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Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM)

Terry Lyons, Andrew D. McLeod

TL;DR

HOLGRIM advances sparse interpolation of $Lip(\\gamma)$-regular functions by refining GRIM with a data-driven extension plus thinning via recombination, enabling sparse representations of complex linear combinations on finite data. It links pointwise values to finite dual functionals and demonstrates that the resulting sparsity is controlled by packing numbers of the data, with higher regularity $\\gamma$ reducing data concentration requirements. The work provides a concrete algorithmic framework, complexity bounds, and convergence guarantees, including extensions to compact domains with $Lip(\\eta)$-norm control, broadening applicability to kernel quadrature and regularity-aware learning. Altogether, HOLGRIM offers a scalable, provably accurate approach for sparse Lip$(\\gamma)$ interpolation in high dimensions, with potential impact on numerical quadrature, kernel methods, and data-efficient learning.

Abstract

In this paper we introduce the Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM) for finding sparse approximations of Lip$(γ)$ functions, in the sense of Stein, given as a linear combination of a (large) number of simpler Lip$(γ)$ functions. HOLGRIM is developed as a refinement of the Greedy Recombination Interpolation Method (GRIM) in the setting of Lip$(γ)$ functions. HOLGRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques in a data-driven fashion. The dynamic growth is driven by a greedy selection algorithm in which multiple new points may be selected at each step. The thinning reduction is carried out by recombination, the linear algebra technique utilised by GRIM. We establish that the number of non-zero weights for the approximation returned by HOLGRIM is controlled by a particular packing number of the data. The level of data concentration required to guarantee that HOLGRIM returns a good sparse approximation is decreasing with respect to the regularity parameter $γ> 0$. Further, we establish complexity cost estimates verifying that implementing HOLGRIM is feasible.

Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM)

TL;DR

HOLGRIM advances sparse interpolation of -regular functions by refining GRIM with a data-driven extension plus thinning via recombination, enabling sparse representations of complex linear combinations on finite data. It links pointwise values to finite dual functionals and demonstrates that the resulting sparsity is controlled by packing numbers of the data, with higher regularity reducing data concentration requirements. The work provides a concrete algorithmic framework, complexity bounds, and convergence guarantees, including extensions to compact domains with -norm control, broadening applicability to kernel quadrature and regularity-aware learning. Altogether, HOLGRIM offers a scalable, provably accurate approach for sparse Lip interpolation in high dimensions, with potential impact on numerical quadrature, kernel methods, and data-efficient learning.

Abstract

In this paper we introduce the Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM) for finding sparse approximations of Lip functions, in the sense of Stein, given as a linear combination of a (large) number of simpler Lip functions. HOLGRIM is developed as a refinement of the Greedy Recombination Interpolation Method (GRIM) in the setting of Lip functions. HOLGRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques in a data-driven fashion. The dynamic growth is driven by a greedy selection algorithm in which multiple new points may be selected at each step. The thinning reduction is carried out by recombination, the linear algebra technique utilised by GRIM. We establish that the number of non-zero weights for the approximation returned by HOLGRIM is controlled by a particular packing number of the data. The level of data concentration required to guarantee that HOLGRIM returns a good sparse approximation is decreasing with respect to the regularity parameter . Further, we establish complexity cost estimates verifying that implementing HOLGRIM is feasible.
Paper Structure (10 sections, 13 theorems, 215 equations)

This paper contains 10 sections, 13 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Lipschitz Functions and Sandwich Theorems
  3. Sparse Approximation Problem Formulation
  4. Extension to Compact Domains
  5. Pointwise Values Via Linear Functionals
  6. The HOLGRIM Algorithm
  7. Complexity Cost
  8. Theoretical Guarantees Inherited from GRIM
  9. Point Separation Lemma
  10. The HOLGRIM Convergence Theorem

Key Result

Theorem 2.9

Let $V$ and $W$ be Banach spaces, and assume that the tensor powers of $V$ are all equipped with admissible norms (cf. Definition admissible_tensor_norm). Assume that ${\cal M} \subset V$ is non-empty and closed. Let $\varepsilon, K_0 > 0$, and $\gamma > \eta > 0$ with $k,q \in {\mathbb Z}_{\geq 0}$ Suppose $\psi = \left(\psi^{(0)} ,\ldots ,\psi^{(k)}\right) , \phi = \left( \phi^{(0)} , \ldots , \

Theorems & Definitions (43)

  • Definition 2.1: Admissible Norms on Tensor Powers
  • Definition 2.2: ${\mathrm{Lip}}(\gamma,{\cal M},W)$ functions
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9: Lipschitz Sandwich Theorem 3.1 in LM24
  • Theorem 2.10: Pointwise Lipschitz Sandwich Theorem 3.11 in LM24
  • ...and 33 more