Core Concepts
The author establishes explicit representer theorems for solutions of learning problems in reproducing kernel Banach spaces, promoting sparsity and efficient learning.
Abstract
The paper focuses on sparse learning methods in reproducing kernel Banach spaces. It introduces the concept of sparsity and its importance in machine learning. By establishing representer theorems, the authors aim to provide a clear understanding of how RKBS can promote sparsity for learning solutions. They analyze minimum norm interpolation (MNI) and regularization problems, proposing sufficient conditions that transform explicit representations into sparse kernel representations with fewer terms than observed data points. The study highlights specific RKBSs like sequence space ℓ1(N) and measure space that exhibit sparse representer theorems for MNI and regularization models. The content delves into mathematical formulations, convex functions, subdifferential sets, extreme points, and dual problems to support their arguments.
Stats
For each x ∈ X, K(x, ·) ∈ B∗.
Sparsity level of f under kernel representation.
L(f) = y.
∂φ(g) - ∂φ(f) ≥ ⟨ν, g - f⟩B.
co(A) = co(ext(A)).
Lˆνα = y.
Quotes
"Sparsity of a learning solution is a desirable feature in machine learning."
"A solution of the regularization problem is a linear combination of the n kernel sessions."
"RKBSs are appropriate hypothesis spaces for sparse learning methods."