Riemannian Geometry Classifier (RGC)

• Riemannian Geometry-Based Classifier treats covariance matrices as points on an SPD manifold, leveraging intrinsic curvature for enhanced classification.
• It employs key manifold operations such as matrix logarithm and exponential mappings to compute geodesic distances for robust decision-making.
• The online MDRM variant uses sliding windows and directional curve verification to achieve rapid and accurate BCI classification even under noise.

A Riemannian Geometry-Based Classifier (RGC) is an approach to supervised learning that treats input data—such as covariance matrices from EEG or other signal modalities—as points on a Riemannian manifold, rather than vectors in a Euclidean space. This paradigm exploits geometric structure intrinsic to symmetric positive-definite (SPD) matrices, leveraging manifold operations (e.g., geodesic distance, exponential and logarithmic maps) to achieve robustness and precision in classification, particularly for brain-computer interface (BCI) applications.

1. Riemannian Manifolds and the Representation of SPD Matrices

RGC begins by formalizing the data geometry: EEG trials or similar signals are encoded as covariance matrices, which are by construction SPD: $$S^+(C) = \{ P \in \mathbb{R}^{C \times C} : P = P^\top,\ x^\top P x > 0,\ \forall x \in \mathbb{R}^C \setminus \{0\} \}$$ Unlike points in Euclidean space, these matrices lie on a non-flat, curved Riemannian manifold. This structure is crucial: Euclidean metrics applied to SPD matrices can produce erroneous areal relationships; in contrast, manifold-aware techniques such as RGC employ the affine-invariant Riemannian metric: $$d(P_1, P_2) = \|\log(P_1^{-1/2} P_2 P_1^{-1/2})\|_F$$ where $\log(\cdot)$ is the matrix logarithm and $\|\cdot\|_F$ denotes the Frobenius norm. This quantifies geodesic distance—respecting curvature and affinely invariant properties—enabling comparisons robust to electronic, electrical, and biological signal variabilities.

2. Covariance Estimation and Manifold Operations

RGC's accuracy depends on the reliable estimation of SPD covariance matrices. The paper explores several estimators: the Sample Covariance Matrix (SCM), shrinkage estimators like Ledoit-Wolf and Schäfer, and the Fixed Point estimator. These are assessed for ensuring positive-definiteness and numerical conditioning (eigenvalue spread), as covariance quality critically impacts downstream geometric computations.

Key manifold operations underpin RGC's workflows:

• Matrix logarithm and exponential mapping: Linearizes manifold neighborhoods for Euclidean operation and projects back to the manifold:
  • Logarithmic map at $P$:

  $$\log_P(Q) = P^{1/2} \log(P^{-1/2} Q P^{-1/2}) P^{1/2}$$ - Exponential map at $P$ (for mapping tangent vectors to the manifold):

  $$\exp_P(S) = P^{1/2} \exp(P^{-1/2} S P^{-1/2}) P^{1/2}$$

These facilitate data aggregation, interpolation, and mean computation in the tangent (Euclidean) space, followed by remapping to SPD(n).

A critical construct is the Riemannian mean (geometric mean) of $\{P_1, …, P_I\}$: $$G(P_1, ..., P_I) = \underset{P}{\mathrm{argmin}} \sum_{i} d^2(P, P_i)$$ This minimizer, typically computed iteratively, serves as a class prototype for minimum-distance classification.

3. Classification Algorithms: Offline and Online MDRM

The main discriminator described is the Minimum Distance to Riemannian Mean (MDRM) classifier. For each class, the geometric mean of training covariances is computed. New observations are compared to each class center using manifold geodesic distances; the nearest center determines class membership.

The paper extends MDRM to an online, asynchronous regime via a novel strategy:

• Data segmentation: Continuous data undergo overlapping windowed segmentation.
• Covariance extraction: For each window, covariances are computed (after domain-specific preprocessing such as frequency band extraction for SSVEP).
• Occurrence probability $\rho(k)$ and curve direction decision**: The sequence of predictions over a decision buffer of D epochs is tallied, and only if a single class exceeds a threshold is it considered. A curve direction check is then performed by summing gradients (distance changes to the candidate class mean). A valid decision must show a net negative trend (the trajectory of covariances is approaching the class mean), providing robustness to transient misclassifications:

  $\Delta\delta = \sum (\delta(j) - \delta(j-1)) < 0$

where $\delta(j)$ is the normalized distance at epoch $j$.

4. Experimental Validation in SSVEP-based BCI

The RGC is empirically validated using SSVEP experiments, where subjects fixate on stimuli flashing at specific frequencies. EEG data are bandpass-filtered, and covariance matrices computed across rearranged channels (e.g., 8 channels $\times$ 3 frequencies = 24).

The classifier's performance is evaluated both offline (using full 6-second trials) and online (with asynchronous, real-time sliding windows). Key reported metrics include:

• Classification accuracy
• Information Transfer Rate (ITR), in bits/min
• Decision delay

The online, curve-verified RGC achieves higher accuracy and significantly lowers decision times (to $\sim$1.2 seconds on average) compared to offline or conventional MDRM approaches.

5. Mathematical Underpinnings

The classifier’s formulation is mathematically rigorous:

• SPD Manifold: $S^+(C)$, with SPD constraints
• Affinely-invariant Riemannian metric
• Iterative computation of the geometric mean
• Mapping between the manifold and tangent space via log/exponential maps
• Classification in online mode via thresholds on occurrence probabilities and gradient-summed normalized distances

All steps rely on explicit, closed-form expressions for logarithmic maps and geodesic distances, with no reliance on parametric learning or data modeling assumptions beyond covariance estimation.

6. Practical Advantages, Robustness, and Applicability

RGC offers several practical benefits:

• Noise resilience: By leveraging the manifold’s affine invariance, irrelevant variances (e.g., sensor drift, inter-subject variability) are suppressed.
• Rapid, data-efficient adaptation: The online implementation can reach decisions after short windows, ideal for asynchronous BCIs demanding fast responsiveness.
• Extendibility: While demonstrated on SSVEP, the methodology is suitable for any BCI paradigm with covariance-based features (e.g., ERP, motor imagery).
• Computational tractability: The iterative mean calculation and log/exponential operations are efficient for moderate channel counts, suitable for real-time applications.

The method’s effectiveness and mathematical soundness position it as a robust and versatile tool for real-world BCI systems.

7. Summary and Outlook

A Riemannian Geometry-Based Classifier, as instantiated for SSVEP-based BCIs in this work, encodes all EEG trials as covariance matrices on the SPD manifold and classifies them using geodesic distance to class means. The introduction of an online, curve-direction–verified algorithm further increases real-time robustness and decision speed. The approach is grounded in information geometry, exploits recent advances in matrix manifold methods, and is broadly applicable to noisy, variable neurophysiological data. Experimentation confirms its efficacy and efficiency, suggesting a strong foundation for future work in online, adaptive brain signal decoding and broader covariance-based pattern recognition tasks.