diffcb 0.1.1

Differentiable Critical Bandwidth: Silverman's modality test as a differentiable PyTorch layer with IFT backward pass.

DCB — Differentiable Critical Bandwidth

A PyTorch package that makes Silverman's critical bandwidth test (1981) fully differentiable, enabling end-to-end gradient-based optimization over the modal structure of continuous distributions.

Overview

The critical bandwidth h_crit is the minimum KDE bandwidth at which a distribution appears to have at most m modes — a classical nonparametric statistic for modality testing. DCB replaces every non-differentiable operation in its computation with a smooth surrogate, then uses the Implicit Function Theorem to compute exact gradients through the root-finding step at O(1) memory cost.

import torch
from dcb import DCBLayer

X = torch.randn(256, requires_grad=True)   # 1D samples
layer = DCBLayer(target_modes=1)
h_crit = layer(X)                          # differentiable scalar
h_crit.backward()                          # exact IFT gradients

Installation

pip install diffcb

Or from source:

git clone https://github.com/ryZhangHason/differentiable-critical-bandwidth
cd differentiable-critical-bandwidth
pip install -e ".[dev]"

Paper

Ruiyu Zhang. "Differentiable Critical Bandwidth: Making Silverman's Modality Test End-to-End Trainable." Journal of Machine Learning Research, 2026 (in preparation).

Confirmed Experimental Results

All results produced on Kaggle GPU (T4 / P100) — see experiments/ and outputs/.

Experiment	Result	Criterion
Validation (m≥2)	R²=0.91, MAE=0.07, Spearman ρ=0.89	R²≥0.85, MAE≤0.10 ✓
Speedup vs scipy (n=8192)	10.5× on T4	≥3× ✓
GAN mode preservation	h_crit=1.232 >> 0.3	h_crit>0.3 ✓
Anomaly AUC (KDDCup99)	DCB=0.9982 vs IF=0.9867	DCB≥IF ✓

Repository Structure

dcb/            Core PyTorch package (layer.py, solver.py, kde.py, utils.py)
experiments/    Reproduction scripts for all paper figures and tables
  phase1_validation.py   Figure 1: DCB vs reference h_crit scatter
  phase1_speedup.py      Figure 2: GPU speedup benchmark
  phase1_ablation.py     Figures S1–S2: ε/τ sensitivity heatmaps
  phase2_gan.py          Figure 3: GAN mode-collapse prevention
  phase3_anomaly.py      Table 2 + Figure 5: anomaly detection benchmark
tests/          Unit tests (pytest, 35/35 passing)
outputs/        All generated figures and tables (PDFs, PNGs, CSVs)
notebooks/      Quickstart and demo notebooks

Reproducing Paper Results

# Phase 1: validation, speedup, ablation
python experiments/phase1_validation.py
python experiments/phase1_speedup.py
python experiments/phase1_ablation.py

# Phase 2: GAN mode collapse experiment
python experiments/phase2_gan.py

# Phase 3: anomaly detection benchmark
python experiments/phase3_anomaly.py

For GPU runs, use the provided Kaggle kernels:

• Phase 1–2: hsingle/dcb-full-experiments
• Phase 3: hsingle/dcb-phase-3-anomaly-detection

Kaggle GPU Notes

Kaggle may assign a P100 (sm_60) instead of T4. The Phase 3 kernel handles this automatically by installing torch==2.2.2+cu118 (the earliest PyTorch release with both Python 3.12 and sm_60 support) when P100 is detected.

License

MIT — see LICENSE.