GREEN

1 GREEN

GREEN (Gabor Riemann EEGNet) is a lightweight, interpretable-by-design neural network that combines learnable complex Gabor wavelet convolutions, SPD covariance pooling, trainable shrinkage, BiMap, and a reference-centered log-Euclidean map for end-to-end EEG biomarker prediction.

Ref: @paillardGREENLightweight2024

Overview

Problem: deep EEG models lack interpretability and interoperability with established neuroscience concepts (band-pass filtering, phase coupling) and scale unsustainably; non-deep Riemannian pipelines are interpretable but rely on handcrafted filterbanks and sequential optimization. GREEN integrates both: learnable Gabor wavelets (interpretable as band-pass filters), Riemannian geometry, and deep learning in one lightweight architecture for subject-level prediction from resting-state EEG.

• Tasks/datasets (five tasks, three datasets, >5,000 participants):
  • TUAB (TUH (Temple University Hospital) abnormal subset): 2,993 patients, 24–36 electrodes → 21 common 10-20 channels; pathology classification (all subjects) and age regression (1,278 healthy subjects); recordings cropped to 15 min.
  • TDBRAIN: 1,274 recordings, 26 electrodes (10-10); eyes-open vs eyes-closed classification (ICA-based ocular-artifact removal added: fastPICARD + ICLabel) and sex classification (620 female / 654 male).
  • CAU (Chung-Ang University Hospital): 1,155 patients, 19 electrodes (10-20); three-way diagnosis classification (normal / mild cognitive impairment / dementia); includes photic-stimulation periods.
  • Common preprocessing: average referencing, band-pass 1–100 Hz, autoreject artifact removal; TUAB/TDBRAIN resampled to 125 Hz, CAU to 140 Hz. Each sample = E = 10 randomly drawn 10-second windows.
• Contributions: modular, neuroscience-informed architecture; state-of-the-art-competitive performance with orders of magnitude fewer parameters than large DL models (breaking the log-linear accuracy–parameter scaling of Kim et al. 2023 models on CAU); learnable sparse filterbanks (2–3 wavelets suffice for some tasks vs 49 in the baseline); interpretability demonstrations (Berger effect, EEG-power topographies, learned wavelet frequencies); modular extension with pairwise PLV pooling.
• Baseline: the non-deep pipeline of Bomatter et al. (2024): pre-defined Morlet filterbank (49 wavelets, central frequencies log-spaced 1–64 Hz) → sample covariance → PCA projection to SPD manifold (Sabbagh et al. 2019) → tangent-space projection → RidgeCV linear model.

Architecture

Pipeline: raw EEG windows (E×C×T) → parametrized Gabor convolution (E×F×C×T) → pooling (P×F·C×F·C) → Shrinkage → BiMap → Log∘ReEig → tangent vectors → FC head.

1. Parametrized convolution (WaveletConv). Complex Gabor wavelet kernel with learnable carrier frequency $f$ and Gaussian-window standard deviation ${\sigma }_t$:

$${\varphi }_f(t)=\frac{1}{\sqrt{2\pi }{\sigma }_t}\exp \left(-\frac{t^2}{2{\sigma }_t^2}\right)\exp (2\pi ift).$$

Both parameters are learned by gradient descent; wavelets are interpretable as band-pass filters, give shift-invariance after the modulus, and (unlike Morlet) share equal lengths, enabling cross-frequency block structure.
2. Pooling layers. For wavelet-transformed signals $X_i,X_j\in {\mathbb{C}}^{C\times T}$, the covariance block is

$$C_{i,j}=\frac{1}{T}\mathrm{Re}\left(X_i{X}_j^{\top }\right),$$

giving diagonal blocks = within-frequency channel covariances and off-diagonal blocks = cross-frequency interactions. Optionally, pairwise PLV matrices from the complex signals are concatenated.
3. Shrinkage layer. Trainable Ledoit–Wolf-style shrinkage with a trace-preserving property:

$$X_{\mathrm{shrunk}}=(1-\alpha )X+\alpha \frac{\mathrm{Tr}(X)}{p}X,$$

with trainable $\alpha$ — the Shrinkage primitive; preserves total power and down-weights off-diagonals.
4. BiMap layer. $BiMap(X)=WX{W}^{\top }$ with $W\in {\mathbb{R}}^{Q\times C}$; motivated by the formal result of Sabbagh et al. (2019) that projecting rank-deficient covariance matrices to a common subspace enables valid SPD regression — the BiMap primitive acting as spatial filtering. Semi-orthogonality via manifold optimization was tested but dropped in the main experiments (unconstrained $W$ still yields strictly positive feature maps, with cheaper computation).
5. Combined ReEig + LogMap layer. Eigenvalues are rectified and logged in a single diagonalization step (ReEig + LogEig). Key modification vs standard LogEig: the reference point is a running log-Euclidean mean instead of the identity:

