ITSA: Individual Tangent Space Alignment for EEG

• Individual Tangent Space Alignment is a pre-alignment strategy that normalizes covariance matrices per subject to preserve the geometric structure in EEG data.
• It projects EEG trial covariances to tangent space and applies norm rescaling to equalize feature dispersion between source and target domains.
• ITSA uses a supervised rotational alignment via SVD to align class anchors, significantly improving cross-subject and cross-montage EEG decoding.

Searching arXiv for papers on “Individual Tangent Space Alignment” and closely related tangent-space alignment methods. Search query: "Individual Tangent Space Alignment EEG" Individual Tangent Space Alignment (ITSA) is a pre-alignment strategy for cross-subject and cross-montage EEG transfer learning that operates on covariance-derived tangent-space features. In the formulation introduced for auditory-cued gait adaptation decoding, ITSA consists of three stages—subject-specific recentering, distribution matching via feature rescaling, and supervised rotational alignment—and is designed to improve leave-one-subject-out generalisation while preserving the geometric structure of symmetric positive definite covariance matrices (Lai-Tan et al., 11 Aug 2025). The acronym is ambiguous in the broader literature: for example, “ITSA” in stereo matching denotes “Information-Theoretic Shortcut Avoidance,” which is unrelated to Individual Tangent Space Alignment (Chuah et al., 2022).

1. Terminology and conceptual scope

In the EEG-transfer setting, ITSA denotes an individual tangent-space alignment procedure because each subject is first aligned independently before source data are pooled. The term “individual” therefore refers to subject-specific normalisation at the covariance-manifold level, rather than to a generic manifold-learning algorithm that aligns arbitrary samplewise tangent spaces. The proposed use case is cross-subject Brain-Computer Interface transfer, including a leave-one-subject-out regime and a cross-montage regime in which training and testing can involve different electrode configurations (Lai-Tan et al., 11 Aug 2025).

The method is built on a standard Riemannian view of EEG trial covariances. EEG trials are converted to covariance matrices, these matrices are treated as points on the manifold of symmetric positive definite matrices, and tangent-space projection is then used to obtain Euclidean features suitable for alignment and classification. ITSA is therefore neither a purely Euclidean preprocessing method nor a purely manifold-space classifier; it spans covariance-space recentering, tangent-space projection, and Euclidean alignment in the projected feature space.

A central distinguishing claim is that prior tangent-space alignment methods can lose subject-specific structure if all source subjects are treated as one aggregated source domain from the outset. ITSA addresses that issue by preserving individuality during the initial recentering stage and only then combining source data. This design places ITSA close to Riemannian transfer methods, but with a specific emphasis on subject-preserving pre-alignment before pooled training.

2. Problem setting in cross-subject and cross-montage EEG transfer

The motivating problem is supervised transfer learning for EEG decoding when a new subject has little labelled data and when inter-subject variability is substantial. In the reported study, the application is an auditory-cued gait adaptation task evaluated under leave-one-subject-out cross-validation. The paper attributes poor cross-subject generalisation to inter-subject anatomical variability, electrode position shifts, inter-subject differences in brain function, movement artefacts and muscle activity during walking, and motor planning differences. These factors induce covariate shift between source and target EEG distributions, and the paper further notes that class-conditional structure can also differ across subjects (Lai-Tan et al., 11 Aug 2025).

The data are segmented EEG trials

$E \in \mathbb{R}^{t \times e \times s},$

where $t$ is the number of trials, $e$ the number of channels, and $s$ the number of time samples per trial. For trial $i$, the spatial covariance matrix is

$C_i = \frac{1}{s-1} E_i E_i^\top.$

This covariance representation is the basis for both the Riemannian and RCSP components of the full decoding pipeline.

The experimental task uses a publicly available dataset reported as OpenNeuro accession ds00197144, originally with 108 EEG channels and 20 healthy participants, with subjects 19 and 20 excluded due to artefact contamination, yielding 18 subjects. Participants walked on a treadmill while synchronising heel strikes to rhythmic auditory cues that could either increase (Advance tempo) or decrease (Delay tempo). The two classes are defined as adaptive heel strikes, taken as the first three heel strikes after tempo change, and non-adaptive heel strikes, taken as the middle three heel strikes in the trial. EEG was segmented around

$[S_x - 256 : S_{x+2} + 256],$

followed by a sliding window with window length $s=100$ and 50\% overlap.

The cross-montage setting introduces an additional shift. Training is performed on the 108-channel data, while testing is simulated on lower-density subsets corresponding to a 10-10 montage with 60 electrodes and a 10-20 montage with 19 electrodes. The paper does not use interpolation or an explicit spatial projection to a common montage; instead, it uses PCA so that training and testing features have compatible dimensionality before classification.

