**footnotetext: Equal contribution

# Multi-Source and Test-Time Domain Adaptation on Multivariate Signals using
Spatio-Temporal Monge Alignment

Théo Gnassounou*<EMAIL_ADDRESS>
Université Paris-Saclay
Inria, CEA
Palaiseau, 91120, France Antoine Collas*<EMAIL_ADDRESS>
Université Paris-Saclay
Inria, CEA
Palaiseau, 91120, France Rémi Flamary<EMAIL_ADDRESS>
CMAP, UMR 7641
Institut Polytechnique de Paris
Palaiseau, 91120, France Karim Lounici<EMAIL_ADDRESS>
CMAP, UMR 7641
Institut Polytechnique de Paris
Palaiseau, 91120, France Alexandre Gramfort<EMAIL_ADDRESS>
Université Paris-Saclay
Inria, CEA
Palaiseau, 91120, France

###### Abstract

Machine learning applications on signals such as computer vision or biomedical
data often face significant challenges due to the variability that exists
across hardware devices or session recordings. This variability poses a Domain
Adaptation (DA) problem, as training and testing data distributions often
differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA) to
mitigate these variabilities. This Optimal Transport (OT) based method adapts
the cross-power spectrum density (cross-PSD) of multivariate signals by
mapping them to the Wasserstein barycenter of source domains (multi-source
DA). Predictions for new domains can be done with a filtering without the need
for retraining a model with source data (test-time DA). We also study and
discuss two special cases of the method, Temporal Monge Alignment (TMA) and
Spatial Monge Alignment (SMA). Non-asymptotic concentration bounds are derived
for the mappings estimation, which reveals a bias-plus-variance error
structure with a variance decay rate of $\mathcal{O}(n_{\ell}^{-1/2})$ with
$n_{\ell}$ the signal length. This theoretical guarantee demonstrates the
efficiency of the proposed computational schema. Numerical experiments on
multivariate biosignals and image data show that STMA leads to significant and
consistent performance gains between datasets acquired with very different
settings. Notably, STMA is a pre-processing step complementary to state-of-
the-art deep learning methods.

##  1 Introduction

Machine learning approaches have led to impressive results across many
applications, from computer vision and biology to audio and language
processing. However, these approaches have known limitations in the presence
of distribution shifts between training and evaluation datasets. Indeed,
performance drops in the presence of distribution shifts have been observed in
different fields such as computer vision [1], clinical data [2], and tabular
data [3].

##### Domain Adaptation (DA) from single to multi-domain

This problem of data shift is well-known in machine learning and has been
investigated in the DA community [4, 5, 6, 7]. Given source domain data with
access to labels, DA methods aim to learn a model that can adequately predict
on a target domain where no label is available, assuming the existence of a
distribution shift between the two domains. Traditionally, DA methods try to
adapt the source to be closer to the target using reweighting methods [4, 8],
mapping estimation [9, 10] or dimension reduction [11]. Inspired by the
successes of deep learning in computer vision, modern methods aim to reduce
the shift between the embeddings of domains learned by a feature extractor. To
achieve this, most methods attempt to minimize a divergence between the
features of the source and target data. Several divergences have been proposed
in the literature: correlation distance [12], adversarial method [1], maximum
mean discrepancy distance [13] or Optimal Transport (OT) [14, 15].

With the rise of portable devices and open data, the problem of DA has evolved
from a single-source setting where only one source is known to a multi-source
setting where each source domain can have a different shift. This leads to a
multi-source DA problem [16], where the variability across multiple domains
can help training better predictors. For instance, one can learn domain-
specific batch normalization [17, 18], compute weights to give more importance
to some domains [19], or use moment matching to align the features [20].
Another method to deal with multiple sources is to create an intermediate
domain between sources and target. [21] propose to compute a Wasserstein
barycenter of the sources and then project this barycenter to the target using
classical OT methods.

An additional challenge with traditional DA methods [15, 1, 12] is that they
require using both the source and target domains to train the predictor. This
means one needs access to the source data to train a model for new domains.
However, the source data may not always be available due to privacy concerns
or memory limitations. Test-time DA is a branch of DA that aims to adapt a
predictor to the target data without access to the source data. In [22], the
authors propose SHOT, which trains the classifier on source data and matches
the target features to the fixed classifier using Information Maximization
(IM). IM aims to produce predictions that are individually confident and
globally diverse. [23] enhance SHOT with multi-source information, allowing
the selection of the optimal combination of sources by learning the weights of
each source model. [24] uses the local structure clustering technique on the
target feature to adapt the model. Note that these methods are often complex
and require high computational resources since they require training a new
estimator for each new target domain.

##### DA for multivariate signals and biomedical data

This paper focuses on multi-source and test-time DA for multivariate signals,
which are common in applications where data is collected from multiple
devices. For instance, in the biomedical field, signals such as
Electroencephalogram (EEG), Electrocardiogram (ECG), or Electromyogram (EMG)
are often multivariate and can exhibit a shift between subjects, sessions, or
devices. Several applications of DA in biosignals have been proposed for pairs
of datasets [25, 26]. However, in clinical datasets, we often have access to
multiple subjects or hospital data, which can be considered as separate
domains. In this paper, we are interested in two biosignal applications with
their own specificities: sleep staging and Brain-Computer Interface (BCI).
Sleep staging is the process of categorizing sleep into five stages based on
the electrical activity of the brain, i.e., EEG [27]. BCI motor imagery
classifies mental visualization of motor movements using EEG signals. In both
applications, datasets can exhibit diverse population distributions because of
variations in factors such as age, gender, diseases, or individual
physiological responses to stimuli [28]. All these variabilities introduce
data shifts between different subjects or datasets (domains) that limit
performances of traditional supervised learning methods [29, 26].

To take into account those shifts, or more precisely to try to mitigate them,
simple normalization techniques have proven useful for BCI and sleep staging,
leading to easy computation and application at test-time. For instance,
researchers proposed Riemannian Procrustes Analysis (RPA) [30] to reduce the
shift in BCI. RPA matches data distributions of source and target domains
through recentering, stretching, and rotation, while operating on a Riemanian
manifold of covariance matrices. When using deep learning, people typically
only use the recentering part proposed in RPA and recenter the signal using
Euclidean alignment [31, 32] or Riemannian Alignment (RA) [33, 29].
Alternatively, in sleep staging, a study by [34] utilized a convolutional
Monge mapping normalization method to align each signal to a Wasserstein
barycenter. This approach improved sleep staging significantly, compared to
classical normalization techniques, such as the unit standard deviation
scaling [35, 36].

Figure 1: Illustration of Monge Alignment. At train-time the barycenter (grey
ellipse) is estimated from 3 source distributions (colored ellipses). The
predictor is learned on normalized data. At test-time the same barycenter is
used to align the unlabeled target distribution (orange ellipse) and predict.

##### Contributions

This paper proposes a general method for multi-source and test-time DA for
multivariate signals. This method denoted as Monge Alignment (MA), uses OT
modeling of stationary signals to estimate a mapping. The latter aligns the
different domains onto a barycenter of the domains before learning a
predictor, attenuating the shift (Figure 1 top). At test time, one only needs
to align the target domain on the barycenter and predict with the trained
model (Figure 1 bottom). This method can be seen as a special case of [21],
where the authors propose to align signals using the Wasserstein barycenter,
but where the modeling of multivariate signals allows for more efficient
barycenter computation and mapping estimation. It is also a generalization of
the preliminary work of [34] where a Convolutional Monge Mapping Normalization
was proposed to deal very efficiently with temporal shifts, but the different
signals in the observations are assumed to be independent. The novel
contributions of this paper are:

  * •

A general method for multi-source and test-time DA for multivariate signals,
using OT between Gaussian signals, denoted as Spatio-Temporal Monge Alignment
(STMA) as well as Temporal and Spatial Monge Alignments (TMA and SMA
respectively) tailored for specific shifts assumptions.

  * •

Efficient algorithms for estimations of the mapping and the barycenter of the
signals using Fast Fourier Transform (FFT) and block-diagonalization of the
covariances matrices.

  * •

Non-asymptotic concentration bounds of STMA, TMA and SMA estimation which
present the interactions between the number of samples, the number of domains,
the dimensionality of the signals, and the size of the filters.

  * •

Novel numerical experiments on two biosignal tasks, Sleep staging, and BCI
motor imagery, that show the benefits of the proposed method over state-of-
the-art methods and illustrate the relative importance of spatial and temporal
filtering.

In addition to the contributions discussed above, the proposed multi-source
method is predictor-agnostic. It can be used for shallow or deep learning
methods, as it acts as a pre-processing step. The estimated parameters can be
very efficiently computed for new domains without the need for the source data
or refitting of the learned model at test-time. Table 1 summarizes the
specificities of the proposed method compared to other existing DA methods.

##### Paper outline

After quickly recalling OT between Gaussian distributions, we introduce the
spatio-temporal Monge mapping for Gaussian multivariate signals in subsection
3.1. In section 4, we detail the Spatio-Temporal Monge Alignment (STMA) method
for multi-source and test-time DA and its special cases using only temporal
(TMA) or spatial (SMA) alignments. This section also provides non-asymptotic
concentration bounds which reveals a bias-plus-variance error structure. In
section 5, we apply MA to two real-life biosignal tasks using EEG data: sleep
stage classification and BCI motor imagery and provide an illustrative example
on images (2D signals).

Methods |  Multi-source |  Test-time |  No need to refit |  Method agnostic |  Temporal shift |  Spatial shift
---|---|---|---|---|---|---
Divergence minimization [12] | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
Moment Matching [20] | ✓ | ✗ | ✗ | ✗ | ✓ | ✓
SHOT [22] | ✗ | ✓ | ✗ | ✗ | ✓ | ✓
DECISION [23] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓
AdaBN [17] | ✓ | ✓ | ✓ | ✗ | ✓ | ✓
Riemanian Alignment [29] | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Temporal MA [34] | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
Spatial MA (ours) | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Spatio-Temp MA (ours) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

Table 1: Comparison of DA methods with various specificities. “Multi-source
indicates the ability to handle multiple sources, “Test-time” means that the
method does not require source data at test-time, “No need to refit” signifies
whether refitting is required for a new domain at test-time, “Method agnostic”
refers to independence from a specific type of predictors (e.g., deep
learning), while “Temporal shift” and “Spatial shift” indicate the capability
to handle temporal and spatial shifts. Only STMA handles all the
specificities.

