Multivariate intensity estimation via hyperbolic wavelet selection

Akakpo, Nathalie

doi:10.1016/j.jmva.2017.07.005

Cited by 1 publication

(2 citation statements)

References 53 publications

(52 reference statements)

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“…, and κ depends on linearly on µ(X ) 1 . Then by integrating according to t, we obtain that for any ε ∈ (0, 1],…”

Section: Improved Risk Bounds For Least Squares Contrastsmentioning

confidence: 99%

“…To our knowledge, the minimax rates of convergence over mixed Sobolev spaces are unknown for regression. However, the results of [29] for Gaussian white noise model as well as the results of [1] for density estimation suggest that these rates should be of the order of n − 2r 1+2r , up to a logarithmic term. In fact, the minimax rate can not be obtained by our strategy since the rate of approximation error in O(c − 2r 3 ) (up to logarithmic terms) is not the optimal rate of convergence which is in O(c −2r ) (up to logarithmic terms), the latter rate being achieved by hyperbolic cross approximation [15].…”

Section: Multivariate Functionsmentioning

confidence: 99%

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Learning with tree tensor networks: complexity estimates and model selection

Michel,

Nouy

2020

Preprint

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In this paper, we propose and analyze a model selection method for tree tensor networks in an empirical risk minimization framework. Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in numerical analysis and data science. They correspond to sum-product neural networks with a sparse connectivity associated with a dimension partition tree T , widths given by a tuple r of tensor ranks, and multilinear activation functions (or units). The approximation power of these model classes has been proved to be near-optimal for classical smoothness classes. However, in an empirical risk minimization framework with a limited number of observations, the dimension tree T and ranks r should be selected carefully to balance estimation and approximation errors. In this paper, we propose a complexity-based model selection strategy à la Barron, Birgé, Massart. Given a family of model classes, with different trees, ranks and tensor product feature spaces, a model is selected by minimizing a penalized empirical risk, with a penalty depending on the complexity of the model class. After deriving bounds of the metric entropy of tree tensor networks with bounded parameters, we deduce a form of the penalty from bounds on suprema of empirical processes. This choice of penalty yields a risk bound for the predictor associated with the selected model. For classical smoothness spaces, we show that the proposed strategy is minimax optimal in a least-squares setting. In practice, the amplitude of the penalty is calibrated with a slope heuristics method. Numerical experiments in a least-squares regression setting illustrate the performance of the strategy for the approximation of multivariate functions and univariate functions identified with tensors by tensorization (quantization).

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“…, and κ depends on linearly on µ(X ) 1 . Then by integrating according to t, we obtain that for any ε ∈ (0, 1],…”

Section: Improved Risk Bounds For Least Squares Contrastsmentioning

confidence: 99%

Section: Multivariate Functionsmentioning

confidence: 99%