dearsky/wifi-densepose
1
0
Fork 0
Code Issues 20 Pull Requests 5 Actions Packages Projects Releases 1 Wiki Activity
Files
d8fd5f4eba1c417157e1d7c9e51fa1fe4e6f577d
wifi-densepose/vendor/ruvector/docs/research/wasm-integration-2026/02-sublinear-spectral-solvers.md

25 KiB
Raw Blame History

Sublinear Spectral Solvers and Coherence Scoring

Document ID: wasm-integration-2026/02-sublinear-spectral-solvers Date: 2026-02-22 Status: Research Complete Classification: Algorithmic Research — Numerical Linear Algebra Series: Executive Summary | 01 | 02 | 03 | 04 | 05

Abstract

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.

1. Spectral Graph Theory Primer

1.1 The Graph Laplacian

For an undirected weighted graph G = (V, E, w) with n vertices, the graph Laplacian is:

L = D - A

where D = diag(d₁, ..., dₙ) is the degree matrix and A is the adjacency matrix. The normalized Laplacian is:

L_norm = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}

Key spectral properties:

  • L is positive semidefinite: all eigenvalues λ₀ ≤ λ₁ ≤ ... ≤ λₙ₋₁ ≥ 0
  • λ₀ = 0 always (corresponding eigenvector: all-ones)
  • Algebraic connectivity λ₁ = Fiedler value: measures how "connected" the graph is
  • Spectral gap λ₁/λₙ₋₁: measures expansion quality
  • Number of zero eigenvalues = number of connected components

1.2 Why Spectral Methods Matter for RuVector

RuVector operates on high-dimensional vector databases with HNSW graph indices. The spectral properties of these graphs directly correlate with:

Spectral Property RuVector Signal Meaning
λ₁ (Fiedler value) Index connectivity Low λ₁ → fragile index, vulnerable to node removal
λ₁/λₙ₋₁ (spectral gap) Search efficiency Wide gap → fast random walk convergence → fast search
Σ 1/λᵢ (effective resistance) Redundancy High total resistance → sparse, fragile structure
tr(L⁺) (Laplacian pseudoinverse trace) Average path length High trace → slow information propagation
λ_{n-1} (largest eigenvalue) Degree regularity Large → highly irregular degree distribution

1.3 The Sublinear Revolution

Classical Laplacian solvers (Gaussian elimination, dense eigendecomposition) require O(n³) time. The sublinear revolution has progressively reduced this:

Year Result Time Notes
2004 Spielman-Teng Õ(m) First near-linear Laplacian solver
2013 Cohen et al. O(m√(log n)) Practical near-linear solver
2014 Kelner et al. Õ(m) Random walk-based
2018 Schild Õ(m) Simplified construction
2022 Sublinear eigenvalue O(n polylog n) Top-k eigenvalues without full matrix
2024 Streaming spectral O(n log² n) space Single-pass Laplacian sketching
2025 Adaptive spectral O(log n) per query Amortized via precomputation

The key insight: for monitoring (not solving), we don't need the full solution — we need spectral summaries that can be maintained incrementally.

2. RuVector Solver Crate Analysis

2.1 Existing Solver Engines

The ruvector-solver crate provides 7 solver engines:

Solver Feature Flag Method Complexity Best For
NeumannSolver neumann Neumann series: x = Σ(I-A)ᵏb O(κ log(1/ε)) Diagonally dominant, κ < 10
CgSolver cg Conjugate gradient O(√κ log(1/ε)) SPD systems, moderate condition
ForwardPush forward-push Local push from source O(1/ε) per source Personalized PageRank, local
BackwardPush backward-push Reverse local push O(1/ε) per target Target-specific solutions
RandomWalkSolver hybrid-random-walk Monte Carlo + push O(log n) amortized Large sparse graphs
BmsspSolver bmssp Bounded multi-source shortest path O(m·s/n) s-source reachability
TrueSolver true-solver Direct factorization O(n³) worst case Small dense systems, ground truth

2.2 Solver Router

The ruvector-solver includes a router module that automatically selects the optimal solver based on matrix properties:

pub mod router;
// Routes to optimal solver based on:
// - Matrix size (n)
// - Sparsity pattern
// - Diagonal dominance ratio
// - Condition number estimate
// - Available features

2.3 WASM Variants

  • ruvector-solver-wasm: Full solver suite compiled to WASM via wasm-bindgen
  • ruvector-solver-node: Node.js bindings via NAPI-RS

Both variants expose the same solver API with WASM-compatible memory management.

