# Riemannian Geometry-Based EEG Approaches: A Literature Review

Imad Eddine Tibermacine1
Samuele Russo2
Ahmed Tibermacine3 Abdelaziz Rabehi4 Bachir Nail4 Kamel Kadri5 Christian
Napoli1,6,7

###### Abstract

The application of Riemannian geometry in the decoding of brain-computer
interfaces (BCIs) has swiftly garnered attention because of its
straightforwardness, precision, and resilience, along with its aptitude for
transfer learning, which has been demonstrated through significant
achievements in global BCI competitions. This paper presents a comprehensive
review of recent advancements in the integration of deep learning with
Riemannian geometry to enhance EEG signal decoding in BCIs. Our review updates
the findings since the last major review in 2017, comparing modern approaches
that utilize deep learning to improve the handling of non-Euclidean data
structures inherent in EEG signals. We discuss how these approaches not only
tackle the traditional challenges of noise sensitivity, non-stationarity, and
lengthy calibration times but also introduce novel classification frameworks
and signal processing techniques to reduce these limitations significantly.
Furthermore, we identify current shortcomings and propose future research
directions in manifold learning and riemannian-based classification, focusing
on practical implementations and theoretical expansions, such as feature
tracking on manifolds, multitask learning, feature extraction, and transfer
learning. This review aims to bridge the gap between theoretical research and
practical, real-world applications, making sophisticated mathematical
approaches accessible and actionable for BCI enhancements.

1Department of Computer, Automation and Management Engineering, Sapienza
University of Rome, Rome, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Department of Psychology, Sapienza University of Rome, Rome, Italy
<EMAIL_ADDRESS>
3Department of Computer Science, University of Biskra, Biskra, Algeria
<EMAIL_ADDRESS>
4Faculty of Science and Technology, University of Djelfa, Djelfa, Algeria
<EMAIL_ADDRESS><EMAIL_ADDRESS>
5Polytechnic Military School, Algiers, Algeria<EMAIL_ADDRESS>}
6Institute for Systems Analysis and Computer Science, Italian National
Research Council, Italy
7Department of Computational Intelligence, Czestochowa University of
Technology, Poland

_Keywords_ EEG $\cdot$ Riemannian Geometry $\cdot$ Deep Learning $\cdot$
Classification

##  1 Introduction

Brain-computer interfaces [9] revolutionize the field of human-machine
interactions by enabling direct communication pathways between the brain and
external devices, bypassing traditional muscular outputs[1]. Predominantly,
BCIs utilize electroencephalography (EEG) due to its cost-effectiveness and
minimal computational demands[2]. EEG-based BCIs convert brain activity,
measured through EEG signals, into computer commands, leveraging
classification algorithms like the common spatial pattern (CSP) to recognize
signal patterns associated with specific brain activities such as motor
imagery[3]. Despite their promise, EEG-based BCIs face critical challenges
including high signal variability, noise, non-stationarity, and the high
dimensionality of data spaces[4].

The domain of BCI has expanded its utility beyond basic research, offering
profound applications in rehabilitation, assistive technologies, and adaptive
human-computer interfaces that respond to the user’s mental states[5]. Yet,
despite these advancements, BCIs are predominantly still in the prototype
stage, rarely deployed outside laboratory settings due to their low robustness
and the extensive calibration required for each user[6]. Current BCIs suffer
from poor reliability, often misinterpreting user commands, which
significantly reduces accuracy and information transfer rates even under
controlled conditions[7]. This sensitivity to environmental noise and the
inherent non-stationarity of EEG signals further complicates their practical
application[8].

In the rapidly evolving field of BCIs, deep learning has emerged as a
transformative force, enabling sophisticated methods for classifying EEG
data[10]. These advancements hold the promise of revolutionizing the
interaction between humans and machines by interpreting neural activities
directly through EEG signals[11]. Despite these advances, the development of
BCIs faces significant challenges, notably the limited availability of large,
annotated training datasets, which are crucial for training robust deep
learning models[12]. In response, researchers have increasingly turned to
generative modeling, a burgeoning area within machine learning, to augment
existing datasets through the synthesis of synthetic EEG data[13]. This
approach not only enhances the diversity of training examples but also
addresses the gap caused by scarce data resources[14].

Achieving practical usability of BCIs requires that they be robust across
different contexts, users, and over time, all while maintaining minimal
calibration requirements[15][24]. Addressing these multifaceted challenges
necessitates a comprehensive strategy: identifying new, reliable
neurophysiological markers at the neuroscience level[16]; improving user
training to stabilize EEG pattern generation at the human interaction
level[17]; and advancing signal processing techniques to enhance feature
extraction[18] and classifier robustness at the computational level[19][25].
Prominently, the application of Riemannian geometry[20] to handle EEG
covariance matrices has shown considerable promise[21]. This geometric
approach not only accommodates the non-Euclidean characteristics of EEG data
but also leverages the intrinsic geometric properties of these data to enhance
classification accuracy[22][23].

Traditional approaches like Fourier analysis[26] frequently fail due to the
inherently nonstationary nature of EEG signals. This limitation has paved the
way for the adoption of time-frequency analysis techniques, such as the
wavelet transform, which are now being integrated with spatial filtering[27]
methods like the CSP to improve the localizability of rhythmic EEG components.
Such advancements are critical for enhancing the efficacy of motor imagery
(MI) EEG classifiers, which are vital for real-time BCI applications[28].

Moreover, a significant aspect of reducing signal variability and enhancing
classifier performance in BCIs[29] involves focusing on covariance matrices,
typically handled as elements within Euclidean spaces. However, considering
these matrices in their naturally occurring Riemannian manifold[30], which
better reflects their symmetric and positive definite (SPD) properties[31],
can substantially optimize the computational processes. Techniques such as
Principal Components Analysis (PCA)[32] and Canonical Correlation Analysis
(CCA)[33] have traditionally exploited covariance estimates for spatial
filtering in BCIs[34]. By refining how these matrices are
treated—acknowledging their Riemannian structure—we can significantly enhance
the spatial and temporal resolution of EEG signal analysis[35].

This paper presents a detailed review focused on the applications of
Riemannian Geometry and deep learning-based Riemannian geometry in the domains
of BCIs and EEG. We explore how these advanced methodologies enhance BCI
design by addressing core challenges inherent in traditional systems. Our
discussion extends to the integration of geometric deep learning with
generative modeling and sophisticated signal processing techniques. We propose
new research directions and outline strategic advancements necessary for
developing BCIs that are not only efficacious in controlled settings but also
robust and accessible for practical, everyday applications.

##  2 Brain-Computer Interface

Brain-computer interfaces that utilize oscillatory neural activity are a
significant branch of non-invasive systems for neurocommunication and
control[36]. These interfaces typically leverage the modulation of brain
rhythms captured via EEG, and are commonly employed in motor imagery-based
BCIs[37]. Such BCIs extensively use the CSP algorithm combined with Linear
Discriminant Analysis for signal classification, making them highly effective
for decoding user intentions based on neural activity patterns[38].

The CSP algorithm is pivotal for maximizing the variance of signals from one
mental state while simultaneously minimizing it for another[39]. This is
achieved by designing spatial filters that enhance the discriminability
between two contrasting classes of brain activity. Mathematically, the
optimization of CSP filters, represented as column vectors $\mathbf{w}$, is
framed as an eigenvalue problem aimed at maximizing the following objective
function:

$J_{\text{CSP}}(\mathbf{w})=\frac{\mathbf{w}^{T}C_{1}\mathbf{w}}{\mathbf{w}^{T}C_{2}\mathbf{w}}$
(1)

Here, $C_{1}$ and $C_{2}$ are the averaged covariance matrices of the EEG
signals filtered for two different classes, which are computed as:

$C_{j}=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\mathbf{S}_{j}^{i}=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\mathbf{Z}_{j}^{i}(\mathbf{Z}_{j}^{i})^{T}$
(2)

where $N_{j}$ is the number of trials for class $j$, and $\mathbf{Z}_{j}^{i}$
represents the $i$-th trial from class $j$ matrix in a channel-filtered EEG
dataset[40].

The CSP algorithm essentially reduces the multi-channel EEG data into a lower-
dimensional space[41], where the spatial filters $\mathbf{w}$ corresponding to
the largest (or smallest) eigenvalues capture the most (or least)
discriminative features of the EEG signals.

Upon obtaining the optimal spatial filters, the transformed feature $f_{i}$
for each trial is calculated using:

$f_{i}=\log(\mathbf{w}^{T}S_{i}\mathbf{w})$ (3)

where $S_{i}$ is the covariance matrix of the current trial. These
logarithmically transformed variances are then fed into an LDA classifier,
which constructs a hyperplane to distinguish between the classes by solving:

$a=\mathbf{C}^{-1}(\mu_{1}-\mu_{2}),\quad
b=-\frac{1}{2}(\mu_{1}+\mu_{2})^{T}a$ (4)

where $\mu_{1}$ and $\mu_{2}$ are the mean feature vectors of each class, and
$\mathbf{C}$ is the pooled covariance matrix of the features. LDA assumes
Gaussian class conditional densities with equal covariance matrices for both
classes[42], which simplifies the computation of the decision
boundaries[43][44].

