Graph-CSPNet

1 Graph-CSPNet

Graph-CSPNet is the graph-based companion of Tensor-CSPNet that treats the tensor of EEG spatial covariance matrices (time windows × filter bands) as the nodes of a time-frequency graph and learns CSP-like filters with manifold-valued graph convolution layers.

Ref: @juGraphCSPNetGraphBased2023

Code available at https://github.com/GeometricBCI/Tensor-CSPNet-and-Graph-CSPNet.

Overview

The paper (IEEE TNNLS 2023, DOI 10.1109/TNNLS.2023.3307470; arXiv 2211.02641) enhances the geometric classifier Tensor-CSPNet from a time-frequency-analysis perspective. Problem: fixed segmentation in Tensor-CSPNet is rigid; Graph-CSPNet replaces it with a flexible, non-uniform time-frequency segmentation whose local structure is captured by a graph, motivated by the Heisenberg–Gabor uncertainty principle (low frequencies: high spectral / low temporal resolution; high frequencies: the opposite).

• Contribution: (i) the time-frequency graph (nodes = SPD covariance matrices, edges = local graph topology built with a modified ε-neighborhoods approach); (ii) a new manifold-valued graph convolution, the graph BiMap layer; (iii) the formal coining of the term Geometric Classifiers; (iv) a spectrum-perturbation theorem (Theorem 1) bounding eigenvalue change ratio by node degree; (v) evaluation on five public MI datasets, near-optimal in 9/11 scenarios.
• Datasets (Table III): KU (Korea University Dataset (MI-KU or Lee2019 or OpenBMI), 54 subjects, 20 of 62 channels selected, 2 classes, 200 trials/session, imagery 1–3.5 s, 1,000 Hz, 2 sessions); Cho2017 (Cho2017 or GigaDB, 49 of 52 subjects, 20 of 64 channels, 200 trials, imagery 3–6 s, 512 Hz, 1 session); BNCI2014001 (BCI Competition IV Dataset 2a, 9 subjects, 22 Ag/AgCl + 3 EOG, 4 classes, 288 trials/session, imagery 2–6 s, 250 Hz, 2 sessions); BNCI2014002 (BNCI2014_002, 13 participants in the text / 14 in Table III, 15 electrodes, 2 classes right hand/feet, 160 trials, imagery 3–8 s, 512 Hz, 1 session); BNCI2015001 (BNCI2015_001, 12 subjects, 13 channels, 2 classes right hand/both feet, 200 trials/session, imagery 3–8 s, 512 Hz, 2 sessions; 4 subjects had an excluded 3rd session).
• Preprocessing: all signals filtered with Chebyshev Type II filters in 4 Hz intervals (nine bands covering 4–40 Hz), max passband loss 3 dB, min stopband attenuation 30 dB; trials assumed band-pass filtered, centered, scaled (inherited from Tensor-CSPNet).
• Splits: subject-specific: within-session shuffled 10-fold CV (class-balanced folds, repeated ten times) on both sessions where available; holdout S1→S2 for KU, T→E for BNCI2014001, A→B for BNCI2015001; single-session datasets (Cho2017, BNCI2014002) with 10-fold CV only.

Architecture

Pipeline for a trial:

1. Time-frequency distribution. Given a segmentation plan $\{\Delta t_i\times \Delta f_i{\}}_{i\in I}$, each segment yields a spatial covariance node $S(\Delta t_i\times \Delta f_i)={\bar{X}}_i{\bar{X}}_i^{\top }\in {\mathbb{S}}_{++}^{n_C}$ (a CovLayer-like operation on the band-passed, centered, scaled segment).
2. Graph construction (Local Graph Topology, LGT). Edges connect two nodes $S_i,S_j$ iff they fall in a box $B_{{ϵ}_1,{ϵ}_2}$ of the time-frequency domain:

$$0\le \mathrm{mid}(\Delta t_i)-\mathrm{mid}(\Delta t_j)\le {ϵ}_1,|\mathrm{mid}(\Delta f_i)-\mathrm{mid}(\Delta f_j)|\le {ϵ}_2,$$

with time evolution only in the forward direction. The weighted adjacency is a Gaussian RBF kernel on the AIRM Riemannian distance,