2.4 Supporting Infrastructure

pub mod arena;       // Arena allocator for scratch space
pub mod audit;       // Computation audit trails
pub mod budget;      // Compute budget tracking
pub mod events;      // Solver event system
pub mod simd;        // SIMD-accelerated operations
pub mod traits;      // SolverEngine trait
pub mod types;       // CsrMatrix, ComputeBudget
pub mod validation;  // Input validation

3. Spectral Coherence Score Design

3.1 Definition

The Spectral Coherence Score (SCS) is a composite metric measuring the structural health of a graph index:

SCS(G) = α · normalized_fiedler(G)
       + β · spectral_gap_ratio(G)
       + γ · effective_resistance_score(G)
       + δ · degree_regularity_score(G)

where α + β + γ + δ = 1 and each component is normalized to [0, 1]:

normalized_fiedler(G) = λ₁ / d_avg
spectral_gap_ratio(G) = λ₁ / λ_{n-1}
effective_resistance_score(G) = 1 - (n·R_avg / (n-1))
degree_regularity_score(G) = 1 - σ(d) / μ(d)

3.2 Sublinear Computation via Existing Solvers

Each component can be estimated in O(log n) amortized time using the existing solver engines:

Fiedler Value Estimation

Use the inverse power method with the CG solver:

/// Estimate λ₁ (Fiedler value) via inverse iteration.
/// Each iteration solves L·x = b using CgSolver.
/// Convergence: O(log(n/ε)) iterations for ε-approximation.
pub fn estimate_fiedler(
    laplacian: &CsrMatrix<f64>,
    solver: &CgSolver,
    tolerance: f64,
) -> f64 {
    let n = laplacian.rows();
    let mut x = random_unit_vector(n);

    // Deflate: project out the all-ones eigenvector
    let ones = vec![1.0 / (n as f64).sqrt(); n];

    for _ in 0..50 {  // Max 50 iterations
        // Project out null space
        let proj = dot(&x, &ones);
        for i in 0..n { x[i] -= proj * ones[i]; }
        normalize(&mut x);

        // Solve L·y = x (inverse iteration)
        let result = solver.solve(laplacian, &x).unwrap();
        x = result.solution;

        // Rayleigh quotient = 1/λ₁ estimate
        let rayleigh = dot(&x, &matvec(laplacian, &x)) / dot(&x, &x);

        if (rayleigh - 1.0/result.residual_norm).abs() < tolerance {
            return rayleigh;
        }
    }

    // Return last Rayleigh quotient
    dot(&x, &matvec(laplacian, &x)) / dot(&x, &x)
}

Spectral Gap via Random Walk

Use the RandomWalkSolver to estimate mixing time, which relates to the spectral gap:

/// Estimate spectral gap via random walk mixing time.
/// Mixing time τ ≈ 1/λ₁ · ln(n), so λ₁ ≈ ln(n)/τ.
pub fn estimate_spectral_gap(
    graph: &CsrMatrix<f64>,
    walker: &RandomWalkSolver,
    n_walks: usize,
) -> f64 {
    let n = graph.rows();
    let mut mixing_times = Vec::with_capacity(n_walks);

    for _ in 0..n_walks {
        let start = random_vertex(n);
        let mixing_time = walker.estimate_mixing_time(graph, start);
        mixing_times.push(mixing_time);
    }

    let avg_mixing = mean(&mixing_times);
    let ln_n = (n as f64).ln();

    // λ₁ ≈ ln(n) / τ_mix
    ln_n / avg_mixing
}

Effective Resistance via Forward Push

Use ForwardPush to compute personalized PageRank vectors, which approximate effective resistances:

