Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements

Adcock, Ben; Sui, Yi

doi:10.1007/s00365-019-09467-0

Cited by 48 publications

(8 citation statements)

References 37 publications

(95 reference statements)

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“…An example is the discrete-inspace-continuous-in-time model, which can occur when sensors in physical space take continuous recordings a time-dependent function f (y,t). Another problem, which arises commonly in uncertainty quantification (see [13,44,76] and references therein), is the problem where one measures both the function f (y) and its gradient ∇ f (y) simultaneously at a sample point y. We anticipate that many of the key results of this work can be extended to substantially more general sampling models.…”

Section: Conclusion and Challengesmentioning

confidence: 95%

“…The application of 1 -minimization for computing sparse polynomial approximations of functions was first considered in [18,39,65,79,97]. This led to substantial amounts of subsequent research, including [3,4,13,14,19,26,26,43,44,52,55,57,63,74,76,78,81,86,90,91,94,95,[98][99][100][101][102]. Specific extensions to weighted and lower sparsity models were developed in [1-3, 5, 9, 25, 75, 80, 99].…”

Section: Related Literaturementioning

confidence: 99%

“…We first describe the computation of the approximation (13). Since any p ∈ P S;V h can be expressed p = ∑ ι∈S c ι φ ι with c ι ∈ V h , we can rewrite (13) as…”

Section: Computation Of the Least-squares Approximationmentioning

confidence: 99%

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Towards optimal sampling for learning sparse approximation in high dimensions

Adcock¹,

Cardenas²,

Dexter³

et al. 2022

Preprint

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In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector-or even Hilbert space-valued. Our main objective is to study how the sampling strategy affects the sample complexity -that is, the number of samples that suffice for accurate and stable recovery -and to use this insight to obtain optimal or near-optimal sampling procedures. We consider two settings. First, when a target sparse representation is known, in which case we present a near-complete answer based on drawing independent random samples from carefully-designed probability measures. Second, we consider the more challenging scenario when such representation is unknown. In this case, while not giving a full answer, we describe a general construction of sampling measures that improves over standard Monte Carlo sampling. We present examples using algebraic and trigonometric polynomials, and for the former, we also introduce a new procedure for function approximation on irregular (i.e., nontensorial) domains. The effectiveness of this procedure is shown through numerical examples. Finally, we discuss a number of structured sparsity models, and how they may lead to better approximations.

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Section: Conclusion and Challengesmentioning

confidence: 95%

Section: Related Literaturementioning

confidence: 99%

See 1 more Smart Citation

Towards optimal sampling for learning sparse approximation in high dimensions

Adcock¹,

Cardenas²,

Dexter³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For m = 1 this hypothesis is confirmed numerically in Figure 4. For an application in the setting of weighted sparsity we refer to the recent work [36]. Note that this does not have to be the case in general.…”

Section: Dependence On the Seminormmentioning

confidence: 99%

Convergence bounds for empirical nonlinear least-squares

2022

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We consider best approximation problems in a nonlinear subset [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . 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 nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results 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.

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“…Sparse approximation by possibly multivariate functions of data that includes gradient information is a highly investigated subject, from local piecewise spline interpolation to global compressive sparsification by ℓ 1 -norm optimization [1]. The Prony algorithm, which was originally designed for sums of exponentials [25] and used for sparse polynomials over finite fields to decode the 1959 BCH digital error correction code, is suitable for floating point data [6,7].…”

Section: Relation To Previous Workmentioning

confidence: 99%

Sparse Polynomial Hermite Interpolation

Kaltofen

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial has terms, our algorithms, require argument/value triples ( , ( ), ′ ( )) for = 0, . . . , + ⌈( +1)/2⌉ −1, where is randomly sampled and the probability of a correct output is determined from a degree bound for . With ′ we denote the derivative of . Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound ≥ for the number of terms is given, our algorithms use a randomly selected and, with high probability, ⌈ /2⌉ + triples, but then never return an incorrect output.The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2 + 1 and + values ( ), respectively, they need more arguments . The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from + 1 values of and 2 − 1 values of ′ . CCS CONCEPTS• Mathematics of computing → Interpolation.

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Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements

Cited by 48 publications

References 37 publications

Towards optimal sampling for learning sparse approximation in high dimensions

Towards optimal sampling for learning sparse approximation in high dimensions

Convergence bounds for empirical nonlinear least-squares

Sparse Polynomial Hermite Interpolation

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