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# Curse of dimensionality

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  1. arXiv

    Simulation-based inference stops training only the posterior net: summary statistics take the lead, and training size grows only polynomially in observation dimension

    A simulation-based inference framework jointly learns a low-dimensional summary statistic and an amortized posterior estimator with non-asymptotic guarantees: in exponential-family and smooth-model settings, the training size needed for a target accuracy grows only polynomially with observation dimension, not exponentially as in classical nonparametric conditional distribution estimation.

    Hong Kong University of Science and Technology
    Rui Li（First author）
    Simulation-Based InferenceSliced Maximum Mean DiscrepancyABC-MCMC Local RefinementOracle InequalityCurse of dimensionality
  2. Advances in Computational Mathematics

    ReLU CNNs in Korobov spaces lift the approximation order from second order to order m+1, with far weaker dimensional growth than the Sobolev case

    This work studies the Lp error of approximating higher-order Korobov functions f∈Km+1_p(Ω) by deep ReLU convolutional neural networks (CNNs), proving that for depth L≤Csd4m3N(log_2 N) there exists a network with inf‖f−f_L‖Lp(Ω)≤Cm,d‖Dm+1f‖Lp(Ω)N−m−1(log_2 N)(m+2)(d−1), i.e. it improves the classical second-order mixed-derivative rate O(L−2+1/p) to order (m+1) up to a logarithmic factor, and concludes that the higher-order expressivity of CNNs does not severely suffer from the curse of dimensionality.

    Zhejiang University
    Yuwen Li（First author）Guozhi Zhang（Corresponding author）
    Sparse grid interpolationKorobov spaceConvolutional Neural NetworksCurse of dimensionality

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