MAtt: A Manifold Attention Network for EEG Decoding
Anonymous Author(s)
Abstract
Recognition of electroencephalographic (EEG) signals highly affect the efficiencyof non-invasive brain-computer interfaces (BCIs).
While recent advances of deeplearning (DL)-based EEG decoders offer improved performances, the developmentof geometric learning (GL) has attracted much attention for offering exceptionalrobustness in decoding noisy EEG data.
However, there is a lack of studies on themerged use of deep neural networks (DNNs) and geometric learning for EEG decoding.
We herein propose a manifold attention network (mAtt), a novel geometricdeep learning (GDL)-based model, featuring a manifold attention mechanism thatcharacterizes spatiotemporal representations of EEG data fully on a Riemanniansymmetric positive definite (SPD).
The evaluation of the proposed mAtt on bothtime-synchronous and -asyncronous EEG datasets suggests its superiority overother leading DL methods for general EEG decoding.
Furthermore, analysis ofmodel interpretation reveals the capability of mAtt in capturing informative EEGfeatures and handling the non-stationarity of brain dynamics.
Introduction and related works
A brain-computer interface (BCI) is a type of human-machine interaction that bridges a pathway frombrain to external devices.
Electroencephalogram (EEG), a non-invasive neuromonitoring modalitywith high portability and affordability, has been widely used to explore practical applications of BCI inthe real world [40, 19, 37].
For instance, disabled users can type messages through an EEG-based BCIthat recognizes the steady-state visual evoked potential (SSVEP) induced by flickering visual targetspresented on a screen [1, 44, 12].
Stroke patients who need restoration of motor function undergomotor-imagery (MI) BCI-controlled rehabilitation as an active training [3, 2].
Most EEG-based BCIsystems are designed to detect/recognize reproducible time-asynchronous or time-synchronous EEGpatterns of interest, depending on the schemes of BCI [28].
For example, the MI EEG pattern is anendogenous oscillatory perturbation without an explicit onset time sources from the motor cortex[15].
On the other hand, a time-synchronous EEG pattern is time-locked to a specific event.
Forexample, the pattern of SSVEP is synchronized to the change of brightness on a flickering visualtarget.
The efficiency of BCI systems largely relies on the accuracy and robustness of the EEGdecoder.
However, due to the low signal-to-noise ratio (SNR) [20] and non-stationarity [17] of EEG,translating perplexing EEG signals into meaningful information has been a grand technology andscientific challenge in the field.
Recent advances in deep learning (DL) have contributed to the rapid development of DL-based EEGdecoding techniques [35].
DL models are capable of extracting features automatically according to given training data.
Convolutional neural network (CNN) is one type of the most common DL modelsand has achieved remarkable performance in tasks such as image recognition and object detection[26, 16, 39].
CNN models newly designed for EEG decoding use convolutional kernels that serveas spatial and temporal filters but with extra flexibility to optimize the transformation of EEG dataautomatically through model training [25, 36, 45].
In addition to the fast growth of DL-based EEGdecoders, geometric learning (GL) approaches, mostly based on Riemannian geometry (RG), havebeen adopted in the field of BCI [27].
RG is a type of non-Euclidean geometry that has a differentinterpretation of Euclid’s fifth postulate (i.e. parallel postulate) [41].
In GL, geodesic between pointson the manifold is a critical feature for classification tasks in BCI.
The power and spatial distributionof a segment of multi-channel EEG signals can be coded into a covariance matrix that is symmetricpositive definite (SPD) in general.
This allows mapping of EEG data directly onto a Riemannianmanifold where Riemannian metric is insensitive to extreme outliers and noise [14, 27].
In 2010,
Barachant et al.
[6] proposed Minimum Distance to Mean (MDM) that maps target EEG data onto the SPD manifold to find the nearest class center.
Later on, they developed TSLDA [7] that projects datafrom the manifold to a specific tangent space where Euclidean classifiers are applicable.
RG-basedclassification for EEG decoding has shown extra robustness as the relationship between data samplescan be stably preserved, leading to success in recent data competitions in the BCI field such as’DecMEG2014’ 1 and the ’BCI challenge’ 2 .