$$A_{ij}=\{\begin{array}{ll}{e}^{-d_{g_{\mathrm{AIRM}}}^2(S_i,S_j)/t},& S_i,S_j\text{ adjacent};\\ 0,& \text{otherwise},\end{array}$$

with preset kernel width $t>0$. Edge weights are computed from the average Riemannian distances across the training set; each frequency band component (θ, µ, β, γ) forms a separate connected component. Numerical values of ${ϵ}_1,{ϵ}_2$ are not reported.
3. Graph BiMap layers (manifold-valued graph convolution):

$$H^{(l+1)}\leftarrow \mathrm{RBN}\left(\mathrm{ReEig}\right(W^{(l)}({\bar{D}}^{-1}{\bar{A}}^{(l)})H^{(l)}{W}^{(l)\top }\left)\right),$$

with $\bar{A}:=A+I_N$, ${\bar{D}}_{ii}:={\sum }_j{\bar{A}}_{ij}$, and $H^{(l)}\in {\mathbb{R}}^{|V|\times n_C^2}$ a node function; $H^{(0)}=(S_1,\dots ,S_N)$. ${\bar{A}}^{(l)}:=I_N$ for $l\ge 1$ (see Theorem 1), $W^{(l)}$ a full-row-rank (Stiefel) transform — i.e., a BiMap congruence per node with graph aggregation. Row-normalized ${\bar{D}}^{-1}\bar{A}$ is used to normalize only along the time direction. RBN = Riemannian batch normalization via parallel transport (the SPDBatchNormMeanVar analogue). The ReEig step drops the smallest eigenvalues with lower bound $\beta =10^{-6}$. Theorem 1 bounds the spectrum perturbation ratio:

$$1-N_i{C}_i\le \frac{\lambda ({\bar{S}}_i)}{\lambda (S_i)}\le 1+N_i{C}_i,$$

where $N_i$ is node degree and $C_i={\max }_{(i,j)\in E}\{\exp ({\lambda }_{ij})\}$ with ${\lambda }_{ij}$ the largest eigenvalue of $\log (S_i^{-1/2}{S}_j{S}_i^{-1/2})$. Because the ratio grows with depth, the graph adjacency is only used in the first layer (${\bar{A}}^{(l)}=I_N$, $l\ge 1$).
4. LOG layer and classifier. The LogEig log map at the identity maps SPD matrices to the tangent space; the flattened vectors feed a Euclidean linear classifier trained with cross-entropy.

Relation to Tensor-CSPNet: identical backbone (BiMaps, RBN, ReEig, LOG, linear head) and underlying space $({\mathbb{S}}_{++},g_{\mathrm{AIRM}})$ with Riemannian optimization; the difference is that CNNs for temporal dynamics are replaced by time-frequency graph processing via graph BiMaps.

Model Parameters

Segmentation plans (Table IV; non-overlapping, non-uniform). Window length in seconds per 4-Hz band:

Dataset	θ 4–8	µ 8–12	β 12–16…24–28	γ 28–32…36–40
KU / Cho2017	0.5	0.5	0.5	0.25
BNCI2014001 (2a)	0.25	0.25	0.25	0.125
BNCI2014002 / BNCI2015001	1	1	1	0.5

Graph sizes: 60 nodes for KU/BNCI2014002/BNCI2015001 (= 5 windows × 6 bands + 10 windows × 3 bands), 48 for BNCI2014001 (θ/µ/β/γ components have 4/4/16/24 nodes), 33 for Cho2017.

Graph BiMap blocks (expand→reduce): KU & Cho2017: 20→30→20; BNCI2014001: 22→36→22; BNCI2014002: 15→30→15; BNCI2015001: 13→30→13.

Parameter counts (two-layer graph BiMap): total $N{o}_2(o_1+o_3)+(cN+1)o_3^2$. For BNCI2014001 ($N=48,c=4,o_1=22,o_2=36,o_3=22$): 169,444 parameters — ≈ 6× the 27,104 of 1-CSPNet(9,1,1) and ≈ 2/3 of the 232,360 of 5-CSPNet(9,3,1) (Tensor-CSPNet).