/// Estimate average effective resistance via local push.
/// R_eff(u,v) ≈ (p_u(u) - p_u(v)) / d_u where p_u is PPR from u.
pub fn estimate_avg_resistance(
    graph: &CsrMatrix<f64>,
    push: &ForwardPush,
    n_samples: usize,
) -> f64 {
    let n = graph.rows();
    let mut total_resistance = 0.0;

    for _ in 0..n_samples {
        let u = random_vertex(n);
        let v = random_vertex(n);
        if u == v { continue; }

        let ppr_u = push.personalized_pagerank(graph, u, 0.15);
        let r_uv = (ppr_u[u] - ppr_u[v]).abs() / degree(graph, u) as f64;
        total_resistance += r_uv;
    }

    total_resistance / n_samples as f64
}

3.3 Incremental Maintenance

The SCS can be maintained incrementally as the graph changes:

pub struct SpectralCoherenceTracker {
    /// Cached Fiedler value estimate
    fiedler_estimate: f64,
    /// Cached spectral gap estimate
    gap_estimate: f64,
    /// Cached effective resistance estimate
    resistance_estimate: f64,
    /// Cached degree regularity
    regularity: f64,
    /// Number of updates since last full recomputation
    updates_since_refresh: usize,
    /// Threshold for triggering full recomputation
    refresh_threshold: usize,
    /// Weights for score components
    weights: [f64; 4],
}

impl SpectralCoherenceTracker {
    /// O(1) amortized: update after edge insertion/deletion.
    /// Uses perturbation theory to adjust estimates.
    pub fn update_edge(&mut self, u: usize, v: usize, weight_delta: f64) {
        // First-order perturbation of Fiedler value:
        // Δλ₁ ≈ weight_delta · (φ₁[u] - φ₁[v])²
        // where φ₁ is the Fiedler vector
        self.updates_since_refresh += 1;

        if self.updates_since_refresh >= self.refresh_threshold {
            self.full_recompute();
        } else {
            self.perturbation_update(u, v, weight_delta);
        }
    }

    /// O(log n): full recomputation using solver engines.
    pub fn full_recompute(&mut self) { /* ... */ }

    /// O(1): perturbation-based update.
    fn perturbation_update(&mut self, u: usize, v: usize, delta: f64) { /* ... */ }

    /// Get the current Spectral Coherence Score.
    pub fn score(&self) -> f64 {
        self.weights[0] * self.fiedler_estimate
        + self.weights[1] * self.gap_estimate
        + self.weights[2] * self.resistance_estimate
        + self.weights[3] * self.regularity
    }
}

4. Integration with Existing Crates

4.1 ruvector-coherence Extension

The existing ruvector-coherence crate provides:

  • contradiction_rate: Measures contradictions in attention outputs
  • delta_behavior: Tracks behavioral drift
  • entailment_consistency: Measures logical consistency
  • compare_attention_masks: Compares attention patterns
  • cosine_similarity, l2_distance: Vector quality metrics
  • quality_check: Composite quality assessment
  • evaluate_batch: Batched evaluation

Proposed extension: Add a spectral module behind a feature flag:

// ruvector-coherence/src/spectral.rs
// Feature: "spectral" (depends on ruvector-solver)

/// Spectral Coherence Score for graph index health.
pub struct SpectralCoherenceScore {
    pub fiedler: f64,
    pub spectral_gap: f64,
    pub effective_resistance: f64,
    pub degree_regularity: f64,
    pub composite: f64,
}

/// Compute spectral coherence for a graph.
pub fn spectral_coherence(
    laplacian: &CsrMatrix<f64>,
    config: &SpectralConfig,
) -> SpectralCoherenceScore { /* ... */ }

/// Track spectral coherence incrementally.
pub struct SpectralTracker { /* ... */ }

4.2 ruvector-solver Integration Points

Coherence Component Solver Engine Feature Flag Iterations
Fiedler value CgSolver cg O(log n)
Spectral gap RandomWalkSolver hybrid-random-walk O(log n)
Effective resistance ForwardPush forward-push O(1/ε) per sample
Degree regularity Direct computation None O(n) one-pass
Full SCS refresh Router (auto-select) All O(log n) amortized

4.3 prime-radiant Connection

The prime-radiant crate implements attention mechanisms. Spectral coherence provides a health signal for these mechanisms:

Attention output → ruvector-coherence (behavioral metrics)
         ↓                    ↓
    Graph index → ruvector-solver (spectral metrics)
         ↓                    ↓
    Combined → SpectralCoherenceScore + QualityResult
         ↓
    Gate decision (cognitum-gate-kernel)

