Concentration Of Measure Phenomenon Research Articles

Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2018), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lov\'{a}sz local lemma and the concentration of measure phenomenon on the sphere in place of Tao's use of a quantitative homotopy argument.

Robust in practice: Adversarial attacks on quantum machine learning

State-of-the-art classical neural networks are observed to be vulnerable to small crafted adversarial perturbations. A more severe vulnerability has been noted for quantum machine learning (QML) models classifying Haar-random pure states. This stems from the concentration of measure phenomenon, a property of the metric space when sampled probabilistically, and is independent of the classification protocol. To provide insights into the adversarial robustness of a quantum classifier on real-world classification tasks, we focus on the adversarial robustness in classifying a subset of encoded states that are smoothly generated from a Gaussian latent space. We show that the vulnerability of this task is considerably weaker than that of classifying Haar-random pure states. In particular, we find only mildly polynomially decreasing robustness in the number of qubits, in contrast to the exponentially decreasing robustness when classifying Haar-random pure states and suggesting that QML models can be useful for real-world classification tasks.

Drawing together control landscape and tomography principles

The ability to control quantum systems using shaped fields as well as to infer the states of such controlled systems from measurement data are key tasks in the design and operation of quantum devices. Here we associate the success of performing both tasks to the structure of the underlying control landscape. We relate the ability to control and reconstruct the full state of the system to the absence of singular controls, and show that for sufficiently long evolution times singular controls rarely occur. Based on these findings, we describe a learning algorithm for finding optimal controls that makes use of measurement data obtained from partially accessing the system. Open challenges stemming from the concentration of measure phenomenon in high dimensional systems are discussed.

On the tightness of Gaussian concentration for convex functions

The concentration of measure phenomenon in Gauss’ space states that every L-Lipschitz map f on ℝn satisfies $${\gamma _n}\left({\left\{{x:| {f(x) - {M_f}|\,\geqslant t} } \right\}} \right)\,\leqslant 2{e^{- {{{t^2}} \over {2{L^2}}},}}\quad t > 0,$$ where γn is the standard Gaussian measure on ℝn and Mf is a median of f. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when f is additionally assumed to be convex. In particular, we show that if the variance Var(f) (with respect to γn) satisfies $$\alpha L\leqslant \sqrt {{\rm{Var(}}f{\rm{)}}} $$ for some 0 0,$$ where c, C > 0 are constants depending only on α.

Fate of the Hoop Conjecture in Quantum Gravity.

We consider a closed region R of 3D quantum space described via SU(2) spin networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary ∂R and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interpret such phenomenon as a pregeometric analogue of Thorne's "hoop conjecture," at the core of the formation of a horizon in general relativity.

Errors-in-variables models with dependent measurements

Suppose that we observe $y\in\mathbb{R}^{n}$ and $X\in\mathbb{R}^{n\times m}$ in the following errors-in-variables model: \begin{eqnarray*}y&=&X_{0}\beta^{*}+\epsilon\\X&=&X_{0}+W\end{eqnarray*} where $X_{0}$ is an $n\times m$ design matrix with independent subgaussian row vectors, $\epsilon\in\mathbb{R}^{n}$ is a noise vector and $W$ is a mean zero $n\times m$ random noise matrix with independent subgaussian column vectors, independent of $X_{0}$ and $\epsilon$. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its $n$ observations. Such error structures appear in the science literature when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $\beta^{*}\in\mathbb{R}^{m}$ from the model given a single observation matrix $X$ and the response vector $y$. We establish consistency in estimating $\beta^{*}$ and obtain the rates of convergence in the $\ell_{q}$ norm, where $q=1,2$ for the Lasso-type estimator, and for $q\in [1,2]$ for a Dantzig-type Conic programming estimator. We show error bounds which approach that of the regular Lasso and the Dantzig selector in case the errors in $W$ are tending to 0. We analyze the convergence rates of the gradient descent methods for solving the nonconvex programs and show that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to a neighborhood of the global minimizers: the size of the neighborhood is bounded by the statistical error in the $\ell_{2}$ norm. Our analysis reveals interesting connections between computational and statistical efficiency and the concentration of measure phenomenon in random matrix theory. We provide simulation evidence illuminating the theoretical predictions.

