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."