. Far reaching results were obtained and various powerful techniques have been developed, which rather often have a probabilistic flavor. The purpose of this workshop was to explored these new perspectives, to reach out to other areas concerned with high-dimensional problems, and to bring together researchers having different angles on high-dimensional phenomena.

Geometric Analysis-Based Cluster Head Selection for Sectorized Wireless Powered Sensor Networks

In this letter, we consider a sectorized wireless powered sensor network (WPSN), wherein sensor nodes, which are clustered based on sectors, transmit data to the cluster head (CH) using energy harvested from a hybrid access point. We construct a system model for this sectorized WPSN and formulate an optimization problem to obtain an optimal CH that maximizes the achievable transmission rate of sensing data. To obtain the optimal CH with low overhead and complexity, we perform an asymptotic geometric analysis (GA) that determines the optimal coordinates of the CH in the considered area of the sector such that only a small number of sensor nodes near this optimal coordinates need to be investigated. Simulation results show that the performance of the proposed GA-based CH selection method is close to the optimal performance exhibited by exhaustive search; moreover, the proposed method substantially reduces the channel feedback overhead.

Book Review: Alice and Bob meet Banach: The interface of asymptotic geometric analysis and quantum information theory

Book Review: Alice and Bob meet Banach: The interface of asymptotic geometric analysis and quantum information theory

The curse of dimensionality for numerical integration on general domains

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal ψ2-estimate. In particular, we obtain the result for the important class of volume-normalized ℓpd-balls in the complete regime 2≤p≤∞. This extends a result in a work of Hinrichs et al. (2014) to the whole range 2≤p≤∞, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.

Some Open Problems in Asymptotic Geometric Analysis

Some Open Problems in Asymptotic Geometric Analysis

Mini-Workshop: Perspectives in High-Dimensional Probability and Convexity

Understanding the geometric structure of systems involving a huge amount of parameters is a central problem in mathematics and applied sciences today. Here, geometric and analytical ideas meet in a non-trivial way and powerful probabilistic tools play a key role in many discoveries. Two essentially independent areas of mathematics concerned with high-dimensional problems are asymptotic geometric analysis and information-based complexity. In this Mini-Workshop we brought together researchers from both fields to explore the connections and form synergies to develop new perspectives.

Asymptotic Geometric Analysis

The workshop was dedicated to new developments in Asymptotic Geometric Analysis, the study of high-dimensional geometric objects by analytic and probabilistic means, and its interplay with other fields such as functional analysis, convex geometry, probability and graph theory. A central aspect concerned volume inequalities for sections and projections of high-dimensional convex sets.

Book Review: Asymptotic geometric analysis, Part I

Book Review: Asymptotic geometric analysis, Part I

Convex Geometry and its Applications

The past 30 years have not only seen substantial progress and lively activity in various areas within convex geometry, e.g., in asymptotic geometric analysis, valuation theory, the L_p -Brunn-Minkowski theory and stochastic geometry, but also an increasing amount and variety of applications of convex geometry to other branches of mathematics (and beyond), e.g. to PDEs, statistics, discrete geometry, optimization, or geometric algorithms in computer science. Thus convex geometry is a flourishing and attractive field, which is also reflected by the considerable number of talented young mathematicians at this meeting.

Revisiting Additivity Violation of Quantum Channels

We prove additivity violation of minimum output entropy of quantum channels by straightforward application of \epsilon-net argument and L\'evy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm-like structures - positive homogeneity and triangle inequality. Then, a connection between Hastings' method and ours is also discussed. Besides, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels.

Around the circular law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a n×n random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

Mini-Workshop: Geometry of Quantum Entanglement

The workshop aimed at developing interactions between researchers from quantum information theory and from asymptotic geometric analysis. A central notion discussed was the phenomenon of quantum entanglement, which naturally leads to geometric considerations in high-dimensional vector spaces. In these spaces, phenomena such as concentration of measure become prominent and may invalidate our low-dimensional intuition.

Empirical Processes with a Bounded Ψ 1 Diameter

We study the empirical process $${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}$$ , where F is a class of mean-zero functions on a probability space (Ω, μ), and $${(X_{i})_{i =1}^N}$$ are selected independently according to μ. We present a sharp bound on this supremum that depends on the Ψ 1 diameter of the class F (rather than on the Ψ 2 one) and on the complexity parameter γ 2(F,Ψ 2). In addition, we present optimal bounds on the random diameters $${{\rm sup}_{f \in F} {\rm max}_{|I|=m}\left(\sum_{i \in I} f^{2}(X_{i})\right)^{1/2}}$$ using the same parameters. As applications, we extend several well-known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on $${\mathbb{R}^n}$$ .

The $M$-ellipsoid, symplectic capacities and volume

In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body by its volume radius.

Kahane-Khinchin type averages

We prove a Kahane-Khinchin type result with a few random vectors, which are distributed independently with respect to an arbitrary logconcave probability measure on R n . This is an application of a small ball estimate and Chernoff's method, that has been recently used in the context of Asymptotic Geometric Analysis.

Using Rademacher permutations to reduce randomness

It is shown how a special family of unitary operators, called the Rademacher permutations and related to the Clifford algebra, can be used to reduce the level of randomness in several results in asymptotic geometric analysis.

Simplicity v/s Complexity in the Framework of Geometric Asymptotic Analysis and Some New Applications of the Concentration Phenomenon

We introduce a concept of Simplicity which in a sense corresponds to the reverse direction to the concept of Complexity. Many problems of Asymptotic Geometric Analysis rotate around this notion. We also describe the concept of Concentration and suggest a new direction of possible applications of the Concentration Phenomenon.