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].
Let G be an undirected graph with m edges and d vertices. We show that d-dimensional Ising models on G can be learned from n i.i.d. samples within expected total variation distance some constant factor of min{1, (m + d)/n}, and that this rate is optimal. We show that the same rate holds for the class of d-dimensional multivariate normal undirected graphical models with respect to G. We also identify the optimal rate of min{1, m/n} for Ising models with no external magnetic field.
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is effective, but the underlying probabilistic machinery can be daunting. "Encoding arguments" provide an alternative presentation in which probabilistic reasoning is encapsulated in a "uniform encoding lemma". This lemma provides an upper bound on the probability of an event using the fact that a uniformly random choice from a set of size $n$ cannot be encoded with fewer than $\log_2 n$ bits on average. With the lemma, the argument reduces to devising an encoding where bad objects have short codewords. In this expository article, we describe the basic method and provide a simple tutorial on how to use it. After that, we survey many applications to classic problems from discrete mathematics and computer science. We also give a generalization for the case of non-uniform distributions, as well as a rigorous justification for the use of non-integer codeword lengths in encoding arguments. These latter two results allow encoding arguments to be applied more widely and to produce tighter results.Comment: 50 pages, 7 figure
We investigate the size of vertex confidence sets for including part of (or the entirety of) the seed in seeded uniform attachment trees, given knowledge of some of the seed's properties, and with a prescribed probability of failure. We also study the problem of identifying the leaves of a seed in a seeded uniform attachment tree, given knowledge of the positions of all internal nodes of the seed.
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