Classification of covariance matrices using a Riemannian-based kernel for BCI applications

TLDR: A new kernel is derived by establishing a connection with the Riemannian geometry of symmetric positive definite matrices, effectively replacing the traditional spatial filtering approach for motor imagery EEG-based classification in brain-computer interface applications.

About: This article is published in Neurocomputing.The article was published on 2013-07-01 and is currently open access. It has received 326 citations till now. The article focuses on the topics: Kernel method & Radial basis function kernel.

Figures

Table 2: Average time in second for each method in training (training the classifier) and test (applying the classifier) stage.

Figure 5: p-values for the six pairs of mental tasks in a subject-independent manner (see text for details).

Figure 4: Comparison of classification accuracy for each individual pair of mental tasks. From the left to the right: RK-SVM versus CSP, ARK-SVM versus CSP classification and RK-SVM with ARK-SVM.

Figure 3: Evolution of mean classification accuracy when an increasing number of trials is used to estimate the geometric mean of the test session. From 15 trials (10% of the test set), the performance becomes better than the performance obtained without adaptation.

Figure 2: A relevant choice of Cref leads to a quasi invariant feature space across sessions. Left: representations of the feature distribution in the tangent plane for the two most discriminant dimensions, selected with a t-test, for two classes in two different sessions without adaptation of the reference point. The reference point used for the second session is estimated on the data of the first session (standard way). Right: the reference point of the second session is adapted using the data of the second session. Data from subject 1 from the BCI Competition IV, dataset IIa (see section 4.1 for dataset description)

Figure 1: Manifold M and the corresponding local tangent space TCM at C. The Logarithmic map LogC(.) projects the matrix Ci ∈ M into the tangent space. The Exponential map ExpC(.) projects the element of the tangent space Si back to the manifold.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Classification of covariance matrices using a riemannian-based kernel for bci applications" ?

The authors demonstrate that this new approach outperforms significantly state of the art results, effectively replacing the traditional spatial filtering approach. 

Q2. What have the authors stated for future works in "Classification of covariance matrices using a riemannian-based kernel for bci applications" ?

Future work will investigate the online use of this algorithm and the use of the adaptive kernel as a possible way to deal with inter-subjects variability. 

Q3. What is the way to use a covariance matrix in a classifier?

If one is interested in using a covariance matrix as a feature in a classifier, a natural choice consists in vectorizing it in order to process this quantity as a vector and then use any vector-based classification algorithms. 

Q4. What is the advantage of the present method?

The distinct advantage of the present method is that it can be applied directly, avoiding the need of spatial filtering (Barachant et al., 2010b). 

Q5. What is the way to handle the variability between sessions?

A simple way to handle the variability between sessions is to estimate at the beginning of the session a new reference point, by arithmetic or geometric mean, projects data from the second session to a new tangent space and apply the classifier with unchanged parameters αp and b. 

Q6. What is the way to overcome this problem?

To overcome this problem, the authors can use trials from the beginning of the test session to estimate the reference point, or use an iterative estimation of this point. 

Q7. What is the average performance of the subjects?

Since CSP is designed for binary classification, the authors have evaluated the average performance per subject, for all 6 possible pairs of mental tasks: {LH/RH, LH/BF, LH/TO, RH/BF, RH/TO, BF/TO}. 

Q8. What is the spatial covariance matrix of the EEG random signal?

For each trial Xp of known class yp ∈ {−1, 1}, one can estimate the spatial covariance matrix of the EEG random signal by the E × E sample covariance matrix (SCM) : Cp = 1/(T−1) XpXTp . 

Q9. What branch of differential geometry is used to manipulate spatial covariance matrix?

the correct manipulation of these matrices relies on a special branch of differential geometry, namely the Riemannian geometry (Berger, 2003).