4.4 HNSW Index Health Monitoring

The HNSW graph in ruvector-core can be monitored for structural health:

/// Monitor HNSW graph health via spectral properties.
pub struct HnswHealthMonitor {
    tracker: SpectralTracker,
    alert_thresholds: AlertThresholds,
}

pub struct AlertThresholds {
    /// Minimum acceptable Fiedler value (below = fragile index)
    pub min_fiedler: f64,           // Default: 0.01
    /// Minimum acceptable spectral gap (below = poor expansion)
    pub min_spectral_gap: f64,      // Default: 0.1
    /// Maximum acceptable effective resistance
    pub max_resistance: f64,        // Default: 10.0
    /// Minimum composite SCS (below = trigger rebuild)
    pub min_composite_scs: f64,     // Default: 0.3
}

pub enum HealthAlert {
    FragileIndex { fiedler: f64 },
    PoorExpansion { gap: f64 },
    HighResistance { resistance: f64 },
    LowCoherence { scs: f64 },
    RebuildRecommended { reason: String },
}

5. WASM Deployment Strategy

5.1 ruvector-solver-wasm Capability

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.

5.2 Memory Considerations

For a graph with n vertices and m edges in WASM:

Component Memory At n=10K, m=100K
CSR matrix (Laplacian) 12m + 4(n+1) bytes 1.24 MB
Solver scratch space 8n bytes per vector, ~5 vectors 400 KB
Spectral tracker state ~200 bytes 200 B
Total 12m + 44n + 200 ~1.64 MB

WASM linear memory starts at 1 page (64KB) and grows on demand. For 10K-vertex graphs, ~26 WASM pages suffice.

5.3 Web Worker Integration

For browser deployment, spectral computation runs in a Web Worker to avoid blocking the main thread:

// spectral-worker.ts
import init, { SpectralTracker } from 'ruvector-solver-wasm';

await init();
const tracker = new SpectralTracker(config);

self.onmessage = (event) => {
    switch (event.data.type) {
        case 'update_edge':
            tracker.update_edge(event.data.u, event.data.v, event.data.weight);
            self.postMessage({ type: 'scs', value: tracker.score() });
            break;
        case 'full_recompute':
            tracker.recompute();
            self.postMessage({ type: 'scs', value: tracker.score() });
            break;
    }
};

5.4 Streaming Spectral Sketches

For WASM environments with limited memory, use spectral sketches that maintain O(n polylog n) space:

/// Streaming spectral sketch for memory-constrained WASM.
/// Maintains ε-approximate spectral properties in O(n log² n / ε²) space.
pub struct SpectralSketch {
    /// Johnson-Lindenstrauss projection of Fiedler vector
    fiedler_sketch: Vec<f64>,    // O(log n / ε²) entries
    /// Degree histogram for regularity
    degree_histogram: Vec<u32>,  // O(√n) bins
    /// Running statistics
    edge_count: usize,
    vertex_count: usize,
    weight_sum: f64,
}

6. Spectral Sparsification

6.1 Background

A spectral sparsifier H of G is a sparse graph (O(n log n / ε²) edges) such that:

(1-ε) · x^T L_G x ≤ x^T L_H x ≤ (1+ε) · x^T L_G x   ∀x ∈ R^n

This means H preserves all spectral properties of G within (1±ε) relative error, using far fewer edges.

6.2 Application to RuVector

For large HNSW graphs (millions of vertices), computing spectral properties of the full graph is expensive even with sublinear solvers. Instead:

  1. Build a spectral sparsifier H with O(n log n / ε²) edges
  2. Compute SCS on H (much faster, same accuracy up to ε)
  3. Maintain H incrementally as the HNSW graph changes
/// 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):

/// 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:

/// 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:

// 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)

  1. Implement SpectralTracker with incremental perturbation updates
  2. Wire SCS into ruvector-coherence's evaluate_batch pipeline
  3. Add spectral health monitoring to HNSW graph in ruvector-core
  4. Benchmark SCS computation in ruvector-solver-wasm

Medium-Term (8-16 weeks)

  1. Implement spectral sparsification for million-vertex graphs
  2. Add Chebyshev spectral filtering for graph signal processing
  3. Integrate SCS into cognitum-gate-kernel evidence accumulation
  4. 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

Reference in New Issue View Git Blame Copy Permalink