21 research outputs found
Convergence bounds for local least squares approximation
We consider the problem of approximating a function in a general nonlinear
subset of , when only a weighted Monte Carlo estimate of the -norm
can be computed. Of particular interest in this setting is the concept of
sample complexity, the number of sample points that are necessary to achieve a
prescribed error with high probability. Reasonable worst-case bounds for this
quantity exist only for particular model classes, like linear spaces or sets of
sparse vectors. For more general sets, like tensor networks or neural networks,
the currently existing bounds are very pessimistic. By restricting the model
class to a neighbourhood of the best approximation, we can derive improved
worst-case bounds for the sample complexity. When the considered neighbourhood
is a manifold with positive local reach, its sample complexity can be estimated
by means of the sample complexities of the tangent and normal spaces and the
manifold's curvature.Comment: 17 pages, 4 figures, text overlap with arXiv:2108.0523
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Convergence bounds for empirical nonlinear least-squares
We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds
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Convergence bounds for empirical nonlinear least-squares
We consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ by minimizing the empirical norm ∥u − v∥n. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations
Weighted sparsity and sparse tensor networks for least squares approximation
Approximation of high-dimensional functions is a problem in many scientific
fields that is only feasible if advantageous structural properties, such as
sparsity in a given basis, can be exploited. A relevant tool for analysing
sparse approximations is Stechkin's lemma. In its standard form, however, this
lemma does not allow to explain convergence rates for a wide range of relevant
function classes.
This work presents a new weighted version of Stechkin's lemma that improves
the best -term rates for weighted -spaces and associated function
classes such as Sobolev or Besov spaces. For the class of holomorphic
functions, which occur as solutions of common high-dimensional
parameter-dependent PDEs, we recover exponential rates that are not directly
obtainable with Stechkin's lemma.
Since weighted -summability induces weighted sparsity, compressed
sensing algorithms can be used to approximate the associated functions. To
break the curse of dimensionality, which these algorithms suffer, we recall
that sparse approximations can be encoded efficiently using tensor networks
with sparse component tensors. We also demonstrate that weighted
-summability induces low ranks, which motivates a second tensor train
format with low ranks and a single weighted sparse core. We present new
alternating algorithms for best -term approximation in both formats.
To analyse the sample complexity for the new model classes, we derive a novel
result of independent interest that allows the transfer of the restricted
isometry property from one set to another sufficiently close set. Although they
lead up to the analysis of our final model class, our contributions on weighted
Stechkin and the restricted isometry property are of independent interest and
can be read independently.Comment: 39 pages, 5 figure
Convergence bounds for empirical nonlinear least-squares
We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds
Convergence bounds for empirical nonlinear least-squares
We consider best approximation problems in a nonlinear subset
of a Banach space of functions . The norm is assumed
to be a generalization of the -norm for which only a weighted Monte Carlo
estimate can be computed. The objective is to obtain an
approximation of an unknown function by
minimizing the empirical norm . In the case of linear subspaces
it is well-known that such least squares approximations can
become inaccurate and unstable when the number of samples is too close to
the number of parameters . We review this
statement for general nonlinear subsets and establish error bounds for the
empirical best approximation error. Our results are based on a restricted
isometry property (RIP) which holds in probability and we show that is sufficient for the RIP to be satisfied with high probability. Several
model classes are examined where analytical statements can be made about the
RIP. Numerical experiments illustrate some of the obtained stability bounds.Comment: 32 pages, 18 figures; major revision
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Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations
A statistical learning approach for parametric PDEs related to
Uncertainty Quantification is derived. The method is based on the
minimization of an empirical risk on a selected model class and it is shown
to be applicable to a broad range of problems. A general unified convergence
analysis is derived, which takes into account the approximation and the
statistical errors. By this, a combination of theoretical results from
numerical analysis and statistics is obtained. Numerical experiments
illustrate the performance of the method with the model class of hierarchical
tensors
Efficient approximation of high-dimensional exponentials by tensornetworks
In this work a general approach to compute a compressed representation of the
exponential of a high-dimensional function is presented. Such
exponential functions play an important role in several problems in Uncertainty
Quantification, e.g. the approximation of log-normal random fields or the
evaluation of Bayesian posterior measures. Usually, these high-dimensional
objects are intractable numerically and can only be accessed pointwise in
sampling methods. In contrast, the proposed method constructs a functional
representation of the exponential by exploiting its nature as a solution of an
ordinary differential equation. The application of a Petrov--Galerkin scheme to
this equation provides a tensor train representation of the solution for which
we derive an efficient and reliable a posteriori error estimator. Numerical
experiments with a log-normal random field and a Bayesian likelihood illustrate
the performance of the approach in comparison to other recent low-rank
representations for the respective applications. Although the present work
considers only a specific differential equation, the presented method can be
applied in a more general setting. We show that the composition of a generic
holonomic function and a high-dimensional function corresponds to a
differential equation that can be used in our method. Moreover, the
differential equation can be modified to adapt the norm in the a posteriori
error estimates to the problem at hand
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Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations
A statistical learning approach for parametric PDEs related to
Uncertainty Quantification is derived. The method is based on the
minimization of an empirical risk on a selected model class and it is shown
to be applicable to a broad range of problems. A general unified convergence
analysis is derived, which takes into account the approximation and the
statistical errors. By this, a combination of theoretical results from
numerical analysis and statistics is obtained. Numerical experiments
illustrate the performance of the method with the model class of hierarchical
tensors
Variational Monte Carlo - Bridging concepts of machine learning and high dimensional partial differential equations
A statistical learning approach for parametric PDEs related to
Uncertainty Quantification is derived. The method is based on the
minimization of an empirical risk on a selected model class and it is shown
to be applicable to a broad range of problems. A general unified convergence
analysis is derived, which takes into account the approximation and the
statistical errors. By this, a combination of theoretical results from
numerical analysis and statistics is obtained. Numerical experiments
illustrate the performance of the method with the model class of hierarchical
tensors