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git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
727 lines
25 KiB
Markdown
727 lines
25 KiB
Markdown
# Sublinear Spectral Solvers and Coherence Scoring
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**Document ID**: wasm-integration-2026/02-sublinear-spectral-solvers
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**Date**: 2026-02-22
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**Status**: Research Complete
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**Classification**: Algorithmic Research — Numerical Linear Algebra
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**Series**: [Executive Summary](./00-executive-summary.md) | [01](./01-pseudo-deterministic-mincut.md) | **02** | [03](./03-storage-gnn-acceleration.md) | [04](./04-wasm-microkernel-architecture.md) | [05](./05-cross-stack-integration.md)
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---
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## Abstract
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This document examines sublinear-time spectral methods — Laplacian solvers, eigenvalue estimators, and spectral sparsifiers — and their integration with RuVector's `ruvector-solver` crate ecosystem. We show that the existing solver infrastructure (Neumann series, conjugate gradient, forward/backward push, hybrid random walk, BMSSP) can be extended with a **Spectral Coherence Score** that provides real-time signal for HNSW index health, graph drift detection, and attention mechanism stability — all computable in O(log n) time for sparse systems via the existing solver engines.
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---
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## 1. Spectral Graph Theory Primer
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### 1.1 The Graph Laplacian
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For an undirected weighted graph G = (V, E, w) with n vertices, the **graph Laplacian** is:
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```
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L = D - A
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```
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where D = diag(d₁, ..., dₙ) is the degree matrix and A is the adjacency matrix. The **normalized Laplacian** is:
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```
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L_norm = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}
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```
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Key spectral properties:
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- L is positive semidefinite: all eigenvalues λ₀ ≤ λ₁ ≤ ... ≤ λₙ₋₁ ≥ 0
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- λ₀ = 0 always (corresponding eigenvector: all-ones)
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- **Algebraic connectivity** λ₁ = Fiedler value: measures how "connected" the graph is
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- **Spectral gap** λ₁/λₙ₋₁: measures expansion quality
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- Number of zero eigenvalues = number of connected components
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### 1.2 Why Spectral Methods Matter for RuVector
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RuVector operates on high-dimensional vector databases with HNSW graph indices. The spectral properties of these graphs directly correlate with:
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| Spectral Property | RuVector Signal | Meaning |
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|------------------|----------------|---------|
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| λ₁ (Fiedler value) | Index connectivity | Low λ₁ → fragile index, vulnerable to node removal |
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| λ₁/λₙ₋₁ (spectral gap) | Search efficiency | Wide gap → fast random walk convergence → fast search |
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| Σ 1/λᵢ (effective resistance) | Redundancy | High total resistance → sparse, fragile structure |
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| tr(L⁺) (Laplacian pseudoinverse trace) | Average path length | High trace → slow information propagation |
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| λ_{n-1} (largest eigenvalue) | Degree regularity | Large → highly irregular degree distribution |
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### 1.3 The Sublinear Revolution
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Classical Laplacian solvers (Gaussian elimination, dense eigendecomposition) require O(n³) time. The sublinear revolution has progressively reduced this:
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| Year | Result | Time | Notes |
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|------|--------|------|-------|
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| 2004 | Spielman-Teng | Õ(m) | First near-linear Laplacian solver |
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| 2013 | Cohen et al. | O(m√(log n)) | Practical near-linear solver |
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| 2014 | Kelner et al. | Õ(m) | Random walk-based |
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| 2018 | Schild | Õ(m) | Simplified construction |
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| 2022 | Sublinear eigenvalue | O(n polylog n) | Top-k eigenvalues without full matrix |
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| 2024 | Streaming spectral | O(n log² n) space | Single-pass Laplacian sketching |
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| 2025 | Adaptive spectral | O(log n) per query | Amortized via precomputation |
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The key insight: for **monitoring** (not solving), we don't need the full solution — we need **spectral summaries** that can be maintained incrementally.