The nascent field of geometric deep learning (GDL) has expanded by emerging techniques togeneralize the use of deep neural networks to non-Euclidean spaces.
Efforts have been madeto transitioning useful operations from Euclidean to Riemannian spaces, including convolution[18, 11, 8], activation function [18, 11], batch normalization [9, 13], that facilitate the ongoingdevelopment of GDL tools.
SPDNet [18] is a Riemannian network for non-linear SPD-based learningon Riemannian manifolds using bi-linear mapping that mimics Euclidean convolution for visualclassification tasks.
ManifoldNet [11] offers high performance in medical image classification withmanifold autoencoder.
[30] characterizes 3D movement via the manifold polar coordinate witha geodesic CNN.
[32] performs convolution on the manifold as a generalization of local graphor manifold pseudo-coordinate for vertex classification on graph and shape correspondence task.
In contrast of the vast develop of geometric deep learning (GDL) in many other scientific fields,only few studies focus on decoding EEG data with a merge use of GL and DL.
[46] proposed anetwork architecture that integrates fusion of Euclidean-based module and manifold-based modulewith multiple LSTM and attention structures to extract spatiotemporal information of EEG.
[38]proposes a Riemannian-embedding-banks method that separates the entire embeddings into multiplesub-problems for learning spatial patterns of MI EEG signals based on the features extracted fromthe SPDNet.
[21] combines federated learning and transfer learning on Riemannian manifold usingthe spatial information of EEG.
[22] proposes deep optimal transport on the manifold to minimizethe cost of domain adaptation from the source domain to the target domain.
[23] extracts multi-viewrepresentations of EEG.
These studies have established cornerstones toward the field of future GDLfor EEG decoding, but the increment of performance is yet marginal.
Most of the above-mentionedtechniques can not map the temporal information of EEG onto the manifold, or still rely on Euclideantools to handle EEG features.
We herein propose a manifold attention network, a novel GDLframework, which maps EEG features on a Riemannian SPD manifold where the spatiotemporal
EEG patterns are fully characterized.
The main contributions of the present study are the following:
• a manifold attention network proposed for decoding general types of EEG data.
• a lightweight, interpretable, and efficient GDL framework that is capable of capturing
spatiotemporal EEG features across Euclidean and Riemannian spaces.
• an empirical validation of our proposed model demonstrating its generalizable superiority
over leading DL approaches in EEG decoding.
• neuroscientific insights interpreted by the model that not only echo prior knowledge but also
offer a new look into the dynamical brain.
This article is organized as follows: we first brief the essential background of RG and manifoldattention mechanism; next, we describe the proposed mAtt architecture with details of model designand training; we then validate our proposed model experimentally; lastly, we interpret our proposedmodel with neuroscientific insights.
Preliminary
A manifold is considered as the expansion of curve and surface in Euclidean space.
It is a topologicalspace that can locally regarded as an open set in Hilbert space.
Suppose a manifold is endowed witha differential structure (i.e. a collection of charts satisfying transition mapping, which is definedon the overlap of charts), it is then the so-called differential manifold [24].
Riemannian geometryis a differential manifold equipped with Riemannian metric.
We consider the Symmetric positivedefinite (SPD) manifold, which allows us to manipulate manifold-valued data on the manifold directly.
The spatial information of EEG signal can be represented as a specific covariance matrix, whichrecords the relationship between channels, and is a critical index for us to understand EEG signal.
However, the solution of the Riemannian mean doesn’t have a close form, thus we need to calculatethe approximate mean in an iteration manner [6, 11] until convergence conditions are satisfied.
Riemannian mean is not suitable for being applied in deep learning because of the high computationalcomplexity.
Therefore, we have the Log-Euclidean metric below.
Notations
GL ( n, R ) :
= { A ∈ R n × n | determinant ( A ) ̸ = 0 } is a general linear group, which is the set of allreal non-singular sqaure matrices.
( M , g ) denotes connected Riemannian manifold.
Sym ( n ) :=
{ S ∈ M n × n ( R ) | S T = S } is the space of all n × n real symmetric matrices, where M n × n ( R )specifies the space of all real square matrices, ( . )
T is the transpose operator, and Sym + ( n ) :
={ P ∈ M n × n ( R )
| P = P T , v T Pv > 0 , ∀ v ∈ R n − { 0 }} is the set of all n × n symmetric positivedefinite(SPD) matrices.
