Module spectral 

Spectral Methods for Graph Analysis

Chebyshev polynomials and spectral graph theory for efficient diffusion and filtering without eigendecomposition.

Key Capabilities

• Chebyshev Graph Filtering: O(Km) filtering where K is polynomial degree
• Graph Diffusion: Heat kernel approximation via Chebyshev expansion
• Spectral Clustering: Efficient k-way partitioning
• Wavelet Transforms: Multi-scale graph analysis

Integration with Mincut

Spectral methods pair naturally with mincut:

• Mincut identifies partition boundaries
• Chebyshev smooths attention within partitions
• Spectral clustering provides initial segmentation hints

Mathematical Background

Chebyshev polynomials T_k(x) satisfy:

• T_0(x) = 1
• T_1(x) = x
• T_{k+1}(x) = 2x·T_k(x) - T_{k-1}(x)

This recurrence enables O(K) evaluation of degree-K polynomial filters.

Structs

ChebyshevExpansion

Chebyshev expansion of a function f(x) ≈ Σ c_k T_k(x)

ChebyshevPolynomial

Chebyshev polynomial of the first kind

ClusteringConfig

Spectral clustering configuration

GraphFilter

Graph filter that applies spectral operations

GraphWavelet

Graph wavelet at specific vertex

ScaledLaplacian

Scaled Laplacian for Chebyshev approximation L_scaled = 2L/λ_max - I (eigenvalues in [-1, 1])

SpectralClustering

Spectral clustering

SpectralFilter

Spectral graph filter using Chebyshev approximation

SpectralWaveletTransform

Spectral Wavelet Transform

WaveletScale

Wavelet scale configuration

Enums

FilterType

Type of spectral filter

LaplacianNorm

Normalized Laplacian (symmetric or random walk)