Graph hyperparameters: forward “time direction” and “frequency direction” lattice vectors; in all main experiments the forward time flow is set to the nearest integer values of half its maximum. ReEig bound $\beta =10^{-6}$; adjacency re-initialized per CV fold (10 times), so the network is individual-specific. ${ϵ}_1,{ϵ}_2$ and Gaussian kernel width $t$: not numerically reported.

Training Parameters

• Loss: cross-entropy.
• Optimizer: Riemannian optimization — first-order Riemannian adaptive methods (Adam, AdaGrad) adapted to the manifold constraints, with retraction and parallel transport; graph BiMap/BiMap weights live on Stiefel manifolds, RBN parameters on the SPD manifold.
• Learning rate, batch size, epochs, early stopping: not reported in this paper (the companion Tensor-CSPNet paper reports lr 0.01 with decay, batch 28, max 60 epochs, patience 15).
• Reporting: “for each scenario, we select the best result among multiple runs.”
• Statistical test for the graph-topology ablation: one-tailed Wilcoxon signed-rank test with Bonferroni–Holm correction, $\alpha =0.05$.

Results

Average accuracy % (SD). Baselines re-run by the authors; KU/BNCI2014001 CV indices were re-initialized relative to the Tensor-CSPNet paper, so baseline numbers differ by ≈ 1%.

| Dataset | Scenario | FBCSP | FBCNet | Tensor-CSPNet | Graph-CSPNet |
|---|---:|---:|---:|---:|
| KU | 10-Fold CV (S1) | 64.33 (15.43) | 73.36 (13.71) | 73.28 (15.10) | 72.51 (15.31) |
| KU | 10-Fold CV (S2) | 66.20 (16.29) | 73.68 (14.97) | 74.16 (14.50) | 74.44 (15.52) |
| KU | Holdout (S1→S2) | 59.67 (14.32) | 67.74 (14.52) | 69.50 (15.15) | 69.69 (14.72) |
| Cho2017 | 10-Fold CV | 61.75 (13.26) | 65.34 (11.14) | 67.30 (12.94) | 67.51 (12.89) |
| BNCI2014001 | 10-Fold CV (T) | 71.29 (16.20) | 75.48 (14.00) | 75.11 (12.68) | 77.55 (15.63) |
| BNCI2014001 | 10-Fold CV (E) | 73.39 (15.55) | 77.16 (12.77) | 77.36 (15.27) | 78.82 (13.40) |
| BNCI2014001 | Holdout (T→E) | 66.13 (15.54) | 71.53 (14.86) | 73.61 (13.98) | 71.95 (13.36) |
| BNCI2014002 | 10-Fold CV | 76.07 (13.29) | 79.64 (12.77) | 80.58 (11.87) | 81.65 (11.74) |
| BNCI2015001 | 10-Fold CV (A) | 79.46 (14.16) | 82.62 (13.11) | 81.29 (14.78) | 84.62 (12.38) |
| BNCI2015001 | 10-Fold CV (B) | 81.96 (11.14) | 84.92 (10.30) | 85.29 (10.54) | 88.00 (7.87) |
| BNCI2015001 | Holdout (A→B) | 73.46 (14.09) | 74.50 (16.01) | 79.04 (14.67) | 79.75 (14.63) |

Graph-CSPNet is best in 9/11 scenarios (not KU S1 CV → FBCNet; not 2a T→E → Tensor-CSPNet).

Ablations/hyperparameters (BNCI2014001, simplified depth-1 graph BiMap, in/out dims 20):

• Non-graph baseline “Time Direction (0,0,0,0)” (i.e., $A^{(0)}:=I_N$) is significantly worse than any graph topology — the graph structure itself is what helps.
• Time Direction (2,2,2,4) with fixed frequency direction (1,1,4,3) gives the best performance; significant vs the non-graph case in 10-fold CV, but not significant in the holdout experiment.
• Example graphs contain 60 nodes and 390 edges for direction (2,2,2,4); larger forward time flow ⇒ more edges.
• The LGT averages each node with its graph neighbors, shifting the spectral distribution (t-SNE and discrete-spectrogram analyses).