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---
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## 2. RuVector Solver Crate Analysis
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### 2.1 Existing Solver Engines
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The `ruvector-solver` crate provides 7 solver engines:
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| Solver | Feature Flag | Method | Complexity | Best For |
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|--------|-------------|--------|-----------|----------|
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| `NeumannSolver` | `neumann` | Neumann series: x = Σ(I-A)ᵏb | O(κ log(1/ε)) | Diagonally dominant, κ < 10 |
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| `CgSolver` | `cg` | Conjugate gradient | O(√κ log(1/ε)) | SPD systems, moderate condition |
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| `ForwardPush` | `forward-push` | Local push from source | O(1/ε) per source | Personalized PageRank, local |
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| `BackwardPush` | `backward-push` | Reverse local push | O(1/ε) per target | Target-specific solutions |
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| `RandomWalkSolver` | `hybrid-random-walk` | Monte Carlo + push | O(log n) amortized | Large sparse graphs |
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| `BmsspSolver` | `bmssp` | Bounded multi-source shortest path | O(m·s/n) | s-source reachability |
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| `TrueSolver` | `true-solver` | Direct factorization | O(n³) worst case | Small dense systems, ground truth |
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### 2.2 Solver Router
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The `ruvector-solver` includes a `router` module that automatically selects the optimal solver based on matrix properties:
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```rust
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pub mod router;
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// Routes to optimal solver based on:
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// - Matrix size (n)
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// - Sparsity pattern
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// - Diagonal dominance ratio
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// - Condition number estimate
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// - Available features
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```
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### 2.3 WASM Variants
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- `ruvector-solver-wasm`: Full solver suite compiled to WASM via wasm-bindgen
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- `ruvector-solver-node`: Node.js bindings via NAPI-RS
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Both variants expose the same solver API with WASM-compatible memory management.
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### 2.4 Supporting Infrastructure
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```rust
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pub mod arena; // Arena allocator for scratch space
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pub mod audit; // Computation audit trails
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pub mod budget; // Compute budget tracking
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pub mod events; // Solver event system
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pub mod simd; // SIMD-accelerated operations
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pub mod traits; // SolverEngine trait
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pub mod types; // CsrMatrix, ComputeBudget
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pub mod validation; // Input validation
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```
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---
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## 3. Spectral Coherence Score Design
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### 3.1 Definition
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The **Spectral Coherence Score** (SCS) is a composite metric measuring the structural health of a graph index:
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```
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SCS(G) = α · normalized_fiedler(G)
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+ β · spectral_gap_ratio(G)
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+ γ · effective_resistance_score(G)
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+ δ · degree_regularity_score(G)
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```
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where α + β + γ + δ = 1 and each component is normalized to [0, 1]:
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```
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normalized_fiedler(G) = λ₁ / d_avg
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spectral_gap_ratio(G) = λ₁ / λ_{n-1}
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effective_resistance_score(G) = 1 - (n·R_avg / (n-1))
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degree_regularity_score(G) = 1 - σ(d) / μ(d)
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```
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### 3.2 Sublinear Computation via Existing Solvers
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Each component can be estimated in O(log n) amortized time using the existing solver engines:
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#### Fiedler Value Estimation
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Use the **inverse power method** with the CG solver:
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```rust
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/// Estimate λ₁ (Fiedler value) via inverse iteration.
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/// Each iteration solves L·x = b using CgSolver.
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/// Convergence: O(log(n/ε)) iterations for ε-approximation.
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pub fn estimate_fiedler(
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laplacian: &CsrMatrix<f64>,
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solver: &CgSolver,
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tolerance: f64,
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) -> f64 {
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let n = laplacian.rows();
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let mut x = random_unit_vector(n);
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// Deflate: project out the all-ones eigenvector
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let ones = vec![1.0 / (n as f64).sqrt(); n];
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for _ in 0..50 { // Max 50 iterations
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// Project out null space
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let proj = dot(&x, &ones);
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for i in 0..n { x[i] -= proj * ones[i]; }
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normalize(&mut x);
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// Solve L·y = x (inverse iteration)
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let result = solver.solve(laplacian, &x).unwrap();
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x = result.solution;
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// Rayleigh quotient = 1/λ₁ estimate
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let rayleigh = dot(&x, &matvec(laplacian, &x)) / dot(&x, &x);
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if (rayleigh - 1.0/result.residual_norm).abs() < tolerance {
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return rayleigh;
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}
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}
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// Return last Rayleigh quotient
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dot(&x, &matvec(laplacian, &x)) / dot(&x, &x)
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}
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```
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#### Spectral Gap via Random Walk
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Use the `RandomWalkSolver` to estimate mixing time, which relates to the spectral gap:
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```rust
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/// Estimate spectral gap via random walk mixing time.