##### Notations

Vectors are denoted by small cap boldface letters (e.g., $\mathbf{x}$),
matrices by large cap boldface letters (e.g., $\mathbf{X}$). The element-wise
product is $\odot$, and the element-wise power of $n$ is $\cdot^{\odot n}$.
$\left\llbracket 1,K\right\rrbracket\,$ denotes $\\{1,\ldots,K\\}$. The
absolute value is $|.|$. The discrete convolution operator is $*$.
$\mathcal{S}_{n}$ and $\mathcal{S}_{n}^{++}$ denote the sets of symmetric and
symmetric positive definite matrices of size $n\times n$. $\mathcal{H}_{n}$
and $\mathcal{H}_{n}^{++}$ denote the sets of hermitian and hermitian positive
definite matrices of size $n\times n$. $\mathcal{C}_{n}$ is the set of real-
valued circulant matrices of size $n\times n$. We denote by
$\lambda_{\max}(\mathbf{A})$ the maximum eigenvalue of the matrix
$\mathbf{A}$. We also define the effective rank by
$\mathbf{r}(\mathbf{A})=\frac{\text{tr}(\mathbf{A})}{\lambda_{\max}(\mathbf{A})}$
where $\text{tr}(\mathbf{A})$ is the trace of $\mathbf{A}$. $\mathbb{N}^{*}$
is the integer set excluding $0$, and $\mathbb{R}^{+*}$ is the set of strictly
positive real values. $\mathbf{X}_{k}$ and $\mathbf{X}_{t}$ relate to source
domains $k\in\left\llbracket 1,n_{d}\right\rrbracket\,$ and the target domain,
respectively. $\operatorname{vec}:\mathbb{R}^{n_{c}\times
n_{\ell}}\to\mathbb{R}^{n_{c}n_{\ell}}$ concatenates rows of a time series
into a vector, and
$\operatorname{vec}^{-1}:\mathbb{R}^{n_{c}n_{\ell}}\to\mathbb{R}^{n_{c}\times
n_{\ell}}$ is the reciprocal. $\left\lceil{.}\right\rceil$ and
$\left\lfloor{.}\right\rfloor$ denote the smallest integer greater than or
equal to, and the greatest integer less than or equal to, respectively.
$(\mathbf{x})_{l}$ refers to the $l^{\text{th}}$ element of $\mathbf{x}$, and
$(\mathbf{X})_{lm}$ denotes the element of $\mathbf{X}$ at the $l^{\text{th}}$
row and $m^{\text{th}}$ column. $\mathbf{X}^{\mathsf{H}}$ is the conjugate
transpose of $\mathbf{X}$. $\otimes$ is the Kronecker operator.

##  2 Signal adaptation with Optimal Transport

In this section, we first briefly introduce OT between centered Gaussian
distributions. Then, we describe how [34] has used this mapping between
distributions to align univariate signals on a barycenter. Finally, we
introduce approximations on circulant matrices, which explain how the
structure of covariance matrices is efficiently used.

###  2.1 Optimal Transport between Gaussian distributions

####  2.1.1 Monge mapping for Gaussian distributions

Let two Gaussian distributions
$\mu_{s}=\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and
$\mu_{t}=\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$, where
${\boldsymbol{\Sigma}}_{s}$ and ${\boldsymbol{\Sigma}}_{t}$ are symmetric
positive definite matrices. OT between Gaussian distributions is one of the
rare cases where a closed-form solution exists.

The OT mapping, also called Monge mapping, is the following affine function
[37, 38, 39]:

$m(\mathbf{x})=\mathbf{A}\mathbf{x}\quad\text{with}\quad\mathbf{A}={\boldsymbol{\Sigma}}_{s}^{-\frac{1}{2}}\left({\boldsymbol{\Sigma}}_{s}^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{t}{\boldsymbol{\Sigma}}_{s}^{\frac{1}{2}}\right)^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{s}^{-\frac{1}{2}}=\mathbf{A}^{\mkern-1.5mu\mathsf{T}}\;.$
(1)

In practical applications, one can estimate the covariance matrices of the two
distributions to get an estimation of the Wasserstein distance and its
associated mapping [40].

####  2.1.2 Wasserstein barycenter between Gaussian distributions

The Wasserstein barycenter $\overline{\mu}$ defines a notion of averaging of
probability distributions $\\{\mu_{k}\\}_{1\leq k\leq n_{d}}$ which is the
solution of a convex program involving Wasserstein distances:

$\overline{\mu}\triangleq\underset{\mu}{\text{arg
min}}\;\frac{1}{n_{d}}\sum_{k=1}^{n_{d}}\mathcal{W}_{2}^{2}(\mu,\mu_{k})\;.$
(2)

Interestingly, when $\mu_{k}$ are Gaussian distributions, the barycenter is
still a Gaussian distribution
$\overline{\mu}=\mathcal{N}(\mathbf{0},\overline{{\boldsymbol{\Sigma}}})$
[41]. There is, in general, no closed-form expression for computing the
covariance matrix $\overline{{\boldsymbol{\Sigma}}}$. But the first order
optimality condition of this convex problem shows that
$\overline{{\boldsymbol{\Sigma}}}$ is the unique positive definite fixed point
of the map

$\overline{{\boldsymbol{\Sigma}}}=\Psi\left(\overline{{\boldsymbol{\Sigma}}},\left\\{{\boldsymbol{\Sigma}}_{k}\right\\}_{k=1}^{n_{d}}\right)\quad\text{where}\quad\Psi\left(\mathbf{A},\left\\{\mathbf{B}_{k}\right\\}_{k=1}^{n_{d}}\right)=\frac{1}{n_{d}}\sum_{k=1}^{n_{d}}\left(\mathbf{A}^{\frac{1}{2}}\mathbf{B}_{k}\mathbf{A}^{\frac{1}{2}}\right)^{\frac{1}{2}}.$
(3)

Iterating this fixed-point map, i.e.,

$\overline{{\boldsymbol{\Sigma}}}^{(\ell+1)}=\Psi\left(\overline{{\boldsymbol{\Sigma}}}^{(\ell)},\left\\{{\boldsymbol{\Sigma}}_{k}\right\\}_{k=1}^{n_{d}}\right),$
(4)

converges in practice to the solution $\overline{{\boldsymbol{\Sigma}}}$ [42].

###  2.2 OT mapping of stationary signals using circulant matrices

OT emerges as a natural tool for addressing distribution shift and has found
significant application in DA, as evidenced by works such as [43], [15], and
[19]. Another line of methods leverages the Gaussian assumption to
computationally simplify the estimation of Wasserstein mapping and barycenter,
as illustrated in [42] and [40]. In the context of this work, the focus is on
the analysis of time-series data. Under certain judicious assumptions, the
computational efficiency of Monge mapping and barycenter computation is
further enhanced, as proposed by [34].

####  2.2.1 Circulant matrices for stationary and periodic signals

(a) $\mathbf{A}\in{\mathcal{C}_{5}\cap\mathcal{S}_{5}^{++}}$ with coefficients
$\color[rgb]{0.1484375,0.5234375,0.8984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1484375,0.5234375,0.8984375}\mdblksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\color[rgb]{0.98046875,0.74609375,0.31640625}\definecolor[named]{pgfstrokecolor}{rgb}{0.98046875,0.74609375,0.31640625}\mdblksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\color[rgb]{0.109375,0.77734375,0.5234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.109375,0.77734375,0.5234375}\mdblksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\in\mathbb{R}$

(b)
$\mathcal{P}_{f}(\mathbf{A})\in{\mathcal{E}_{3,5}\cap\mathcal{S}_{5}^{++}}$
approximates $\mathbf{A}$

Figure 2:  A Monge mapping $\mathbf{A}$ (cf. Equation 1) that is a $5\times 5$
symmetric positive definite and circulant matrix, i.e., belonging to
$\mathcal{C}_{5}\cap\mathcal{S}_{5}^{++}$, and its approximation in
$\mathcal{E}_{3,5}\cap\mathcal{S}_{5}^{++}$ as defined in Equation 9. In this
case, for $\mathbf{x}\in\mathbb{R}^{n_{\ell}}$,
$\mathcal{P}_{f}(\mathbf{A})\mathbf{x}=\mathbf{h}*\mathbf{x}$ with
$\mathbf{h}=[\color[rgb]{0.98046875,0.74609375,0.31640625}\definecolor[named]{pgfstrokecolor}{rgb}{0.98046875,0.74609375,0.31640625}\mdblksquare\color[rgb]{0.1484375,0.5234375,0.8984375}\definecolor[named]{pgfstrokecolor}{rgb}{0.1484375,0.5234375,0.8984375}\mdblksquare\color[rgb]{0.98046875,0.74609375,0.31640625}\definecolor[named]{pgfstrokecolor}{rgb}{0.98046875,0.74609375,0.31640625}\mdblksquare\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
A common assumption in signal processing is that if the signal is long enough,
then the signal can be considered periodic and stationary, and the covariance
matrix becomes a symmetric positive definite (i.e.,
$\mathcal{S}_{n_{\ell}}^{++}$) and circulant (i.e., $\mathcal{C}_{n_{\ell}}$)
matrix i.e., a symmetric positive definite matrix with cyclically shifted
rows. 2(a) presents an example of such a matrix with an univariate signal of
length $n_{\ell}=5$. The utility of this assumption lies in its
diagonalization through the Fourier basis, simplifying analyses and
computations on such time series. Indeed, for
$\mathbf{A}\in{\mathcal{C}_{n_{\ell}}\cap\mathcal{S}_{n_{\ell}}^{++}}$ and
denoting the power spectrum density (PSD) by
$\mathbf{q}\in(\mathbb{R}^{+*})^{{n_{\ell}}}$, we have the decomposition [44,
Theorem 3.1]

$\mathbf{A}=\mathbf{F}_{n_{\ell}}\operatorname{diag}(\mathbf{q})\mathbf{F}_{n_{\ell}}^{\mathsf{H}}\;,$
(5)

with the Fourier basis of elements

$(\mathbf{F}_{n_{\ell}})_{lm}\triangleq\frac{1}{\sqrt{n_{\ell}}}\exp\left(-2i\pi\frac{(l-1)(m-1)}{n_{\ell}}\right)\;,$
(6)

for $l,m\in\left\llbracket 1,n_{\ell}\right\rrbracket\,$ and
$n_{\ell}\in\mathbb{N}^{*}$.