It is crucial to note that while CSP and LDA provide a robust framework for
feature extraction and classification[45], their efficacy depends on
accurately estimated covariance matrices, which require sufficient trial data
to avoid overfitting. Moreover, the assumption of linear separability[46] by
LDA may not always hold in complex brain activity patterns, suggesting the
need for more sophisticated nonlinear classifiers or deep learning approaches
in more advanced BCI systems[47][48].

The mathematical rigor involved in optimizing CSP and employing LDA
classifiers in oscillatory activity-based BCIs[49] showcases the intricate
balance between statistical assumptions and practical EEG data
characteristics[50]. Enhancements in computational strategies and model
robustness are continually evolving, aiming to improve the adaptability and
accuracy of BCIs in real-world applications[51].

##  3 Riemannian Geometry Concepts

In this section, we introduce the mathematical notations and basic definitions
fundamental to our study on EEG preprocessing. Scalars are denoted by
lowercase letters (e.g., $a$), vectors by boldface lowercase (e.g.,
$\mathbf{v}$), representing column vectors unless stated otherwise, matrices
by boldface uppercase (e.g., $\mathbf{M}$), and tensors of order three or
higher by Euler script letters (e.g., $\mathcal{T}$). The space
$\mathbb{R}^{n}$ represents the $n$-dimensional real vector space, and
$\mathbb{R}^{n\times n}$ denotes all $n\times n$ real matrices. Vector
$\mathbf{e}_{n}$, comprising all ones, belongs to $\mathbb{R}^{n}$, and
$\mathbf{I}_{n}$ denotes the identity matrix in $\mathbb{R}^{n\times n}$. For
any vector $\mathbf{x}$, the matrix $\text{diag}(\mathbf{x})$ refers to a
diagonal matrix with the elements of $\mathbf{x}$ on its main diagonal.

Figure 1: The SPD Manifold pipeline[76]

We denote by $\text{vec}(\mathbf{A})$ the operation that converts a matrix
$\mathbf{A}$ into a vector by stacking its columns. In the special case where
$\mathbf{A}$ is symmetric, the vectorization $\text{vec}(\mathbf{A})$ is
defined such that it takes only the unique entries due to symmetry. The norms
$\|\cdot\|_{p}$ and $\|\cdot\|_{F}$ correspond to the $L_{p}$ and Frobenius
norms of a vector or matrix, respectively.

Definition 1.1. A matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ is deemed
Symmetric Positive Definite (SPD) if it holds that $\mathbf{A}=\mathbf{A}^{T}$
and $\mathbf{x}^{T}\mathbf{A}\mathbf{x}>0$ for all non-zero vectors
$\mathbf{x}\in\mathbb{R}^{n}$. The eigenvalues of such a matrix $\mathbf{A}$,
denoted by $\lambda(\mathbf{A})$, are guaranteed to be positive.

Definition 1.2. A matrix $\mathbf{A}$ is considered orthogonal if its columns
comprise an orthogonal unit vector set, implying
$\mathbf{A}^{T}\mathbf{A}=\mathbf{I}_{n}$.

Definition 1.3. The exponential and logarithmic functions of a matrix
$\mathbf{A}\in\mathbb{R}^{n\times n}$, denoted by $\exp(\mathbf{A})$ and
$\log(\mathbf{A})$, are defined through its eigenvalue decomposition. If
$\mathbf{A}$ is expressible as
$\mathbf{U}\text{diag}(\lambda_{1},\ldots,\lambda_{n})\mathbf{U}^{T}$, where
$\mathbf{U}$ is orthogonal and $\lambda_{i}$ are the eigenvalues of
$\mathbf{A}$, then:

$\exp(\mathbf{A})=\mathbf{U}\text{diag}(\exp(\lambda_{1}),\ldots,\exp(\lambda_{n}))\mathbf{U}^{T},$
(5)
$\log(\mathbf{A})=\mathbf{U}\text{diag}(\log(\lambda_{1}),\ldots,\log(\lambda_{n}))\mathbf{U}^{T}.$
(6)

A manifold, in the simplest terms, refers to a space that, at a local level,
resembles the Euclidean space of dimension $n$, known as $\mathbb{R}^{n}$.
This property of local similarity to $\mathbb{R}^{n}$ qualifies it as a
topological manifold [52]. When such a space is equipped with a differential
structure, it ascends to become a differentiable manifold, thereby enabling
the attachment of a tangent space at any point [53]. The tangent space
encapsulates all potential vectors that are tangential to any conceivable
curve passing through that point on the manifold.

Within the manifold paradigm, a Riemannian manifold stands out as a
differentiable manifold that is further enhanced with a smoothly varying inner
product on its tangent spaces, constituting the Riemannian metric tensor [54].
This metric tensor facilitates the measurement of angles between vectors
within the same tangent space, the magnitude of vectors, and the distance
between vectors. It allows for the calculation of curve lengths on the
manifold, including the geodesic distance which is the shortest distance
between two points on the manifold determined by the paths of curves [55].

Considering $\mathcal{M}$ as a Riemannian manifold and $T_{X}\mathcal{M}$ as
its tangent space at point $X$, for any two vectors $V,W\in T_{X}\mathcal{M}$,
the Riemannian metric $g_{X}(V,W)$ provides the inner product [56]. When
discussing the manifold of SPD matrices, our focus is particularly on the
tangent space $T_{X}\mathcal{M}$ and the Riemannian metric at that point.

For a smooth curve $\gamma:[0,1]\rightarrow\mathcal{M}$, parametrized with
respect to a metric on $\mathcal{M}$, there are numerous ways to describe its
trajectory, each with a different speed of traversal. Of particular interest
are those curves that are parametrized by arc length, termed naturally
parametrized curves [57]. These curves progress with a uniform rate and are
defined in relation to the Riemannian metric. The curve’s length from
$\gamma(0)$ to $\gamma(1)$ is computed by integrating the metric-induced
velocity along the curve.

For two distinct points $X_{1},X_{2}\in\mathcal{M}$, and a naturally
parametrized curve $\gamma$ that connects these points with $\gamma(0)=X_{1}$
and $\gamma(1)=X_{2}$, the length $L(\gamma)$ of this curve is expressed by
the integral[52]:

$L(\gamma)=\int_{0}^{1}\sqrt{g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))}\,dt$
(7)

The optimal path on a manifold that minimizes the distance between two points
is called a geodesic. While geodesics represent the paths of least distance,
they are uniquely characterized by a uniform rate of traversal and are not
necessarily the shortest paths for manifolds that are not simply connected.
These paths, particularly on a spherical surface, might have multiple
representations, such as the numerous geodesics connecting the poles [55]. The
distance along a geodesic, known as the Riemannian distance between two points
$X_{1}$ and $X_{2}$, is essential in determining the manifold’s completeness.
Here, we concentrate on manifolds that are geometrically complete [54].

The tangent space at point $X$ on a manifold $\mathcal{M}$, denoted
$T_{X}\mathcal{M}$, acts as a local linear approximation to $\mathcal{M}$
within a specific region around $X$, usually within a certain radius allowing
for a bijection via the exponential map [58]. For every point $X^{\prime}$
within this neighborhood, the exponential map is the bridge connecting
$X^{\prime}$ to $X$ through a unique geodesic.

For any smooth scalar function $f$ defined on $\mathcal{M}$, the Riemannian
gradient at $X$ is given as $\nabla f(X)$ in the direction that maximizes the
rate of increase of $f$. If $\gamma$ is a geodesic with $\gamma(0)=X$ and
$\dot{\gamma}(0)=V$, then $\nabla f\circ\gamma$ describes the change of $f$
along $\gamma$, and $\nabla f(X)$ is that vector in $T_{X}\mathcal{M}$ such
that the inner product $\langle V,\nabla f(X)\rangle$ equals the derivative of
$f$ along the curve at $t=0$, defined as[58]:

$\langle V,\nabla f(X)\rangle=\frac{d}{dt}f(\gamma(t))\bigg{|}_{t=0}.$ (8)

The Riemannian gradient facilitates the computation of directional derivatives
on manifolds, linking classical differential calculus to the geometry of the
manifold. The exponential map, denoted as
$\text{Exp}_{X}:T_{X}\mathcal{M}\rightarrow\mathcal{M}$, projects a tangent
vector at $X$ onto $\mathcal{M}$, and conversely, the logarithm map, or
$\text{Log}_{X}$, maps points back to $T_{X}\mathcal{M}$, providing a method
to traverse between a tangent space and its manifold [59].

Assigning a set $\mathcal{M}$ of square matrices the structure of a Riemannian
manifold incorporates a local Euclidean geometry, thereby enriching it with a
rigorous mathematical underpinning. Suppose we have
$M^{\prime}\in\mathcal{M}$, a manifold of dimension $K$, and
$\xi_{M^{\prime}}=M^{\prime}-M$ representing a tangent vector at $M$. This
vector is part of a higher-dimensional tangent space $T_{M}\mathcal{M}$
associated with $M$.