<
A, B > F means the Frobenius inner product, defined as Tr ( A T B ) , where
Tr ( . ) is the trace operator.
Log ( . ) and Exp ( . ) are the principle logrithm operator for SPD matrix[31]and exponential operator for symmetric matrix respectively.
Both of them can be computed usingthe orthogonal diagonalization .
Log :
Sym + ( n ) (cid:55)→
Sym ( n ) is an operator that maps a SPD matrix
P ∈
Sym + ( n ) to Sym ( n ) by:
[Equation]
where U is the matrix of eigenvectors of P , since P ∈
Sym + ( n ) , σ i > 0 , i = 1 , ..., n
The inverse projection is Exp of symmetric matrix:
Exp : Sym ( n ) (cid:55)→
Sym + ( n ) , an operator mapsa symmetric matrix S ∈
Sym ( n ) to Sym ( n ) by:
[Equation]
where V is the matrix of eigenvectors of S .
2.2 Log-Euclidean metric
Log-Euclidean metric (LEM) offers a more simple, similar, and efficient generalization to calculatethe center on the SPD manifold than AIM.
LEM is a bi-invariant metric on the Lie group on the SPDmanifold [4].
The geodesic distance from P 1 to P 2 on the Sym + ( n ) is also given by [4]:
[Equation]
Furthermore, we can also define the Log-Euclidean mean( G ) via the Log-Euclidean distance: [Equation] P
[Equation]
P l ) where P 1 , ...
[Figure]
Fortunately, the solution to the formula above has a closed form to follow, given by [5]:
[Equation]
In this work, we are going to use the weighted Log-Euclidean mean that is endowed with differentweights in different P l .
We denote the weight of each P l as w l , where ∀ l ∈ { 1 , 2 , ..., k } .
Here,{ w l } kl =1 satisfies the convexity constraint definition (i.e. k (cid:88) l = w l = 1 , and w l > 0 ).
The definitionand the corresponding weighted Log-Euclidean mean can be defined and derived as:
[Equation]
and
[Equation]
respectively.
Methodology
As shown in Figure 1(a), the architecture of mAtt includes components of the feature extraction(FE), the manifold attention module, transitioning from Euclidean to Riemannian space (E2R), andtransitioning from Riemannian to Euclidean space (R2E).
3.1 Feature extraction of EEG signals
We use two convolutional layers to extract information of the raw EEG signals, where the firstconvolutional layer performs spatial filtering to the multi-channel EEG signals and the secondconvolutional layer extracts spatiotemporal features.
Our parameter setting follows [45].
3.2 From Euclidean space to SPD manifold(E2R operation)
As illustrated in 1 (b), we convert the embeddings from the feature extraction stage to the SPD dataand map the feature embeddings from Euclidean space to the SPD manifold.
Suppose ˜ f denotes theembeddings after the feature extraction stage, we divide the whole embeddings into several epochs˜ f 1 , ˜ f 2 , ..., ˜ f m , and calculate the sample covariance matrix(SCM) of each ˜ f
i , ∀ i ∈ { 1 , 2 , ..., m } .
Bydoing so, we get a sequence of covariance matrices that present the temporal information of theembeddings ˜
f
in the form of SPD data, called SCM ˜ f 1 , SCM ˜ f 2 , ..., SCM ˜ f m .
After we get somedatapoints, we do trace-normalization and add a small number ϵ on each main diagonal element for
[Figure]
each SCM ˜ f
i (i.e.
SCM ˜ fi tr (cid:16) SCM ˜ fi (cid:17) + ϵI ) where i ∈ { 1 , 2 , ..., m } , I is the identity matrix, and we set ϵ as1e-5 in our source code.
The resulting SPD sequence is denoted as ˜ X =
[ ˜ x 1 , ˜ x 2 , ..., ˜ x m ] .
We add asmall identity matrix on them to promise ˜ x i to be a well-defined SPD matrix.
3.3 Manifold attention module
The input of this layer is a sequence of SPD data.
The overview of the manifold attention module isillustrated in Figure2 (a).