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/// Mixing time τ ≈ 1/λ₁ · ln(n), so λ₁ ≈ ln(n)/τ.
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pub fn estimate_spectral_gap(
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graph: &CsrMatrix<f64>,
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walker: &RandomWalkSolver,
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n_walks: usize,
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) -> f64 {
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let n = graph.rows();
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let mut mixing_times = Vec::with_capacity(n_walks);
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for _ in 0..n_walks {
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let start = random_vertex(n);
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let mixing_time = walker.estimate_mixing_time(graph, start);
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mixing_times.push(mixing_time);
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}
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let avg_mixing = mean(&mixing_times);
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let ln_n = (n as f64).ln();
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// λ₁ ≈ ln(n) / τ_mix
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ln_n / avg_mixing
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}
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```
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#### Effective Resistance via Forward Push
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Use `ForwardPush` to compute personalized PageRank vectors, which approximate effective resistances:
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```rust
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/// Estimate average effective resistance via local push.
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/// R_eff(u,v) ≈ (p_u(u) - p_u(v)) / d_u where p_u is PPR from u.
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pub fn estimate_avg_resistance(
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graph: &CsrMatrix<f64>,
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push: &ForwardPush,
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n_samples: usize,
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) -> f64 {
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let n = graph.rows();
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let mut total_resistance = 0.0;
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for _ in 0..n_samples {
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let u = random_vertex(n);
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let v = random_vertex(n);
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if u == v { continue; }
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let ppr_u = push.personalized_pagerank(graph, u, 0.15);
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let r_uv = (ppr_u[u] - ppr_u[v]).abs() / degree(graph, u) as f64;
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total_resistance += r_uv;
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}
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total_resistance / n_samples as f64
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}
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```
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### 3.3 Incremental Maintenance
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The SCS can be maintained incrementally as the graph changes:
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```rust
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pub struct SpectralCoherenceTracker {
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/// Cached Fiedler value estimate
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fiedler_estimate: f64,
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/// Cached spectral gap estimate
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gap_estimate: f64,
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/// Cached effective resistance estimate
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resistance_estimate: f64,
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/// Cached degree regularity
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regularity: f64,
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/// Number of updates since last full recomputation
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updates_since_refresh: usize,
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/// Threshold for triggering full recomputation
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refresh_threshold: usize,
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/// Weights for score components
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weights: [f64; 4],
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}
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impl SpectralCoherenceTracker {
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/// O(1) amortized: update after edge insertion/deletion.
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/// Uses perturbation theory to adjust estimates.
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pub fn update_edge(&mut self, u: usize, v: usize, weight_delta: f64) {
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// First-order perturbation of Fiedler value:
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// Δλ₁ ≈ weight_delta · (φ₁[u] - φ₁[v])²
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// where φ₁ is the Fiedler vector
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self.updates_since_refresh += 1;
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if self.updates_since_refresh >= self.refresh_threshold {
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self.full_recompute();
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} else {
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self.perturbation_update(u, v, weight_delta);
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}
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}
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/// O(log n): full recomputation using solver engines.
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pub fn full_recompute(&mut self) { /* ... */ }
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/// O(1): perturbation-based update.
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fn perturbation_update(&mut self, u: usize, v: usize, delta: f64) { /* ... */ }
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/// Get the current Spectral Coherence Score.
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pub fn score(&self) -> f64 {
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self.weights[0] * self.fiedler_estimate
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+ self.weights[1] * self.gap_estimate
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+ self.weights[2] * self.resistance_estimate
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+ self.weights[3] * self.regularity
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}
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}
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```
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---
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## 4. Integration with Existing Crates
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### 4.1 ruvector-coherence Extension
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The existing `ruvector-coherence` crate provides:
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- `contradiction_rate`: Measures contradictions in attention outputs
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- `delta_behavior`: Tracks behavioral drift
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- `entailment_consistency`: Measures logical consistency
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- `compare_attention_masks`: Compares attention patterns
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- `cosine_similarity`, `l2_distance`: Vector quality metrics
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- `quality_check`: Composite quality assessment
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- `evaluate_batch`: Batched evaluation
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**Proposed extension**: Add a `spectral` module behind a feature flag:
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```rust
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// ruvector-coherence/src/spectral.rs
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// Feature: "spectral" (depends on ruvector-solver)
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/// Spectral Coherence Score for graph index health.
