Greedy Recombination Interpolation Method (GRIM) Development for Sparse Function Approximations

Core Concepts

The authors introduce the Greedy Recombination Interpolation Method (GRIM) to find sparse function approximations using dynamic growth and thinning techniques.

Abstract

The Greedy Recombination Interpolation Method (GRIM) is developed to provide sparse approximations of functions by combining dynamic growth-based interpolation and thinning-based reduction techniques. The method utilizes recombination outside the setting of measure support reduction, controlling sparsity based on data concentration. GRIM matches contemporary kernel quadrature techniques' performance, offering a novel approach to finding sparse function approximations efficiently.
Key points include the comparison with existing methods like CoSaMP, LASSO, and GEIM, the application of GRIM in various fields like image processing and machine learning, and the theoretical analysis of its convergence and complexity cost.
The Banach GRIM algorithm involves dynamically growing linear functionals from data subsets and applying recombination for accurate approximations. The algorithm optimizes over multiple permutations to enhance accuracy.
Recombination Thinning Lemma 3.1 details how recombination can be used to find an approximation that coincides with the target function throughout a given subset of data. The Banach GRIM Convergence Theorem establishes theoretical guarantees for the algorithm's performance.

Stats

A consequence of this difference is that certain aspects of GEIM are not necessarily ideal for our task.
For each j ∈ {1, . . . , s} we do the following.
Let D := dim (ker(A)) ≥ N − M. Take e(1), . . . , e(N − D) ∈ {1, . . . , N} to be the indices i ∈ {1, . . . , N} for which x′i > 0.
Then u ∈ Span(F) is returned as our approximation of φ that satisfies, for every σ ∈ Σ, that |σ(φ − u)| ≤ ε0.
This ensures that a1, . . . , aN > 0 whilst leaving the expansion φ = PN i=1 aifi unaltered.

Quotes

"The growth in GRIM is data-driven rather than feature-driven."
"Recombination preserves convexity benefits enjoyed by convex kernel quadrature."
"GRIM dynamically grows linear functionals while applying recombination for accurate approximations."

Key Insights Distilled From

Greedy Recombination Interpolation Method (GRIM)

by Terry Lyons,... at arxiv.org 03-11-2024

https://arxiv.org/pdf/2205.07495.pdf

Deeper Inquiries

How does GRIM compare to other sparse approximation algorithms like Matching Pursuit

GRIM, the Greedy Recombination Interpolation Method, differs from other sparse approximation algorithms like Matching Pursuit in its approach to finding sparse approximations of functions. While Matching Pursuit greedily grows a collection of non-zero weights one-by-one to construct the approximation, GRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques. In GRIM, the growth is data-driven rather than feature-driven, allowing for more flexibility in selecting features and improving overall performance. Additionally, GRIM uses recombination to find an element that coincides with the target function throughout a given subset of linear functionals. This hybrid approach sets GRIM apart from traditional methods like Matching Pursuit.

What implications does the use of recombination have on computational efficiency in practice

The use of recombination in practice can have significant implications on computational efficiency. Recombination allows for reducing the number of non-zero components in a solution while preserving convexity and ensuring that certain properties are maintained during optimization processes. In the context of algorithms like GRIM, recombination helps find sparse approximations efficiently by dynamically growing subsets based on data information and then applying thinning techniques through recombination to refine these subsets further. By leveraging recombination effectively within iterative steps as outlined in the Banach Recombination Step, computational efficiency can be enhanced by optimizing over multiple permutations and shuffles to improve accuracy without sacrificing speed.

How can GRIM's hybrid approach benefit applications beyond kernel quadrature tasks

The hybrid approach employed by GRIM has implications beyond kernel quadrature tasks and can benefit various applications requiring sparse representations or approximations. By combining dynamic growth strategies with thinning-based reduction techniques facilitated by recombination, GRIM offers a versatile method for finding accurate yet computationally efficient solutions across different domains such as compressed sensing, image processing, machine learning inference acceleration tasks among others.
This versatility stems from its ability to adaptively select features based on data characteristics rather than predetermined choices seen in some traditional methods. The use of recombination ensures that sparsity is controlled effectively while maintaining accuracy levels required for diverse applications outside kernel quadrature tasks.