Motivated by [18] and [43], we capture the spatiotemporal information onthe manifold.
Suppose the module takes a sequence of SPD matrices [ ˜ x 1 , ˜ x 2 , ..., ˜ x m ] , denoted as ˜ X .
Here we have the query, key, and value in the form of SPD matrices on the manifold.
We convertthe ˜ x
i to the q i , k i , and v i via bilinear mapping and exploit non-linear and valid features from eachsegment.
Suppose the shape of ˜ x
i is d c × d c , and h q , h k , and h v is the mapping from ˜ x
i to q i , k i ,and v i respectively:
[Equation]
[Equation]
[Equation]
where ˜ x i ∈
Sym + ( d c ) , W q , W k , and W v ∈ R d u × d c ( d u < d c ) denotes transformation matrices.
Tomake sure the output q
i , k i , and v i are also SPD matrices, transition matrices W q , W k , and W v arerow-full rank matrices.
After we got q i , k i , and v i by bilinear mapping, we define the similarity for measuring the q i and k j
SPD matrices.
In Euclidean space, there are several ways to define the similarity.
A most commonway is to use dot-product[43] to measure the similarity of query and key.
However, our query, key,and value are SPD matrices instead of vectors as regular attention.
We define the similarity basedon the Log-Euclidean distance (equation(2)) between query and key.
Suppose we have q i and k j ,for some i, j ∈ { 1 , ..., m } .
The similarity sim ( . ) is a strictly decreasing function of distance[0 , ∞ )
(cid:55)→ [0 , 1] and is defined as: sim ( q i , k j ) = [Equation] ,k [Equation]
: = α ij .
Then, the attentionmatrix is:
[Equation]
We then use Softmax function to shrink the range along the row direction, making values in rowhave convexity constraint property.
The final attention probability matrix A ′ is:
[Equation]
=
[ α
[Equation]
ij
[Equation]
m [Equation] mwhere α ′ ij = exp ( α ij ) (cid:80) mk =1 exp ( α ik ) , ∀ i, j ∈ 1 , · · · , m .
Finally, we combine the attention probability matrixand v 1 , v 2 , ..., v m to get the final output v ′ 1 , v ′ 2 , ..., v ′ m and define the output v ′
i ( ∀ i = 1 , 2 , ..., m ) via
Log-Euclidean mean as: v ′ i
=
Exp (cid:32) m (cid:88) l = α ′ il Log ( v l ) (cid:33)
The whole mAtt procedure is illustarted in Algorithm1. Require:
A sequence of SPD data { ˜ x i } mi =1 , transformation matrices W q , W k , W v 1: for i = 1: m do 2: q i = W q ˜ x i W Tq 3: k i = W k ˜ x
i W Tk 4: v i = W v ˜ x i W Tv 5: end for 6: ∀ i, j ∈ { 1 , · · · , m } , A := [ α ij ] m × m = 1 1+ log (1+ δ L ( q i ,k j )) 7: A ′ =
Softmax ( A ) 8: for i = 1: m do 9: v ′ i = Exp (cid:32) m (cid:88) l = α ′ il Log ( v l ) (cid:33) 10: end for 11: return a sequence of SPD data { v ′ i } mi = As shown in Figure2(b), suppose i = 1 , m = 3 , we compute the similarity α ′ 11 , α ′ 12 , α ′ 13 via thedistance from k 1 to q 1 , k 2 to q 1 , and k 3 to q 1 respectively. The points on the SPD manifold M areaveraged after being projected into the tangent space based on an identity matrix I (denoted as T I ),and then the weighted sum is mapped back to the SPD manifold.3.4 From Riemannian manifold to Euclidean space(R2E) and loss layers
After passing through attention module, ReEig layer is used to imitate the ReLU function.
But it isdifferent from the ReLU function that sets the threshold to the value of input, ReEig sets a thresholdto the eigenvalue of the input, can be defined in [18].
R2E operation aims to map the SPD databack to the Euclidean space for executing the final classification, which is composed of a Log layerand regular flatten layer in [18] sequentially.
Log layer is the most common skill in geometric deeplearning to project the SPD data to the Euclidean space.
