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Table Of Integrals Series And Products

On robust estimation of the location parameter

Stochastic differential equations and diffusion processes

Objective: Most current Electroencephalography (EEG)-based Brain-Computer Interfaces (BCIs) are based on machine learning algorithms. There is a large diversity of classifier types that are used in this field, as described in our 2007 review paper. Now, approximately 10 years after this review publication, many new algorithms have been developed and tested to classify EEG signals in BCIs. The time is therefore ripe for an updated review of EEG classification algorithms for BCIs. Approach: We surveyed the BCI and machine learning literature from 2007 to 2017 to identify the new classification approaches that have been investigated to design BCIs. We synthesize these studies in order to present such algorithms, to report how they were used for BCIs, what were the outcomes, and to identify their pros and cons. Main results: We found that the recently designed classification algorithms for EEG-based BCIs can be divided into four main categories: adaptive classifiers, matrix and tensor classifiers, transfer learning and deep learning, plus a few other miscellaneous classifiers. Among these, adaptive classifiers were demonstrated to be generally superior to static ones, even with unsupervised adaptation. Transfer learning can also prove useful although the benefits of transfer learning remain unpredictable. Riemannian geometry-based methods have reached state-of-the-art performances on multiple BCI problems and deserve to be explored more thoroughly, along with tensor-based methods. Shrinkage linear discriminant analysis and random forests also appear particularly useful for small training samples settings. On the other hand, deep learning methods have not yet shown convincing improvement over state-of-the-art BCI methods. Significance: This paper provides a comprehensive overview of the modern classification algorithms used in EEG-based BCIs, presents the principles of these Review of Classification Algorithms for EEG-based BCI 2 methods and guidelines on when and how to use them. It also identifies a number of challenges to further advance EEG classification in BCI.

https://iopscience.iop.org/article/10.1088/1741-2552/aab2f2/pdf

A Review of Classification Algorithms for EEG-based Brain-Computer Interfaces: A 10-year Update

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

Objective: This paper tackles the problem of transfer learning in the context of electroencephalogram (EEG)-based brain–computer interface (BCI) classification. In particular, the problems of cross-session and cross-subject classification are considered. These problems concern the ability to use data from previous sessions or from a database of past users to calibrate and initialize the classifier, allowing a calibration-less BCI mode of operation. Methods: Data are represented using spatial covariance matrices of the EEG signals, exploiting the recent successful techniques based on the Riemannian geometry of the manifold of symmetric positive definite (SPD) matrices. Cross-session and cross-subject classification can be difficult, due to the many changes intervening between sessions and between subjects, including physiological, environmental, as well as instrumental changes. Here, we propose to affine transform the covariance matrices of every session/subject in order to center them with respect to a reference covariance matrix, making data from different sessions/subjects comparable. Then, classification is performed both using a standard minimum distance to mean classifier, and through a probabilistic classifier recently developed in the literature, based on a density function (mixture of Riemannian Gaussian distributions) defined on the SPD manifold. Results: The improvements in terms of classification performances achieved by introducing the affine transformation are documented with the analysis of two BCI datasets. Conclusion and significance: Hence, we make, through the affine transformation proposed, data from different sessions and subject comparable, providing a significant improvement in the BCI transfer learning problem.

/pdf/transfer-learning-a-riemannian-geometry-framework-with-2nrivr7o0h.pdf

Transfer Learning: A Riemannian Geometry Framework With Applications to Brain–Computer Interfaces

Data, which lie in the space $\mathcal {P}_{m\,}$ , of $m \times m$ symmetric positive definite matrices, (sometimes called tensor data ), play a fundamental role in applications, including medical imaging, computer vision, and radar signal processing. An open challenge, for these applications, is to find a class of probability distributions, which is able to capture the statistical properties of data in $\mathcal {P}_{m\,}$ , as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on $\mathcal {P}_{m\,}$ . Distributions of this kind were first considered by Pennec in 2006. However, the present paper gives an exact expression of their probability density function for the first time in existing literature. This leads to two original contributions. First, a detailed study of statistical inference for Riemannian Gaussian distributions, uncovering the connection between the maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and the implementation of an expectation-maximisation algorithm, for the estimation of mixtures of Riemannian Gaussian distributions. The paper applies this new algorithm, to the classification of data in $\mathcal {P}_{m\,}$ , (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.