The inner product defined by the Riemannian metric on $T_{M}\mathcal{M}\times
T_{M}\mathcal{M}\rightarrow\mathbb{R}$ induces a norm on the tangent space,
given by
$\|\xi_{M^{\prime}}\|_{M}=\sqrt{\langle\xi_{M^{\prime}},\xi_{M^{\prime}}\rangle_{M}}$,
and facilitates the computation of geodesic distances $d(M,M^{\prime})$ on
$\mathcal{M}$ [59]. These distances enable the formulation of mean values of
points on the manifold as:

$\text{Mean}(\\{M_{1},\dots,M_{N}\\})=\underset{M\in\mathcal{M}}{\arg\min}\sum_{i=1}^{N}d(M,M_{i})^{2}.$
(9)

The exponential mapping at $M$ in $\mathcal{M}$, denoted as $\text{Exp}_{M}$,
and its inverse, the logarithm mapping $\text{Log}_{M}$, are both essential in
preserving the manifold’s structure, especially when mapping to and from the
tangent space. $\text{Exp}_{M}(\xi_{M})$ approximates $M^{\prime}$ when
$\xi_{M}$ is small, and the distance between $M$ and $M^{\prime}$ is given by
the norm of $\xi_{M}$ in $\mathbb{R}^{K}$ via $\text{Log}_{M}$. This
relationship allows the introduction of vectorization for $\mathcal{M}$,
$\mathcal{P}_{M}:\mathcal{M}\rightarrow\mathbb{R}^{K}$, defined as
$\mathcal{P}_{M}(M^{\prime})=\phi(\text{Log}_{M}(M^{\prime}))$ [59].

For a compact subset of $\mathcal{M}$ where the matrices $\\{M_{i}\\}$ lie,
the mean $M$ can be approximated, resulting in a simplified geodesic distance
expression:

$d(M,M^{\prime})\approx\|\mathcal{P}_{M}(M)-\mathcal{P}_{M}(M^{\prime})\|.$
(10)

In the realm of regression on manifolds, the vectorization $\mathcal{P}_{M}$
is pivotal for leveraging machine learning algorithms. It adapts matrix points
in $\mathcal{M}$ onto $\mathbb{R}^{K}$, facilitating their use in regression
techniques that typically assume a Euclidean data structure. For instance,
with a collection $\\{M_{i}\\}$ and corresponding response variables
$\\{y_{i}\\}$, one first computes the mean of the samples $M$ and applies
vectorization to obtain $\\{v_{i}\\}$. Linear regression methods, such as
ridge regression, can then be applied, presupposing a linear relationship
$y_{i}\approx v_{i}^{T}\beta$, with $\beta$ representing the regression
coefficients [60].

###  3.1 The Covariance Matrix of EEG

Consider $\mathbf{X}\in\mathbb{R}^{M\times T}$, representing an EEG signal
that has undergone band-pass filtering, where $M$ denotes the number of
channels and $T$ represents the number of temporal samples. To analyze the
statistical properties of the EEG signal, we construct the covariance matrix
$\mathbf{P}$ as follows:

$\mathbf{P}=\frac{1}{T-1}\mathbf{X}\mathbf{X}^{T}$ (11)

This matrix $\mathbf{P}$ is not only symmetric but also empirically confirmed
to be SPD, encapsulating important statistical information about the EEG
signals. The symmetry of $\mathbf{P}$ arises because
$(\mathbf{X}\mathbf{X}^{T})^{T}=\mathbf{X}\mathbf{X}^{T}$, and it attains its
positive-definiteness under the condition that $\mathbf{X}$ has full row rank,
which is typically satisfied if $T>M$ and the EEG data are sufficiently
diverse [61].

Properties of Symmetric Positive-Definite Matrices: A matrix
$\mathbf{A}\in\mathbb{R}^{n\times n}$ is deemed positive-definite if for any
non-zero vector $\mathbf{v}\in\mathbb{R}^{n}$, it holds that:

$\mathbf{v}^{T}\mathbf{A}\mathbf{v}>0$ (12)

Applying this to our covariance matrix $\mathbf{P}$[62], we observe:

$\mathbf{v}^{T}\mathbf{P}\mathbf{v}=\mathbf{v}^{T}\left(\frac{1}{T-1}\mathbf{X}\mathbf{X}^{T}\right)\mathbf{v}=\frac{1}{T-1}\|\mathbf{X}^{T}\mathbf{v}\|^{2}>0$
(13)

where $\|\mathbf{X}^{T}\mathbf{v}\|$ represents the Euclidean norm of
$\mathbf{X}^{T}\mathbf{v}$, which is greater than zero unless $\mathbf{v}$ is
orthogonal to all rows of $\mathbf{X}$. This ensures that $\mathbf{P}$ retains
all the characteristics of an SPD matrix, including its utility in defining a
metric space on the manifold of such matrices [63].

Geometric Interpretation in EEG Analysis: As an element of the manifold of SPD
matrices, $\mathbf{P}$ can be further analyzed using tools of Riemannian
geometry. The diagonal elements of $\mathbf{P}$ represent the variance of the
filtered EEG signal at each channel, indicating the power spectrum. The off-
diagonal elements provide a measure of covariance between different channels,
reflecting how signal components vary together over time—a crucial aspect for
understanding functional connectivity in the brain [64]. Distances between
such matrices, computed via geodesic paths on this manifold, offer a natural
measure of similarity between different EEG states or conditions, enhancing
applications like classification and clustering in brain-computer interfaces
[63].

Incorporating these mathematical insights enriches our understanding of how
the covariance matrix functions within EEG signal analysis and underscores the
significance of employing a Riemannian framework to advance analytical
capabilities.

###  3.2 Riemannian Metrics on SPD Manifolds

The choice of Riemannian metric on the manifold of SPD matrices significantly
influences the analysis and processing of data represented by these matrices,
such as EEG signals. The Log-Euclidean Metric (LEM) and the Affine-Invariant
Metric (AIM) are two predominant metrics used in this context [63, 65].

Log-Euclidean Metric (LEM): The Log-Euclidean Metric (LEM) offers an efficient
and robust computational method for handling the manifold of SPD matrices.
This metric is particularly valued for preserving the bi-invariance property
within the Lie group structure of SPD matrices, making it suitable for various
applications, including image processing and medical imaging analysis, where
maintaining the structure of data during transformations is crucial [65].

Under the LEM, the geodesic distance between two points $P_{1}$ and $P_{2}$ on
the manifold of SPD matrices is defined as:

$\delta_{L}(P_{1},P_{2})=\|\operatorname{Log}(P_{1})-\operatorname{Log}(P_{2})\|_{F},$
(14)

where $\operatorname{Log}$ denotes the matrix logarithm, transforming matrices
to a space where the Euclidean tools can be applied. The norm $\|\cdot\|_{F}$
represents the Frobenius norm, which measures matrix entries’ absolute
differences, thus providing a natural distance metric in the logarithmic
domain [65].

The Log-Euclidean mean of a set of matrices, crucial for statistical analysis
on manifolds, is calculated using:

$G=\operatorname{Exp}\left(\frac{1}{k}\sum_{i=1}^{k}\operatorname{Log}(P_{i})\right),$
(15)

This formulation allows the mean of matrices to be computed efficiently and
without the convergence issues that may arise with other Riemannian metrics.
Additionally, when considering a weighted mean where each matrix $P_{i}$ is
assigned a weight $w_{i}$, fulfilling the conditions $\sum_{i=1}^{k}w_{i}=1$
and $w_{i}>0$, the weighted mean is given by:

$G=\operatorname{Exp}\left(\sum_{i=1}^{k}w_{i}\operatorname{Log}(P_{i})\right).$
(16)

Affine-Invariant Metric (AIM): The AIM is another highly regarded metric for
SPD manifolds, especially valued for its invariant properties under affine
transformations. The geodesic distance under this metric between $P_{1}$ and
$P_{2}$ is computed as:

$\delta_{A}(P_{1},P_{2})=\left\|\operatorname{Log}(P_{1}^{-1/2}P_{2}P_{1}^{-1/2})\right\|_{F},$
(17)

reflecting the minimal invariant distance under congruent transformations
[63]. The mean under AIM, often used for central tendency estimation in
statistics on manifolds, does not have a closed-form solution and is typically
computed numerically [66].

Other Important Riemannian Metrics:

1\. Fisher-Rao Metric: Originally developed in the context of information
geometry, the Fisher-Rao metric has been adapted to the geometry of SPD
manifolds. It provides a statistical interpretation of the distance,
correlating with the intrinsic curvature of the data space, making it suitable
for probabilistic interpretations of data [67].

2\. Cholesky Decomposition-based Metric: This metric leverages the Cholesky
decomposition of SPD matrices, enabling an alternative parametrization of the
manifold. It simplifies certain computational procedures, particularly in
optimization contexts, by linearizing the manifold using lower triangular
matrices [68].

3\. Procrustes Metric: Focused on shape analysis, the Procrustes metric
measures the similarity between shapes after transformations like translation,
scaling, and rotation. When adapted to SPD matrices, it helps analyze shape
changes in data structures characterized by these matrices [60].

Each of these metrics offers unique advantages depending on the specific
requirements of the application, such as computational efficiency, robustness
to transformations, or interpretability in statistical analysis. Choosing the
appropriate metric is crucial for effectively analyzing and interpreting the
geometric structure of data residing on SPD manifolds.