By doing so, we reduce the manifold to a flatspace by Log ( . ) operation.
We take Log ( . ) operation on the output from the attention module layerv ′ 1 , v ′ 2 , ..., v ′ m ∈
Sym + ( d u ) .
Denote the whole R2E operation as h L :
Sym + ( d u ) (cid:55)→ R d u × ( d u +1) / 2 :h L ( v ′ i )
= flatten ( Log ( v ′ i ))
= flatten (cid:0) S ( diag ( log ( σ 1 ) , · · · , log ( σ d u )))
S T (cid:1)where S is the eigenvector-matrix of v
′ i , and σ 1 , · · · , σ d u are the eigenvalues of v
i .
The Log ( . )operation is the same as equation(1), and the flatten ( A ) operation flatten the upper triangle of thearbitrary symmetric matrix A .
Finally, we set a fully connected layer and regular softmax operation on embeddings after R2Eoperation.
Suppose the output from the whole model stream is ˆ y, the groundtruth is y, we define theloss L as the cross-entropy loss of ˆ y and y .
3.5 Evaluation
Time-asynchronous EEG data : We employ the BCIC-IV-2a as a representative dataset providingtime-asynchronous EEG data.
The BCIC-IV-2a dataset is one of the most commonly used public EEG dataset released for the BCI Competition IV in 2008 [10]. It contains EEG recordings in afour-class motor-imagery task from nine subjects with two repeated session each on different days.
During the task, the subjects were instructed to perform imagination of one of the four types ofmovements (right hand, left hand, feet, and tongue) for four seconds after an instructional cue. Each session consists of a total of 288 trials with 72 trials for each type of the motor imagery. The EEGsignals were recorded by 22 Ag/AgCl EEG electrodes at the central and surrounding regions in asampling rate of 250 Hz. We performed standard preprocessing procedures for the 22-channel EEGsignals, including 1) Down-sampling from 256 Hz to 128 Hz, 2) Band-pass filtering at 4-38 Hz, and3) Segmenting EEG signals at 0.5s-4s of the onset of cue for each trial.
Time-synchronous EEG data: The MAMEM-SSVEP-II dataset was incoporated as the representative dataset of time-synchronous EEG data.
It consists of EEG recordings from 11 subjectsperforming a SSVEP-based task where the subjects were instructed to gaze at one of the five visualstimuli flickering at different frequencies (6.66, 7.50, 8.57, 10.00, and 12.00 Hz) for five seconds. Each subject executed five runs of five cue-based trials for each of the five stimulation frequencies.
The EEG signals were acquired using the EGI 300 Geodesic EEG System (GES 300) equipped with256 channels in a sampling rate of 250 Hz. The preprocessing procedures for this dataset were 1) Band-pass filtering at 1-50 Hz, 2) Selecting eight channels (PO7, PO3, PO, PO4, PO8, O1, Oz, and O2) in the occipital area, the location of visual cortex, and 3) Segmenting each trial into four 1-secondsegments at 1s-5s of the onset of cue, yielding a total of 500 trials of 1-second 8-channel SSVEPsignals for each subject.
Model training and validation: A series of experiments were conducted to evaluate the performances of the mAtt and other EEG decoders with the scenario of real-world BCI usage into account.
In the real-world usage of BCI, a user usually needs to go through a training session to collect asufficient amount of individual EEG data for training the decoding model before executing the BCIsystem.
To stick with the practical scenario, we performed an individual training scheme where achunk of trials within a subject are assigned to training data and the left-over trials within the samesubject are used for testing [45, 29].
For the BCIC-IV-2a dataset, we used the first session of a subjectto the training set where one out of eight was used for validation for mAtt with m = 3 .
The modelwith the lowest validation loss within 350 iterations was used for testing on the second session of thesame subject.
Similarly, for the MAMEM-SSVEP-II dataset, we assigned the first four sessions of asubject to the training set where one out of four was used for validation for mAtt with m = 7 .
Themodel with the lowest validation loss within 180 iterations was used for testing on the fifth session ofthe same subject.
As all models were initialized with random weights, the classification performancewas estimated by the mean across ten repeats for each subject.