####  2.2.2 Convolutional Monge Mapping and filter size

In the paper by [34], the authors use the circularity structure to diagonalize
all the covariance matrices with the Fourier basis. This leads to a
simplification of the Monge mapping Equation 1. Let us consider two centered
Gaussian distributions: a source distribution
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and a target distribution
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$, both with covariance
matrices in $\mathcal{C}_{n_{\ell}}\cap\mathcal{S}_{n_{\ell}}^{++}$, and a
realization $\mathbf{x}\in\mathbb{R}^{n_{\ell}}$ of the source distribution.
Using Fourier-based eigen-factorization of the covariance matrices, we can map
between these two Gaussian distributions, characterized by their power
spectral densities (PSDs)
$\mathbf{q}_{s}\triangleq\operatorname{diag}(\mathbf{F}_{n_{\ell}}^{\mathsf{H}}{\boldsymbol{\Sigma}}_{s}\mathbf{F}_{n_{\ell}})$
and
$\mathbf{q}_{t}\triangleq\operatorname{diag}(\mathbf{F}_{n_{\ell}}^{\mathsf{H}}{\boldsymbol{\Sigma}}_{t}\mathbf{F}_{n_{\ell}})$,
with the Monge mapping from Equation 1. This mapping $\mathbf{A}$ belongs to
$\mathcal{C}_{n_{\ell}}\cap\mathcal{S}_{n_{\ell}}^{++}$ and hence can be
expressed as the following convolution [44, Chapter 3]

$m(\mathbf{x})=\mathbf{h}*\mathbf{x}\;,\quad\text{with}\quad\mathbf{h}\triangleq\frac{1}{\sqrt{n_{\ell}}}\mathbf{F}_{n_{\ell}}^{\mathsf{H}}\left(\mathbf{q}_{t}^{\odot\frac{1}{2}}\odot\mathbf{q}_{s}^{\odot-\frac{1}{2}}\right)\in\mathbb{R}^{n_{\ell}}\;.$
(7)

Utilizing the same factorization, [34] introduce a novel closed-form
expression for barycenter computation. Subsequently, the authors propose the
Convolutional Monge Mapping Normalization (CMMN) method, wherein a barycenter
is computed, and all domains are aligned to this central point, constituting a
test-time DA approach. Notably, this method exhibits significant performance
gains in practical applications.

A nuanced aspect of this technique lies in choosing the filter size in
Equation 7. In practice, $\mathbf{h}$ is computed with a size $f$ instead of
$n_{\ell}$. Opting for a filter size equivalent to the signal size (i.e.,
$f=n_{\ell}$) results in a perfect mapping of domains to the barycenter,
potentially eliminating class-discriminative information. Selecting $f=1$
scales the entire signal, equivalent to a standard z-score operation. Despite
its significance, the impact of $f$ is not studied by [34]. Here, we explore
the approximations made to incorporate the filter size $f$ into the
theoretical framework.

####  2.2.3 Approximation of circulant matrices

Matrices in $\mathcal{C}_{n_{\ell}}$, and their associated filter, have
$n_{\ell}$ parameters, i.e., the signal length. To reduce this parameter
number, we propose approximating matrices in $\mathcal{C}_{n_{\ell}}$ by
linear mappings with only $f\ll n_{\ell}$ parameters. First, we define the set
of circulant matrices with $f$ parameters as

$\mathcal{E}_{f,n_{\ell}}\triangleq\left\\{\mathbf{A}\in\mathcal{C}_{n_{\ell}}\mid(\mathbf{A})_{1l}=0,\;l\in\left\llbracket\left\lceil{f/2}\right\rceil+1,n_{\ell}-\left\lfloor{f/2}\right\rfloor\right\rrbracket\right\\}\;.$
(8)

Then, we approximate every $\mathbf{A}\in\mathcal{C}_{n_{\ell}}$ by its
nearest element in $\mathcal{E}_{f,n_{\ell}}$, i.e.,

$\mathcal{P}_{f}(\mathbf{A})\triangleq\mathop{\rm
arg\,min}_{\mathbf{\Gamma}\in\mathcal{E}_{f,n_{\ell}}}\,\|\mathbf{\Gamma}-\mathbf{A}\|_{\text{Fro}}$
(9)

where $\|.\|_{\text{Fro}}$ is the Frobenius norm. Thus,
$\mathcal{P}_{f}(\mathbf{A})$ is the orthogonal projection of $\mathbf{A}$
onto $\mathcal{E}_{f,n_{\ell}}$ and its elements are
$(\mathcal{P}_{f}(\mathbf{A}))_{1l}=(\mathbf{A})_{1l}$ for
$l\in\left\llbracket
1,\left\lceil{f/2}\right\rceil\right\rrbracket\,\cup\llbracket
n_{\ell}-\left\lfloor{f/2}\right\rfloor+1,n_{\ell}\rrbracket$. An example in
$\mathcal{E}_{3,5}$ is presented in 2(b). With this approximation, we get that
$\mathcal{P}_{f}(\mathbf{A})$ applies a convolution with the filter
$\mathbf{h}\in\mathbb{R}^{f}$ that contains the non-zero elements of the first
row of $\mathbf{A}$,

$\mathcal{P}_{f}(\mathbf{A})\;\mathbf{x}=\mathbf{h}*\mathbf{x}\enspace.$ (10)

Since $\mathcal{P}_{f}(\mathbf{A})\in\mathcal{C}_{n_{\ell}}$, it admits the
decomposition
$\mathcal{P}_{f}(\mathbf{A})=\mathbf{F}_{n_{\ell}}\operatorname{diag}(\mathbf{q})\mathbf{F}_{n_{\ell}}^{\mathsf{H}}$
with $\mathbf{q}\in\mathbb{C}^{n_{\ell}}$ that will be interpreted latter as a
cross-PSD. An important property of $\mathbf{h}$ is its computation from a
sub-sampling***In the following, we assume that the signal length ($n_{\ell}$)
is a multiple of the filter size ($f$). of $\mathbf{q}$, denoted
$\mathbf{p}\in{\mathbb{C}^{f}}$ and of elements

$(\mathbf{p})_{l}\triangleq(\mathbf{q})_{\frac{(l-1)n_{\ell}}{f}+1}$ (11)

for $l\in\left\llbracket 1,f\right\rrbracket\,$. We denote by $g_{f}$ the
function that does this operation, i.e.,

$g_{f}(\mathbf{q})\triangleq\mathbf{p}\;.$ (12)

The following proposition bridges the gap between filtering,
$\mathcal{E}_{f,n_{\ell}}$ and $g_{f}$. This way, we can sub-sample a given
cross-PSD to get a smaller one of size $f$. The associated filter is computed
with an inverse Fourier transform and the mapping is achieved with a
convolution. Overall, the number of parameters is $f$ and the complexity to
compute the filter and the mapping are $\mathcal{O}(f\log(f))$ and
$\mathcal{O}(n_{\ell}\log(n_{\ell}))$ respectively thanks to the Fast Fourier
Transform (FFT).

######  Proposition 1 ($\mathcal{E}_{f,n_{\ell}}$ and filtering)

Let $\mathbf{F}_{n_{\ell}}\in\mathbb{C}^{n_{\ell}\times n_{\ell}}$ and
$\mathbf{F}_{f}\in\mathbb{C}^{f\times f}$ be the Fourier bases. Given
$\mathbf{A}=\mathbf{F}_{n_{\ell}}\operatorname{diag}(\mathbf{q})\mathbf{F}_{n_{\ell}}^{\mathsf{H}}\in\mathcal{E}_{f,n_{\ell}}$
with $\mathbf{q}\in\mathbb{C}^{n_{\ell}}$, and $\mathbf{h}\in\mathbb{R}^{f}$
the filter containing the non-zero elements of the first row of $\mathbf{A}$,
then for every $\mathbf{x}\in\mathbb{R}^{n_{\ell}}$, we have

$\mathbf{A}\mathbf{x}=\mathbf{h}*\mathbf{x}$

where $*$ is the convolution operator and $\mathbf{h}$ is the inverse Fourier
transform of $\mathbf{p}=g_{f}(\mathbf{q})\in{\mathbb{C}^{f}}$, i.e.,

$\mathbf{h}=\frac{1}{\sqrt{f}}\mathbf{F}_{f}^{\mathsf{H}}\mathbf{p}\;.$

##  3 Spatio-Temporal Monge mapping and barycenter for Gaussian signals

This section introduces a new method for mapping multivariate signals by
seamlessly incorporating the filter size parameter and using the previously
presented approximation of circulant matrices. We first describe the structure
of the spatio-temporal covariance matrix. Then, we define the $f$-Monge
mapping and compute its closed-form expression for the spatio-temporal
structure as well as the associated Wasserstein barycenter. Next, we establish
two mappings and their barycenters when only the temporal or spatial structure
is considered. In the following, we consider multivariate signals
$\mathbf{X}\triangleq[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$ with $n_{c}$ channels of length $n_{\ell}$, and assume that the
vectorized signal follows a centered Gaussian distribution, i.e.,
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}})$
where ${\boldsymbol{\Sigma}}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}$ is the
spatio-temporal covariance matrix.

###  3.1 Spatio-temporal covariance matrix

First, the spatio-temporal covariance matrix is expressed as

${\boldsymbol{\Sigma}}=\mathbb{E}\left[\operatorname{vec}(\mathbf{X})\operatorname{vec}(\mathbf{X})^{\mkern-1.5mu\mathsf{T}}\right]=\begin{pmatrix}{\boldsymbol{\Sigma}}_{1,1}&\dots&{\boldsymbol{\Sigma}}_{1,n_{c}}\\\
\dots&\dots&\dots\\\
{\boldsymbol{\Sigma}}_{n_{c},1}&\dots&{\boldsymbol{\Sigma}}_{n_{c},n_{c}}\end{pmatrix}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}\;,$
(13)

where ${\boldsymbol{\Sigma}}_{l,m}$ corresponds to the temporal covariance
matrix between the signals of the lth and the mth channels. We add a simple
assumption on the ${\boldsymbol{\Sigma}}_{l,m}$ that will lead to a simple
Monge mapping later in this section.

