High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more It is the first to integrate theory, key tools, and modern applications of high-dimensional probability Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression

High-Dimensional Probability: An Introduction with Applications in Data Science

We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

/pdf/symmetric-informationally-complete-quantum-measurements-4v18eptkry.pdf

Symmetric informationally complete quantum measurements

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

/pdf/symmetric-informationally-complete-quantum-measurements-4q4incjacv.pdf

Symmetric Informationally Complete Quantum Measurements

Probabilistic methods in the geometry of Banach spaces

The survey is devoted to the modern state of the theory of interpolation of linear operators acting in Banach spaces. Principal attention is devoted to real and complex methods and applications of the theory of interpolation to analysis.

Interpolation of Linear Operators

Convex bodies: Classical geometric inequalities Classical positions of convex bodies Isomorphic isoperimetric inequalities and concentration of measure Metric entropy and covering numbers estimates Almost Euclidean subspaces of finite dimensional normed spaces The $\ell$-position and the Rademacher projection Proportional theory $M$-position and the reverse Brunn-Minkowski inequality Gaussian approach Volume distribution in convex bodies Elementary convexity Advanced convexity Bibliography Subject index Author index

Asymptotic Geometric Analysis

Let Sj: (Ω, P) → S1 ⊂ ℂ be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S1. We determine the best constants ap in the Khintchine-type inequality 

$${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$

for 0 < p < 1, verifying a conjecture of U. Haagerup that 

$${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$

. Both expressions are equal for p = p0 }~ 0.4756. For p ≥ 1 the best constants ap have been known for some time. The result implies for a norm 1 sequence x ∈ ℂn, ‖x‖2 = 1, that 

$${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$

, answering a question of A. Baernstein and R. Culverhouse.

On the best constants in the Khintchine inequality for Steinhaus variables

This chapter discusses the isometric theory of Banach spaces that was born and developed in inseparable connection with other areas of the Banach space theory. The classical direction, initiated by the work of Banach, is the characterization of the isometries of Banach spaces. Surjective isometries of Banach spaces are now well understood in a long sequence of the results culminating in complete characterization of such isometries for all rearrangement invariant spaces by Zaidenberg, Kalton, and Randrianantoanina. The results that belong to the intersection of the isometric theory, harmonic analysis, probability, and combinatorics are discussed in this chapter. Recently, discovered connections between convexity and the Fourier transform are also discussed and are explained through a complete analytic solution to the Busemann–Petty problem on the section of convex bodies that was considered one of the most important isometric questions in convexity.

Chapter 21 - Aspects of the Isometric Theory of Banach Spaces

The isotropy constants of the schatten classes are bounded

For every prime numberk, we give an explicit construction of a complexk-dimensional spaceXk with projection constantγ(Xk) = √k − 1/√k + 1/k. Moreover, there are realk-dimensional spacesXk withγ(xK) ≧ √k − 1 for a subsequence of integersk. Hence in both casesγ(Xk)/√k → 1 which is the maximal possible value sinceγ(Xk) ≦ √k is generally true.

Hermann König

Papers

Asymptotic Geometric Analysis

On the best constants in the Khintchine inequality for Steinhaus variables

Chapter 21 - Aspects of the Isometric Theory of Banach Spaces

The isotropy constants of the schatten classes are bounded

Spaces with large projection constants