###  3.3 Tangent Space at a Point on a Manifold

The concept of tangent space is central to understanding the local geometry of
manifolds and plays a critical role in the analysis of data that lie on these
manifolds, such as covariance matrices of EEG signals. A tangent space at a
point on a manifold provides a linear approximation of the manifold near that
point, facilitating operations like vector addition and scalar multiplication
which are not inherently defined on the manifold itself [52].

Let $\mathcal{M}$ be a smooth manifold and $p$ a point on $\mathcal{M}$. The
tangent space to $\mathcal{M}$ at $p$, denoted as $T_{p}\mathcal{M}$, is a
vector space consisting of the tangent vectors to all possible curves through
$p$ on the manifold. Formally, if $\gamma:(-\epsilon,\epsilon)\to\mathcal{M}$
is a smooth curve with $\gamma(0)=p$, then the derivative
$\gamma^{\prime}(0)$, which represents the velocity vector of $\gamma$ at $p$,
is a tangent vector at $p$ [53].

For practical computation, particularly in applications involving SPD
matrices, we express tangent vectors in coordinates. If $\mathcal{M}$ is
parameterized locally around $p$ by a coordinate system
$\phi:U\subset\mathbb{R}^{n}\to\mathcal{M}$, where $U$ is an open set in
$\mathbb{R}^{n}$, the tangent vectors can be expressed as:

$\mathbf{v}=\sum_{i=1}^{n}v^{i}\frac{\partial}{\partial x^{i}}\bigg{|}_{p}$
(18)

where $v^{i}$ are components of $\mathbf{v}$ in the coordinate basis
$\left\\{\frac{\partial}{\partial x^{i}}\bigg{|}_{p}\right\\}$ [52].

In the context of SPD matrices, which form an open subset of the space of
symmetric matrices, the tangent space at any point $P\in\mathcal{SPD}(n)$ can
be identified with the space of symmetric matrices. If $P$ is an SPD matrix
and $\mathbf{S}$ is a symmetric matrix, then a curve on $\mathcal{SPD}(n)$
passing through $P$ with direction $\mathbf{S}$ at $t=0$ can be represented
as:

$\gamma(t)=P^{1/2}\exp(tP^{-1/2}\mathbf{S}P^{-1/2})P^{1/2}$ (19)

where $\exp$ denotes the matrix exponential. The derivative of this curve at
$t=0$ gives a tangent vector at $P$, which is precisely $\mathbf{S}$ [63].

Understanding the tangent space of SPD matrices is crucial for developing
algorithms in EEG signal processing. For example, the process of projecting
EEG covariance matrices onto the tangent space allows for the linearization of
the manifold structure, simplifying computations such as averaging or
classification. This projection involves mapping SPD matrices to their tangent
spaces at a chosen reference point, usually the mean covariance matrix, and
performing linear operations in this vector space before projecting back to
the manifold [69].

Figure 2: Illustration of the gradient descent algorithms on the Riemannian
product manifold[125].

###  3.4 Geodesic Distances on Riemannian Manifolds

An SPD matrix of size $M\times M$ resides on a manifold denoted as
$\mathcal{P}(M)$, characterized by its smooth and curved structure. On such
Riemannian manifolds, the concept of geodesics plays a crucial role in
understanding and quantifying the intrinsic geometric properties of the space
[70].

Geodesics are the generalization of "straight lines" to Riemannian manifolds
and represent the shortest path between two points on the manifold.
Mathematically, a geodesic $\gamma(t)$ on a manifold $\mathcal{M}$ satisfies
the geodesic equation:

$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0,$ (20)

where $\nabla$ denotes the Levi-Civita connection associated with the
manifold’s Riemannian metric, and $\dot{\gamma}(t)$ is the tangent vector to
$\gamma$ at $t$. This equation ensures that the acceleration of $\gamma$ is
zero, signifying that the path has no deviation other than what is necessary
to remain on the manifold [54].

To compute a geodesic path under the Log-Euclidean framework, one uses the
exponential and logarithmic maps. For matrices $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$, the geodesic connecting them can be parameterized as:

$\gamma(t)=\operatorname{Exp}\left((1-t)\operatorname{Log}(\mathbf{P}_{1})+t\operatorname{Log}(\mathbf{P}_{2})\right)$
(21)

for $t\in[0,1]$, where $\operatorname{Exp}$ is the matrix exponential. This
expression provides an explicit parametric form for the geodesic, connecting
two points on the manifold in a manner that respects the curvature and
topological constraints [65].

The curvature of the manifold, which affects the shape of geodesics, can be
quantified through the Riemann curvature tensor $R$, defined by:

$R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$ (22)

for vector fields $X,Y,$ and $Z$. The curvature informs us about how the
metric space bends locally and impacts geodesic deviation [70].

The ability to measure and analyze distances between EEG covariance matrices
using geodesics[71] is invaluable for tasks such as classification and
clustering within brain-computer interfaces[72]. Understanding the manifold’s
geometric structure[73] allows for more nuanced insights into brain
connectivity patterns and functional dynamics, facilitating advanced
neuroscientific analyses[74][75].

##  4 Riemannian Geometry-Based EEG Approaches

This article presents a synthesis of discoveries from a meticulously curated
collection of 42 publications, manually chosen based on their pertinence to
the fusion of Riemannian geometry and deep learning in brain-computer
interfaces (BCIs). The selection of articles was made through a focused
exploration conducted on Google Scholar and PubMed. The exploration, executed
with the objective of encompassing the most recent and pertinent research up
to May 3rd, 2024, utilized a search query formulated as: (“Riemannian
geometry” OR “manifold learning” Or “SPD Manifolds”) AND (“brain computer
interface” OR “EEG”). This curation procedure encompassed both peer-reviewed
publications and preprints, ensuring a thorough incorporation of the newest
advancements and dialogues in the domain.

To furnish a structured and comprehensive analysis of the vast body of
literature concerning the utilization of Riemannian geometry in brain-computer
interfaces, the chosen studies have been classified into five distinct
domains: Feature Extraction, Classification, Manifold Learning, Tangent Space
Methods, and Transfer Learning. This classification was selected to mirror the
foundational phases of BCI signal processing and the specific obstacles that
each phase poses. The core elements of BCI systems are encapsulated in feature
extraction and classification, with a focus on extracting meaningful patterns
from raw EEG data and subsequently utilizing these patterns for making
predictions or decisions. Manifold learning and tangent space methods are
segregated due to their significance in addressing the intrinsic non-Euclidean
nature of EEG data, enabling a more intricate examination of how Riemannian
geometry can enhance these procedures. Notably, transfer learning is
delineated as a distinct category owing to its escalating importance in
augmenting the adaptability and efficiency of BCIs across various subjects and
sessions, showcasing the pragmatic implications of Riemannian techniques in
real-world scenarios.

Table 1: Overview of the reviewed related works Category | Papers
---|---
Feature Extraction |  [104],[109],[90],[117]
Classification Approaches |  [94],[80],[79],[98],[85],[123],[122],[77],[102],[101],[91],[116],[113]
Manifold Learning |  [83],[82],[86],[87],[96],[86],[97] ,[115],[124],[100],[99],[106]
Tangent Space Approaches |  [108],[93],[92],[95],[81],[119],[78]
Transfer Learning Approaches |  [84],[110],[107],[89],[114]

###  4.1 Feature Extraction Methods

Feature extraction is a SOTA necessary step after collecting EEG data. It
allows reducing the dimensionality of the data and making it more comprehended
by machine learning and deep learning models. Various studies have explored
the use of Riemannian geometry for feature extraction and discriminant
analysis in EEG-based BCIs. In [104], the authors used Covariance Matrices as
EEG Signal Descriptors and exploring various dissimilarity metrics on SPD
matrices and introducing a novel feature, HORC, which combines different
relevance matrices under a tensor framework for improved classification
accuracy. A Riemannian Spatial Pattern (RSP) method was proposed in [109] to
extract spatial patterns from Riemannian classification for motor imagery
tasks. The RSP method uses a backward channel selection procedure and compares
it with the Common Spatial Pattern (CSP) approach. The RSP method provides
precise mapping and clustering of imagined motor movements, which is
especially useful in differentiating finger flexions. The spatial pattern
extraction can be described using:

$C=\frac{1}{N}\sum_{i=1}^{N}x_{i}x_{i}^{T}$ (23)

where $C$ is the covariance matrix, and $x_{i}$ represents the EEG signals.

