EE(G)-SPDNet

1 EE(G)-SPDNet

EE(G)-SPDNet is an end-to-end Deep Riemannian Network for EEG that learns a filterbank with temporal convolutions (channel-specific or channel-independent), computes spatial covariance matrices, and classifies through an SPDNet, with a Bayesian-optimized filterbank variant (BO-SPDNet) as companion.

Ref: @wilsonDeepRiemannianNetworks2022

Code available at https://github.com/dcwil/eegspdnet.

Overview

Problem: DRNs for EEG so far used handcrafted filterbanks (Tensor-CSPNet 9-band, MRC-MLP 43-band, RFNet 50-band, TSMNet 4-band conv), and it was unclear how network size, filterbank learning, and end-to-end ability affect performance. The study proposes two DRNs with learnable filterbanks, systematically compared over filterbank size (1–8 bands), channel specificity, and interband covariance, against state-of-the-art ConvNets.

• EE(G)-SPDNet: a second-order convolutional network — temporal convolution (mimicking band-pass filtering, learned end-to-end via backprop) → sample covariance pooling → SPDNet. Optimized jointly with classification; flexible across sampling rates; can learn band-/notch-/multiband filters; channel-specific mode gives one filter per electrode per band.
• BO-SPDNet: identical SPDNet, but the filterbank (bandpass cut-off pairs) is searched by Bayesian optimization using rSVM cross-validation accuracy (Log-Euclidean kernel) as the objective; restricted to Sinc-like bandpass responses.
• Datasets/tasks (exactly two): High Gamma Dataset (HGD) — motor movement, 14 participants, ~1,000 trials/subject (≈880 validation, ~160 final evaluation), 44 selected electrodes of 128, four classes (rest; movement of right hand, left hand, tongue/feet), 250 Hz after resampling. BCI Competition IV Dataset 2a — motor imagery, 9 participants, 22 channels, 2 sessions × 288 trials, four classes; session 1 for validation splits, session 2 as final evaluation.
• Preprocessing (matching Schirrmeister et al. 2017): channel selection (44-channel HGD montage; 22 channels for 2a), clipping at 800 µV, resampling to 250 Hz; HGD band-pass 4–125 Hz, BCIC-IV-2a band-pass 4–38 Hz. Validation phase uses an 80:20 train/test split of the training data as a proxy for the holdout.
• Contributions: first systematic comparison of DRN filterbank design; end-to-end DRNs outperform Deep4/ShallowFBCSP ConvNets using physiologically plausible frequencies (10–20, 20–35, 65–90 Hz); layer-by-layer (LBL) analysis revealing loss of Riemannian information through the SPDNet; multi-band filter learning analysis.

Architecture

Pipeline: EEG trial → Conv (learnable filterbank) → SCM pooling → SPDNet [3 × (BiMap → ReEig)] → LogEig → Vect → Linear.

1. Convolution. Temporal kernels (length 25 samples) applied per band, either channel-independent (one filter per band for all electrodes) or channel-specific (one filter per electrode per band). For channel-independent multi-band models the per-band covariances are concatenated into a block-diagonal matrix, optionally including the interband covariance (off-diagonal blocks computed by stacking band-filtered signals along the electrode axis before covariance estimation) — three model sub-types: Spec, Ind, Ind [RMINT].
2. Covariance pooling. Sample Covariance Matrix over each (filtered) trial (the CovLayer primitive):

$$C=\frac{1}{N_T-1}X{X}^{\top },X\in {\mathbb{R}}^{N_e\times N_T}.$$

Multi-band concatenation is provably SPD (block-diagonal of SPD blocks). Regularization $M_{\mathrm{reg}}=M+\lambda I$ applied only when matrices are non-PD (needed for < 0.01% of trials); BiMap/ReEig also force PD.
3. SPDNet (Huang & Van Gool 2017): three BiMap–ReEig pairs, each BiMap roughly halving the SPD dimension:

$$X_k=W_k^{\top }{X}_{k-1}{W}_k,W_k\in \mathrm{Stiefel}(d_{k-1}\times d_k);$$

ReEig:

$$X_k=U_{k-1}\max (ϵI,{\Sigma }_{k-1})U_{k-1}^{\top },$$

with rectification threshold $ϵ$ (value not reported); then LogEig (Log-Euclidean metric, identity reference):