###### Assumption 2

For every $l,m\in\left\llbracket 1,n_{c}\right\rrbracket\,^{2}$, we assume
that ${\boldsymbol{\Sigma}}_{l,m}$, defined in Equation 13, belongs to
$\mathcal{C}_{n_{\ell}}$, i.e.,

${\boldsymbol{\Sigma}}_{l,m}=\mathbf{F}_{n_{\ell}}\operatorname{diag}(\mathbf{q}_{l,m})\mathbf{F}_{n_{\ell}}^{\mathsf{H}},$

where $\mathbf{q}_{l,m}\in\mathbb{C}^{n_{\ell}}$ is such that the cross-PSD
$\mathbf{Q}_{\ell}\triangleq\left[(\mathbf{q}_{l,m})_{\ell}\right]_{l,m=1,1}^{n_{c},n_{c}}\in\mathcal{H}_{n_{c}}^{++}$,
for $\ell\in\left\llbracket 1,n_{\ell}\right\rrbracket\,$.

Figure 3:  Block diagonalization of the covariance matrix
${\boldsymbol{\Sigma}}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}$ following the
Assumption 2 with $n_{\ell}=5$ and $n_{c}=3$.
$\mathbf{F}=\operatorname{diag}(\mathbf{F}_{n_{\ell}},\dots,\mathbf{F}_{n_{\ell}})\in\mathbb{C}^{n_{c}n_{\ell}}$
and $\mathbf{U}$ is a permutation matrix.

Under the Assumption 2, ${\boldsymbol{\Sigma}}$ is re-written

${\boldsymbol{\Sigma}}=\mathbf{F}\begin{pmatrix}\operatorname{diag}(\mathbf{q}_{1,1})&\dots&\operatorname{diag}(\mathbf{q}_{1,n_{c}})\\\
\dots&\dots&\dots\\\
\operatorname{diag}(\mathbf{q}_{n_{c},1})&\dots&\operatorname{diag}(\mathbf{q}_{n_{c},n_{c}})\end{pmatrix}\mathbf{F}^{\mathsf{H}}\;,$
(14)

with
$\mathbf{F}\triangleq\operatorname{diag}(\mathbf{F}_{n_{\ell}},\dots,\mathbf{F}_{n_{\ell}})\in\mathbb{C}^{n_{c}n_{\ell}\times
n_{c}n_{\ell}}$. Hence, there exists a permutation matrix
$\mathbf{U}\in\mathbb{R}^{n_{c}n_{\ell}\times n_{c}n_{\ell}}$ that block-
diagonalizes ${\boldsymbol{\Sigma}}$, i.e.,

${\boldsymbol{\Sigma}}=\mathbf{F}\mathbf{U}\mathbf{Q}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}\;,$
(15)

with
$\mathbf{Q}\triangleq\operatorname{diag}(\mathbf{Q}_{1},\dots,\mathbf{Q}_{n_{\ell}})\in{\mathcal{H}_{n_{c}n_{\ell}}^{++}}$
and $\mathbf{Q}_{\ell}$ defined in Assumption 2. An example of this
decomposition for $n_{\ell}=5$ and $n_{c}=3$ is given in Figure 3. Notably,
$\mathbf{Q}$ is the cross-PSD and can be sub-sampled by extending the function
$g_{f}$ to block-diagonal matrices, i.e.,

$g_{f}(\mathbf{Q})\triangleq\operatorname{diag}\left(\mathbf{Q}_{1},\mathbf{Q}_{\frac{n_{\ell}}{f}+1},\dots,\mathbf{Q}_{\frac{(f-1)n_{\ell}}{f}+1}\right)\in{\mathcal{H}_{n_{c}f}^{++}}.$
(16)

Thus, we can sub-sample the cross-PSDs of the blocks of
${\boldsymbol{\Sigma}}$,

$\begin{pmatrix}\operatorname{diag}(\mathbf{p}_{1,1})&\dots&\operatorname{diag}(\mathbf{p}_{1,n_{c}})\\\
\dots&\dots&\dots\\\
\operatorname{diag}(\mathbf{p}_{n_{c},1})&\dots&\operatorname{diag}(\mathbf{p}_{n_{c},n_{c}})\end{pmatrix}=\mathbf{V}g_{f}(\mathbf{Q})\mathbf{V}^{\mkern-1.5mu\mathsf{T}}\in{\mathcal{H}_{n_{c}f}^{++}}$
(17)

with $\mathbf{p}_{l,m}=g_{f}(\mathbf{q}_{l,m})\in{\mathbb{C}^{f}}$ and
$\mathbf{V}\in\mathbb{R}^{n_{c}f\times n_{c}f}$ a permutation matrix.

###  3.2 Spatio-Temporal $f$-Monge mapping and barycenter

The Assumption 2 implies a specific structure on the Monge mapping from
Equation 1. Indeed, given $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$
and $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$ source and target
centered Gaussian distributions respectively and following the Assumption 2,
the Monge mapping, introduced in the Equation 1, has the structure

$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{1,1}&\dots&\mathbf{A}_{1,n_{c}}\\\
\dots&\dots&\dots\\\
\mathbf{A}_{n_{c},1}&\dots&\mathbf{A}_{n_{c},n_{c}}\end{pmatrix}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}\quad\text{with}\quad\mathbf{A}_{l,m}\in\mathcal{C}_{n_{\ell}}$

for every $l,m\in\left\llbracket 1,n_{c}\right\rrbracket\,$. This motivates
the definition of the $f$-Monge mapping which leverages the approximation of
circulant matrices defined in subsection 2.2.

###### Definition 3

Given $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$ source and target centered
Gaussian distributions respectively and following the Assumption 2, the
$f$-Monge mapping on $\mathbf{X}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ is
defined as

$m_{f}(\mathbf{X})=\operatorname{vec}^{-1}\left(\widetilde{\mathbf{A}}\operatorname{vec}\left(\mathbf{X}\right)\right)$

where

$\widetilde{\mathbf{A}}=\begin{pmatrix}\mathcal{P}_{f}(\mathbf{A}_{1,1})&\dots&\mathcal{P}_{f}(\mathbf{A}_{1,n_{c}})\\\
\dots&\dots&\dots\\\
\mathcal{P}_{f}(\mathbf{A}_{n_{c},1})&\dots&\mathcal{P}_{f}(\mathbf{A}_{n_{c},n_{c}})\end{pmatrix}\quad\text{and}\quad\mathbf{A}={\boldsymbol{\Sigma}}_{s}^{-\frac{1}{2}}\left({\boldsymbol{\Sigma}}_{s}^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{t}{\boldsymbol{\Sigma}}_{s}^{\frac{1}{2}}\right)^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{s}^{-\frac{1}{2}}$

is the Monge mapping from Equation 1 and $\mathcal{P}_{f}$ is the function
which sub-samples the cross-PSDs and defined in Equation 9.

(a) Cross-PSD alignment

(b) Bures-Wasserstein distance

Figure 4: Illustration of the Spatio-Temporal Monge mapping on sleep data. (a)
Cross-PSD alignment from source cross-PSD (green dotted line) to target cross-
PSD (blue dotted line) for different filter sizes. (b) Bures-Wasserstein
distance between the aligned signal and the source signal (green line) and the
target line (blue line).

We now explore the properties of the $f$-Monge mapping. The next proposition
establishes the spatio-temporal mapping, showing that it is a sum of
convolutions and thus fast to compute.

######  Proposition 4 (Spatio-Temporal mapping)

Let $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$ be source and target
centered Gaussian distributions respectively and following the Assumption 2,
i.e., for $d\in\\{s,t\\}$,
${\boldsymbol{\Sigma}}_{d}=\mathbf{F}\mathbf{U}\mathbf{Q}_{d}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$
as defined in Equation 15. Let $f\leq n_{\ell}$,
$\mathbf{P}_{d}=g_{f}(\mathbf{Q}_{d})$ and

$\begin{pmatrix}\operatorname{diag}(\mathbf{p}_{1,1})&\dots&\operatorname{diag}(\mathbf{p}_{1,n_{c}})\\\
\dots&\dots&\dots\\\
\operatorname{diag}(\mathbf{p}_{n_{c},1})&\dots&\operatorname{diag}(\mathbf{p}_{n_{c},n_{c}})\end{pmatrix}\triangleq\mathbf{V}\mathbf{P}_{s}^{-\frac{1}{2}}\left(\mathbf{P}_{s}^{\frac{1}{2}}\mathbf{P}_{t}\mathbf{P}_{s}^{\frac{1}{2}}\right)^{\frac{1}{2}}\mathbf{P}_{s}^{-\frac{1}{2}}\mathbf{V}^{\mkern-1.5mu\mathsf{T}}\in{\mathcal{H}_{n_{c}f}^{++}}$

with $\mathbf{V}\in\mathbb{R}^{n_{c}f\times n_{c}f}$ the permutation matrix
from Equation 17. Given a signal
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$ such that
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$,
the $f$-Monge mapping introduced in the Definition 3 is a sum of convolutions

$m_{f}\left(\mathbf{X}\right)=\left[\sum_{j=1}^{n_{c}}\mathbf{h}_{1,j}*\mathbf{x}_{j},\dots,\sum_{j=1}^{n_{c}}\mathbf{h}_{n_{c},j}*\mathbf{x}_{j}\right]^{\mkern-1.5mu\mathsf{T}}$

where
$\mathbf{h}_{i,j}=\frac{1}{\sqrt{f}}\mathbf{F}_{f}^{\mathsf{H}}\mathbf{p}_{i,j}\in\mathbb{R}^{f}$.

The Wasserstein barycenter of multiple Gaussian distributions is the solution
of a fixed-point equation as seen in subsubsection 2.1.2. It is possible to
exploit the structure of the spatio-temporal covariances to reduce the
computational complexity of the barycenter by dealing with block diagonal
matrices.