Here we evaluate the proposed mAtt using both time-asynchronous and time-synchronous EEG datato giveempirical evidence of the advantages. The performance in a general use for EEG decodingis compared against leading GL- and DL-based techniques. We incorporate the BCI Competition
IV 2a Dataset (BCIC-IV-2a) [10] to assess the performance on time-asynchronous motor-imagery EEG decoding and the MAMEM EEG SSVEP Dataset II (MAMEM-SSVEP-II) [33] to assess theperformance on time-synchronous SSVEP EEG decoding. Previous and current state-of-the-art DL-based models listed for comparison with mAtt include FBCNet[29], SCCNet[45], EEGNet[25],and ShallowConvNet[36].
4.1 Performance comparison
We show the performance comparison on MI EEG decoding.
As shown in Table 1, mAtt outperformsall other SOTA DL methods on both time-synchronous (SSVEP) and -asynchronous (MI) EEGdecoding.
We conclude that mAtt has a generalizable superiority in decoding EEG for all types of
BCI systems.
The classification result at the level of individual subject is provided in the appendix.
4.2 Model interpretation
Through analysis of model interpretation, we are able to uncover the underlying characteristics learntfrom the data.
Figure 3 illustrates the gradient response for MI EEG decoding across channel and
[Table]
[Figure]
across time.
We can see left/right hand MI responses are strong at C4 and C3 corresponding toright/left motor cortices that control the lateral motor functions of the contralateral side of the body[34].
Both feet and tongue MI, that are not lateral movements, presents strong responses at CPzabove the midline of motor cortex.
The spatial distribution clearly exhibit asymmetric pattern for and tongue).
Strong response of motor imagery is marked by dark red at specific frequency bands and time intervals. Increased response of motor imagery is found at mu band ( 10 Hz) for all classes. The strong response of left/right hand motor imagery occurs at 1-2 seconds, the feet motor imagery is most vivid at 0.5seconds, and the tongue motor imagery induced two peaks at 1 second and 3 seconds.
left/right hand MI and symmetric pattern for feet/tongue MI.
The temporal information is available in Figure 4, where all four types of MI induce strong response at mu band around 10 Hz.
This result isin line with the well-known association between motor function and mu rhythm in EEG recordings[34]
The visualization of our model for SSVEP decoding the appendix.
[Figure] Figure 5 depicts the distribution of attention scores across epochs within the (a) MI and (b) SSVEP EEG signals.
For MI EEG signals, attention score is the highest at the first epoch and decreases inthe following epochs, which implies that the beginning of the motor imagery may contribute a higherimportance determined by the manifold attention module.
The profile of attention score for SSVEP EEG signals presents a similar traits that the earlier epochs relate to higher importance.
This analysisreveals the capability of mAtt in handling the non-stationarity of the dynamical brain activity.
4.3 Limitation
In our framework, vacuum permittivity ϵ is added on all main diagonal elements of covariance
cx
i x
Tito ensure the rigor of SPD matrix.
But the operation may cause the repeated singular value ϵ in S i .
Therefore, we proposed possible solutions for this issue:1) Let m < n when dividing the embeddingsinto several time segments, reducing the possibility of getting low-rank S i ; 2) Let ϵ be randomlydrawn from a specific distribution, such as Uniform (1 e − 8 , 1 e − 4) to solve this issue, which isalso a practicable solution; 3) Use the derivative of a low-rank matrix[42] to cope with this issue.
Conclusion
We propose a manifold attention network as a novel GDL framework for decoding both timesynchronous and -asynchronous EEG decoding.
Using back propagation based on the Stiefelmanifold, the proposed mAtt is capable of mapping EEG features onto a Riemannian manifold,where spatiotemporal EEG patterns are captured and characterized, within a lightweight architecture.
The experimental results suggest the superiority of mAtt over current SOTA DL methods for both time-synchronous and -asynchronous EEG decoding.
With the interpretability of mAtt, we visualize the spatial and temporal EEG patterns, which are in line with prior neuroscientific knowledge and shed light on potential possibility of tracking the brain dynamics.
In sum, our privileged method, mAtt, improves the SOTA performance of EEG decoding and is expected to impact on GDL-based
EEG processing with generalizable efficiency and robustness for future development of various BCIsystems.
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