Huang et al. [90] introduce the Common Amplitude-Phase Measurement (CAPM)
method, designed to simultaneously analyze the amplitude and phase information
of EEG signals on a Riemannian manifold. This dual consideration promises to
enhance classification accuracy for BCI applications significantly. The
initial step involves extracting the amplitude and phase from EEG signals
using the wavelet transform, specifically applying a complex Morlet wavelet.
The transformation is mathematically represented as:

$\hat{x}_{c}(t,f)=x_{c}(t)*\psi(t,f)=a_{c}(f,t)e^{i\theta_{c}(f,t)}$ (24)

where $\hat{x}_{c}(t,f)$ denotes the complex signal representation at channel
$c$, time $t$, and frequency $f$. Here, $a_{c}(f,t)$ is the amplitude,
$\theta_{c}(f,t)$ represents the phase, and $\psi(t,f)$ is the Morlet wavelet
function. To dimensionally reduce while preserving discriminative features,
the authors devise a Riemannian graph embedding technique. The adjacency
matrix $S$, based on the Riemannian distance between the SPD covariance
matrices of EEG trials, is defined by:

$s_{nr}=\begin{cases}\exp\left(-\frac{\delta_{nr}}{\sigma}\right),&\text{if
}Y_{n}\in C_{z}(Y_{r})\text{ and }Y_{r}\in C_{z}(Y_{n})\\\
0,&\text{otherwise}\end{cases}$ (25)

where $\delta_{nr}$ is the Riemannian distance between covariance matrices
$P_{n}$ and $P_{r}$:

$\delta_{nr}=\delta_{R}(P_{n},P_{r})=\left\|\log(P_{n}^{-1}P_{r})\right\|=\left(\sum_{l=1}^{L}\log^{2}\beta_{l}\right)^{1/2}$
(26)

The low-dimensional data $\hat{Y}_{n}$ is used to compute a new covariance
matrix $\hat{P}_{n}=W^{H}P_{n}W$. Classification is performed using the
Minimum Distance to Riemannian Mean (MDRM):

$z_{n}=\text{sign}\left[\left(\sum_{m=1}^{M}\log^{2}\eta_{m}\right)^{1/2}-\left(\sum_{m=1}^{M}\log^{2}\rho_{m}\right)^{1/2}\right]$
(27)

where $\eta_{m}$ and $\rho_{m}$ are the eigenvalues of
$\hat{P}_{-1}^{-1}\hat{P}_{n}$ and $\hat{P}_{+1}^{-1}\hat{P}_{n}$
respectively. A regularized linear regression model is then applied to
optimize the classification parameter $b$:

$\min_{b}\frac{1}{2}\left\|z-Db\right\|^{2}+\lambda(1-\alpha)\left\|b\right\|_{2}^{2}+\lambda\alpha\left\|b\right\|_{1}$
(28)

where $D$ is the matrix of input vectors $d_{n}$ and $z$ is the corresponding
label vector.

The CAPM method effectively captures the intrinsic amplitude and phase
information of EEG signals, optimizing the spatial-spectral filters and
enhancing classification performance through Riemannian geometry-based
regularization. Experimental results demonstrate significant improvements in
classification accuracy on BCI competition datasets.

Gurve et al. [117] proposed a framework for classifying motor imagery EEG data
using covariance matrices as descriptors and investigating various
dissimilarity metrics on the manifold of SPD matrices. The study compares the
performance of Log-Euclidean distance, Stein divergence, Kullback–Leibler
divergence, and Von Neumann divergence for classification. Additionally, the
paper introduces a new feature, Heterogeneous Orders Relevance Composition
(HORC), combining different relevance matrices (Covariance, Mutual
Information, or Kernel Matrix) under a tensor framework and multiple kernel
fusion. The framework is further refined using Neighborhood Component Feature
Selection (NCFS) to optimize the feature subset.

[117] also introduced a method to improve MI classification performance by
employing Non-negative Matrix Factorization (NMF) for EEG channel selection
and using the Riemannian geometry of covariance matrices for feature
extraction. The method reduces the dimensionality of the EEG data, mitigates
overfitting, and enhances classification accuracy by selecting subject-
specific channels. The NMF is used to decompose the covariance matrix as
follows:

$C\approx WH$ (29)

where $C$ is the covariance matrix, and $W$ and $H$ are non-negative matrices.
The neighborhood component feature selection (NCFS) algorithm is then applied
to select the most important features, further refined by:

$Dw(f_{i},f_{j})=\sum w_{k}^{2}|f_{ik}-f_{jk}|$ (30)

where $w_{k}$ is the weighting vector for the $k$-th feature.

These studies collectively advance the application of Riemannian geometry in
EEG feature extraction and discriminant analysis, offering improved accuracy,
robustness, and computational efficiency in BCI systems.

###  4.2 Classficiation Approaches

Recent approaches of BCI classification have notably shifted from traditional
Euclidean metrics to employing Riemannian geometry, better reflecting the
complex data structures. [94] and [80] demonstrated the use of Riemannian
distance-based kernels within SVMs to significantly enhance classification
accuracy for motor imagery tasks, surpassing traditional methods without
extensive spatial filtering. Similarly, [79] documents substantial performance
improvements in steady-state visually evoked potential (SSVEP) classification
by aligning methods more closely with intrinsic data geometry. The integration
of Riemannian geometry with decision tree frameworks in [98] improves
classification by capturing non-linear data relationships. Additionally, [85]
introduces an expectation-maximization algorithm for robust covariance
estimation in the presence of incomplete EEG data, outperforming traditional
imputation techniques. Furthermore, [123] develops new Riemannian geometry-
based metrics to monitor and enhance user performance during BCI training,
ensuring a more accurate and reliable progress assessment.

The work [122] explores the application of graph neural networks (GNNs) on SPD
manifolds to classify motor imagery EEG signals. This approach leverages the
time-frequency characteristics of EEG data, which are represented on SPD
manifolds. The mathematical formulation involves constructing a graph where
each node represents an EEG channel, and edges are weighted by a function of
the Riemannian distance between SPD matrices. The SPD matrices are derived
from the covariance of time-frequency representations of the EEG signals,
capturing both spatial and spectral information.

The key mathematical components include the computation of the Riemannian
distance between two SPD matrices $\Sigma_{1}$ and $\Sigma_{2}$:

$\delta_{R}(\Sigma_{1},\Sigma_{2})=\left\|\log(\Sigma_{1}^{-1/2}\Sigma_{2}\Sigma_{1}^{-1/2})\right\|_{F},$
(31)

where $\log$ denotes the matrix logarithm and $\left\|\cdot\right\|_{F}$ is
the Frobenius norm. This distance metric is crucial for defining the weights
of the edges in the graph neural network, ensuring that the intrinsic geometry
of the data is preserved.

The GNN model processes the SPD matrices using graph convolutional layers
specifically designed for manifold-valued data. Let
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph with vertices $\mathcal{V}$
corresponding to EEG channels and edges $\mathcal{E}$ weighted by the
Riemannian distance. The signal at each vertex $v$ is represented by an SPD
matrix $\Sigma_{v}$. The graph convolution operation on SPD manifolds is
defined as:

$H^{(l+1)}\leftarrow\text{RBN}\left(\text{ReEig}\left(W^{(l)}\left(\tilde{D}^{-1}\tilde{A}\right)H^{(l)}W^{(l)T}\right)\right)$
(32)

where $\mathbf{H}^{(l)}$ is the feature matrix at layer $l$,
$\mathbf{W}^{(l)}$ is the trainable weight matrix, $\mathcal{N}(v)$ denotes
the neighbors of vertex $v$, and $\sigma$ is a non-linear activation function.
This operation ensures that the convolution respects the manifold structure of
the data.

[113] describes a method that combines Independent Component Analysis (ICA)
with Riemannian geometry to enhance emotion recognition from EEG signals; this
involves projecting covariance matrices onto the Riemannian manifold and
integrating these features into a deep learning model, resulting in superior
performance compared to traditional methods. Similarly, [77] applies
Riemannian geometry decoding algorithms to large-scale EEG datasets, where the
baseline minimum distance to Riemannian mean approach yields the highest
classification accuracy for motor imagery and execution tasks, underscoring
the scalability of these methods. Additionally, [102] explores the feasibility
of imagined speech classification using EEG signals, employing covariance
matrix descriptors on the Riemannian manifold with a relevance vector machine
classifier to achieve high accuracy, revealing promising potential for BCI
applications in speech imagery.

In the paper [101] the authors propose a novel brain-ventilator interface
(BVI) framework that detects patient-ventilator disharmony from EEG signals.
This work leverages the spatial covariance matrices of EEG signals and
utilizes Riemannian geometry to classify different respiratory states. The
approach is robust against the non-stationarity and noise inherent in EEG
signals. Mathematically, the framework involves calculating the spatial
covariance matrix $C$ for the EEG signals, which is SPD. The Riemannian mean
$C_{k}$ of these matrices is computed using:

$C_{k}=\arg_{C}\min\sum_{C_{i}\in S_{k}}d(C,C_{i})$ (33)

The features are then projected onto the tangent space using the logarithmic
map:

$\log_{Q}(P)=S_{Q}=Q^{1/2}logm(Q^{{-1/2}}PQ^{1/2})Q^{1/2}$ (34)

These features are used in a classifier to detect respiratory states, thus
enabling the BVI system.

The work on [91] introduces an innovative method combining Riemannian geometry
with sparse optimization and Dempster-Shafer theory for enhanced motor imagery
classification. The method, known as RSODSF, extracts features by first
calculating the covariance matrices from segmented EEG signals, projecting
them into Riemannian tangent space, and applying sparse optimization:

$w=\arg\min_{w}\frac{1}{2}\left\|y-Fw\right\|_{2}^{2}+\lambda\left\|w\right\|_{1}$
(35)

where $F$ is the matrix of features, $y$ is the label vector, $w$ is the
weight vector, and $\lambda$ is the regularization parameter. The
probabilistic outputs of a support vector machine (SVM) classifier are fused
using Dempster-Shafer theory to improve classification accuracy:

$m_{1,2}(EA)=\begin{cases}\frac{\sum_{EB\cap EC=EA}m_{1}(EB)\times
m_{2}(EC)}{1-\sum_{EB\cap EC=\emptyset}m_{1}(EB)\times m_{2}(EC)}&\text{if
}EA\neq\emptyset\\\ 0&\text{if }EA=\emptyset\end{cases}$ (36)

The proposed method demonstrates significant accuracy improvements on BCI
competition datasets.