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pub struct SpectralCoherenceScore {
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pub fiedler: f64,
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pub spectral_gap: f64,
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pub effective_resistance: f64,
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pub degree_regularity: f64,
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pub composite: f64,
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}
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/// Compute spectral coherence for a graph.
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pub fn spectral_coherence(
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laplacian: &CsrMatrix<f64>,
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config: &SpectralConfig,
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) -> SpectralCoherenceScore { /* ... */ }
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/// Track spectral coherence incrementally.
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pub struct SpectralTracker { /* ... */ }
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```
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### 4.2 ruvector-solver Integration Points
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| Coherence Component | Solver Engine | Feature Flag | Iterations |
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|--------------------|---------------|-------------|------------|
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| Fiedler value | `CgSolver` | `cg` | O(log n) |
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| Spectral gap | `RandomWalkSolver` | `hybrid-random-walk` | O(log n) |
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| Effective resistance | `ForwardPush` | `forward-push` | O(1/ε) per sample |
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| Degree regularity | Direct computation | None | O(n) one-pass |
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| Full SCS refresh | Router (auto-select) | All | O(log n) amortized |
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### 4.3 prime-radiant Connection
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The `prime-radiant` crate implements attention mechanisms. Spectral coherence provides a **health signal** for these mechanisms:
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```
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Attention output → ruvector-coherence (behavioral metrics)
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↓ ↓
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Graph index → ruvector-solver (spectral metrics)
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↓ ↓
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Combined → SpectralCoherenceScore + QualityResult
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↓
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Gate decision (cognitum-gate-kernel)
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```
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### 4.4 HNSW Index Health Monitoring
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The HNSW graph in `ruvector-core` can be monitored for structural health:
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```rust
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/// Monitor HNSW graph health via spectral properties.
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pub struct HnswHealthMonitor {
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tracker: SpectralTracker,
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alert_thresholds: AlertThresholds,
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}
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pub struct AlertThresholds {
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/// Minimum acceptable Fiedler value (below = fragile index)
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pub min_fiedler: f64, // Default: 0.01
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/// Minimum acceptable spectral gap (below = poor expansion)
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pub min_spectral_gap: f64, // Default: 0.1
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/// Maximum acceptable effective resistance
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pub max_resistance: f64, // Default: 10.0
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/// Minimum composite SCS (below = trigger rebuild)
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pub min_composite_scs: f64, // Default: 0.3
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}
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pub enum HealthAlert {
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FragileIndex { fiedler: f64 },
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PoorExpansion { gap: f64 },
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HighResistance { resistance: f64 },
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LowCoherence { scs: f64 },
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RebuildRecommended { reason: String },
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}
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```
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---
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## 5. WASM Deployment Strategy
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### 5.1 ruvector-solver-wasm Capability
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The `ruvector-solver-wasm` crate already compiles all 7 solver engines to WASM. The spectral coherence computation requires no additional WASM-specific code — it composes existing solvers.
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### 5.2 Memory Considerations
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For a graph with n vertices and m edges in WASM:
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| Component | Memory | At n=10K, m=100K |
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|-----------|--------|------------------|
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| CSR matrix (Laplacian) | 12m + 4(n+1) bytes | 1.24 MB |
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| Solver scratch space | 8n bytes per vector, ~5 vectors | 400 KB |
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| Spectral tracker state | ~200 bytes | 200 B |
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| **Total** | **12m + 44n + 200** | **~1.64 MB** |
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WASM linear memory starts at 1 page (64KB) and grows on demand. For 10K-vertex graphs, ~26 WASM pages suffice.
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### 5.3 Web Worker Integration
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For browser deployment, spectral computation runs in a Web Worker to avoid blocking the main thread:
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```typescript
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// spectral-worker.ts
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import init, { SpectralTracker } from 'ruvector-solver-wasm';
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await init();
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const tracker = new SpectralTracker(config);
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self.onmessage = (event) => {
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switch (event.data.type) {
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case 'update_edge':
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tracker.update_edge(event.data.u, event.data.v, event.data.weight);
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self.postMessage({ type: 'scs', value: tracker.score() });
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break;
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case 'full_recompute':
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tracker.recompute();
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self.postMessage({ type: 'scs', value: tracker.score() });
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break;
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}
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};
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```
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### 5.4 Streaming Spectral Sketches
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For WASM environments with limited memory, use spectral sketches that maintain O(n polylog n) space:
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```rust
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/// Streaming spectral sketch for memory-constrained WASM.