$${\varphi }_{\mathrm{LE}}^{X_{\mathrm{ref}}}(X)=\mathrm{vect}(\mathrm{Log}(X)-\mathrm{Log}(X_{\mathrm{ref}})),X_{\mathrm{ref}}\leftarrow \mathrm{Exp}((1-\theta )\mathrm{Log}(X_B)+\theta \mathrm{Log}(X_{\mathrm{ref}})),$$

where $X_B$ is the batch log-Euclidean mean and $\theta$ the momentum (value not reported). This denoises the spatially structured volume-conduction/background components; Riemannian batch normalization was tried and not retained (no observed benefit).
6. FC head. Two fully connected layers (64 and 32 hidden units) with batch normalization, dropout (p = 0.333), and GELU non-linearity; MSE loss for regression, cross-entropy for classification.

Model variants: baseline (predefined bank, no BiMap, no Adam); G1 (BiMap with $Q=C-1$, LogMap, linear head, Adam); G1P (+ PLV pooling); G2 (+ hidden layer, 32 units); G3 (learnable filterbank of $F=10$ wavelets, two BiMaps with $Q=64$ and $Q=32$, hidden layer); G3P (G3 + PLV).

Model Parameters

• Wavelets: $F=10$ learned Gabor wavelets (G3/G3P); baseline: 49 Morlet wavelets (1–64 Hz, log-spaced, fixed).
• Pooling features: $P$ = 1 (covariance) or 2 (covariance + PLV).
• BiMap outputs: $Q=C-1$ (G1/G1P/G2) or $64\to 32$ (G3/G3P).
• E = 10 windows of 10 s; FC head 64→32, dropout p = 0.333, GELU, batch norm.
• Shrinkage parameter $\alpha$: trainable; reference-mean momentum $\theta$: not reported.
• Full parameter counts per variant: not tabulated; the lean design (2 params per wavelet) places the models orders of magnitude below large DL models on the CAU complexity–accuracy plot.

Training Parameters

• Optimizer: Adam (per the variant table).
• Loss: mean-square error (age regression) / cross-entropy (all classification tasks).
• Learning rate, batch size, number of epochs, weight decay, early stopping: not reported.
• Protocol: Monte-Carlo cross-validation with 100 random splits, test size 20%, Nadeau–Bengio corrected resampled t-test; the CAU dementia benchmark uses the exact single train/test split of Kim et al. (2023) for comparability.

Results

Median scores (balanced accuracy; R² for age):

Task	Dataset	Model	Median score	t99	p-value
Age regression	TUAB	G3	0.7211 (R²)	2.9906	0.0018
Age regression	TUAB	G2	0.6879 (R²)	0.8650	0.1946
Pathology classification	TUAB	G3	0.8379	1.7064	0.0455
Pathology classification	TUAB	G2	0.8344	2.4251	0.0086
EO vs EC	TDBRAIN	G3	0.8623	2.2305	0.0140
EO vs EC	TDBRAIN	G2	0.8223	0.2184	0.4138
Sex prediction	TDBRAIN	G3	0.8325	−0.0491	0.4804
Sex prediction	TDBRAIN	G2	0.8141	1.9805	0.0252
Dementia diagnosis	CAU	G3	0.6045	3.2383	0.0008
Dementia diagnosis	CAU	G2	0.5893	1.5949	0.0570

• Age regression: relative R² improvements over the baseline — G1 ≈ 0.68 (p = 0.340), G2 ≈ 0.69 (p = 0.195), G3 ≈ 0.72 (p = 0.002). The hidden layer alone is not significant; the learnable filterbank gives the largest gain.
• Task dependence: G2 significantly improves sex prediction and pathology decoding; G3 significantly improves EO/EC, pathology, and dementia; a predefined bank can match (pathology) or beat (sex) the learned bank.
• CAU dementia benchmark: G3/G3P outperform the best single model of Kim et al. (2023) and match the ensemble (“bag”) performance with orders of magnitude fewer parameters. G1P > G1 significantly; G3P > G3 slightly — PLV carries complementary information but is not indispensable once wavelets are learned.
• Filterbank size: minimal banks of only 2–3 wavelets reach asymptotic performance for pathology and EO/EC; age and sex need more wavelets. Learned wavelet frequencies: low frequencies 1–2 Hz and the alpha band (8–12 Hz) chosen across tasks.
• Interpretability (Berger effect): on EO/EC, G3 places a wavelet at ~9 Hz (minimal frequency-domain standard deviation) and uses low/high-frequency wavelets (0.5 Hz, ~30 Hz) as reference points; power topographies from the covariance diagonal show the characteristic occipital alpha pattern. A 0.5 Hz wavelet is consistently selected despite 1 Hz high-pass filtering.

Code

https://github.com/Roche/neuro-green. spdlearn implements the architecture as the Green model (WaveletConv, Shrinkage, BiMap, LogEig, BatchReNorm).