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Spectral Analysis for Sparse Matrix Computation: Insights and Potential

Ruifeng Zhang, Xipeng Shen

arXiv:2608.29362v1cs.PFcs.LG

TL;DR

Sparse computations are difficult to optimize because diverse nonzero patterns affect performance, while spatial statistics may miss global structure. The paper treats sparse matrices as 2-D signals, extracts FFT-based spectral features, and combines them with spatial features for kernel selection. Across SuiteSparse and pruned-LLM settings, the combined features improve prediction and selected-kernel runtime, including 1.035–1.245× speedups for pruned LLM decoding.

  • Problem

    Sparse computation performance depends on diverse nonzero patterns, but spatial characterizations may miss global structures relevant to selecting storage formats and kernels.

  • Method

    The paper treats sparse matrices as binary 2-D images, applies 2-D FFT analysis, and combines spectral descriptors with spatial features for SpMV kernel selection.

  • Results

    1.035–1.245× kernel speedups are achieved on pruned LLM decoding when spectral features are added to the kernel-selection process.

  • Takeaways & Limitations

    Spectral features provide complementary information for sparse-matrix characterization and practical SpMV performance optimization across SuiteSparse and pruned-LLM matrices.

  • Takeaways & Limitations

    Density-map normalization reduces resolution and can lose fine detail, although dominant directional and low-frequency signatures remain identifiable.

Abstract

from arXiv · show

Sparse computations are fundamental to scientific computing, graph analytics, and machine learning, yet their performance is highly sensitive to the diverse sparsity and patterns. This is because cache reuse, memory coalescing, and load balancing depend critically on the sparsity patterns. This work gives the first known exploration of the connections between sparse matrix computation and spectral analysis by treating sparse matrices as two-dimensional signals and analyzing their frequency-domain representations through Fast Fourier Transform. We show that spectral signatures uncover global structural characteristics that are not sufficiently captured by conventional spatial statistics and provide complementary information for understanding sparse computation performance. Experiments on incorporating spectral features into machine-learning-based SpMV format selection demonstrate the usefulness of such spectral analysis over a state-of-the-art spatial-only model. By uncovering the principled connections between spectral characteristics and sparse matrix computations, this work introduces a novel analytical perspective into sparse computation, and provides a new approach to enhancing the current sparse structure characterization and optimization. On pruned LLM decoding, adding spectral features improves kernel selection and yields 1.035--1.245$\times$ kernel speedups.

1 Introduction

Sparse computation performance depends strongly on irregular nonzero patterns, while spatial summaries can miss global structure. This work introduces FFT-based spectral characterization to complement spatial features and support sparse-kernel optimization.

  • Irregular nonzero distributions cause memory-access irregularity, poor coalescing, and workload imbalance in sparse computation.These effects make sparse-pattern characterization important for performance on parallel architectures.
  • Storage-format and kernel selection strongly influence sparse-computation efficiency, especially on GPUs.The appropriate choices are critically determined by matrix sparsity patterns.
  • Spatial features such as dimensions, sparsity, and row/column distributions can miss global bands, blocks, and mixed structures.Matrices with similar row-length distributions may nevertheless differ in locality, memory coalescing, and load balance.
  • The work treats sparse matrices as binary 2-D images and applies FFT-based spectral analysis to reveal periodicity, orientation, and scale.The approach is motivated by frequency-domain representations being potentially more expressive than pure spatial statistics.
  • The study investigates how spectral signatures reflect sparse patterns, relate to real-world matrices, and improve sparse-computation analysis.Its contributions include a first 2-D FFT-based view, density-map normalization, interpretable spectral signatures, and kernel-selection evaluation.

2 Background

The paper uses 2-D FFT representations to characterize sparse matrices in the frequency domain. Compact descriptors summarize spectral scale, orientation, and energy concentration for interpreting sparse structure.

