SPDIM: Source-Free Unsupervised Conditional and Label Shift Adaptation in EEG

The non-stationary nature of electroencephalography (EEG) introduces distribution shifts across domains (e.g., days and subjects), posing a significant challenge to EEG-based neurotechnology generalization. Without labeled calibration data for target domains, the problem is a source-free unsupervised domain adaptation (SFUDA) problem. For scenarios with constant label distribution, Riemannian geometry-aware statistical alignment frameworks on the symmetric positive definite (SPD) manifold are considered state-of-the-art. However, many practical scenarios, including EEG-based sleep staging, exhibit label shifts. Here, we propose a geometric deep learning framework for SFUDA problems under specific distribution shifts, including label shifts. We introduce a novel, realistic generative model and show that prior Riemannian statistical alignment methods on the SPD manifold can compensate for specific marginal and conditional distribution shifts but hurt generalization under label shifts. As a remedy, we propose a parameter-efficient manifold optimization strategy termed SPDIM. SPDIM uses the information maximization principle to learn a single SPD-manifold-constrained parameter per target domain. In simulations, we demonstrate that SPDIM can compensate for the shifts under our generative model. Moreover, using public EEG-based brain-computer interface and sleep staging datasets, we show that SPDIM outperforms prior approaches.

💡 Research Summary

Electroencephalography (EEG) signals are notoriously non‑stationary: the statistical distribution of recorded epochs changes across recording sessions, days, and subjects. In brain‑computer interface (BCI) and clinical applications such as sleep staging, this non‑stationarity forces practitioners to collect labeled calibration data for every new session, which is costly and limits scalability. The problem can be cast as source‑free unsupervised domain adaptation (SF‑UDA): a model trained on one or more source domains must be deployed on unlabeled target domains without any further supervision.

State‑of‑the‑art SF‑UDA methods for EEG rely on Riemannian geometry of symmetric positive‑definite (SPD) covariance matrices. By aligning first‑order (Fréchet mean) and second‑order (variance) statistics on the SPD manifold, approaches such as recentering (RCT) combined with tangent‑space mapping (TSM) can compensate for linear mixing and conditional shifts introduced by different forward models. However, these methods assume that class priors are identical across domains. In many realistic scenarios—most notably sleep staging—the proportion of each sleep stage varies dramatically between subjects and nights, a phenomenon known as label shift. Aligning marginal distributions under label shift can actually increase classification error, a limitation that has not been formally addressed for EEG.

The authors introduce a realistic generative model that captures both conditional shifts (domain‑specific linear mixing matrices (A_j)) and label shifts (domain‑specific class priors (\pi_j)). The model assumes zero‑mean latent sources (z_i) whose spatial covariance (E_i) follows a log‑linear relationship with the label: (s_i = B(1_{y_i} - \pi_j) + \varepsilon_i), where (B) is a sparse matrix that encodes label information only in the first (D) dimensions. The observed EEG epoch is (x_i = A_j z_i), and its empirical covariance (C_i = A_j E_i A_j^\top) serves as an SPD feature.

Two theoretical results are proved. First, under the generative model the Fréchet mean of the covariances in any domain converges to the identity matrix as the number of samples grows, which justifies the use of RCT. Second, RCT + TSM perfectly removes the conditional shift introduced by (A_j) but leaves a residual bias proportional to the domain‑specific prior (\pi_j). Consequently, when label shift is present, RCT + TSM yields domain‑invariant representations only up to an additive term that depends on (\pi_j), leading to systematic mis‑classification.

To overcome this, the paper proposes SPDIM (SPD‑constrained Information Maximization). SPDIM augments the RCT + TSM pipeline with a single domain‑specific SPD bias matrix (\Phi_j). The transformed feature becomes
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