Project 28: Functional Analysis and RKHS Kernel Lab

Build a kernel-method lab that makes Hilbert-space concepts computational and testable.

Quick Reference

Attribute Value
Difficulty Level 5: Master (The First-Principles Wizard)
Time Estimate 3 weeks
Main Programming Language Python
Alternative Programming Languages Julia, MATLAB, Scala
Coolness Level Level 5: Pure Magic (Super Cool)
Business Potential 2. The “Micro-SaaS / Pro Tool” (Solo-Preneur Potential)
Knowledge Area Functional Analysis / Kernel Methods
Main Book “Learning with Kernels” by Schölkopf and Smola

1. Learning Objectives

  1. Connect normed/Hilbert-space concepts to kernel ML workflows.
  2. Validate kernel PSD properties and representer-theorem consequences.
  3. Implement dual-form kernel ridge predictions with diagnostics.
  4. Explain regularization as function-space norm control.

2. All Theory Needed (Per-Concept Breakdown)

Concept A: Hilbert Spaces and Operators

Fundamentals Hilbert spaces generalize Euclidean geometry to function spaces via inner products.

Deep Dive into the concept Many learning problems become linear in suitable function spaces. Operator perspective clarifies stability, projection, and regularization.

Concept B: RKHS and Kernels

Fundamentals RKHS couples evaluation functionals with a reproducing kernel.

Deep Dive into the concept Kernel choice encodes geometry and prior assumptions. PSD conditions guarantee valid inner-product interpretation.

Concept C: Representer Theorem

Fundamentals Solutions under broad regularization classes lie in finite kernel spans.

Deep Dive into the concept This makes infinite-dimensional optimization computationally finite.

3. Build Blueprint

  1. Implement kernel matrix builder and PSD checks.
  2. Implement dual ridge regression/classification solver.
  3. Verify primal/dual prediction equivalence on toy sets.
  4. Add lambda sweep with norm-vs-error plots.

4. Real-World Outcome (Target)

$ python rkhs_lab.py --kernel rbf --dataset energy.csv

Kernel PSD: PASS
Primal RMSE: 0.421
Dual RMSE:   0.421
Max prediction delta: 6.2e-07
||f||_H: 3.182

5. Core Design Notes from Main Guide

Core Question

“How do infinite-dimensional function-space models become finite practical computations?”

Common Pitfalls

  • Ignoring near-PSD numerical corrections
  • Confusing large RKHS norm with better fit
  • Scaling to dense Gram solves without approximation strategy

Definition of Done

  • Kernel validity diagnostics implemented
  • Dual solver and prediction pipeline working
  • Regularization sweep includes norm/error tradeoff analysis
  • Includes scaling limitations and approximation options

6. Extensions

  1. Add Nyström approximation.
  2. Add random Fourier feature baseline.
  3. Add Gaussian-process interpretation notes.