The paper [116] suggested an expansion of the generalized learning vector
quantization (GLVQ) to the curved Riemannian manifold of SPD matrices. The
scholars introduce a technique known as Generalized Learning Riemannian Space
Quantization (GLRSQ), which adjusts the GLVQ framework to operate within the
suitable Riemannian metric, thereby notably improving classification accuracy
for tasks involving EEG-based motor imagery. The cost function for GLRSQ on a
Riemannian manifold is given by:

$E(W)=\sum_{i=1}^{m}\phi\left(\mu(X_{i},W)\right)$ (37)

where $\phi$ is a monotonically increasing function, and $\mu(X_{i},W)$ is
defined as:

$\mu(X_{i},W)=\frac{\delta(X_{i},W_{J})-\delta(X_{i},W_{K})}{\delta(X_{i},W_{J})+\delta(X_{i},W_{K})}$
(38)

with $W_{J}$ and $W_{K}$ being the closest correct and incorrect prototypes,
respectively. The gradients of the cost function with respect to the
prototypes are derived as:

$\displaystyle\Delta W_{J}$
$\displaystyle=-\alpha(t)\nabla_{W_{J}}E=-\alpha(t)\phi^{\prime}\frac{4\delta_{K}}{(\delta_{J}+\delta_{K})^{2}}\text{Log}_{W_{J}}(X_{i}),$
(39) $\displaystyle\Delta W_{K}$
$\displaystyle=\alpha(t)\nabla_{W_{K}}E=\alpha(t)\phi^{\prime}\frac{4\delta_{J}}{(\delta_{J}+\delta_{K})^{2}}\text{Log}_{W_{K}}(X_{i}),$

where $\alpha(t)$ is the learning rate, $\phi^{\prime}$ is the derivative of
$\phi$, and $\text{Log}_{W_{J}}(X_{i})$ denotes the logarithmic map of $X_{i}$
at $W_{J}$. The exponential and logarithmic maps are used to update the
prototypes on the SPD manifold:

$\displaystyle\text{Exp}_{W}(V)$
$\displaystyle=W^{1/2}\exp(W^{-1/2}VW^{-1/2})W^{1/2},$ (40)
$\displaystyle\text{Log}_{W}(X)$
$\displaystyle=W^{1/2}\log(W^{-1/2}XW^{-1/2})W^{1/2}.$

The update rules for the prototypes $W_{J}$ and $W_{K}$ are given by:

$\displaystyle W_{J}$
$\displaystyle\leftarrow\text{Exp}_{W_{J}}\left(-\alpha(t)\nabla_{W_{J}}E\right),$
(41) $\displaystyle W_{K}$
$\displaystyle\leftarrow\text{Exp}_{W_{K}}\left(-\alpha(t)\nabla_{W_{K}}E\right).$

This innovative approach ensures that the prototypes remain within the SPD
manifold, leveraging the geometric properties of the manifold to enhance
classification performance. The experimental results demonstrate that GLRSQ
significantly outperforms traditional Euclidean-based GLVQ methods and shows
competitive performance with state-of-the-art techniques in EEG classification
for motor imagery tasks.

###  4.3 manifold learning

Integration of deep learning with Riemannian geometry-based methods was a tren
in recent years. In [83], the proposed EEG-SPDNet incorporates Riemannian
geometry into deep network architectures, enhancing the decoding by exploiting
the physiological plausibility of frequency regions. Concurrently, [82]
introduces a method for robust representation of EEG signals through spatial
covariance matrices, capturing homogeneous segments effectively. Furthermore,
[86] develops a model that optimizes subject-specific frequency band selection
for motor imagery classification by constructing multiple Riemannian graphs
and applying advanced graph embedding and fusion techniques, tailoring the
approach to individual variations in EEG signal patterns.

[87] introduces a novel method to address the limitations of existing SPD
matrix-based Riemannian networks. They propose Riemannian Embedding Banks
(REB), which partition the problem of learning common spatial patterns into
multiple subproblems, each modeled separately and then combined within SPD
neural networks. The REB method utilizes the SPDNet framework, which includes
layers such as BiMap and ReEig for transformation and feature extraction. The
BiMap layer performs a bilinear mapping to generate more discriminative and
compact SPD matrix features. The transformation is given by:

$X_{k}=W_{s}X_{s-1}W_{s}^{T},$ (42)

where $W_{s}$ is the transformation matrix ensuring the output $X_{k}$ remains
in the form of an SPD matrix.

Similar to the ReLU layer in traditional neural networks, the ReEig layer
rectifies SPD matrices with small positive eigenvalues:

$X_{s}=U_{s-1}\max(\epsilon I,\Sigma_{s-1})U_{s-1}^{T},$ (43)

where $U_{s-1}$ and $\Sigma_{s-1}$ are obtained from the eigenvalue
decomposition $X_{s-1}=U_{s-1}\Sigma_{s-1}U_{s-1}^{T}$, and $\epsilon$ is a
threshold parameter. The REB approach optimizes the embedding by assigning
features to clusters and ensuring that the samples within each cluster are
informative. The clustering is achieved by minimizing:

$L_{A}(X_{i};f_{p})=\sum_{X_{j}\in f_{p}(X_{i})}\delta_{R}^{2}(X_{i},X_{j}),$
(44) $L_{P}(X_{i},y_{i};f_{p})=-\sum_{X_{j}\in
f_{p}(X_{i}),y_{j}=y_{i}}\delta_{R}^{2}(X_{i},X_{j}),$ (45)

and

$L_{N}(X_{i};f_{p})=\sum_{X_{m}\in f_{p}(X_{i}),y_{m}\neq
y_{i}}\log(\min(10^{-4},Q(D(X_{i},X_{m}))),$ (46)

where $Q(D)$ is a distribution function ensuring negative samples are
uniformly distributed based on their distances. The final embedding is
constructed by concatenating the sub-embeddings produced by individual
learners:

$f=[f_{d}(X_{i};f_{1}),f_{d}(X_{i};f_{2}),\ldots,f_{d}(X_{i};f_{K})],$ (47)

and the loss function for training is given by:

$L_{k}=\lambda_{1}\sum_{X_{i}\in
C_{k}}L_{C_{k}}(X_{i};f_{p})+\lambda_{2}\sum_{X_{i}\in
C_{k}}L_{f_{k}}(X_{i};f_{k}).$ (48)

The experimental results on public EEG datasets demonstrate the superiority of
the proposed REB approach in learning common spatial patterns of EEG signals,
increasing convergence speed, and maintaining generalization despite the non-
stationary nature of the data.

The paper [96] leverages the graph structure of EEG trials, incorporating node
influence properties and an ensemble of semantic graphs into the neural
structured learning framework. This approach improves classification
performance by capturing the complex relationships between EEG trials and
integrating manifold regularities into the learning process. The optimization
equation includes supervised and neighbor loss functions:

$\text{loss}=\sum_{i=1}^{D}L(y_{i},\hat{y}_{i})+\alpha\sum_{i=1}^{D}L_{N}(y_{i},x_{i},N(x_{i})),$
(49)

where $L(y_{i},\hat{y}_{i})$ is the supervised loss,
$L_{N}(y_{i},x_{i},N(x_{i}))$ is the neighbor loss, and $\alpha$ is the graph
regularization parameter.

[118] presents a novel approach using geodesic correlation on the SPD
manifold. The geodesic correlation measure $G(X,Y)$ for two SPD matrices $X$
and $Y$ is given by:

$G(X,Y)=\sum_{i=1}^{d}\log^{2}\lambda_{i}$ (50)

where $\lambda_{i}$ are the generalized eigenvalues of the pair $(X,Y)$. This
framework enhances geodesic correlation for multi-view, self-supervised
learning.

[86] proposes the MRGF model to optimize frequency band selection and extract
spatial and spectral features from EEG signals. The framework constructs
multiple Riemannian graphs and employs graph embedding and fusion techniques.
The bilinear mapping $W$ for dimensionality reduction is optimized by
preserving distance structures between the high-dimensional manifold and low-
dimensional embedding:

$\min_{W}\sum_{P_{i},P_{j}\in
C}\left|\delta_{R}(P_{i},P_{j})-\delta_{R}(WP_{i}W^{T},WP_{j}W^{T})\right|$
(51)

where $\delta_{R}$ is the Riemannian distance, and $C$ is the experimental
dataset.