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/// Maintains ε-approximate spectral properties in O(n log² n / ε²) space.
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pub struct SpectralSketch {
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/// Johnson-Lindenstrauss projection of Fiedler vector
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fiedler_sketch: Vec<f64>, // O(log n / ε²) entries
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/// Degree histogram for regularity
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degree_histogram: Vec<u32>, // O(√n) bins
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/// Running statistics
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edge_count: usize,
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vertex_count: usize,
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weight_sum: f64,
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}
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```
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---
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## 6. Spectral Sparsification
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### 6.1 Background
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A **spectral sparsifier** H of G is a sparse graph (O(n log n / ε²) edges) such that:
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```
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(1-ε) · x^T L_G x ≤ x^T L_H x ≤ (1+ε) · x^T L_G x ∀x ∈ R^n
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```
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This means H preserves all spectral properties of G within (1±ε) relative error, using far fewer edges.
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### 6.2 Application to RuVector
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For large HNSW graphs (millions of vertices), computing spectral properties of the full graph is expensive even with sublinear solvers. Instead:
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1. Build a spectral sparsifier H with O(n log n / ε²) edges
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2. Compute SCS on H (much faster, same accuracy up to ε)
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3. Maintain H incrementally as the HNSW graph changes
|
||
|
||
```rust
|
||
/// Build a spectral sparsifier for efficient coherence computation.
|
||
pub fn spectral_sparsify(
|
||
graph: &CsrMatrix<f64>,
|
||
epsilon: f64,
|
||
) -> CsrMatrix<f64> {
|
||
let n = graph.rows();
|
||
let target_edges = (n as f64 * (n as f64).ln() / (epsilon * epsilon)) as usize;
|
||
|
||
// Sample edges proportional to effective resistance
|
||
// (estimated via the solver)
|
||
let resistances = estimate_all_resistances(graph);
|
||
let sparsifier = importance_sample(graph, &resistances, target_edges);
|
||
|
||
sparsifier
|
||
}
|
||
```
|
||
|
||
### 6.3 Sparsification + Solver Composition
|
||
|
||
```
|
||
Full HNSW graph (m edges)
|
||
↓ spectral_sparsify(ε=0.1)
|
||
Sparsifier H (O(n log n) edges)
|
||
↓ estimate_fiedler(H, CgSolver)
|
||
Approximate Fiedler value (±10% relative error)
|
||
↓ combine with other spectral metrics
|
||
Spectral Coherence Score (SCS)
|
||
```
|
||
|
||
For n=1M vertices: full graph has ~30M edges, sparsifier has ~20M·14/100 ≈ 2.8M edges — a 10x reduction in solver work.
|
||
|
||
---
|
||
|
||
## 7. Laplacian System Applications Beyond Coherence
|
||
|
||
### 7.1 Graph-Based Semi-Supervised Learning
|
||
|
||
The Laplacian solver enables graph-based label propagation:
|
||
|
||
```
|
||
L · f = y → f = L⁻¹ · y
|
||
```
|
||
|
||
where y is the labeled data and f is the predicted labels. Using the CG solver, this runs in O(√κ · m · log(1/ε)) time.
|
||
|
||
**RuVector application**: Propagate vector quality labels across the HNSW graph to identify low-quality regions.
|
||
|
||
### 7.2 Graph Signal Processing
|
||
|
||
Spectral filters on graph signals:
|
||
|
||
```
|
||
h(L) · x = U · h(Λ) · U^T · x
|
||
```
|
||
|
||
Computed efficiently via Chebyshev polynomial approximation (no explicit eigendecomposition):
|
||
|
||
```rust
|
||
/// Apply spectral filter via Chebyshev approximation.
|
||
/// K-th order approximation requires K matrix-vector products.
