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

Abstract

We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted \(\ell ^1\) minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the \(L^2\)-norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity s and at most logarithmic in the dimension d, thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.

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Notes

  1. 1.

    This is not to be confused with expansions in Hermite polynomials, which we do not address in this paper. See [31] for some work in this direction.

  2. 2.

    As we discuss in Sect. 6, we use a slightly different method of proof to remove the factor \(\lambda \) in the error bound, at the expense of a slightly increased log factor.

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Acknowledgements

This work is supported in part by the NSERC Grant 611675 and an Alfred P. Sloan Research Fellowship. Yi Sui also acknowledges support from an NSERC PGSD scholarship.

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Correspondence to Ben Adcock.

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Adcock, B., Sui, Y. Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements. Constr Approx 50, 167–207 (2019). https://doi.org/10.1007/s00365-019-09467-0

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Keywords

  • Multivariate approximation
  • Orthogonal polynomials
  • Nonlinear approximation
  • Compressed sensing

Mathematics Subject Classification

  • 41A25
  • 41A05
  • 41A10
  • 65N15