https://hal.archives-ouvertes.fr/hal-01248573/document

Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

We construct a two-parameter family of actions 
$\omega_{k,a}$ of the Lie algebra $sl(2,\mathbb R)$ by differential-difference operators
on $\mathbb R^N \setminus \{0\}$.
Here, $k$ is a multiplicity-function for the Dunkl
operators, 
and $a>0$ arises from the interpolation of the two $sl(2,\mathbb R)$ actions on the Weil representation of
$Mp(N,\mathbb R)$ and the minimal unitary representation of $O(N+1,2)$.
We prove that this action $\omega_{k,a}$ lifts 
to a unitary representation of the universal covering of
$SL(2,\mathbb R)$, 
and can even be extended to a holomorphic semigroup $\Omega_{k,a}$.
In the $k\equiv 0$ case,
our semigroup generalizes the Hermite semigroup
studied by R. Howe ($a=2$) and the Laguerre semigroup by
the second author with G. Mano ($a=1$).
One boundary value of our semigroup $\Omega_{k,a}$ provides us with 
\textit{$(k,a)$-generalized Fourier transforms} $\mathcal{F}_{k,a}$,
which includes the Dunkl transform $\mathcal{D}_k$
($a=2)$ and a new unitary operator $\mathcal{H}_k$
($a=1$), namely a Dunkl--Hankel transform.
We establish the inversion formula, 
and a generalization of the Plancherel theorem, 
the Hecke
identity, the Bochner identity, and a Heisenberg uncertainty
relation for $\mathcal{F}_{k,a}$.
We also find kernel functions for $\Omega_{k,a}$ and $\mathcal {F}_{k,a}$
for $a=1,2$ in terms of Bessel functions and the
Dunkl intertwining operator.

/pdf/laguerre-semigroup-and-dunkl-operators-13cywjjfq1.pdf

Laguerre semigroup and Dunkl operators

In this paper, we consider the extension of the Fourier transform to biquaternion-valued signals. We introduce a transform that we call the biquaternion Fourier transform (BiQFT). After giving some general properties of this transform, we show how it can be used to generalize the notion of analytic signal to complex-valued signals. We introduce the notion of hyperanalytic signal. We also study the Hermitian symmetries of the BiQFT and their relation to the geometric nature of a biquaternion-valued signal. Finally, we present a fast algorithm for the computation of the BiQFT. This algorithm is based on a (complex) change of basis and four standard complex FFTs.

Fast Complexified Quaternion Fourier Transform

We construct a two-parameter family of actions \omega_{k,a} of the Lie algebra sl(2,R) by differential-difference operators on R^N \setminus {0}. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the Weil representation of Mp(N,R) and the minimal unitary representation of O(N+1,2) keeping smaller symmetries. 
We prove that this action \omega_{k,a} lifts to a unitary representation of the universal covering of SL(2,R), and can even be extended to a holomorphic semigroup \Omega_{k,a}. In the k\equiv 0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the second author with G. Mano (a=1). 
One boundary value of our semigroup \Omega_{k,a} provides us with (k,a)-generalized Fourier transforms F_{k,a}, which includes the Dunkl transform D_k (a=2) and a new unitary operator H_k (a=1), namely a Dunkl-Hankel transform. 
We establish the inversion formula, and a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

/pdf/laguerre-semigroup-and-dunkl-operators-2bj3vogpxi.pdf

Salem Said

Papers

Transfer Learning: A Riemannian Geometry Framework With Applications to Brain–Computer Interfaces

Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

Laguerre semigroup and Dunkl operators

Fast Complexified Quaternion Fourier Transform

Laguerre semigroup and Dunkl operators