1 Phase-SPDNet

Phase-SPDNet (SPDNet $ψ$ ) augments spatial covariance matrices with a Takens delay-embedding phase-space reconstruction so that a small SPDNet with only three motor-cortex electrodes (C3, Cz, C4) outperforms state-of-the-art DL decoders on nearly 100 subjects of MOABB motor-imagery data.

Ref: @carraraGeometricNeuralNetwork2024

arXiv:2403.05645; published version: J. Neural Eng. 2024 (the Zhou2016 audit reads the published PDF; version differences are flagged in Notes).

Overview

Problem: DL for MI-BCI suffers from limited data, low SNR, and inter/intra-subject variability; large electrode setups hurt user comfort; coherence alone is unstable.
Idea: recover dynamics lost with few electrodes via time-delay embedding (Takens’ theorem), compute covariance on the reconstructed phase space (Augmented Covariance Method, ACM), and feed these larger SPD matrices to SPDNet, with ( $τ$ , $ψ$ ) selected per fold by MDOP.
Contributions: SPDNet $ψ$ pipeline; few-electrode (3) validation on ~100 subjects / six MOABB datasets; computational and CO2 analysis; GradCam++ explainability.
Datasets (six, MOABB): BNCI2014001 (BCI Competition IV Dataset 2a, 9 subj, 22 ch, 250 Hz, 2 sessions, 4 tasks, 144 trials/class, epoch [2,6] s); BNCI2014004 (BCI Competition IV Dataset 2b under its MOABB name, 9 subj, 3 ch, 250 Hz, 5 sessions, 2 tasks, 360 trials/class, [3,7.5] s); Cho2017 or GigaDB (52 subj, 64 ch, 512 Hz, 1 session, 2 tasks, 100 trials/class, [0,3] s); Schirrmeister2017 (High Gamma Dataset (HGD), 14 subj, 128 ch, 500 Hz, 1 session, 4 tasks, 120 trials/class, [0,4] s); Weibo2014 (10 subj, 60 ch, 200 Hz, 1 session, 7 tasks, 80 trials/class, [3,7] s); Zhou2016 (4 subj, 14 ch, 250 Hz, 3 sessions, 3 tasks, 160 trials/class, [0,5] s).
Preprocessing: band-pass 8-32 Hz (overlap-add), artifact rejection, per-channel zero-mean/unit-std standardization (also mitigates ReEig’s scale dependence, $ϵ = 1 0^{- 4}$ ), full epoch used.
Task/protocol: binary left vs right hand, intra-subject within-session, 5-fold CV per session, on 3 electrodes (C3, Cz, C4); metric ROC-AUC (%).

Architecture

Pipeline: $f_{SPDNet ψ} = f_{delay} \circ f_{Cov} \circ f_{BiMap} \circ f_{ReEig} \circ f_{LogEig} \circ f_{MLP}$ .

Phase-space reconstruction ( $f_{delay}$ ): for embedding dimension $Ψ$ and delay $τ$ , the lag function $s_{p} (x) = x (t + p τ, t + T - (Ψ - p) τ)$ builds the embedded signal

f_{delay} (X) = [s_{0} (x_{1}), \dots, s_{Ψ - 1} (x_{1}), \dots, s_{0} (x_{C}), \dots, s_{Ψ - 1} (x_{C})]^{⊤},

expanding $C$ channels to $C \times Ψ$ “virtual” channels. $τ$ and $ψ$ are chosen per epoch with MDOP (Maximizing Derivatives On Projection, false-nearest-neighbors criterion) and averaged over epochs.
2. Covariance ( $f_{Cov}$ ): sample covariance of the phase-space signal — an augmented CovLayer-style operation: input SPD dimension grows from $S_{++}^{C}$ to $S_{++}^{C \times Ψ}$ .
3. BiMap: congruence transform $Z_{k} = W_{k} Z_{k - 1} W_{k}^{⊤}$ with $W_{k}$ on the compact Stiefel manifold; for SPDNet $ψ$ the subspace is set to half the input dimension; the standard SPDNet baseline keeps the same dimension (rotation-only) for explainability.
4. ReEig: $Z_{k} = U_{k - 1} max (ϵ I, Σ_{k - 1}) U_{k - 1}^{⊤}$ , $ϵ = 1 0^{- 4}$ , no trainable parameters.
5. LogEig: $Z_{k} = U_{k - 1} lo g (Σ_{k - 1}) U_{k - 1}^{⊤}$ , no trainable parameters.
6. MLP: one linear layer + softmax classification; cross-entropy loss.

Optimization: BiMap weights use RiemannAdam from the geoopt library (Stiefel manifold); MLP uses Adam.