[97] proposes a novel manifold attention module designed to handle SPD
matrices. This network employs bilinear mappings to convert input SPD matrices
into query, key, and value matrices while retaining their SPD structure.
Specifically, for a given input $\tilde{x}_{i}$, the query, key, and value
mappings are formulated as follows:

$q_{i}=h_{q}(\tilde{x}_{i};W_{q})=W_{q}\tilde{x}_{i}W_{q}^{T}$ (52)

$k_{i}=h_{k}(\tilde{x}_{i};W_{k})=W_{k}\tilde{x}_{i}W_{k}^{T}$ (53)

$v_{i}=h_{v}(\tilde{x}_{i};W_{v})=W_{v}\tilde{x}_{i}W_{v}^{T}$ (54)

where $W_{q}$, $W_{k}$, and $W_{v}$ are transformation matrices ensuring that
the output remains in the SPD manifold. The similarity between $q_{i}$ and
$k_{j}$ is computed using the Log-Euclidean distance:

$\text{sim}(q_{i},k_{j})=\frac{1}{1+\log(1+\delta_{1}(q_{i},k_{j}))}:=\alpha_{ij}$
(55)

Here, $\delta_{1}$ denotes the Log-Euclidean distance. The attention matrix
$\mathbf{A}=[\alpha_{ij}]_{m\times m}$ is normalized using the softmax
function, and the final output $v_{i}^{\prime}$ is computed as:

$v_{i}^{\prime}=\text{Exp}\left(\sum_{l=1}^{m}\alpha_{il}\text{Log}(v_{l})\right)$
(56)

The backward procedure for gradient descent parameter updating on the
Riemannian manifold involves projecting Euclidean gradients onto the tangent
space using Stiefel gradients and retraction operations. The Stiefel gradient
$\widetilde{\nabla}_{W_{v}}\mathcal{L}$ is computed as:

$\widetilde{\nabla}_{W_{v}}\mathcal{L}=\nabla_{W_{v}}\mathcal{L}-\pi_{N}(\nabla_{W_{v}}\mathcal{L})$
(57)

$\widetilde{\nabla}_{W_{v}}\mathcal{L}=\nabla_{W_{v}}\mathcal{L}-W_{v}\left(\frac{W_{v}^{T}\nabla_{W_{v}}\mathcal{L}+(\nabla_{W_{v}}\mathcal{L})^{T}W_{v}}{2}\right)$
(58)

Finally, the weight update is performed as:

$W_{v}^{(new)}=\Gamma(W_{v}-\lambda\widetilde{\nabla}_{W_{v}}\mathcal{L})$
(59)

where $\Gamma$ is the retraction operation.

In [115], the authors propose a novel method for modeling EEG covariance
matrix distributions using Riemannian spectral clustering (RiSC). The
framework allows for both unimodal and multimodal distributions on the SPD
manifold, facilitating outlier detection and robust classification. The core
of the method involves clustering EEG data points based on the Affine-
Invariant Riemannian Metric (AIRM). For classification, the authors introduce
the multimodal classifier mcRiSC, which computes the distance of a new
observation $P$ to cluster centroids and assigns it to the nearest cluster.
The prediction is formulated as:

$\hat{b}=\arg\min_{b\in\\{1,\ldots,B\\}}\delta_{R}\left(P,\bar{P}^{(b)}\right),$
(60)

where $\bar{P}^{(b)}$ denotes the centroid of cluster $b$. This method allows
flexible application without prior knowledge of the data distribution,
significantly enhancing classification performance.

The study on [124] demonstrates how projecting covariance matrices onto the
Riemannian manifold, followed by their integration into a deep learning model,
can yield superior performance compared to traditional methods. Similarly, the
research on [100] investigates the application of geometric neural networks,
leveraging phase space representations of EEG signals to improve
classification accuracy. Furthermore, [99] employs a manifold-based GCN
diffusion model to effectively capture complex spatial-temporal dynamics in
EEG data, thereby enhancing the robustness and accuracy of BCI systems.
Lastly, the work on [106] introduces an ensemble learning framework that
combines Riemannian geometric features with machine learning classifiers,
which significantly improves cross-session motor imagery classification
accuracy.

These manifold learning approaches collectively advance the field of EEG
signal processing by leveraging the mathematical properties of Riemannian
manifolds to provide more effective and robust tools for feature extraction,
data representation, and classification.

###  4.4 Tangent Space Approaches

Recent advancements in tangent space approaches for EEG signal processing have
introduced innovative methods to enhance feature extraction and classification
in BCIs. These methods leverage the properties of Riemannian geometry to map
covariance matrices into the tangent space, allowing for effective
manipulation and analysis of EEG data. [108] introduced a feature extraction
method combining filter banks and Riemannian tangent space for multi-category
MI classification, addressing frequency variance and noise interference. [93]
revisited Riemannian geometry-based EEG decoding through approximate joint
diagonalization, simplifying Riemannian geometry concepts and reducing
computational complexity. [92] proposed a Riemannian distance-based channel
selection and feature extraction method combining discriminative time-
frequency bands and Riemannian tangent space for MI-BCIs. [95] presented a
spatio-temporal EEG representation learning framework on Riemannian manifold
and Euclidean space, combining spatial and temporal information for improved
classification performance.

[81] proposed a method for motor imagery classification that utilizes
Riemannian geometry with a median absolute deviation (MAD) strategy. This
approach involves calculating the average sample covariance matrices (SCMs) to
select optimal reference metrics in a tangent space mapping (TSM)-based MI-EEG
framework. The features are extracted using TSM, where the data from SCMs are
projected according to a reference matrix representing the feature vector.
Principal component analysis (PCA) and analysis of variance (ANOVA) are then
applied to reduce dimensions and select optimal features for classification
using linear discriminant analysis (LDA). The mathematical formulation of the
Riemannian distance is given by:

$d_{R}(C_{i},C_{j})=\left\|\log(C_{i})-\log(C_{j})\right\|_{F}$ (61)

where $d_{R}$ represents the Riemannian distance, $C_{i}$ and $C_{j}$ are
covariance matrices, and $\left\|\cdot\right\|_{F}$ denotes the Frobenius
norm. Miah et al. demonstrated that this method provides better accuracy
compared to more sophisticated methods.

[119] introduced a manifold regression approach to predict from MEG/EEG
signals without source modeling. This method uses the Riemannian geometry of
rank-reduced covariance matrices and vectorizes them through projection into a
tangent space. The Wasserstein distance and affine-invariant geometric
distance are employed to handle rank-reduced data and improve prediction
accuracy. The experimental results showed that this method outperforms sensor-
space estimators and approaches the performance of source-localization models.

In detail, the covariance matrices of EEG signals are first computed as:

$C_{i}=\frac{1}{N}\sum_{t=1}^{N}x_{i}(t)x_{i}(t)^{T}$ (62)

where $C_{i}$ is the covariance matrix for the $i$-th trial, $N$ is the number
of time samples, and $x_{i}(t)$ is the EEG signal at time $t$.

These covariance matrices are then mapped to the tangent space of the
Riemannian manifold. For a reference point $C_{0}$ on the manifold, the
tangent space projection is given by:

$\phi(C_{i})=\log(C_{0}^{-1/2}C_{i}C_{0}^{-1/2})$ (63)

where $\log$ is the matrix logarithm, and $\phi(C_{i})$ represents the
projected covariance matrix in the tangent space.

For regression, the features are vectorized:

$\phi(C_{i})=\mathrm{vec}\left(\log(C_{0}^{-1/2}C_{i}C_{0}^{-1/2})\right)$
(64)

where $\mathrm{vec}$ denotes the vectorization operation.

The regression model is formulated as:

$\hat{y}=W\cdot\phi(C)$ (65)

where $\hat{y}$ is the predicted value, $W$ is the weight matrix, and
$\phi(C)$ represents the feature vector obtained by projecting the covariance
matrix $C$ into the tangent space.

[78] explored the use of multiple tangent space projections for motor imagery
EEG classification. The method involves projecting covariance matrices from
their native Riemannian space to multiple class-dependent tangent spaces,
enhancing the information provided to the classifier. This approach
significantly improves model accuracy by utilizing different projections
tailored to specific classes. The study demonstrated that multiple tangent
space projections could effectively capture the discriminative features for
better classification performance. The tangent space projection is
mathematically expressed as:

$\phi(C)=\mathrm{vec}\left(\log(C)-\log(C_{0})\right)$ (66)

where $\phi(C)$ is the feature vector, $\log(C)$ is the matrix logarithm of
the covariance matrix $C$, and $C_{0}$ is the reference matrix.

These studies collectively advance the field of tangent space approaches for
EEG signal processing, offering innovative solutions to enhance feature
extraction, reduce dimensionality, and improve classification accuracy in
BCIs.

###  4.5 Transfer Learning Approaches

Transfer learning approaches in EEG signal processing have made significant
strides in reducing calibration time and improving classification accuracy by
leveraging data from multiple subjects. These methods utilize Riemannian
geometry and tangent space alignment to handle the variability between
subjects and sessions effectively.

[84] proposed the Riemannian Geometric Instance Filtering (RGIF) framework to
address the challenge of inter-subject variability in EEG signals. The
framework includes two core components: a stable inter-subject similarity
metric based on Riemannian geometry and a convolutional neural network for
feature extraction. The similarity metric measures the similarity between a
few trials from the target subject and abundant trials from source subjects,
focusing on removing data from significantly different subjects. The geodesic
distance between two SPD matrices $P_{1}$ and $P_{2}$ is defined as:

$\delta_{R}(P_{1},P_{2})=\left\|\log(P_{1}^{-1}P_{2})\right\|_{F}=\left(\sum_{i=1}^{n}\log^{2}\lambda_{i}(P_{1}^{-1}P_{2})\right)^{1/2}$
(67)

where $\lambda_{i}$ are the eigenvalues of $P_{1}^{-1}P_{2}$. The Riemannian
mean matrix $G_{k}$ for the target subject is computed as:

$G_{k}=\operatorname*{arg\,min}_{G}\sum_{i=1}^{N_{k}}\delta_{R}^{2}(G,P_{i})$
(68)

This framework significantly reduces calibration time while maintaining high
classification accuracy by combining data from similar subjects and the
powerful feature extraction capabilities of CNNs.