######  Lemma 5 (Spatio-Temporal barycenter)

Let $n_{d}$ centered Gaussian distributions
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$ with
${\boldsymbol{\Sigma}}_{k}=\mathbf{F}\mathbf{U}\mathbf{Q}_{k}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$
following the Assumption 2. The Wasserstein barycenter of the $n_{d}$
distributions is the unique centered Gaussian distribution
$\mathcal{N}(\mathbf{0},\overline{{\boldsymbol{\Sigma}}})$ satisfying

$\overline{{\boldsymbol{\Sigma}}}=\mathbf{F}\mathbf{U}\overline{\mathbf{Q}}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}\quad\text{and}\quad\overline{\mathbf{Q}}=\Psi\left(\overline{\mathbf{Q}},\left\\{\mathbf{Q}_{k}\right\\}_{k=1}^{n_{d}}\right)\;,$

where $\Psi$ is defined in Equation 3.

###  3.3 Special cases: Temporal and Spatial $f$-Monge mappings

####  3.3.1 Pure temporal $f$-Monge mapping and barycenter

In some other scenarios, it may be possible that spatial correlation is less
relevant in the learning process. In this context, we assume the signals are
uncorrelated between the different channels so that the covariance matrix for
each channel is circulant. This setting leads to the mapping and barycenter
formulas studied in [34], here considering $n_{c}$ uncorrelated sensors. To
detail this, we introduce a new structure for ${\boldsymbol{\Sigma}}$ in the
following assumption.

###### Assumption 6

We assume that the different channels are uncorrelated and that the covariance
matrix for each channel is circulant, i.e.,

${\boldsymbol{\Sigma}}=\mathbf{F}\operatorname{diag}(\mathbf{q}_{1},\dots,\mathbf{q}_{n_{c}})\mathbf{F}^{\mathsf{H}}\in\mathcal{S}_{n_{c}n_{\ell}}^{++},$

where $\mathbf{q}_{c}\in(\mathbb{R}^{+*})^{n_{\ell}}$ are PSDs.

This enables a novel reformulation of the $f$-Monge mapping, specifically
tailored to accommodate data that are only temporally correlated.

######  Proposition 7 (Temporal mapping)

Let $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$ be source and target
centered Gaussian distributions respectively and following the Assumption 6,
i.e., for $d\in\\{s,t\\}$,
${\boldsymbol{\Sigma}}_{d}=\mathbf{F}\operatorname{diag}(\mathbf{q}_{1,d},\dots,\mathbf{q}_{n_{c},d})\mathbf{F}^{\mathsf{H}}$.
Let $f\leq n_{\ell}$, and $\mathbf{p}_{c,d}=g_{f}(\mathbf{q}_{c,d})$. Given a
signal
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$ such that
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$,
the $f$-Monge mapping introduced in Definition 3 is the convolution

$m_{f}(\mathbf{X})=\left[\mathbf{h}_{1}*\mathbf{x}_{1},\dots,\mathbf{h}_{n_{c}}*\mathbf{x}_{n_{c}}\right]^{\top}$

where
$\mathbf{h}_{c}=\frac{1}{\sqrt{f}}\mathbf{F}_{f}^{\mathsf{H}}\left({\mathbf{p}_{c,t}}^{\odot\frac{1}{2}}\odot{\mathbf{p}_{c,s}}^{\odot-\frac{1}{2}}\right)\in\mathbb{R}^{f}$.

Furthermore, in this case we recover a closed-form solution to the Wasserstein
barycenter fixed-point equation, which enables its computation without the
need for iterative methods. This leads to significantly improved computational
efficiency.

######  Lemma 8 (Temporal barycenter)

Let $n_{d}$ centered Gaussian distributions
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$ with
${\boldsymbol{\Sigma}}_{k}=\mathbf{F}\operatorname{diag}(\mathbf{q}_{1,k},\dots,\mathbf{q}_{n_{c},k})\mathbf{F}^{\mathsf{H}}$
following the Assumption 6. The Wasserstein barycenter of the $n_{d}$
distributions is the centered Gaussian distribution
$\mathcal{N}(\mathbf{0},\overline{{\boldsymbol{\Sigma}}})$ with

$\overline{{\boldsymbol{\Sigma}}}=\mathbf{F}\operatorname{diag}(\overline{\mathbf{q}}_{1},\dots,\overline{\mathbf{q}}_{n_{c}})\mathbf{F}^{\mathsf{H}}\quad\text{and}\quad\overline{\mathbf{q}}_{c}=\left(\frac{1}{n_{d}}\sum^{n_{d}}_{k=1}\mathbf{q}_{c,k}^{\odot\frac{1}{2}}\right)^{\odot
2}\;.$

####  3.3.2 Pure spatial Monge mapping and barycenter

Conversely, in some other alternative scenarios, the relevance of temporal
correlation in the learning process may be diminished. Here, we consider only
correlations across different channels, that is one set $f=1$. Adopting a
correlation matrix between channels
${\boldsymbol{\Xi}}=\mathbb{E}(\mathbf{X}\mathbf{X}^{\mkern-1.5mu\mathsf{T}})\in\mathcal{S}_{n_{c}}^{++}$,
this assumption gives rise to a distinctive structure for
${\boldsymbol{\Sigma}}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}$.

###### Assumption 9

The random signal has a spherical covariance matrix
${\boldsymbol{\Sigma}}\in\mathcal{S}_{n_{c}n_{\ell}}^{++}$ w.r.t. time.
Formally for every $l,m\in\left\llbracket 1,n_{c}\right\rrbracket\,$,
${\boldsymbol{\Sigma}}_{l,m}$ defined in Equation 13 belongs to
$\mathcal{E}_{1,n_{\ell}}$, i.e.,

${\boldsymbol{\Sigma}}_{l,m}=\sigma_{l,m}\mathbf{I}_{n_{\ell}}\quad\text{with}\quad\sigma_{l,m}>0\;.$

This implies that the overall covariance matrix is given by a Kronecker
product

${\boldsymbol{\Sigma}}={\boldsymbol{\Xi}}\otimes\mathbf{I}_{n_{\ell}}\quad\text{with}\quad{\boldsymbol{\Xi}}=\begin{pmatrix}\sigma_{1,1}&\dots&\sigma_{1,n_{c}}\\\
\dots&\dots&\dots\\\
\sigma_{n_{c},1}&\dots&\sigma_{n_{c},n_{c}}\end{pmatrix}\in\mathcal{S}_{n_{c}}^{++}\;.$

This new structure leads to the following Monge mapping, specifically designed
to accommodate data which are only spatially correlated.

######  Proposition 10 (Spatial mapping)

Let $\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$ and
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$ be source and target
centered Gaussian distributions respectively and following the Assumption 9,
i.e., for $d\in\\{s,t\\}$,
${\boldsymbol{\Sigma}}_{d}={\boldsymbol{\Xi}}_{d}\otimes\mathbf{I}_{n_{\ell}}$.
Given a signal
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$ such that
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{s})$,
the $f$-Monge mapping introduced in Definition 3 is

$m_{f}(\mathbf{X})={\boldsymbol{\Xi}}_{s}^{-\frac{1}{2}}\left({\boldsymbol{\Xi}}_{s}^{\frac{1}{2}}{\boldsymbol{\Xi}}_{t}{\boldsymbol{\Xi}}_{s}^{\frac{1}{2}}\right)^{\frac{1}{2}}{\boldsymbol{\Xi}}_{s}^{-\frac{1}{2}}\mathbf{X}\;.$

It is interesting to note that when covariance matrices commute, this boils
down to the CORAL mapping proposed by [9], as already observed by [42]. Also
note that Monge mappings are restricted to semi-definite mappings which is not
the case of CORAL. The purely spatial barycenter still requires a fixed-point
algorithm, albeit with reduced complexity since the size of the covariance
matrices is $n_{c}\times n_{c}$.

######  Lemma 11 (Spatial barycenter)

Let $n_{d}$ centered Gaussian distributions
$\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$ with
${\boldsymbol{\Sigma}}_{k}={\boldsymbol{\Xi}}_{k}\otimes\mathbf{I}_{n_{\ell}}$
following the Assumption 9. The Wasserstein barycenter of the $n_{d}$
distributions is the unique centered Gaussian distribution
$\mathcal{N}(\mathbf{0},\overline{{\boldsymbol{\Sigma}}})$ satisfying

$\overline{{\boldsymbol{\Sigma}}}=\overline{{\boldsymbol{\Xi}}}\otimes\mathbf{I}_{n_{\ell}}\quad\text{and}\quad\overline{{\boldsymbol{\Xi}}}=\Psi\left(\overline{\boldsymbol{\Xi}},\left\\{{\boldsymbol{\Xi}}_{k}\right\\}_{k=1}^{n_{d}}\right)\;,$

where $\Psi$ is defined in Equation 3.

##  4 Multi-source Signal Adaptation with Monge Alignment

In this section, we introduce the Monge Alignment (MA) method that leverages
the Monge mappings and barycenters developed in the previous sections to
perform DA (see Figure 1). Indeed, MA pushes forward the distributions of the
different domains to the barycenter of source domains. By doing so, we
effectively reduce the discrepancy between the source and target domains,
improving generalization of any predictor trained on these adapted source
domains. Notably, the proposed approach can be applied at test-time, by
performing adaptation to new target domains without the need to retrain a
predictor. First, we describe the general algorithm MA at train and test
times. It can be implemented in three specific ways, depending on whether we
consider temporal shifts (TMA), spatial shifts (SMA), or both (STMA). Second,
the computational aspects of these methods are presented with the
computational complexities and the choice of filter size. Third, the
statistical estimation errors made by these algorithms are assessed. In the
following, given $n_{d}$ labeled source domains, we denote by
$\mathbf{X}_{k}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ the data of each domain
$k\in\left\llbracket 1,n_{d}\right\rrbracket\,$, with corresponding labels
$\mathbf{y}_{k}$. Also, we denote by $\mathbf{X}_{t}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$, the target data.

###  4.1 Monge Alignment algorithm

At train-time, MA adapts each source domain to their barycenter with the
$f$-Monge mapping from Definition 3 and then a predictor is trained. At test-
time, MA adapts each target domain to the previously learned barycenter using
the $f$-Monge mapping. Remarkably, this adaption is performed without
accessing the source data. Finally, inference is achieved on the adapted
target data using the trained predictor. These two phases are presented in
algorithm 1 and algorithm 2.

