Project 11: Inner Product and Bilinear Form Geometry Studio

Build a geometry studio that compares projection, orthogonality, and optimization behavior under multiple inner products and bilinear forms.

Quick Reference

Attribute Value
Difficulty Expert (Level 4)
Time Estimate 2 weeks
Language Python (alternatives: Julia, MATLAB)
Prerequisites Inner products, orthogonality, Gram matrices
Key Topics Bilinear forms, Gram-Schmidt, projection theorem

1. Learning Objectives

  1. Implement metric-aware projection and orthogonality checks.
  2. Compare Euclidean and weighted geometries on same data.
  3. Diagnose Gram-Schmidt numerical instability.
  4. Handle indefinite bilinear forms safely.
  5. Explain how metric choice changes optimization interpretation.

2. Theoretical Foundation

2.1 Core Concepts

  • Bilinear form vs inner product distinction.
  • Gram matrix and definiteness.
  • Orthonormalization under custom metrics.
  • Projection theorem in abstract inner-product spaces.

2.2 Why This Matters

Many pipelines assume Euclidean geometry by default. In weighted regression, anisotropic physics, and preconditioned optimization, that assumption is wrong. This project makes geometry explicit and testable.

2.3 Historical Context / Background

From least-squares origins to modern numerical optimization, inner products have been central to defining meaningful error and distance in domain-specific contexts.

2.4 Common Misconceptions

  • Symmetric bilinear form automatically defines distance.
  • Orthogonality is basis-only, not metric-dependent.
  • Classical Gram-Schmidt is always “good enough”.

3. Project Specification

3.1 What You Will Build

A tool that:

  • Registers multiple metrics/forms
  • Computes orthonormal bases under chosen metric
  • Projects vectors/signals onto subspaces
  • Compares residual behavior across metrics

3.2 Functional Requirements

  1. Metric registry (Euclidean, weighted SPD, indefinite form).
  2. Metric-aware Gram-Schmidt or QR pathway.
  3. Projection engine with orthogonality residual checks.
  4. Visualization/report comparing outcomes across metrics.

3.3 Non-Functional Requirements

  • Correctness: metric consistency in every operation.
  • Stability: report orthogonality loss measures.
  • Clarity: outputs explain geometric interpretation.

3.4 Example Usage / Output

$ python geometry_studio.py --metric weighted_l2 --subspace poly_basis
[INFO] metric=weighted_l2
[INFO] gram_eigs=[0.41, 1.27, 4.83]
[INFO] orthogonality_residual=4.2e-07
[INFO] projection_error_metric_norm=1.9e-03
[INFO] wrote comparison report to reports/metric_compare.md

3.5 Real World Outcome

You get a reusable metric-aware projection module and a set of diagnostics that demonstrate how changing geometry changes “best approximation” outcomes.

4. Solution Architecture

4.1 High-Level Design

Metric Registry -> Basis Processor -> Orthonormalizer -> Projection Engine -> Comparator -> Reporter

4.2 Key Components

Component Responsibility Key Decisions
Metric Registry Defines bilinear/inner forms SPD validation policy
Orthonormalizer Builds metric-orthonormal basis modified GS vs QR
Projection Engine Computes metric projection residual orthogonality checks
Comparator Cross-metric analysis consistent test vectors
Reporter Human-readable outcomes include geometric interpretation

4.3 Data Structures

MetricSpec:
  name
  gram_operator
  is_positive_definite

ProjectionReport:
  metric
  basis_quality
  residual_checks

4.4 Algorithm Overview

  1. Select metric and validate properties.
  2. Build/orthonormalize basis under metric.
  3. Compute projection and residual.
  4. Verify residual orthogonality in same metric.
  5. Compare against alternate metrics.

5. Implementation Guide

5.1 Development Environment Setup

Install numerical and symbolic packages
Prepare test vectors and basis fixtures
Enable reproducible random seeds

5.2 Project Structure

P11/
  metrics/
  orthonormalize/
  project/
  compare/
  reports/

5.3 The Core Question You’re Answering

“How does changing the inner product change what counts as a best approximation?”

5.4 Concepts You Must Understand First

  1. Inner-product axioms and positivity
  2. Gram matrix interpretation
  3. Projection theorem and residual orthogonality

5.5 Questions to Guide Your Design

  1. What checks prevent accidental Euclidean fallback?
  2. How will you show instability differences between orthogonalization methods?

5.6 Thinking Exercise

Take one vector and two metrics; predict projection coefficients before computing. Explain differences geometrically.

5.7 The Interview Questions They’ll Ask

  1. Why does metric choice change projection results?
  2. What is the practical difference between bilinear form and inner product?
  3. How do you detect loss of orthogonality numerically?
  4. Why can normal equations be less stable than QR?
  5. What does Gram-matrix definiteness tell you?

5.8 Hints in Layers

  • Hint 1: Implement metric interface before algorithms.
  • Hint 2: Add explicit checks for symmetry/positive-definiteness.
  • Hint 3: Compare classical and modified Gram-Schmidt residuals.
  • Hint 4: Keep projection and residual checks in same metric context.

5.9 Books That Will Help

Topic Book Chapter
Inner products Axler Inner-product chapters
Numerical orthogonalization Golub & Van Loan QR chapters
Practical optimization geometry Boyd/Vandenberghe Least-squares preliminaries

5.10 Implementation Phases

  1. Metric abstraction + validation
  2. Orthonormalization module
  3. Projection module
  4. Cross-metric comparison reports

5.11 Key Implementation Decisions

  • Force explicit metric parameter in public APIs.
  • Treat indefinite forms as separate mode with guardrails.

6. Testing Strategy

6.1 Test Categories

  • Positive-definite metric fixtures
  • Indefinite form edge cases
  • Near-dependent basis stability cases

6.2 Critical Test Cases

  1. Euclidean baseline projection.
  2. Weighted metric projection with known analytic result.
  3. Nearly dependent basis to compare GS variants.

6.3 Test Data

  • Polynomial and vector-space fixtures with known Gram matrices.

7. Common Pitfalls & Debugging

  • Symptom: Projection residual not orthogonal.
    • Why: Mixed metrics between solve and validation.
    • Fix: unify metric context.
    • Quick test: check all <r,u>_G values.
  • Symptom: Orthonormal basis check drifts.
    • Why: classical GS instability.
    • Fix: modified GS/QR fallback.
    • Quick test: compare ||Q^T G Q - I|| by method.

8. Extensions & Challenges

  • Add function-space metrics (integral-based inner products).
  • Add interactive metric slider for geometry visualization.
  • Add KKT-system interpretation for constrained projection.

9. Real-World Connections

  • Weighted regression and generalized least squares.
  • Preconditioned optimization and solver design.
  • Signal processing with custom energy metrics.

10. Resources

11. Self-Assessment Checklist

  • I can explain metric-dependent orthogonality clearly.
  • My tool validates metric assumptions before projection.
  • I can show and interpret orthogonality-loss diagnostics.
  • I can describe one real scenario where Euclidean metric is wrong.

12. Submission / Completion Criteria

  • Metric-aware projection studio with comparison report.
  • Stability comparison between orthonormalization strategies.
  • Written interpretation of at least one indefinite-form case.

This guide expands Project 11 from LINEAR_ALGEBRA_LEARNING_PROJECTS.md.