arXiv subjects

Steven Soojin Kim

Publications and source records attributed to Steven Soojin Kim.

5 recordsLinked to original sources

arXiv2021

An asymptotic thin shell condition and large deviations for random multidimensional projections

Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n \leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in three different asymptotic regimes as $n \rightarrow \infty$: "constant" ($k_n=k$), "sublinear" ($k_n \rightarrow \infty$ but $k_n/n \rightarrow 0$) and "linear" $k_n/n \rightarrow λ$ with $0 < λ\le 1$). When the sequence of random vectors satisfies a certain "asymptotic thin shell condition", we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $\ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $\ell_p^n$ balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a "thin shell" condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $\ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $\mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.

math.PR↗

arXiv2021

Large deviation principles induced by the Stiefel manifold, and random multi-dimensional projections

Given an $n$-dimensional random vector $X^{(n)}$ , for $k < n$, consider its $k$-dimensional projection $\mathbf{a}_{n,k}X^{(n)}$, where $\mathbf{a}_{n,k}$ is an $n \times k$-dimensional matrix belonging to the Stiefel manifold $\mathbb{V}_{n,k}$ of orthonormal $k$-frames in $\mathbb{R}^n$. For a class of sequences $\{X^{(n)}\}$ that includes the uniform distributions on scaled $\ell_p^n$ balls, $p \in (1,\infty]$, and product measures with sufficiently light tails, it is shown that the sequence of projected vectors $\{\mathbf{a}_{n,k}^\intercal X^{(n)}\}$ satisfies a large deviation principle whenever the empirical measures of the rows of $\sqrt{n} \mathbf{a}_{n,k}$ converge, as $n \rightarrow \infty$, to a probability measure on $\mathbb{R}^k$. In particular, when $\mathbf{A}_{n,k}$ is a random matrix drawn from the Haar measure on $\mathbb{V}_{n,k}$, this is shown to imply a large deviation principle for the sequence of random projections $\{\mathbf{A}_{n,k}^\intercal X^{(n)}\}$ in the quenched sense (that is, conditioned on almost sure realizations of $\{\mathbf{A}_{n,k}\}$). Moreover, a variational formula is obtained for the rate function of the large deviation principle for the annealed projections $\{\mathbf{A}_{n,k}^\intercal X^{(n)}\}$, which is expressed in terms of a family of quenched rate functions and a modified entropy term. A key step in this analysis is a large deviation principle for the sequence of empirical measures of rows of $\sqrt{n} \mathbf{A}_{n,k}$, which may be of independent interest. The study of multi-dimensional random projections of high-dimensional measures is of interest in asymptotic functional analysis, convex geometry and statistics. Prior results on quenched large deviations for random projections of $\ell_p^n$ balls have been essentially restricted to the one-dimensional setting.

math.PR↗

arXiv2018

A conditional limit theorem for high-dimensional $\ell^{p}$ spheres

The study of high-dimensional distributions is of interest in probability theory, statistics and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The $\ell^p$ spaces and norms are of particular interest in this setting. In this paper, we establish a limit theorem for distributions on $\ell^p$ spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of $\ell^p$ balls in a high-dimensional Euclidean space.

math.PR↗

arXiv2015

Large deviations for random projections of $\ell^p$ balls

Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is "universal", in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the directions corresponding to the classical Cramér's theorem, and show that those directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of $\ell^p$ balls.

math.PR↗

arXiv2015

Cramér's theorem is atypical

The empirical mean of $n$ independent and identically distributed (i.i.d.) random variables $(X_1,\dots,X_n)$ can be viewed as a suitably normalized scalar projection of the $n$-dimensional random vector $X^{(n)}\doteq(X_1,\dots,X_n)$ in the direction of the unit vector $n^{-1/2}(1,1,\dots,1) \in \mathbb{S}^{n-1}$. The large deviation principle (LDP) for such projections as $n\rightarrow\infty$ is given by the classical Cramér's theorem. We prove an LDP for the sequence of normalized scalar projections of $X^{(n)}$ in the direction of a generic unit vector $θ^{(n)} \in \mathbb{S}^{n-1}$, as $n\rightarrow\infty$. This LDP holds under fairly general conditions on the distribution of $X_1$, and for "almost every" sequence of directions $(θ^{(n)})_{n\in\mathbb{N}}$. The associated rate function is "universal" in the sense that it does not depend on the particular sequence of directions. Moreover, under mild additional conditions on the law of $X_1$, we show that the universal rate function differs from the Cramér rate function, thus showing that the sequence of directions $n^{-1/2}(1,1,\dots,1) \in \mathbb{S}^{n-1},$ $n \in \mathbb{N}$, corresponding to Cramér's theorem is atypical.

math.PR↗