Spectral Embedding Scalability Benchmarking

spectral-embedding-scalability-benchmarking

Summary

Benchmark the matrix-free spectral embedding algorithm (tl.spectral) in SnapATAC2 to validate its claimed linear time and space complexity when applied to single-cell datasets of 10 million or more cells. This skill confirms scalability performance through empirical measurement of runtime and peak memory consumption against the theoretical complexity claim.

When to use

Apply this skill when you have a large single-cell count matrix (≥10 million cells) in CSR format and need to verify whether the matrix-free spectral embedding in SnapATAC2 achieves its documented linear scaling behavior on your hardware and dataset characteristics. Use it as a validation step before committing large-scale analyses or as a performance regression test when upgrading SnapATAC2 versions.

When NOT to use

• Input matrix is already embedded or has been pre-reduced to <10 million cells; use this skill only when validating large-scale scalability is the explicit goal.
• You lack memory profiling tools or cannot isolate the spectral embedding step from the surrounding pipeline; meaningful benchmarking requires isolated, repeatable measurement.
• The research question does not concern algorithmic scalability or performance validation; if you only need the embedding result, apply tl.spectral directly without instrumentation.

Inputs

• Single-cell count matrix with ≥10 million cells in CSR (compressed sparse row) format
• SnapATAC2 AnnData object or compatible matrix container

Outputs

• Wall-clock runtime measurements (seconds) as a function of cell count
• Peak memory usage profile (megabytes or gigabytes) throughout spectral decomposition
• Spectral embedding matrix with weighted eigenvectors
• Performance plot comparing observed time/space complexity to linear scaling expectation

How to apply

Load or generate a single-cell count matrix with ≥10 million cells and initialize it in CSR format compatible with SnapATAC2. Execute tl.spectral using the default cosine similarity metric (as of Release 2.3.0) while capturing wall-clock runtime using Python's time module or system profiler (e.g., cProfile). Monitor peak memory usage throughout spectral decomposition using a memory profiler such as memory_profiler or psutil. Document the number of eigenvectors returned and verify they are weighted by eigenvalues (default behavior in Release 2.3.0) to confirm correct output structure. Finally, plot the measured execution time and peak memory against dataset size and compare the observed slopes to the linear complexity claim; deviation from linearity may indicate algorithmic, hardware, or data-specific constraints.

Related tools

• SnapATAC2 (Provides the matrix-free spectral embedding algorithm (tl.spectral) and CSR matrix infrastructure for large-scale single-cell analysis) — https://github.com/scverse/SnapATAC2
• Python (Runtime environment for executing SnapATAC2 code and instrumenting benchmarks with time module and cProfile)
• Rust (Underlying compiled implementation of SnapATAC2's matrix operations and spectral decomposition for performance) — https://github.com/scverse/SnapATAC2
• memory_profiler or psutil (Tools for monitoring and recording peak memory usage throughout spectral decomposition execution)

Examples

import snapatac2 as snap
import time
import psutil
data = snap.pp.import_data('large_dataset.h5ad')  # ≥10M cells
start = time.time()
snap.tl.spectral(data, n_comps=50)
runtime = time.time() - start
peak_mem = psutil.Process().memory_info().rss / (1024**3)
print(f'Runtime: {runtime}s, Peak Memory: {peak_mem}GB')

Evaluation signals

• Wall-clock runtime exhibits sub-linear or linear growth (slope close to 1 on log-log plot) as cell count increases from 10M to peak tested size
• Peak memory usage scales linearly or sub-linearly with cell count; no exponential memory blowup indicates CSR sparsity is preserved
• Number of eigenvectors returned matches documentation and eigenvalue weighting is applied by default, confirming correct output structure
• Reproducibility: repeated benchmarks on the same dataset show <5–10% variance in runtime and memory, indicating stable performance
• Comparison to dense methods (if available) shows orders-of-magnitude improvement in runtime/memory for sparse 10M+ cell matrices

Limitations

• Scalability is hardware-dependent (RAM, CPU cores, cache topology); results from a shared HPC cluster may not generalize to a laptop or cloud node.
• The cosine similarity metric is the default as of Release 2.3.0; benchmarks with other metrics or custom similarity kernels may show different scaling behavior.
• Wall-clock time is sensitive to system load and background processes; proper statistical controls (multiple runs, isolated environment) are needed for publication-quality results.
• CSR matrix format and sparsity pattern significantly affect memory footprint; very dense or highly irregular data may not achieve claimed linear scaling.

Evidence

• [other] Does the matrix-free spectral embedding algorithm (tl.spectral) in SnapATAC2 achieve linear time and space complexity when applied to datasets of 10 million or more cells?: "Does the matrix-free spectral embedding algorithm (tl.spectral) in SnapATAC2 achieve linear time and space complexity when applied to datasets of 10 million or more cells?"
• [other] SnapATAC2 is capable of scaling to more than 10 million cells, demonstrating its capacity to handle very large single-cell datasets.: "SnapATAC2 is capable of scaling to more than 10 million cells, demonstrating its capacity to handle very large single-cell datasets."
• [other] Initialize the matrix in compressed sparse row (CSR) format compatible with SnapATAC2's backend. Execute tl.spectral with cosine similarity metric (the default similarity metric as of Release 2.3.0) and capture wall-clock runtime using Python's time module or system profiler.: "Initialize the matrix in compressed sparse row (CSR) format compatible with SnapATAC2's backend. Execute tl.spectral with cosine similarity metric (the default similarity metric as of Release 2.3.0)"
• [other] Monitor and record peak memory usage throughout the spectral decomposition using a memory profiler (e.g., memory_profiler or psutil).: "Monitor and record peak memory usage throughout the spectral decomposition using a memory profiler (e.g., memory_profiler or psutil)."
• [other] Document the number of eigenvectors returned and verify they are weighted by eigenvalues (default behavior in Release 2.3.0) to confirm the expected output structure.: "Document the number of eigenvectors returned and verify they are weighted by eigenvalues (default behavior in Release 2.3.0) to confirm the expected output structure."
• [intro] Scale to more than 10 million cells.: "Scale to more than 10 million cells."
• [readme] SnapATAC2: A Python/Rust package for single-cell epigenomics analysis: "SnapATAC2: A Python/Rust package for single-cell epigenomics analysis"