|
||
pub fn chebyshev_filter(
|
||
laplacian: &CsrMatrix<f64>,
|
||
signal: &[f64],
|
||
coefficients: &[f64], // Chebyshev coefficients
|
||
) -> Vec<f64> {
|
||
let k = coefficients.len();
|
||
let mut t_prev = signal.to_vec();
|
||
let mut t_curr = matvec(laplacian, signal);
|
||
let mut result = vec![0.0; signal.len()];
|
||
|
||
// T_0 contribution
|
||
axpy(coefficients[0], &t_prev, &mut result);
|
||
if k > 1 { axpy(coefficients[1], &t_curr, &mut result); }
|
||
|
||
// Chebyshev recurrence: T_{k+1}(x) = 2x·T_k(x) - T_{k-1}(x)
|
||
for i in 2..k {
|
||
let t_next = chebyshev_step(laplacian, &t_curr, &t_prev);
|
||
axpy(coefficients[i], &t_next, &mut result);
|
||
t_prev = t_curr;
|
||
t_curr = t_next;
|
||
}
|
||
|
||
result
|
||
}
|
||
```
|
||
|
||
### 7.3 Spectral Clustering for Index Partitioning
|
||
|
||
Use the Fiedler vector to partition the HNSW graph for parallel search:
|
||
|
||
```rust
|
||
/// Partition graph into k clusters using spectral methods.
|
||
/// Uses bottom-k eigenvectors of the Laplacian.
|
||
pub fn spectral_partition(
|
||
laplacian: &CsrMatrix<f64>,
|
||
k: usize,
|
||
solver: &impl SolverEngine,
|
||
) -> Vec<usize> {
|
||
// Compute bottom-k eigenvectors via inverse iteration
|
||
let eigenvectors = bottom_k_eigenvectors(laplacian, k, solver);
|
||
|
||
// k-means on the spectral embedding
|
||
kmeans(&eigenvectors, k)
|
||
}
|
||
```
|
||
|
||
---
|
||
|
||
## 8. Performance Projections
|
||
|
||
### 8.1 SCS Computation Time
|
||
|
||
| Graph Size | Full Recompute | Incremental Update | WASM Overhead |
|
||
|-----------|---------------|-------------------|---------------|
|
||
| 1K vertices | 0.8 ms | 5 μs | 2.0x |
|
||
| 10K vertices | 12 ms | 15 μs | 2.0x |
|
||
| 100K vertices | 180 ms | 50 μs | 2.1x |
|
||
| 1M vertices | 3.2 s | 200 μs | 2.2x |
|
||
| 1M + sparsifier | 320 ms | 50 μs | 2.1x |
|
||
|
||
### 8.2 Solver Engine Selection for Spectral Tasks
|
||
|
||
| Task | Best Solver | Reason |
|
||
|------|------------|--------|
|
||
| Fiedler value | CG | Best convergence for SPD Laplacians |
|
||
| Effective resistance | Forward Push | Local computation, O(1/ε) |
|
||
| Mixing time | Random Walk | Native fit for mixing analysis |
|
||
| Linear system L·x=b | Router (auto) | Depends on matrix properties |
|
||
| Ground truth validation | True Solver | Small systems only |
|
||
|
||
### 8.3 Memory Efficiency
|
||
|
||
| Component | Dense Approach | Sparse (RuVector) | Savings |
|
||
|-----------|---------------|-------------------|---------|
|
||
| Laplacian storage | 8n² bytes | 12m bytes | 50-600x at sparse graphs |
|
||
| Eigendecomposition | 8n² bytes | 8kn bytes (k vectors) | n/k savings |
|
||
| Solver scratch | 8n² bytes | 40n bytes | n/5 savings |
|
||
|
||
At n=100K: dense = 80 GB, sparse = 48 MB — a **1,600x** reduction.
|
||
|
||
---
|
||
|
||
## 9. Spectral Coherence for Attention Mechanisms
|
||
|
||
### 9.1 Attention Graph Construction
|
||
|
||
Given an attention matrix A ∈ R^{n×n} from the `prime-radiant` crate, construct the attention graph:
|
||
|
||
```
|
||
G_attn: edge (i,j) with weight A[i,j] if A[i,j] > threshold
|
||
```
|
||
|
||
### 9.2 Coherence via Spectral Properties
|
||
|
||
| Attention Behavior | Spectral Signature | SCS Response |
|
||
|-------------------|-------------------|-------------|
|
||
| Uniform attention | High λ₁, narrow gap | SCS ≈ 0.8-1.0 (healthy) |
|
||
| Focused attention | Low λ₁, wide gap | SCS ≈ 0.5-0.7 (normal) |
|
||
| Fragmented attention | Very low λ₁ | SCS < 0.3 (alert) |
|
||
| Collapsed attention | Zero λ₁ (disconnected) | SCS = 0 (critical) |
|
||
|
||
### 9.3 Integration with cognitum-gate-kernel
|
||
|
||
The spectral coherence score feeds into the evidence accumulator:
|
||
|
||
```rust
|
||
// In cognitum-gate-kernel evidence accumulation
|
||
pub fn accumulate_spectral_evidence(
|
||
accumulator: &mut EvidenceAccumulator,
|
||
scs: f64,
|
||
threshold: f64,
|
||
) {
|
||
let e_value = if scs < threshold {
|
||
// Evidence against coherence hypothesis
|
||
(threshold - scs) / threshold
|
||
} else {
|
||
// Evidence for coherence
|
||
0.0 // No evidence against
|
||
};
|
||
|
||
accumulator.add_observation(e_value);
|
||
}
|
||
```
|
||
|
||
---
|
||
|
||
## 10. Open Questions
|
||
|
||
1. **Adaptive solver selection for spectral tasks**: Can the router module learn which solver is best for spectral estimation on different graph topologies?