3. Mathematical construction of ITSA

ITSA begins with a subject-specific recentering step on covariance matrices. The paper gives the Fréchet mean / geometric mean as

$$C_{\text{ref}} = \mathfrak{G}(C_1,\ldots,C_I) = \arg\min_{C} \sum_{i=1}^{I} \delta_R^2(C, C_i),$$

with Riemannian distance

$\delta_R(C_1,C_2) = \left\| \logm(C_1^{-1} C_2) \right\|_{\mathcal F} = \left[\sum_{n=1}^{N} \log^2 \lambda_n \right]^{1/2}.$

In the ITSA procedure itself, the subject mean $t$0 is described as the log-Euclidean mean of that subject’s covariance matrices. Each covariance is then recentred as

$t$1

After recentering, projection to tangent space is performed at the identity, which simplifies the mapping to

$t$2

The paper states that this is followed by vectorisation / half-vectorisation, although the exact operator is not written explicitly.

The second stage is distribution matching by norm rescaling. The rescaled feature is defined as

$t$3

Applied separately to source and target domains, this makes the average norm within each domain equal to $t$4, thereby equalising overall feature dispersion after recentering and tangent projection (Lai-Tan et al., 11 Aug 2025).

The third stage is supervised rotational alignment. The held-out target subject is split into a calibration subset and an evaluation subset. For $t$5 classes, the class-wise means are

$t$6

$t$7

These anchor points are concatenated as

$t$8

Their cross-product matrix is

$t$9

with singular value decomposition

$e$0

The paper then truncates $e$1 and $e$2 to the minimum number of components $e$3 explaining 99.9\% variance, yielding $e$4 and $e$5, and rotates the evaluation features via

$e$6

This SVD-based step is recognisably Procrustes-like. A plausible interpretation is that the method is implementing an orthogonal alignment of target calibration anchors to source anchors, but the paper itself presents the procedure operationally through the anchor matrices, cross-product matrix, and SVD rather than through an explicit optimisation objective.

4. Position within the RCSP–Riemannian decoding pipeline

ITSA is not presented as a standalone classifier. It functions as a pre-alignment block inside a hybrid RCSP + Riemannian architecture. The RCSP component is used to improve class separability and reduce overfitting or noise sensitivity, while the Riemannian component preserves covariance geometry and supplies tangent-space features for statistical learning (Lai-Tan et al., 11 Aug 2025).

The RCSP equations reported in the paper are

$e$7

together with regularised covariances

$e$8

and

$e$9

The manuscript notes inconsistent notation but states that in the actual implementation of diagonal loading, $s$0 and $s$1 is estimated automatically via Ledoit–Wolf.

Two fusion architectures are described. In Sequential RCSP-Riemannian (Seq. RCSP-Rie), spatial filtering occurs first, covariance is then computed on the filtered signals, and ITSA recentering, tangent projection, rescaling, and rotation are applied afterward. In Parallel RCSP-Riemannian (Par. RCSP-Rie), one branch produces RCSP-based spatially filtered features and a second branch produces ITSA/Riemannian tangent-space features; the two feature sets are then concatenated for classification. The paper reports that the parallel fusion architecture performs better than the sequential one.

Classification is performed with a linear SVM using the default regularisation parameter

$s$2

The implementation is reported to use pyRiemann for tangent-space projection and half-vectorisation and scikit-learn for the broader pipeline.

In the cross-montage experiments, PCA is inserted differently depending on whether ITSA is used. Without ITSA, PCA is applied at the end of the processed feature pipeline. With ITSA, PCA is applied after rescaling and before rotation, so that source and target features share the same dimensionality when constructing the rotational alignment. The retention choices reported for montage experiments are 25\% for 10-10 and 1\% for 10-20.

5. Empirical results and reported performance

The evaluation protocol is leave-one-subject-out over the 18 retained subjects. Within each split, subject-specific recentering is performed independently for every subject using only that subject’s data; the 17 source subjects are then concatenated; the target subject is recentred and rescaled separately; and a nested 2-fold CV on the target subject divides the target data into calibration and evaluation subsets, swaps the roles of the two halves, and averages the final F1 score across both folds (Lai-Tan et al., 11 Aug 2025).

The central result is that ITSA substantially improves average LOSO F1 scores over no-alignment baselines in both the sequential and parallel fusion architectures.

Condition	Baseline F1	ITSA F1
Advance, Seq. RCSP-Rie	54.39 ± 11.05	61.15 ± 7.27
Advance, Par. RCSP-Rie	56.23 ± 8.43	61.34 ± 5.49
Delay, Seq. RCSP-Rie	41.96 ± 18.10	57.28 ± 5.65
Delay, Par. RCSP-Rie	42.65 ± 19.52	58.52 ± 5.65

These improvements are reported as statistically significant. After testing normality of paired differences with the Lilliefors test, the paper uses paired $s$3-tests for normal differences and Wilcoxon signed-rank tests for non-normal differences. The reported tests are: $s$4 for sequential Advance,

$s$5

for sequential Delay,

$s$6

for parallel Advance, and

$s$7

for parallel Delay.