Input: Filter size $f$, source data $\\{\mathbf{X}_{k}\\}_{k=1}^{n_{d}}$,
source labels $\\{\mathbf{y}_{k}\\}_{k=1}^{n_{d}}$

1 $\widehat{\boldsymbol{\Sigma}}_{k}\leftarrow$ Compute cov. matrix from
$\mathbf{X}_{k},\forall k$

2 $\widehat{\overline{\boldsymbol{\Sigma}}}$ $\leftarrow$ Learn barycenter

3 $\widehat{m}_{f,k}$ $\leftarrow$ Learn $f$-Monge maps, $\forall k$

4 $\widehat{h}\\!\\!\leftarrow$  Train  on  adapted  data
$\\{\widehat{m}_{f,k}(\mathbf{X}_{k})\\}_{k=1}^{n_{d}}$

5 return Trained predictor $\widehat{h}$, barycenter
$\widehat{\overline{\boldsymbol{\Sigma}}}$

Algorithm 1 MA at Train-Time

Input: Target data $\mathbf{X}_{t}$, trained model $\widehat{h}$, barycenter
$\widehat{\overline{\boldsymbol{\Sigma}}}$ 1
$\widehat{\boldsymbol{\Sigma}}_{t}\leftarrow$ Compute cov. matrix from
$\mathbf{X}_{t}$ 2 $\widehat{m}_{f,t}$ $\leftarrow$ Learn $f$-Monge map  3
$\widehat{\mathbf{y}}_{t}$ $\leftarrow$
$\widehat{h}\left(\widehat{m}_{f,t}(\mathbf{X}_{t})\right)$ return Predictions
$\widehat{\mathbf{y}}_{t}$ Algorithm 2 MA at Test-Time

####  4.1.1 Train-time

MA performs four steps to learn a predictor from adapted source data and their
labels.

##### Covariance matrix computation (algorithm 1, line 1)

The first step involves estimating the covariance matrices
$\widehat{\boldsymbol{\Sigma}}_{k}$ for each source domain
$k\in\left\llbracket 1,n_{d}\right\rrbracket\,$. For each source domain, we
have a multivariate signal
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}$.
Note that, we can estimate either the covariance matrices or their Fourier
domain equivalents, the cross-PSDs. For both STMA and TMA, we employ the Welch
estimator [45] to estimate the cross-PSDs. This involves computing the short-
time Fourier transform of the signals. For all $l\in\left\llbracket
1,n_{\ell}-f+1\right\rrbracket\,,j\in\left\llbracket
1,f\right\rrbracket\,,c\in\left\llbracket 1,n_{c}\right\rrbracket\,$, it is
given by

$\widehat{x}_{c,l,j}=\sum_{k=1}^{f}w_{k}e^{\frac{-2i\pi(j-1)(k-1)}{f}}(\mathbf{X})_{c,l+k-1}\;.$
(18)

where $\mathbf{w}=[w_{1},\dots,w_{f}]^{\mkern-1.5mu\mathsf{T}}$ is the window
function such that $\|\mathbf{w}\|_{2}=1$. For an overlap of $\frac{f}{2}$
samples between adjacent windows, the Welch estimator used for STMA is given
by

$\widehat{\mathbf{P}}=\operatorname{diag}\left(\widehat{\mathbf{P}}_{1},\dots,\widehat{\mathbf{P}}_{f}\right)\in\mathcal{H}_{n_{c}f}^{++}\quad\text{where}\quad\left\\{\begin{aligned}
&\widehat{\mathbf{P}}_{j}=\frac{1}{n_{\ell}-f+1}\widehat{\mathbf{X}}_{j}\widehat{\mathbf{X}}_{j}^{\mathsf{H}}\\\
&(\widehat{\mathbf{X}}_{j})_{c,l}=\widehat{x}_{c,l,j}\;.\end{aligned}\right.$
(19)

In the same manner, the Welch estimator for TMA provides us with the diagonal
$\widehat{\mathbf{p}}$ of the cross-PSDs,

$\widehat{\mathbf{p}}=\left[\widehat{\mathbf{p}}_{1}^{\mkern-1.5mu\mathsf{T}},\dots,\widehat{\mathbf{p}}_{n_{c}}^{\mkern-1.5mu\mathsf{T}}\right]^{\mkern-1.5mu\mathsf{T}}\in(\mathbb{R}^{+*})^{n_{c}f}\quad\text{where}\quad\left\\{\begin{aligned}
&\widehat{\mathbf{p}}_{c}=\frac{1}{n_{\ell}-f+1}\operatorname{diag}\left(\widehat{\mathbf{Z}}_{c}\widehat{\mathbf{Z}}_{c}^{\mathsf{H}}\right)\\\
&(\widehat{\mathbf{Z}}_{c})_{j,l}=\widehat{x}_{c,l,j}\;.\end{aligned}\right.$
(20)

For the SMA method, we employ the classical empirical covariance matrix
estimator, which is suitable since SMA only requires spatial covariance
information, and signals are assumed to be centered,

$\widehat{{\boldsymbol{\Xi}}}=\frac{1}{n_{\ell}}\mathbf{X}{\mathbf{X}}^{\mkern-1.5mu\mathsf{T}}\in\mathcal{S}_{n_{c}}^{++}\;.$
(21)

##### Barycenter computation (algorithm 1, line 2)

Since our data are supposed to be Gaussian-centered, the barycentric
distribution is defined by the covariance $\overline{\boldsymbol{\Sigma}}$,
which can be expressed with the computed covariances. The second step involves
computing the barycentric distribution $\overline{\boldsymbol{\Sigma}}$ from
quantities computed in Equation 19, Equation 20, and Equation 21. For the STMA
and SMA methods and from Lemma 5 and Lemma 11, the computation is performed
with fixed-point iterations presented in Equation 4. In practice, for
iterative methods we compute an approximation of the barycenter by
initializing with the Euclidean mean over the domain’s covariance matrices,
and performing one fixed-point iteration. For the TMA method, the barycenter
is obtained using the closed-form solution from Lemma 8. An example of
barycenter for the STMA method is illustrated in 5(a), where the green line
corresponds to the source cross-PSDs, and the black dotted line represents
their barycenter.

(a) Source cross-PSD

(b) Alignment with $f=32$

(c) Alignment with $f=256$

Figure 5: Illustration of the STMA method on sleep staging data. (a) The
cross-PSD of the barycenter ($\boldsymbol{--}$) is computed from all the
cross-PSD of the source domains. (b) The cross-PSDs of the source are aligned
with the Monge mapping with a small filter size. (c) The cross-PSDs of the
source are aligned with the Monge mapping with a big filter size. The bigger
the filter size is, the more the cross-PSD are well aligned.

#####  $f$-Monge maps computation (algorithm 1, line 3)

The third step consists of computing the $f$-Monge maps $m_{f,k}$ for all
$k\in\left\llbracket 1,n_{d}\right\rrbracket\,$. The latter maps the
$k^{\text{th}}$ source distribution to the barycenter. The formulas of
$m_{f,k}$ are provided in Propositions 4, 7 and 10 for STMA, TMA and SMA
respectively and only involves quantities computed in the first two steps,
i.e., covariance matrices and barycenters. An illustration of the effect of
applying the $f$-Monge maps is shown in 5(b) and 5(c) for different filter
sizes.

##### Predictor training (algorithm 1, line 4)

The final step is to train a supervised predictor $h$, belonging to a function
space $\Omega$, on the aligned source data. The trained model is

$\widehat{h}\in\mathop{\rm
arg\,min}_{h\in\Omega}\;\sum_{k=1}^{n_{d}}\mathcal{L}\left(\mathbf{y}_{k},h\left(\widehat{m}_{f,k}(\mathbf{X}_{k})\right)\right)$
(22)

where $\mathbf{X}_{k}$ is the multivariate signal of the $k^{\text{th}}$
domain and $\mathcal{L}$ is a supervised loss function.

####  4.1.2 Test-time

At test-time, we have access to a new unlabeled target domain
$\mathbf{X}_{t}\in\mathbb{R}^{n_{c}\times n_{\ell}}$. Notably, during this
phase, the source domains are no longer accessible, and we rely solely on both
the source barycenter $\widehat{\overline{\boldsymbol{\Sigma}}}$ and the
trained predictor $\widehat{h}$, both computed at train time. The two first
steps are the computations of the target covariance matrix and the $f$-Monge
map $\widehat{m}_{f,t}$ from the target distribution to the barycenter (see
lines 1 and 2 in algorithm 2). For the three methods, STMA, TMA, and SMA, the
computations employ the same equations as in the train-time phase. Next, we
predict the label using the trained predictor on the aligned target data (see
line 3 in algorithm 2) with
$\widehat{\mathbf{y}}_{t}=\widehat{h}\left(\widehat{m}_{f,t}(\mathbf{X}_{t})\right).$
It should be noted that the proposed test-time adaptation method requires only
a small number of unlabeled samples from the target domain to estimate its
covariance. Furthermore, the barycenter computation and predictor training
occur during the training phase, eliminating the need for retraining during
the test phase. Using MA, the trained predictor can still adapt to new
temporal, spatial, or both shifts.

###  4.2 Computational aspects

We now discuss the computational aspects and numerial complexities of the
proposed method STMA, TMA and SMA. The methods present interesting trade-offs
between the costs of calculation and the shifts considered (spatial, temporal,
or both). Indeed, the more shifts considered, the higher the calculation cost
and conversely. This can motivate the practitioner to use one method rather
than another, depending on the dimensions of the data and the observed shifts
between domains. Then, the choice of the filter size $f$ is discussed.

####  4.2.1 Numerical complexity

Step | STMA | TMA | SMA
---|---|---|---
Covariance matrix computation | $\mathcal{O}\left(n_{d}n_{c}^{2}n_{\ell}f\log{f}\right)$ | $\mathcal{O}\left(n_{d}n_{c}n_{\ell}f\log{f}\right)$ | $\mathcal{O}\left(n_{d}n_{c}^{2}n_{\ell}\right)$
Barycenter computation |  $\mathcal{O}\left(n_{d}n_{c}^{3}f\right)$ (per iter.) | $\mathcal{O}\left(n_{d}n_{c}f\right)$ |  $\mathcal{O}\left(n_{d}n_{c}^{3}\right)$ (per iter.)
Mapping computation | $\mathcal{O}\left(n_{d}(n_{c}^{3}f+n_{c}^{2}f\log f)\right)$ | $\mathcal{O}\left(n_{d}n_{c}f\log f\right)$ | $\mathcal{O}\left(n_{d}n_{c}^{3}\right)$
Mapping application | $\mathcal{O}\left(n_{d}n_{c}^{2}n_{\ell}f\right)$ | $\mathcal{O}\left(n_{d}n_{c}n_{\ell}f\right)$ | $\mathcal{O}\left(n_{d}n_{c}^{2}n_{\ell}\right)$
Table 2:  Computational complexity comparison between STMA (spatio-temporal),
TMA (temporal), and SMA (spatial). The complexity is expressed in terms of
four key parameters: the number of domains $n_{d}$; the number of sensors
$n_{c}$; the length of the signals $n_{\ell}$ and the filter size $f$.

