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Register a new user$L^{q}$-error estimates for approximation of irregular functionals of random vectors
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Added by
Dai Taguchi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Avikainen showed that, for any $p,q in [1,infty)$, and any function $f$ of bounded variation in $mathbb{R}$, it holds that $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensiona
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Let $X$ be a $d$-dimensional random vector and $X_theta$ its projection onto the span of a set of orthonormal vectors ${theta_1,...,theta_k}$. Conditions on the distribution of $X$ are given such that if $theta$ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from $X_theta$ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of $d$, $k$, and the distribution of $X$, allowing consideration not just of fixed $k$ but of $k$ growing with $d$. The results are applied in the setting of projection pursuit, showing that most $k$-dimensional projections of $n$ data points in $R^d$ are close to Gaussian, when $n$ and $d$ are large and $k=csqrt{log(d)}$ for a small constant $c$.
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c`adl`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.
Series expansions of isotropic Gaussian random fields on $mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Lo`eve expansions of isotropic random fields in terms of spherical harmonics. Their multilevel localized structure of basis functions is especially useful in adaptive algorithms. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analyzed.
We study the problem of approximating the eigenspectrum of a symmetric matrix $A in mathbb{R}^{n times n}$ with bounded entries (i.e., $|A|_{infty} leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $A$ up to additive error $pm epsilon n$ using those of a randomly sampled $tilde{O}(frac{1}{epsilon^4}) times tilde O(frac{1}{epsilon^4})$ principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.
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