Abstract. Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process Zx := Ax 2 − √ m x 2 has sub-gaussian increments, that is, Zx − Zy ψ 2 ≤ C x − y 2 for any x, y ∈ R n . Using this, we show that for any bounded set T ⊆ R n , the deviation of Ax 2 around its mean is uniformly bounded by the Gaussian complexity of T . We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, some of which are reviewed in this paper. In particular, we give a new result regarding model selection in the constrained linear model.

We prove a lower bound and an upper bound for the total variation distance between two high-dimensional Gaussians, which are within a constant factor of one another. IntroductionThe Gaussian (or normal) distribution is perhaps the most important distribution in probability theory due to the central limit theorem. For a positive integer d, a vector µ ∈ R d , and a positive definite matrix Σ, the Gaussian distribution with mean µ and covariance matrix Σ is a probability distribution over R d denoted by N (µ, Σ) with density det(2πΣWe denote by N (µ, Σ) a random variable with this distribution. Note that if X ∼ N (µ, Σ) then EX = µ and EXX T = Σ.If the covariance matrix is positive semi-definite but not positive definite, the Gaussian distribution is singular on R d , but has a density with respect to a Lebesgue measure on an affine subspace: let r be the rank of Σ, and let range(Σ) denote the range (also known as the image or the column space) of Σ. Let Π be a d × r matrix whose columns form an orthonormal basis for range(Σ). Then the matrix Σ ′ ≔ Π T ΣΠ has full rank r, and N (µ, Σ) has density given by detwith respect to the r-dimensional Lebesgue measure on µ + range(Σ). The density is zero outside this affine subspace. For general background on high-dimensional Gaussian distributions (also called multivariate normal distributions), see [10,12].

No abstract

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c ∞ (G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. We show that if G is planar then c ∞ (G) = Θ(tw(G)), and there is a constant-factor approximation algorithm for computing c ∞ (G). We also determine, up to constant factors, the value of c ∞ (G) of the Erdős-Rényi random graph G = G(n, p) for all admissible values of p, and show that when the average degree is ω(1), c ∞ (G) is typically asymptotic to the domination number.