  • The three FFT features are illustrated using shifted, log-scaled FFT magnitude spectra.The visualization shows radial energy, directional energy, and spectrum entropy as complementary descriptors.
  • 2-D FFT transforms a spatial matrix into complex frequency coefficients whose magnitude represents energy and phase represents spatial alignment.The resulting magnitude and log-magnitude spectra support frequency-domain visualization and analysis.
  • Radial energy groups frequencies into concentric rings to characterize structural scale.Low-frequency concentration indicates large-scale regularity, whereas high-frequency mass indicates finer or more irregular structure.
  • Directional energy partitions the spectrum into four angular sectors to capture orientation.Row-aligned blocks tend to produce horizontal or vertical energy, while diagonal bands produce 45°/135° ridges.
  • Spectrum entropy measures whether spectral energy is concentrated or diffuse.Regular patterns have low entropy, while noise-like patterns have high entropy.

3 RQ1: Connections between Sparse Patterns and Spectral Features

The study uses 2-D FFT analysis to connect sparse-matrix structures with interpretable spectral signatures. Controlled patterns show that radial and directional spectral features capture block, band, composite, and noise-affected structures, while density-map normalization improves robustness.

  • RQ1: Connections between Sparse Patterns and Spectral Features: Matrix synthesis provides controlled examples for relating sparse patterns and noise to spectral features.The approach enables direct examination of how structural changes affect frequency-domain signatures.
  • Canonical patterns: Centered blocks concentrate energy near zero frequency and produce a horizontal–vertical cross pattern.Changing block width narrows the main lobe in the corresponding frequency direction.
  • Canonical patterns: Block position changes phase but leaves the magnitude spectrum invariant within the matrix boundaries.Thus, magnitude features capture block structure without depending on its spatial location.
  • Canonical patterns: Wider diagonal bands combine an anti-diagonal ridge with an increasingly prominent low-frequency cross pattern.The resulting spectrum blends directional band-like and block-like signatures.
  • Composite patterns: Composite patterns produce approximately combined FFT magnitude signatures, allowing radial and directional features to represent multiple structures.Block-diagonal, heterogeneous-block, and band-plus-block matrices preserve the signatures of their constituent structures.
  • Noise and intra-structure sparsity: 20% in-band sparsity or uniform noise weakens dominant spectral structures, diffuses energy, and elevates spectral entropy.These perturbations can mask the underlying signatures even when the dominant orientation remains visible.
  • Density-map normalization: Density-map normalization preserves dominant orientation and low-frequency concentration while suppressing fine-scale randomness.For a 1024 × 1024 matrix, tiling into 8 × 8 regions produces a 128 × 128 density map; compression reduces resolution and can weaken fine detail.

4 RQ2: Spectral analysis of real-world matrices

The study applies spectral descriptors to 1,314 SuiteSparse matrices and finds that unsupervised clusters capture meaningful differences in scale, orientation, and regularity. These complementary signatures reveal structural diversity relevant to sparse-computation optimization.

  • Dataset and clustering: Unsupervised clustering of SuiteSparse matrices produces structurally meaningful groups in each spectral feature space.The analysis uses 1,314 matrices and standard k-means with k=4.
  • Radial energy: Radial-energy clusters range from block-dominant low-frequency structures to irregular high-frequency patterns.Intermediate clusters contain smaller or less regular structures, including diagonal bands with scattered nonzeros.
  • Directional energy: Directional clusters distinguish anti-diagonal, row- or column-aligned, and wider-band block structures.The directional descriptors separate orientation and apparent structural width.
  • Directional energy: 46.27% of matrices belong to a community dominated by diagonal-oriented energy, indicating that diagonal patterns are common in SuiteSparse.The text associates such near-diagonal patterns with possible benefits from DIA storage.
  • Spectral entropy: Spectral entropy separates highly regular, mixed, and irregular matrices by the concentration or diffusion of spectral energy.Low entropy indicates concentrated orientations, moderate entropy combines structure with irregular regions, and high entropy reflects broadly distributed energy.
  • Combined spectral features: Radial energy, directional energy, and spectral entropy provide complementary axes for describing scale, layout, and overall regularity.A matrix can simultaneously exhibit large-scale structure and diagonal alignment, motivating joint use of the descriptors for kernel selection.