The computational complexity of the four main operations at train and test
times are studied below. The complexities are presented in Table 2 and depend
on the number of sensors ($n_{c}$), signal length ($n_{\ell}$), and filter
size ($f$), and vary depending on the method (STMA, SMA, or TMA). Note that
all methods are linear in the number of domains and signal length. SMA does
not consider temporal correlations, so its complexity is independent of the
filter size $f$ (equivalent to $f=1$), contrary to STMA and TMA. In the same
manner, TMA does not consider spatial correlations, so its complexity is
linear in the number of sensors, contrary to STMA and SMA, which have cubic
complexities. It should be noted that all operations are standard linear
algebra and can be performed on modern hardware (CPU and GPU). A detailed
explanation of the complexities is provided hereafter.

##### Covariance matrix computation

STMA and TMA use the Welch estimator with Fast Fourier Transform (FFT), which
has a complexity of $\mathcal{O}(f\log f)$. TMA method has a linear complexity
in the number of sensors, while the STMA and SMA methods are quadratic since
they consider correlations between sensors.

##### Barycenter computation

This operation applies directly to the covariance matrices, making it
independent of the signal length. TMA has a closed-form barycenter and linear
complexity in the filter size and number of sensors, making it interesting for
data with many sensors. STMA also has a complexity per iteration that is
linear in the filter size. STMA and SMA have a cubic complexity in the sensor
number because of the computations of Singular Value Decomposition (SVDs) to
perform square roots and inverse square roots.

#####  $f$-Monge maps computation

This operation also applies directly to the covariance matrices, making it
independent of the signal length. In the worst case, the complexity per
iteration of STMA is $f\log f$ (filter size). STMA and SMA have a cubic
complexity in the sensor number because of the computations of SVDs. TMA has a
linear complexity in the number of sensors, making it again appealing for data
with many sensors.

#####  $f$-Monge maps application

All the three methods are linear in the signal length. STMA and SMA involve a
quadratic number of operations with respect to the number of sensors contrary
to TMA which has a linear complexity.

####  4.2.2 Filter size selection

The proposed method introduces $f$, the filter size. This key hyper-parameter
serves a dual role. Primarily, it contributes to reducing computation time; a
benefit clearly demonstrated in Table 2 and subsubsection 4.2.1. Secondly, the
filter size dictates the precision with which the PSDs of the domains are
mapped to the barycenter. As visualized in Figure 4 and Figure 5, a larger
filter size yields a more accurate mapping, while a smaller size only subtly
adjusts the PSDs. This flexibility enables a balance between perfectly
aligning the PSDs, which might compromise unique class features, and making
sufficient adjustments to reduce noise and domain-specific variability while
preserving class-specific characteristics.

###  4.3 Concentration bounds of the Monge Alignment

In this subsection, we propose concentration bounds for the estimations of
STMA, TMA, and SMA mappings. These theorems provide a rigorous framework for
understanding the reliability and precision of MA methods. Estimating OT plans
is challenging due to the curse of dimensionality, with OT estimators
generally decaying at a rate of $O(n^{-1/d})$, where $n$ is the number of
observations and $d$ is the data dimension [46]. However, [40] propose a
concentration bound for linear Monge mapping estimation with a faster
convergence rate of $O(n^{-1/2})$, i.e., independent of the data dimension.
This suggests greater accuracy and robustness in high-dimensional settings.
Our proposed bounds account for two key aspects: mapping to the Wasserstein
barycenter and using PSD computation with the Welch estimator. These bounds
differ from [40], which maps to a given empirical covariance matrix. In
particular, we leverage [47] to establish these bounds, highlighting an
induced bias in PSD computation due to the Welch estimator. To introduce the
concentration bound, we first define from [47] the spatial correlation of
delay $\ell$, denoted $\mathbf{R}_{\ell}$, for signal following the Assumption
2, as

$\mathbf{R}_{\ell}=\begin{pmatrix}({\boldsymbol{\Sigma}}_{1,1})_{1,\ell}&\dots&({\boldsymbol{\Sigma}}_{1,n_{c}})_{1,\ell}\\\
\dots&\dots&\dots\\\
({\boldsymbol{\Sigma}}_{n_{c},1})_{1,\ell}&\dots&({\boldsymbol{\Sigma}}_{n_{c},n_{c}})_{1,\ell}\end{pmatrix}\in\mathcal{S}_{n_{c}}^{++}$
(23)

For signals following Assumption 6 only the diagonal,
$(r_{c})_{l}=({\boldsymbol{\Sigma}}_{c,c})_{1,\ell}$ is necessary.

####  4.3.1 STMA concentration bound

The next theorem introduces the general concentration bound, i.e.,
corresponding to STMA.

######  Theorem 12 (STMA concentration bound)

Let $\mathbf{X}_{t}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$.
For $k\in\left\llbracket 1,K\right\rrbracket\,$, let
$\mathbf{X}_{k}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X}_{k})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$.
Let us assume that ${\boldsymbol{\Sigma}}_{t}$ and ${\boldsymbol{\Sigma}}_{k}$
follow the Assumption 2, i.e.,
${\boldsymbol{\Sigma}}_{t}=\mathbf{F}\mathbf{U}\mathbf{Q}_{t}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$
and
${\boldsymbol{\Sigma}}_{k}=\mathbf{F}\mathbf{U}\mathbf{Q}_{k}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$,
that $\|\mathbf{R}_{\ell}\|_{2}\leq\gamma\rho^{|\ell|}$ with $\gamma>0$ and
$\rho\in[0,1)$ and that only fixed point iteration of the barycenter is done.
Then there exist numerical constants $c_{k}>0$, for all $k$, and $C>0$,
independent from $n_{\ell}$, $n_{c}$, $n_{d}$ and $f$, such that with
probability greater than $1-\delta$, we have

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10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptstyle\mathstrut
2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptscriptstyle\mathstrut
2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right)}\right\\}$}}$}}
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2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
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2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
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2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptscriptstyle\mathstrut
2\widetilde{\mathbf{Q}}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2n_{c}}}{\delta}\right)^{32}\right)}\right\\}$}}$}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right)\;,$

with
$\widetilde{\mathbf{Q}}=\max_{d\in\\{1,\dots,n_{d},t\\}}{\|\mathbf{Q}_{d}\|}_{\infty}$.
Bias Variance

We made the choice to limit the theoretical analysis to only one iteration of
the barycenter, allowing to provide this result. In practice, the barycenter
converges fast, leading to good results after one iteration, and we did not
observe performance gains with more iterations in our experiments. In contrast
to the concentration bound proposed by [40], this bound includes a bias term
resulting from the estimation of cross-PSDs using the Welch method. This bias
depends on size of the filter $f$ the temporal correlation of the signal:
higher correlation (large $\rho$) leads to greater bias, as shown on the left
in Figure 6. However, the variance has a classical bound in
$\mathcal{O}(\sqrt{f/n_{\ell}})$. Hence, the filter size can be adjusted to
control the bias-variance trade-off. Figure 6 shows this trade-off with a
dotted line, highlighting the optimal filter size.

Figure 6:  bias-variance estimation tradeoff versus the filter size $f$. For a
given $\rho$, increasing the filter size reduces the bias, but at the cost of
increased variance. A single optimal filter size range balances bias and
variance, yielding a good tradeoff.

####  4.3.2 TMA concentration bound

TMA exhibits a similar concentration bound, i.e., with a bias-variance
tradeoff and is exposed here-after.

######  Theorem 13 (TMA concentration bound)

Let
$\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}]^{\mkern-1.5mu\mathsf{T}}\in\mathbb{R}^{n_{c}\times
n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}})$.
For $k\in\left\llbracket 1,K\right\rrbracket\,$, let
$\mathbf{X}_{k}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X}_{k})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$.
We assume that ${\boldsymbol{\Sigma}}$ and ${\boldsymbol{\Sigma}}_{k}$ follow
Assumption 6, i.e.,
${\boldsymbol{\Sigma}}=\mathbf{F}\operatorname{diag}(\mathbf{q}_{1},\dots,\mathbf{q}_{n_{c}})\mathbf{F}^{\mathsf{H}}$
and
${\boldsymbol{\Sigma}}_{k}=\mathbf{F}\operatorname{diag}(\mathbf{q}_{1}^{k},\dots,\mathbf{q}_{n_{c}}^{k})\mathbf{F}^{\mathsf{H}}$,
and for every $c\in\left\llbracket 1,n_{c}\right\rrbracket\,$
$(r_{c})_{\ell}\leq\gamma\rho^{|\ell|}$ with $\delta>0$ and $\rho\in[0,1)$.
For every $c\in\left\llbracket 1,n_{c}\right\rrbracket\,$, we denote by
$\mathbf{p}_{c}=g_{f}(\mathbf{q}_{c})$,
$\overline{\mathbf{p}}_{c}=g_{f}(\overline{\mathbf{q}}_{c})$, and
$\mathbf{p}_{c,k}=g_{f}(\mathbf{q}_{c,k})$. Then there exists numerical
constants $c_{c,k}^{\prime}>0$, for all $k$ and $C_{c}^{\prime}>0$ independent
from $n_{\ell}$, $n_{c}$, $n_{d}$ and $f$, such that with probability greater
than $1-\delta$, we have

$\|\widehat{\mathbf{A}}-\mathbf{A}\|\lesssim\max_{c}\left(C^{\prime}_{c}+\sum_{k=1}^{n_{d}}c_{c,k}^{\prime}\right)\left(\mathchoice{\leavevmode\hbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\scriptstyle\definecolor{currentcolor}{rgb}{0,0,0}\mathchoice{\hbox{\pagecolor{cyan!17}$\displaystyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
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10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\textstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptscriptstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}$}}
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}\hbox{{$\scriptscriptstyle\definecolor{currentcolor}{rgb}{0,0,0}\mathchoice{\hbox{\pagecolor{cyan!17}$\displaystyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
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10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\textstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}{\hbox{\pagecolor{cyan!17}$\scriptscriptstyle\mathstrut
2\widetilde{\mathbf{p}}_{c}\max\left\\{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right),\sqrt{\frac{5}{2\frac{n_{\ell}}{f}}\log\left(5f^{2}\left(\frac{4\times
10^{2}}{\delta}\right)^{32}\right)}\right\\}$}}$}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right)\;.$

with
$\widetilde{\mathbf{p}}_{c}=\max_{d\in\\{1,\dots,n_{d},t\\}}{\|\mathbf{p}_{c,d}\|}_{\infty}$.
Bias Variance

To the best of our knowledge, this is the first concentration bound of the
method proposed by [34]. This bound includes bias and variance terms as in
STMA concentration bound. The bound has similar behaviors but the independence
of the channels leads to a smaller variance.

