DFouT = coverage
DFouT already places poles over a broad Fourier grid. The remaining problem is selecting which modes matter for each sample.
Active research project
Cubical PH-Guided Sobolev C-Modulation for DFouT
Topo_S4 now frames persistent homology as a frequency selector rather than an extra-capacity module. DFouT supplies broad frequency coverage; 2D log-mel cubical PH localizes persistent time-frequency topology; and a zero-parameter Sobolev C-gate selectively modulates the existing DFouT residues.
S4 / S4D-DFouT
2D Cubical PH
GUDHI Cofaces
Hz/Nyquist Mapping
P96 support=3.0
20% SC35 Results
Zero PH Params
Overview
Earlier Topo_S4 branches focused on 1D FFT PH, group-delay reliability, and learnable PH adapters. The current direction uses 2D log-mel cubical PH because speech evidence is naturally time-frequency structured. Betti curves showed that cubical PH contains class signal, but Betti summaries discard location. GUDHI persistence pair cofaces recover where birth/death events occur, allowing PH to select frequency bands for DFouT pole modulation.
One-line thesis:
DFouT supplies the spectral basis; cubical PH localizes persistent
time-frequency topology; coface saliency converts that topology into
PH-salient frequency pseudo-peaks; and a zero-parameter Sobolev C-gate
selectively modulates DFouT residues.
Cubical PH = topology
Log-mel cubical PH detects persistent 2D time-frequency components and holes rather than raw spectral peaks alone.
Cofaces = location
Birth/death cofaces identify the time-frequency locations of persistent events, which can be projected to frequency.
Method
1. Log-mel cubical filtration
The log-mel spectrogram is normalized and treated as a 2D cubical filtration.
X̃_i(f,t) = (X_i(f,t) − min X_i) / (max X_i − min X_i + ε)
The filtration defines the PH problem. Betti curves are only global summaries of the resulting PH.
2. Persistence pair coface saliency
GUDHI birth/death cofaces locate persistent events on the time-frequency grid.
S_PH(f,t) = Σ_k p_k [K((f,t)−u_k^b) + K((f,t)−u_k^d)]
Time marginalization gives s_PH(f), a PH-salient frequency curve.
3. PH-salient pseudo-peaks
The frequency saliency curve is converted into pseudo-peaks expected by the direct-PH C path.
(pf, pp, sl, sr), P = 96, support_bins = 3.0
P96 means the top 96 frequency pseudo-peaks, not 96 raw GUDHI pairs.
4. Hz/Nyquist pole mapping
Mel-bin coordinates are mapped to physical frequency before DFouT pole alignment.
mel bin → mel center Hz → Hz / Nyquist
This replaces normalized mel-bin coordinates and improves pole-frequency interpretation.
5. PH-to-pole overlap
Each PH support interval softly overlaps DFouT poles through a Lorentzian window.
e_i,n = Σ_k pp_i,k · M_i,k,n
The gate modulates existing poles rather than creating new basis functions.
6. Sobolev C-residue gate
A contrastive high-band score and mild Sobolev factor produce a multiplicative C gate.
C_n^(i) = g_i,n C_n, q_n=(1+ω_n)^β, β=0.5
The PH module has zero learnable parameters.
20% Effective-Data SC35 Results
These are current 20% effective-data results under the official DFouT learning rate lr=0.001. They are not the final full-dataset SC35 claim. Full 100% dataset results will be added separately.
PH-guided beta=0.5, 20% data.
PH-guided beta=0.5, 20% data.
Over 20% DFouT-init baseline.
Over 20% DFouT-init baseline.
| Method | Setting | Best epoch | Train acc | Dev acc | Test acc | Time |
|---|---|---|---|---|---|---|
| DFouT-init baseline | lr=0.001 | 30 | 98.71% | 90.70% | 88.78% | 53.6m |
| PH-guided C/Sobolev | P96, support=3.0, beta=0.5, lr=0.001 | 39 | 99.36% | 91.03% | 89.21% | 107.4m |
| PH-guided C/Sobolev | P96, support=3.0, beta=1.0, lr=0.001 | 34 | 99.24% | 90.78% | 89.10% | 93.6m |
Current interpretation: beta=0.5 gives the best dev and test result. beta=1.0 remains above baseline on test accuracy, but the smaller dev gain suggests that the Sobolev factor should stay mild.
Uniform vs. PH-Guided Sobolev
The claim is not that PH-guided Sobolev maximizes Fisher separation. Instead, PH-guided Sobolev improves downstream accuracy while suppressing much of the within-class variance and noise amplification caused by uniform Sobolev weighting.
Variance and Fisher
| Representation | Within | Between | Fisher |
|---|---|---|---|
| Raw | 15.519 | 0.728 | 0.04688 |
| Uniform Sobolev | 20.017 (+28.98%) | 0.977 (+34.30%) | 0.04881 (+4.12%) |
| PH-guided Sobolev | 17.554 (+13.11%) | 0.819 (+12.52%) | 0.04663 (-0.52%) |
Uniform Sobolev improves Fisher slightly, but by aggressively increasing both between-class separation and within-class variance.
Noise sensitivity
| Representation | MSE | Mean abs |
|---|---|---|
| Raw | 172,076 | 2.468 |
| Uniform Sobolev | 235,691 (+36.97%) | 2.870 (+16.28%) |
| PH-guided Sobolev | 193,368 (+12.37%) | 2.619 (+6.11%) |
PH-guided Sobolev is less aggressive and keeps noise amplification much lower than uniform Sobolev.
P and Support Calibration
P sweep
| P | Mass | HF mass | Cosine | Eff. poles |
|---|---|---|---|---|
| 32 | 0.3436 | 0.3074 | 0.6351 | 12.98 |
| 64 | 0.6127 | 0.5887 | 0.8337 | 20.14 |
| 96 | 0.8373 | 0.8282 | 0.9627 | 25.50 |
| 128 | 1.0000 | 1.0000 | 1.0000 | 28.95 |
P96 keeps most PH/HF mass without becoming the fully dense P128 setting.
Support sweep with P96
| support_bins | Coverage | Interpretation |
|---|---|---|
| 2.0 | 0.645 | too narrow |
| 3.0 | 0.843 | main setting |
| 4.0 | 0.951 | getting broad |
| 5.0+ | ≈1.000 | nearly dense boost |
support=3.0 gives enough pole coverage while preserving PH-specific selectivity.
Current Configuration Target
The exact script name may differ by branch, but the scientific configuration should match this target.
--paper dfout --lr 0.001 --stage C --c_mod_type direct_ph --direct_ph_mode contrastive --direct_ph_mean_mode residual --direct_ph_peak_sobolev_beta 0.5 --ph_source cubical_gudhi --cubical_freq_map hz_nyquist --ph_max_peaks 96 --cubical_pseudo_support_bins 3.0
For 20% effective-data runs, scheduler_total_steps should be corrected from 200000 to approximately 40000.
Roadmap
Near-term
Add full-dataset SC35 results, then run the core ablations: baseline, real PH, global shuffle, within-class shuffle, no Sobolev, residual center, and no-center.
Validation
Repeat the best 20% and full-data settings across multiple seeds and use class-conditional shuffles to separate instance-level PH from class-level topology.