5 RQ3: Improving Sparse Computing through SpMV Kernel Selection

The study evaluates FFT-based spectral features for selecting among SpMV storage formats and GPU kernels across SuiteSparse and pruned-LLM matrices. It uses clustering and kernel-distribution analyses to examine how sparse patterns relate to candidate kernels.

  • Candidate kernels: SpMV format selection is equivalent to kernel selection because different sparse formats require different kernel functions.The study benchmarks 14 kernels spanning general-purpose, diagonal, hybrid, block, and tiled formats.
  • Candidate kernels: The evaluation uses 14 kernels from Ginkgo, TACO, and TileSpMV, covering CSR, COO, ELL, Hybrid, SELL-P, FBCSR, DIA, and tiled formats.These formats span general-purpose, near-diagonal, mixed-regularity, block-locality, and tile-heterogeneity designs.
  • Clustering and kernel distributions: Balanced K-means clusters matrices using combined spectral features, while Figure 6 relates clusters to low-frequency energy, spectral entropy, and 45° directional strength.The low-frequency ratio sums the first four radial-energy bins, and point size represents directional strength in the 45° sector.
  • Clustering and kernel distributions: Figure 7 shows the distribution of fastest kernels across 1,170 SuiteSparse matrices, retaining only kernel classes with at least five samples for training.Each matrix is timed with all 14 candidate kernels to identify its fastest kernel.
  • Datasets: The experiments use SuiteSparse matrices and 1,056 pruned LLM matrices from Qwen2.5-3B, OPT-2.7B, Llama-2-7B, and Mistral-7B.The LLM matrices have 80% or 90% sparsity and arise from attention-layer projections used in batch-one autoregressive decoding.

5.2 Feature sets and model

The model compares base, spatial, and FFT-derived feature sets for multiclass kernel selection. FFT features are extracted across multiple density-map scales, reduced to the 20 most important features, and evaluated with XGBoost.

  • Feature sets: Kernel selection is framed as multiclass classification, with four feature sets compared to isolate the contribution of FFT features.The comparison includes base statistics, WISE spatial features, and FFT-based spectral features.
  • Base features: The baseline uses matrix dimensions, nonzero count, and density, capturing size and workload volume but not nonzero-pattern structure.These features provide workload information without structural descriptors.
  • Spatial features: WISE spatial features characterize size, nonzero skew, and nonzero locality using row, column, tile, and block-distribution statistics.The feature set includes distribution summaries such as means, variances, Gini coefficients, p-ratios, and non-empty counts.
  • Spectral features: FFT features are extracted at full, 1024×1024, and 256×256 density-map scales from shifted, log-scaled magnitude spectra.The resized scales provide medium and coarse views of global structure, while capped compression bounds FFT cost for very large inputs.
  • Spectral features: The spectral feature families include radial, directional, entropy, frequency-band, quadrant, flatness, and bandwidth descriptors.At each scale, the extracted features describe how spectral energy is distributed across frequency, direction, and spatial regions.
  • Feature selection and model: XGBoost selects the top 20 FFT features after accounting for internal correlations and overlapping information among spectral descriptors.The model retains the most important FFT features for SpMV classification and reports the top 10 in Figure 9.
  • Feature selection and model: XGBoost uses multiclass softmax classification with 480 trees, depth 6, learning rate 0.02, regularization coefficients 1.0, and subsampling rate 0.8.SuiteSparse uses five-fold matrix-level cross-validation, while pruned-LLM experiments use leave-one-model-out evaluation.

5.3 Metrics & overhead

The evaluation measures kernel-selection accuracy and speedup, showing that combining spatial and spectral features improves predictions and runtime across SuiteSparse and pruned-LLM matrices.