Model Parameters

Architecture for embedding order $ϕ = 6$ (example input = 3 channels, Cho2017, 1537 samples):

Stage	Layer	Output shape	Parameters
Input	-	[1, 3, 1537]	0
Augmentation	AugmentedDataset	[1, 18, 1393]	0
Covariance	Covariances	[1, 18, 18]	0
SPDNet	BiMap	[1, 9, 9]	162
SPDNet	ReEig	[1, 9, 9]	0
SPDNet	LogEig	[1, 45]	0
Classification	Linear	[1, 2]	45
Total			207

BiMap subspace = 9 = 18/2 (half of the augmented dimension). Total parameters vary with $ϕ$ . MDOP per fold adds the two hyperparameters $τ$ , $ψ$ .

Training Parameters

Common DL config (also confirmed by the Zhou2016 audit with MOABB 1.1.0): epochs 300; batch size 64; validation split 0.1; loss sparse categorical cross-entropy; Adam lr 0.001 (plus RiemannAdam on the Stiefel for BiMap); early stopping patience 75 (monitor validation loss); ReduceLROnPlateau patience 75, factor 0.5. MDOP used per CV fold instead of grid search.

Baselines re-run on the same 3 electrodes, resampled to their design rates: ShallowNet/DeepNet/EEGTCNet at 250 Hz; EEGNet/EEGITNet/EEGNeX at 128 Hz. MOABB version: 1.0 in the arXiv v1 text; 1.1.0 per the Zhou2016 audit.
Hardware for time/CO2 comparisons: Dell C6420 dual-Xeon Cascade Lake SP Gold 6240 @ 2.60 GHz.
Statistics: one-tailed Wilcoxon signed-rank test + Stouffer meta-analysis across datasets, p < 0.05.

Results

AUC-ROC % (mean ± SD), right vs left hand, within-session 5-fold:

Model	BNCI2014001	BNCI2014004	Cho2017	Schirrmeister2017	Weibo2014	Zhou2016
DeepNet	75.80 ± 15.45	72.80 ± 19.48	63.13 ± 14.45	73.04 ± 15.67	73.97 ± 18.07	91.74 ± 7.00
ShallowNet	75.85 ± 15.41	72.17 ± 18.61	64.14 ± 13.03	73.59 ± 15.19	75.36 ± 15.69	88.03 ± 8.55
EEGNet	70.64 ± 19.87	70.27 ± 18.91	60.23 ± 14.98	69.80 ± 16.51	71.94 ± 17.80	88.95 ± 7.84
EEGTCNet	65.98 ± 17.25	66.86 ± 18.41	56.43 ± 12.31	67.87 ± 17.36	65.94 ± 15.69	81.47 ± 11.66
EEGITNet	66.64 ± 13.98	64.93 ± 14.49	54.68 ± 11.97	62.98 ± 16.19	56.97 ± 17.66	72.82 ± 12.78
EEGNeX	68.86 ± 17.27	68.29 ± 17.85	56.64 ± 12.83	64.26 ± 17.58	58.73 ± 19.40	80.25 ± 15.55
SPDNet	71.02 ± 15.64	70.15 ± 16.85	59.95 ± 12.61	67.40 ± 13.01	67.04 ± 17.62	88.85 ± 8.05
SPDNetψ	75.98 ± 17.00	80.46 ± 16.64	66.00 ± 13.29	72.02 ± 13.07	78.01 ± 19.64	94.92 ± 3.34

SPDNetψ is best on all datasets except Schirrmeister2017 (not significant); meta-analysis: SPDNetψ statistically better than all other approaches. Augmentation gives a consistent positive relative gain vs standard SPDNet on every dataset.

Published-version additions (from the Zhou2016 audit §6.8, absent from arXiv v1): Phase-SPDNet OPT 95.62 ± 2.36 on Zhou2016 (best, bold); coherence OPT variants (imag 88.76 ± 5.57; inst 91.14 ± 5.30); DynSpat+EEGNet 87.85 ± 7.01; DynSpat+ShallowNet 87.33 ± 8.87.

Coherence variant (arXiv Table A1, SPDNetψ rows only, AUC %): imaginary coherence 60.89 ± 10.17 (2a), 65.70 ± 14.06 (2b), 56.51 ± 8.60 (Cho2017), 61.69 ± 9.81 (Schirr17), 64.75 ± 11.33 (Weibo2014), 72.16 ± 7.40 (Zhou2016); instantaneous coherence 71.84 ± 16.75, 68.83 ± 17.41, 62.04 ± 12.02, 65.37 ± 12.91, 68.31 ± 14.17, 87.03 ± 6.13.

Computational/environmental (CodeCarbon, Cho2017 top-5 subjects): SPDNetψ 30.16 s avg / 0.22 g CO2-eq vs SPDNet 6.80 s / 0.07 — slowest in wall time (per-fold MDOP) but among the most CO2-friendly.

Code

https://github.com/carraraig/Phase-SPDNet (the paper says code will be made public under BSD-3).

Lucas Heck's Notes

Explorer

Phase-SPDNet