[110] introduced supervised and semi-supervised manifold embedded knowledge
transfer (sMEKT and ssMEKT) algorithms to handle inter-session/subject
variability in MI EEG signals. These algorithms perform domain adaptation by
preserving source domain discriminability and target domain geometric
structure. After Riemannian alignment and tangent space mapping , both sMEKT
and ssMEKT minimize the joint probability distribution shift between the
source and target domains. The alignment process is given by:

$\phi(C_{i})=\log(C_{0}^{-1/2}C_{i}C_{0}^{-1/2})$ (69)

where $\phi(C_{i})$ represents the projected covariance matrix in the tangent
space. The optimization for minimizing the distribution shift is formulated
as:

$\mathcal{L}(\theta)=\sum_{i=1}^{N_{s}}\ell(f_{\theta}(\phi(C_{i}^{s})),y_{i}^{s})+\lambda\sum_{i=1}^{N_{t}}\ell(f_{\theta}(\phi(C_{i}^{t})),y_{i}^{t})$
(70)

where $\ell$ is the loss function, $f_{\theta}$ is the classifier, $C_{i}^{s}$
and $C_{i}^{t}$ are source and target domain samples, respectively, and
$\lambda$ is a regularization parameter. These algorithms demonstrate
significant improvements in classification accuracy with fewer labeled samples
from the target domain.

Other notable contributions in transfer learning approaches include Xu et al.
[89], who proposed a selective cross-subject transfer learning approach based
on Riemannian tangent space for motor imagery BCI, using Riemannian alignment
to bring covariance matrices from different subjects closer and extracting
Riemannian tangent space features for classification. Cai et al. [107]
introduced a framework for motor imagery EEG decoding using manifold embedded
transfer learning, aligning covariance matrices on the SPD manifold and
extracting features from both SPD and Grassmann manifolds, demonstrating
superior performance in transfer tasks. Additionally, [114] presented a
tangent space alignment method for transfer learning in BCIs, aligning EEG
data from different subjects in the tangent space of the positive definite
matrices Riemannian manifold, showing improved classification accuracy.

These studies collectively advance the field of transfer learning approaches
for EEG signal processing, offering innovative solutions to enhance feature
extraction, reduce calibration time, and improve classification accuracy in
BCIs.

##  5 Discussion and Challenges

The integration of Riemannian geometry in EEG-based BCI systems has
significantly progressed feature extraction, classification, and transfer
learning within the respective fields. However, despite these advancements,
various persistent challenges hinder the widespread adoption and efficacy of
these methodologies. A primary obstacle lies in the computational intricacies
associated with managing intricate manifold embeddings and high-dimensional
transformations, particularly problematic in time-sensitive applications where
swift data processing is essential. Furthermore, these approaches often
struggle with generalization across diverse tasks and populations, given the
variability in EEG patterns among individuals presents a notable hurdle.
Additionally, the efficacy of Riemannian geometry-based techniques heavily
relies on the quality of the input EEG data. Factors such as artifacts, noise,
and non-stationarity can notably deteriorate performance, necessitating
advanced preprocessing methods that may not always be practical in real-world
scenarios. Another significant constraint is the lack of interpretability in
these sophisticated mathematical transformations, which can impede clinical
acceptance and further development of the methodologies. Moreover, there
exists a noticeable gap in integrating these approaches with other modalities
like fMRI or MEG, which could potentially enhance diagnostic and therapeutic
capabilities but also introduce additional complexity in data fusion and
analysis.

Figure 3: Classification tasks presented in the current literature

Concentrated efforts in various crucial areas are imperative. Subsequent
research endeavors should prioritize the enhancement of computational
algorithms to achieve higher efficiency, thereby reducing the temporal and
resource demands of existing methodologies, potentially through the
utilization of emerging technologies like GPU acceleration. Additionally,
there exists an urgent requirement for the creation of adaptive models capable
of accommodating the inter-individual variabilities and cognitive state
differences present across a heterogeneous population. Enhancements in
managing noise and artifacts within the Riemannian framework have the
potential to notably augment the quality of EEG data before further
processing. The amalgamation of Riemannian geometry with alternative machine
learning strategies in hybrid models could represent a significant advancement
in enhancing both the efficacy and interpretability of these approaches.
Furthermore, broadening the application of these methodologies to synergize
with other neurological data modalities could offer a more comprehensive
insight into brain function. Lastly, the establishment of open-source
platforms and collaborative frameworks could expedite innovation and
standardize practices in the field, thus rendering these sophisticated
techniques more accessible and adaptable across various applications.

###  5.1 Future Directions in Research

To address the computational complexity of Riemannian geometry-based methods,
future research should focus on optimizing algorithms for faster computation.
This includes developing efficient numerical methods for calculating
Riemannian distances and means, as well as exploring alternative
representations of EEG signals that retain critical information with lower
dimensionality. One potential direction is the use of approximation techniques
or dimensionality reduction methods that preserve the geometric properties of
the data.

Enhancing the scalability of graph neural networks on SPD manifolds is a
critical area for prospective investigation. This task entails the creation of
scalable GNN structures capable of effectively managing extensive EEG
datasets. The exploration of hybrid models amalgamating the merits of
conventional machine learning approaches and deep learning methodologies could
contribute to tackling the scalability challenge. Moreover, delving into
parallel computation and distributed processing methodologies might further
amplify the scalability of these frameworks.

Adapting manifold learning strategies to render them more suitable for real-
time applications holds paramount importance. This objective can be realized
through the formulation of models that are both computationally efficient and
facile to deploy. Subsequent research endeavors should prioritize the
simplification of these models’ intricacy while upholding their efficacy.
Potential strategies may encompass the utilization of approximation
methodologies, streamlined mathematical representations, or innovative model
structures customized for real-time computations.

Improving transfer learning frameworks to effectively manage inter-subject
variability and decrease the requirement for substantial labeled data presents
a significant challenge. Subsequent investigations ought to delve into
resilient transfer learning methodologies capable of functioning efficiently
with limited labeled data. Such endeavors may encompass the formulation of
unsupervised or semi-supervised learning strategies, along with the
utilization of domain adaptation techniques tailored for EEG data. Moreover,
the exploration of more efficient approaches for aligning data across diverse
subjects and sessions within the tangent space of the Riemannian manifold
could contribute to enhancing the efficacy of transfer learning strategies.

To summarize, tackling computational intricacies, enhancing scalability,
streamlining models for practical implementation, and refining transfer
learning frameworks constitute pivotal domains for prospective studies. These
endeavors are poised to facilitate the advancement of more effective, precise,
and user-friendly Brain-Computer Interfaces (BCIs) that can find widespread
utility in both clinical and non-clinical environments.

##  6 Conclusion

This study offers an extensive examination of the fusion of Riemannian
geometry with deep learning methodologies within the domain of brain-computer
interfaces. The unique amalgamation of these sophisticated mathematical
strategies has illustrated substantial potential in augmenting the decoding
capabilities of EEG signals in BCIs. Specifically, Riemannian geometry has
played a pivotal role in the accurate and resilient management of the non-
Euclidean data structures inherent in EEG data, an area where conventional
Euclidean methods often prove inadequate. Our analysis has refreshed the
landscape of EEG signal processing by presenting recent progressions that
exploit deep learning to tackle and alleviate traditional BCI obstacles like
sensitivity to noise, non-stationarity, and prolonged calibration periods.
Through the perspective of Riemannian geometry, a shift has been observed in
the utilization of covariance matrices, transitioning from fundamental
manipulations to sophisticated metric learning and optimization for EEG
classification.

This transition not only enhances existing subspace techniques but also
indicates a shift towards directly manipulating and categorizing covariance
matrices, potentially circumventing traditional methodologies altogether.
Nevertheless, despite the advancements, this analysis also highlights the
necessity for further investigation to surmount the persisting constraints of
BCIs. The infusion of Riemannian geometry into BCI design necessitates novel
EEG representations that are more resilient and classifiers capable of
effectively managing outliers and non-stationarity. Prospective avenues should
also contemplate the formulation of adaptive techniques capable of dynamically
adapting to the user’s cognitive status and environmental variations.
Additionally, delving into the possibilities of Riemannian geometry for
multitask learning, feature extraction, and transfer learning within BCIs
could yield substantial enhancements in both efficacy and usability. In
summary, the convergence of Riemannian geometry and deep learning harbors
significant potential for the trajectory of BCIs.

As this domain progresses, it is imperative that sustained endeavors are
channeled towards not only refining these mathematical frameworks but also
rendering these sophisticated methodologies more accessible for pragmatic BCI
implementations. We anticipate that the advancements expounded in this
analysis will pave the way for more intuitive, efficient, and user-friendly
BCI systems, setting a new benchmark for EEG signal classification and further
bridging the chasm between theoretical exploration and real-world
applications.

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