$$X_l=U_{k-1}\log ({\Sigma }_{k-1})U_{k-1}^{\top },$$

then vectorization $Vect(S)=[S_{1,1},S_{1,2},S_{2,2},S_{1,3},\dots ,S_{n,n}]$ (upper-triangle, redundant off-diagonals kept), then a linear layer with cross-entropy.
4. Meta-optimization (from the adavoudi/spdnet repo): the SPDNet SGD BiMap update is replaced by ${\nabla }_R={\nabla }_E-W_t\mathrm{Sym}(W_t^{\top }{\nabla }_E)$, then $W_{t+1}^{\prime }=Opt(W_t,{\nabla }_R,\lambda )$ (any optimizer, here Adam) and retraction $\Gamma$ back to the Stiefel manifold; matrix backpropagation uses the Ionescu method (identical results to Daleckiĭ–Kreĭn in initial tests).
5. BO-SPDNet: Bayesian optimizer over bandpass cut-off pairs; objective = negative stratified 5-fold CV accuracy of an rSVM (linear kernel, LEM, C = 10) on the SPD matrices; the winning filterbank is then applied and the (identical) SPDNet trained on top.

Model Parameters

• Convolutional kernel length: 25.
• Number of bands $N_B\in \{1,\dots ,8\}$; per band: $N_B$ filters total (Ind) or $N_B$ per electrode (Spec); three sub-types (Spec / Ind / Ind[RMINT]).
• SPDNet: 3 BiMap–ReEig pairs, each BiMap halves the dimension (e.g., 44→22→11→≈6 for HGD input 44×44; 22→11→6→3 for 2a 22×22; exact final sizes not tabulated).
• Depth analysis suggests a minimum final matrix size of ~10–20 is required; “all models consistently favoured $N_B\ge 4$”.
• Random seeds: 4. SPD estimator: SCM. Total parameter counts: not reported.

Training Parameters

• Optimizer: Adam (found best in initial tests; no thorough optimizer/LR search performed).
• Learning rate: 0.001; batch size: 216; LR scheduler: cosine annealing; epochs: 1,000; loss: cross-entropy.
• BO-SPDNet: BO stopping criteria 2,500 iterations or 12 hours; rSVM C = 10, linear kernel; stratified 5-fold CV for the objective.
• Splits: validation 80:20; final evaluation on the holdout (HGD final set ~160 trials/subject; 2a session 2). 2a models re-trained/re-tested without hyperparameter optimization (transferred hyperparameters from HGD validation).
• Software/hardware: Python 3.8, PyTorch, MNE, Braindecode, HyperOpt; bwForCluster NEMO.

Results

Final evaluation-set accuracy (%):

Model	Type (# bands)	HGD	BCIC-IV-2a
EE(G)-SPDNet	Spec (8)	97.0 (p<0.001 vs both ConvNets)	75.2
BO-SPDNet	Ind (8)	94.5	74.7
ConvNet Deep4	-	93.1	73.0
ConvNet ShallowFBCSP	-	94.4	72.9

• EE(G)-SPDNet beats Deep4 and ShallowFBCSP on HGD with statistical significance (Wilcoxon signed-rank); effects reproduced on 2a with decreased significance; BO-SPDNet improvements not significant. Best EE vs best BO on the evaluation set only marginally significant (p = 0.0495).
• Channel specificity: EE(G)-SPDNet strongly favors channel-specific filtering (+2.2% average, p = 0.00781); BO-SPDNet favors channel-independent (+1.7%, p = 0.00781); EE’s preference shrinks as $N_B$ grows.
• Interband covariance: including it gives +2.1% (EE, p = 0.0156) and +0.42% (BO, p = 0.0156) on validation.
• Filterbank learning: BO slightly better in validation (except channel-specific); EE highest on evaluation, more robust from validation to evaluation.
• Learned frequencies: gain spectra peak at 10–20 Hz and 20–35 Hz, plus a high-gamma peak at 65–90 Hz (channel-independent only when interband covariance kept); peaks flatten as $N_B$ increases.
• Depth: number of BiMap–ReEig pairs has little effect as long as the final matrix size ≥ ~10–20.
• Layer-by-layer analysis: an rSVM on intermediate representations often exceeds the final network accuracy (especially early/middle layers for channel-specific models); Euclidean SVMs are much worse than rSVM but improve through the network — evidence that the network optimizes features that lose Riemannian-specific benefit.