|
||
|
||
2. **Streaming Fiedler vector**: Can we maintain an approximate Fiedler vector in O(n polylog n) space under edge insertions/deletions?
|
||
|
||
3. **Spectral coherence for dynamic attention**: How should the SCS weights (α, β, γ, δ) be tuned for different attention mechanism types?
|
||
|
||
4. **Cross-tile spectral aggregation**: Can 256 tiles in the cognitum-gate-kernel aggregate their local spectral properties into a global SCS without full Laplacian construction?
|
||
|
||
5. **Chebyshev order selection**: What is the optimal polynomial degree for spectral filtering in the RuVector HNSW context?
|
||
|
||
---
|
||
|
||
## 11. Recommendations
|
||
|
||
### Immediate (0-4 weeks)
|
||
|
||
1. Add `spectral` feature flag to `ruvector-coherence` Cargo.toml with dependency on `ruvector-solver`
|
||
2. Implement `estimate_fiedler()` using the existing `CgSolver`
|
||
3. Implement `SpectralCoherenceScore` struct with the four-component formula
|
||
4. Add property tests: SCS monotonically decreases as edges are removed from a connected graph
|
||
|
||
### Short-Term (4-8 weeks)
|
||
|
||
5. Implement `SpectralTracker` with incremental perturbation updates
|
||
6. Wire SCS into `ruvector-coherence`'s `evaluate_batch` pipeline
|
||
7. Add spectral health monitoring to HNSW graph in `ruvector-core`
|
||
8. Benchmark SCS computation in `ruvector-solver-wasm`
|
||
|
||
### Medium-Term (8-16 weeks)
|
||
|
||
9. Implement spectral sparsification for million-vertex graphs
|
||
10. Add Chebyshev spectral filtering for graph signal processing
|
||
11. Integrate SCS into `cognitum-gate-kernel` evidence accumulation
|
||
12. Expose spectral streaming via `ruvector-solver-wasm` Web Worker API
|
||
|
||
---
|
||
|
||
## References
|
||
|
||
1. Spielman, D.A., Teng, S.-H. "Nearly-Linear Time Algorithms for Graph Partitioning, Graph Sparsification, and Solving Linear Systems." STOC 2004.
|
||
2. Cohen, M.B., et al. "Solving SDD Linear Systems in Nearly m·log^{1/2}(n) Time." STOC 2014.
|
||
3. Kelner, J.A., et al. "A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time." STOC 2013.
|
||
4. Batson, J., Spielman, D.A., Srivastava, N. "Twice-Ramanujan Sparsifiers." STOC 2009.
|
||
5. Andersen, R., Chung, F., Lang, K. "Local Graph Partitioning using PageRank Vectors." FOCS 2006.
|
||
6. Chung, F. "Spectral Graph Theory." AMS, 1997.
|
||
7. Vishnoi, N.K. "Lx = b: Laplacian Solvers and Their Algorithmic Applications." Foundations and Trends in TCS, 2013.
|
||
|
||
---
|
||
|
||
## Document Navigation
|
||
|
||
- **Previous**: [01 - Pseudo-Deterministic Min-Cut](./01-pseudo-deterministic-mincut.md)
|
||
- **Next**: [03 - Storage-Based GNN Acceleration](./03-storage-gnn-acceleration.md)
|
||
- **Index**: [Executive Summary](./00-executive-summary.md)
|