####  4.3.3 SMA concentration bound

Contrary to STMA and TMA, SMA has a concentration bound with a null bias. This
makes it particularly suitable for short and highly correlated in time
signals. Indeed, SMA ensures accurate estimations without the bias introduced
by the temporal correlation.

######  Theorem 14 (SMA concentration bound)

Let $\mathbf{X}_{t}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X}_{t})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{t})$.
For $k\in\left\llbracket 1,K\right\rrbracket\,$, let
$\mathbf{X}_{k}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ be a realization of
$\operatorname{vec}(\mathbf{X}_{k})\sim\mathcal{N}(\mathbf{0},{\boldsymbol{\Sigma}}_{k})$.
We assume that ${\boldsymbol{\Sigma}}_{t}$ and ${\boldsymbol{\Sigma}}_{k}$
follow Assumption 9, i.e.,
${\boldsymbol{\Sigma}}_{t}=\mathbf{F}\mathbf{U}\mathbf{Q}_{t}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$
and
${\boldsymbol{\Sigma}}_{k}=\mathbf{F}\mathbf{U}\mathbf{Q}_{k}\mathbf{U}^{\mkern-1.5mu\mathsf{T}}\mathbf{F}^{\mathsf{H}}$.
Then there exist numerical constants $c_{k}>0$, for all $k$ and $C>0$
independent from $n_{\ell}$, $n_{c}$, $n_{d}$ and $f$, such that with
probability greater than $1-\delta$, we have

$\|\widehat{\mathbf{A}}-\mathbf{A}\|\lesssim\mathchoice{\leavevmode\hbox
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##  5 Experimental results

This section evaluates MA on different datasets. MA is first tested on two
biosignals classification tasks. The first experiment extends results on sleep
staging proposed in [34] by using multivariate signals (i.e., several
channels) and thus exploiting the spatial information. The second experiment
focuses on Brain-Computer Interface (BCI) Motor Imagery tasks, where spatial
information is crucial to classify. Then, we extend our method to 2D signals
(i.e., images) on a new custom MNIST dataset where each domain corresponds to
a directional blur. We apply MA on this new dataset and show its efficiency to
compensate for the blur.

###  5.1 Biosignal tasks

In this section, we investigate the impact of MA on two critical biosignal
tasks: Sleep stage classification and Brain-Computer Interface (BCI). Each
dataset is introduced alongside the alignment methodology proposed for these
tasks.

####  5.1.1 Biosignal Datasets

##### Sleep Staging Datasets

We utilize four publicly available datasets: MASS [48], HomePAP [49], CHAT
[50], and ABC [51], accessible via the National Sleep Research Resource [28].
Sleep staging is performed using 7-channel EEG signals across all datasets.
The EEG channels considered are F3, F4, C3, C4, O1, O2, and A2, referenced to
FPz. Each night’s data is segmented into 30-second samples with a sampling
frequency of $f_{s}=100$ Hz.

##### Brain-Computer Interface Datasets

We employ five publicly available datasets: BCI Competition IV [52], Weibo2014
[53], PhysionetMI [54], Cho2017 [55], and Schirrmeister2017 [56], accessible
through MOABB [57]. These datasets consists of two classes motor imagery tasks
involving right-hand and left-hand movements. We utilize a common set of 22
channels across the datasets. The length of time series varies across
datasets, and following [33], we extract a uniform segment of 3 seconds from
the middle of each trial for improved consistency with a sampling frequency of
$f_{s}=128$ Hz.

####  5.1.2 Spatial correlation alignment

The selected datasets have been extensively studied in recent years [35, 58,
29]. In particular, several studies have highlighted the critical need to
address the domain shifts in biosignal data [33, 59, 32]. To compensate for
these shifts, they proposed Riemannian Alignment (RA), which is a test-time DA
method tailored for spatial shifts. For a multivariate signal
$\mathbf{X}_{d}\in\mathbb{R}^{n_{c}\times n_{\ell}}$ from the domain $d$, this
alignment is given by the following operation:

$m_{\text{RA}}(X_{d})=\mathbf{M}_{d}^{-\frac{1}{2}}\mathbf{X}_{d}\;,$ (24)

where $\mathbf{M}_{d}\in\mathcal{S}_{n_{c}}^{++}$ is the Riemannian mean of
the covariance matrices of all samples from the domain $d$. In the following
experiments, we compare RA with MA for sleep staging and BCI motor imagery.

####  5.1.3 Deep learning architectures and experimental setup

In our experiments, we used two different architectures and setups for sleep
staging and BCI. For sleep staging, we referred to the approach described in
[35], and for BCI, we followed the setup proposed by [56] as implemented in
MOABB [57]. In the upcoming section, we provide a detailed description of
these two setups.

##### Architecture

Many neural network architectures dedicated to sleep staging have been
proposed [60, 58, 59, 61]. In the following, we choose to focus on the
architecture proposed by [35] that is an end-to-end neural network proposed to
deal with multivariate time series and is composed of two convolutional layers
with non-linear activation functions.

For BCI, different neural network architectures have been developed
specifically [56, 62]. We focus on ShallowFBCSPNet [56], which is an end-to-
end neural network proposed to deal with multivariate time series based on
convolutional layers and filter bank common spatial patterns (FBCSP). Both
architectures are implemented in braindecode package [56].

##### Training setup

The training parameters follows the reference implementations of [35] and [56]
detailed below for reproducibility. For sleep staging experiment, we use the
Adam optimizer with a learning rate of $10^{-3}$, $\beta_{1}=0.9$, and
$\beta_{2}=0.999$. The batch size is set to $128$, and the early stopping is
done on a validation set corresponding to $20\%$ of the subjects in the
training set with a patience of $15$ epochs. We optimize the cross-entropy
with class weight for all methods, which amounts to optimizing for balanced
accuracy (BACC). Each experiment is done ten times with a different seed.

For BCI, we use the Adam optimizer with a cosine annealing learning rate
scheduler starting at $6.25\times 10^{-4}$. The batch size is set to $128$,
and the training is stopped after 200 epochs. No validation set is used. We
optimize the cross-entropy for all methods. We report the average accuracy
score (ACC) across ten different seeds.

##### Filter size and Welch’s parameters

MA requires tuning the filter size $f$ parameters. Yet, experiments show that
this parameter is not critical, as good performances are observed over a large
range of values. We provide a sensitivity analysis of the performance for
different parameters in Figure 7 for sleep staging (left) and BCI (right). It
shows that the value $f=256$ is a good trade-off for the sleep application
while a value of $f=8$ is better for BCI. These parameter values are used in
what follows. For the Welch method, we use a Hann window [63] and an overlap
of $\frac{f}{2}$.

Figure 7: Evolution of the balanced accuracy score for a leave-one-dataset-out
experiment for different values of filter size $f$.

###  5.2 Application to sleep staging

In our investigation, we build on the findings of [34] who showed the benefits
of TMA for sleep staging when using only two sensors. Here, we incorporate
more spatial information by using 7 EEG channels, as detailed in Section
5.1.1. This multidimensional dataset offers a more comprehensive perspective,
thereby demonstrating the utility of STMA and SMA. In what follows, one
subject is considered as one domain.

####  5.2.1 Comparison between different alignments

Historically, sleep staging has relied on neural networks processing raw data
[60]. [35] improved this by applying z-score normalization to 30-second
windows, removing local trends. More recently, [36] introduced session-wide
normalization to eliminate global trends. Among these, window-based
normalization, denoted as “No Align”, has proven most effective in [34] and
will be used as baseline in the following. In our work, we also compare MA
with RA to evaluate performance comprehensively.

Figure 8: Balanced accuracy for sleep stage classification without alignment
(blue), with RA (yellow), and with STMA (green) for different datasets in the
target domain. Each dot represents one subject. STMA outperforms alternative
methods for each target dataset. The number of stars illustrates the level of
significance with a Wilcoxon test (ns: non-significative, $**$: $10^{-2}$,
$***$: $10^{-3}$, or $****$: $10^{-4}$).

Using a leave-one-dataset-out evaluation (4-fold across dataset adaptation),
results in Figure 8 show that standard z-score normalization struggles with
dataset adaptation, with accuracies below 66%, sometimes dropping below 50%
(e.g., CHAT dataset). Spatio-temporal alignment improves accuracy, with STMA
consistently above 70% and even reaching an improvement of 2̃0% for MASS and
CHAT. On the other hand, RA stagnates for three datasets (i.e., ABC, CHAT and
HOMEPAP) and increase the score for only MASS by 5%. This underscores the
importance of temporal information in sleep classification, beyond RA’s
spatial focus. The experiment highlights the benefits of spatiotemporal
alignment: higher accuracy and lower variance, suggesting that underperforming
subjects improve the most. The next section will explore these subjects
further.

####  5.2.2 Study of performance on low-performing domains

In this section, we evaluate the performance of individuals to identify the
subjects that benefit from the most significant improvements. In the Figure 9,
we propose to compare the accuracy without alignment ($x$-axis) and with
spatio-temporal alignment ($y$-axis). A subject (i.e., a dot) above the line
$y=x$ means that the alignment increases its score. For each adaptation, more