The ablation study compares Baseline, Adaptive M, TS, and ITSA. The Adaptive M baseline, adapted from He and Wu (2020), performs subject-wise recentering only in Euclidean space. The TS baseline, based on Bleuzé et al. (2022), includes tangent-space alignment, rescaling, and rotation, but not the subject-specific recentering that defines ITSA. The reported means are:

Setting	Adaptive M	TS	ITSA
Advance, Seq.	57.44 ± 5.35	60.99 ± 6.41	61.15 ± 7.27
Advance, Par.	59.29 ± 4.13	61.00 ± 5.63	61.34 ± 5.49
Delay, Seq.	56.38 ± 6.41	57.13 ± 5.50	57.28 ± 5.65
Delay, Par.	55.08 ± 9.25	58.45 ± 5.65	58.52 ± 5.65

The paper interprets these results as showing that tangent-space methods outperform recentering-only alignment, and that ITSA consistently, though slightly, outperforms TS. This suggests that much of the performance gain comes from tangent-space scaling and rotation, with the extra individual recentering contributing a smaller additional benefit.

At the subject level, ITSA improved 10 of 18 subjects for both sequential and parallel architectures in Advance, and improved 8 subjects in sequential and 11 subjects in parallel in Delay. The paper notes some unsuccessful subjects but states that deterioration was rarely strong.

Cross-montage results are presented as further evidence of robustness. ITSA consistently outperformed the corresponding no-alignment baselines across 108-channel, 10-10, and 10-20 testing. For Advance, performance drops with reduced montage density were small under ITSA: in the sequential architecture only 1.60\% for 10-10 and 1.69\% for 10-20, and in the parallel architecture only 1.86\% for 10-10 and 1.62\% for 10-20. For Delay, smaller montages increased performance even without ITSA, but ITSA still yielded steadier and less variable behaviour across montages. The paper further highlights that ITSA with only 19 or 60 electrodes still outperformed the baseline achieved using the full 108-channel test setup.

The study also reports data-efficiency behaviour under reduced numbers of training subjects. When the number of source subjects is subsampled and averaged across 10 folds, even with more than 50\% reduction in training subjects, such as $s$8, the F1 drop stayed within about 5\% for the representative subjects shown, and performance still exceeded baseline. This suggests robustness to reduced source-subject availability, although the paper does not fully disentangle the contribution of ITSA alone from that of the full fused pipeline.

6. Relation to adjacent methods, common misconceptions, and limitations

ITSA belongs to a broader family of tangent-space and Riemannian transfer methods, but it is not interchangeable with them. A closely related EEG paper, Riemannian Transfer CSP (RTCSP), aligns source-subject covariance representations to a specific target subject in tangent space and then uses the aligned covariances to estimate better CSP spatial filters; however, RTCSP is not named ITSA and uses a different downstream objective, namely target-adapted CSP followed by LDA rather than the ITSA pre-alignment plus RCSP–Riemannian fusion used here (Gunasar et al., 23 Apr 2025). In manifold-learning work, GTSA-PCA is also tangent-space-based, but it aligns local tangent bases or subspaces through a geodesic affinity matrix for spectral embedding rather than performing subject-specific covariance recentering, rescaling, and supervised rotation for EEG transfer (Levada, 20 Apr 2026). The recent LEGO method is likewise relevant because it improves tangent-space estimation under noise, but it is a tangent-estimation front end rather than an ITSA alignment procedure (Kohli et al., 2 Oct 2025).

A first common misconception is that ITSA is unsupervised. It is not. The rotational alignment stage is explicitly supervised and requires labelled calibration data from the target subject. The nested 2-fold split is used precisely to avoid leakage between calibration and evaluation. A second misconception is that the “individual” aspect refers to samplewise tangent-space alignment in the sense used in some manifold-learning literature. In the present setting, “individual” refers to per-subject recentering before source pooling. A third misconception arises from acronym overload: in the stereo-matching literature, ITSA refers to “Information-Theoretic Shortcut Avoidance,” an unrelated method for domain generalisation in synthetic-to-real stereo (Chuah et al., 2022).

The paper also leaves several limitations explicit or implicit. Several equations are malformed or incompletely specified in the manuscript extraction, including the generic tangent-space projection equation; the exact half-vectorisation operator is not defined explicitly; and detailed preprocessing choices such as filtering bands are absent. The gains of ITSA over the TS baseline are modest, which supports the paper’s own interpretation that the largest benefits come from tangent-space scaling and rotation, with the individual recentering step adding a smaller but consistent increment. The cross-montage strategy is pragmatic rather than biophysically principled: it relies on subset channel extraction and PCA-based dimensional compatibility rather than interpolation, forward-model alignment, or a common spatial transform. Finally, the method is demonstrated on a specific auditory-cued gait adaptation dataset with healthy subjects; broader applicability to motor imagery, stroke populations, seated paradigms, or other EEG tasks is plausible but not directly established in the reported experiments (Lai-Tan et al., 11 Aug 2025).

Taken together, the current literature supports a precise characterisation: Individual Tangent Space Alignment is a supervised, subject-preserving, covariance-to-tangent-space transfer method for EEG decoding in which per-subject recentering precedes domain-wise rescaling and class-anchor-based rotational alignment. Its primary empirical strengths are significant gains over no alignment, clear improvements over recentering-only baselines, small but consistent improvements over standard tangent-space alignment, and strong robustness under cross-montage testing when embedded in a parallel RCSP–Riemannian feature-fusion architecture.