Average coherence and its typicality for random pure states

We investigate the generic aspects of quantum coherence guided by the concentration of measure phenomenon. We find the average relative entropy of coherence of pure quantum states sampled randomly from the uniform Haar measure and show that it is typical, i.e., the probability that the relative entropy of coherence of a randomly chosen pure state is not equal to the average relative entropy of coherence (within an arbitrarily small error) is exponentially small in the dimension of the Hilbert space. We find the dimension of a random subspace of the total Hilbert space such that all pure states that reside on it almost always have at least a fixed nonzero amount of the relative entropy of coherence that is arbitrarily close to the typical value of coherence. Further, we show, with high probability, every state (pure or mixed) in this subspace also has the coherence of formation at least equal to the same fixed nonzero amount of the typical value of coherence. Thus, the states from these random subspaces can be useful in the relevant coherence consuming tasks like catalysis in the coherence resource theory. Moreover, we calculate the expected trace distance between the diagonal part of a random pure quantum state and the maximally mixed state, and find that it does not approach to zero in the asymptotic limit. This establishes that randomly chosen pure states are not typically maximally coherent (within an arbitrarily small error). Additionally, we find the lower bound on the relative entropy of coherence for the set of pure states whose diagonal parts are at a fixed most probable distance from the maximally mixed state.

On multivariate quantiles under partial orders

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.

A non-adapted sparse approximation of PDEs with stochastic inputs

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems.

Random polynomials of high degree and Levy concentration of measure

We show that the L^p norms of random sequences {s_N} of L^2 normalized holomorphic sections of increasing powers of an ample line bundle on a compact Kahler manifold are almost surely bounded for 2<p< infinity, and are almost surely O((log N)^{1/2}) for p= infinity. This estimate also holds for almost-holomorphic sections of positive line bundles on symplectic manifolds (in the sense of math.SG/0212180) and we give almost sure bounds for the C^k norms. Our methods involve asymptotics of Bergman-Szego kernels and the concentration of measure phenomenon.

A measure concentration inequality for contracting markov chains

The concentration of measure phenomenon in product spaces means the following: if a subsetA of then'th power of a probability space Χ does not have too small a probability then very large probability is concentrated in a small neighborhood ofA. The neighborhood is in many cases understood in the sense of Hamming distance, and then measure concentration is known to occur for product probability measures, and also for the distribution of some processes with very fast and uniform decay of memory. Recently Talagrand introduced another notion of neighborhood of sets for which he proved a similar measure concentration inequality which in many cases allows more efficient applications than the one for a Hamming neighborhood. So far this inequality has only been proved for product distributions. The aim of this paper is to give a new proof of Talagrand's inequality, which admits an extension to contracting Markov chains. The proof is based on a new asymmetric notion of distance between probability measures, and bounding this distance by informational divergence. As an application, we analyze the bin packing problem for Markov chains.

A new look at independence

The concentration of measure phenomenon is product spaces is a far-reaching abstract generalization of the classical exponential inequalities for sums of independent random variables. We attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon, the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities.

Concentration of measure and isoperimetric inequalities in product spaces

The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work “close” is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.

A simple proof of the majorizing measure theorem

We gieve a completely elementary proof of the existence of majorizing measures for bounded Gaussian processes. The proof relies upon Sudakov's minoration, the concentration of measure phenomenon, and a (somewhat deceptively) simple construction.

A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold

Following closely the Laplace operator method of M. Gromov and V.D. Milman (1983), we slightly improve their general estimate of the concentration functionα(X; e) of a compact Riemannian manifold X by showing that α(X;e)≦C exp(−ce√ϱ0log(1+e√ϱ0)) for numerical constantsC, c > 0, whereρ 0 > 0 is the hypercontractive constant of the Laplacian onX.