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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning 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A Polynomial Chaos Expansion in Dependent Random Variables
Sharif Rahman · 2018-04-13 · via math.ST updates on arXiv.org

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising measure-consistent multivariate orthonormal polynomials in dependent random variables. Unlike existing PCEs, whether classical or generalized, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PCE are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, exploiting whitening transformation for generating orthonormal polynomial bases, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE for a general output variable in terms of the expansion coefficients. An example derived from a stochastic boundary-value problem illustrates the generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables.

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