Alexandre Barachant

Multiclass Brain–Computer Interface Classification by Riemannian Geometry

This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.

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

The use of spatial covariance matrix as a feature is investigated for motor imagery EEG-based classification in brain-computer interface applications. A new kernel is derived by establishing a connection with the Riemannian geometry of symmetric positive definite matrices. Different kernels are tested, in combination with support vector machines, on a past BCI competition dataset. We demonstrate that this new approach outperforms significantly state of the art results, effectively replacing the traditional spatial filtering approach.

Common Spatial Pattern revisited by Riemannian geometry

This paper presents a link between the well known Common Spatial Pattern (CSP) algorithm and Riemannian geometry in the context of Brain Computer Interface (BCI). It will be shown that CSP spatial filtering and Log variance features extraction can be resumed as a computation of a Riemann distance in the space of covariances matrices. This fact yields to highlight several approximations with respect to the space topology. According to these conclusions, we propose an improvement of classical CSP method.

Channel selection procedure using riemannian distance for BCI applications

This article describes a new algorithm to select a subset of electrodes in BCI experiments. It is illustrated on a two-class motor imagery paradigm. The proposed approach is based on the Riemannian distance between spatial covariance matrices which allows to indirectly assess the discriminability between classes. Sensor selection is automatically done using a backward elimination principle. The method is tested on the dataset IVa from BCI competition III. The identified subsets are both consistent with neurophysiological principles and effective, achieving optimal performances with a reduced number of channels.

Riemannian geometry applied to BCI classification

In brain-computer interfaces based on motor imagery, covariance matrices are widely used through spatial filters computation and other signal processing methods. Covariance matrices lie in the space of Symmetric Positives-Definite (SPD) matrices and therefore, fall within the Riemannian geometry domain. Using a differential geometry framework, we propose different algorithms in order to classify covariance matrices in their native space.

Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review

Despite its short history, the use of Riemannian geometry in brain-computer interface (BCI) decoding is currently attracting increasing attention, due to accumulating documentation of its simplicity, accuracy, robustness and transfer learning capabilities, including the winning score obtained in five recent international predictive modeling BCI data competitions. The Riemannian framework is sharp from a mathematical perspective, yet in practice it is simple, both algorithmically and computationally. This allows the conception of online decoding machines suiting real-world operation in adverse conditions. We provide here a review on the use of Riemannian geometry for BCI and a primer on the classification frameworks based on it. While the theoretical research on Riemannian geometry is technical, our aim here is to show the appeal of the framework on an intuitive geometrical ground. In particular, we provide a rationale for its robustness and transfer learning capabilities and we elucidate the link between a simple Riemannian classifier and a state-of-the-art spatial filtering approach. We conclude by reporting details on the construction of data points to be manipulated in the Riemannian framework in the context of BCI and by providing links to available open-source Matlab and Python code libraries for designing BCI decoders.

A fixed-point algorithm for estimating power means of positive definite matrices

Estimating means of data points lying on the Riemannian manifold of symmetric positive-definite (SPD) matrices has proved of great utility in applications requiring interpolation, extrapolation, smoothing, signal detection, and classification. The power means of SPD matrices with exponent p in the interval [–1, 1] interpolate in between the Harmonic mean (p = –1) and the Arithmetic mean (p = 1), while the Geometric (Cartan/Karcher) mean, which is the one currently employed in most applications, corresponds to their limit evaluated at 0. In this paper, we treat the problem of estimating power means along the continuum p ∊ (–1, 1) given noisy observed measurement. We provide a general fixed point algorithm (MPM) and we show that its convergence rate for p = ±0.5 deteriorates very little with the number and dimension of points given as input. Along the whole continuum, MPM is also robust with respect to the dispersion of the points on the manifold (noise), much more than the gradient descent algorithm usually employed to estimate the geometric mean. Thus, MPM is an efficient algorithm for the whole family of power means, including the geometric mean, which by MPM can be approximated with a desired precision by interpolating two solutions obtained with a small ±p value. We also present an approximated version of the MPM algorithm with very low computational complexity for the special case p = ±½. Finally, we show the appeal of power means through the classification of brain–computer interface event-related potentials data.

A special form of SPD covariance matrix for interpretation and visualization of data manipulated with Riemannian geometry

Currently the Riemannian geometry of symmetric positive definite (SPD) matrices is gaining momentum as a powerful tool in a wide range of engineering applications such as image, radar and biomedical data signal processing. If the data is not natively represented in the form of SPD matrices, typically we may summarize them in such form by estimating covariance matrices of the data. However once we manipulate such covariance matrices on the Riemannian manifold we lose the representation in the original data space. For instance, we can evaluate the geometric mean of a set of covariance matrices, but not the geometric mean of the data generating the covariance matrices, the space of interest in which the geometric mean can be interpreted. As a consequence, Riemannian information geometry is often perceived by non-experts as a “black-box” tool and this perception prevents a wider adoption in the scientific community. Hereby we show that we can overcome this limitation by constructing a special form of SPD matr...

Fixed Point Algorithms for Estimating Power Means of Positive Definite Matrices

Estimating means of data points lying on the Riemannian manifold of symmetric positive-definite (SPD) matrices has proved of great utility in applications requiring interpolation, extrapolation, smoothing, signal detection, and classification. The power means of SPD matrices with exponent p in the interval [–1, 1] interpolate in between the Harmonic mean (p = –1) and the Arithmetic mean (p = 1), while the Geometric (Cartan/Karcher) mean, which is the one currently employed in most applications, corresponds to their limit evaluated at 0. In this paper, we treat the problem of estimating power means along the continuum p ∊ (–1, 1) given noisy observed measurement. We provide a general fixed point algorithm (MPM) and we show that its convergence rate for p = ±0.5 deteriorates very little with the number and dimension of points given as input. Along the whole continuum, MPM is also robust with respect to the dispersion of the points on the manifold (noise), much more than the gradient descent algorithm usually employed to estimate the geometric mean. Thus, MPM is an efficient algorithm for the whole family of power means, including the geometric mean, which by MPM can be approximated with a desired precision by interpolating two solutions obtained with a small ±p value. We also present an approximated version of the MPM algorithm with very low computational complexity for the special case p = ±½. Finally, we show the appeal of power means through the classification of brain–computer interface event-related potentials data.