  • Metrics: Selection accuracy measures how often the chosen kernel is fastest, with an optional 5% runtime tolerance.Speedup divides default-kernel runtime by selected-kernel runtime, averaging 50 SpMV repeats.
  • Overhead: Feature-extraction and prediction overhead is excluded because kernel selection targets repeated use of the same sparse matrix.The added spectral-feature overhead ranges from tens to hundreds of iterations, while selection overhead is typically hundreds or thousands of SpMV iterations.
  • SuiteSparse results: Adding spectral features to the spatial baseline improves tolerant accuracy by 3.6 percentage points.Table 1 reports both strict and 5% tolerance accuracy.
  • SuiteSparse results: 79.3% accuracy is achieved by Spatial+Spectral, exceeding the 75.7% Spatial model and the 72.3% Spectral model.Basic size and sparsity features alone achieve 61.5% accuracy.
  • Runtime results: 2.66× speedup over CSR is achieved by Spatial predictions, compared with 2.65× for Spectral; Spatial+Spectral has the highest speedup across fixed baselines.The right panel indicates that spectral predictions tend to avoid severe slowdowns even when the exact optimal kernel is missed.
  • LLM-matrix results: 85.21% accuracy and 1.087x speedup are achieved by Spatial+Spectral across four sparsity and pruning methods.The model also improves end-to-end decode speed by 4.37% for Qwen and 7.47% for OPT under the stated WANDA setting.

5.5 Spectral Feature Importance Analysis

The feature-importance analysis identifies energy, entropy, radial, bandwidth, directional, quadrant, and flatness descriptors as influential FFT signals for SpMV models.

  • Feature importance: Energy_Avg on the full-scale spectrum is the most important FFT feature for SpMV prediction.Energy indicators from the 1024 × 1024 density map follow it in importance.
  • Feature interpretation: Spectral entropy measures energy-distribution uniformity, while radial and bandwidth features emphasize low-frequency energy and large-scale spatial coherence.Directional, quadrant, and flatness features capture orientation-related information.
  • Case studies: Four case studies show Spectral models selecting the fastest kernel where Spatial models miss.The comparison uses the same sparse-matrix/FFT format across the examples.

5.6 Case Study: Effectiveness of Spectral Signals

Case studies show that FFT spectra expose diagonal, block, and irregular structures that guide kernel choices missed by spatial features alone.

  • Sparse matrices: For fxm3_6, TileSpMV is optimal while the spatial model predicts CSR; the spectrum reveals a wide diagonal band and irregular residuals.TileSpMV accommodates both the dominant diagonal structure and irregular components.
  • Sparse matrices: For laser, a strong anti-diagonal spectral ridge indicates diagonal structure aligned with the DIA format.The spatial model lacks explicit features for diagonal alignment.
  • Pruned LLM matrices: For Mistral-7B, Spatial+Spectral recovers CSR and reduces runtime from 17.803 to 7.196 μs, a 2.474× reduction.Spatial instead predicts TileSpMV, while the diffused spectrum indicates irregular sparsity.
  • Pruned LLM matrices: For OPT-2.7B, Spatial+Spectral recovers TileSpMV and reduces runtime from 8.202 to 1.616 μs, a 5.076× reduction.Low-radial concentration indicates large blocks, while diffused spectral entropy indicates varied patterns.

6 Related Work

Prior work applies spectral analysis broadly and in graph computing, but this paper introduces direct 2-D FFT analysis of sparse-matrix nonzero patterns.

  • Scope: Earlier sections cover sparse formats and kernel selection rather than repeating that background here.
  • Novelty: Spectral analysis has not previously been introduced into sparse matrix computing, to the authors’ knowledge.
  • Graph spectral analysis: Graph Fourier Transform methods represent graph signals in eigenbases derived from graph Laplacian or adjacency operators.Graph convolutional methods build on this spectral formulation.
  • Distinction: This approach applies a 2-D FFT directly to the sparse matrix’s nonzero pattern as an image on a regular grid, unlike graph spectral methods.The Fourier basis is fixed and two-dimensional rather than defined by a graph operator.

7 Conclusion

The framework bridges FFT-based spectral signals with sparse computation optimization, complementing traditional spatial metrics for SpMV kernel selection. On pruned LLM decoding, spectral features yield 1.035–1.245× kernel speedups.

  • 1.035–1.245× kernel speedups result from adding FFT-derived spectral features to pruned-LLM SpMV kernel selection.
  • FFT-derived features provide a compact representation that complements traditional spatial metrics across SuiteSparse and pruned-LLM SpMV kernel selection.
  • The framework links performance-relevant spatial structures with corresponding spectral signatures for sparse computation optimization.
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