Shape analysis for time series
Authors: Thibaut Germain
Samuel Gruffaz
Charles Truong
Alain Oliviero Durmus
Laurent Oudre
Venue: NeurIPS 2024 poster (2024)
Keywords: Machine learning for sciences
Machine learning for healthcare
Representation learning for time series
Shape analysis
LDDMM
Kernel methods
Primary area: machine_learning_for_healthcare
TL;DR: This paper introduces an unsupervised representation learning algorithm for time series tailored to biomedical inter-individual studies using tools from shape analysis.

Abstract: Analyzing inter-individual variability of physiological functions is particularly appealing in medical and biological contexts to describe or quantify health conditions. Such analysis can be done by comparing individuals to a reference one with time series as biomedical data.
This paper introduces an unsupervised representation learning (URL) algorithm for time series tailored to inter-individual studies. The idea is to represent time series as deformations of a reference time series. The deformations are diffeomorphisms parameterized and learned by our method called TS-LDDMM. Once the deformations and the reference time series are learned, the vector representations of individual time series are given by the parametrization of their corresponding deformation. At the crossroads between URL for time series and shape analysis, the proposed algorithm handles irregularly sampled multivariate time series of variable lengths and provides shape-based representations of temporal data.
In this work, we establish a representation theorem for the graph of a time series and derive its consequences on the LDDMM framework. We showcase the advantages of our representation compared to existing methods using synthetic data and real-world examples motivated by biomedical applications.

Shapes analysis for time series
Thibaut Germain
Samuel Gruffaz'
Centre Borelli, ENS Paris-Saclay 4 av. des sciences, 91190
Centre Borelli, ENS Paris-Saclay 4 av. des sciences, 91 190
Charles Truong'
Laurent Oudre!
Centre Borelli, ENS Paris-Saclay 4 av. des sciences, 990
Centre Borelli, ENS Paris-Saclay 4 des sciences, 91 190
Alain Durmus
CMAP, CNRS, Ecole polytechnique Institut Polytechnique de Paris 91 120 Palaiseau, France
Abstract
Analyzing inter-individual variability of physiological functions is particularly ap- pealing in medical and biological contexts to describe or quantify health conditions. Such analysis can be done by comparing individuals to reference one with time series as biomedical data. This paper introduces an unsupervised representation learning (URL) algorithm for time series tailored to inter-individual studies. The idea is to represent time series as deformations of a reference time series. The deformations are diffeomorphisms parameterized and learned by our method called TS-LDDMM. Once the deformations and the reference time series are learned, the vector representations of individual time series are given by the parametrization of their corresponding deformation. At the crossroads between URI. for time series and shape analysis, the proposed algorithm handles irregularly sampled multivari- ate time series of variable lengths and provides shape-based representations of temporal data. In this work, we establish a representation theorem for the graph of a time series and derive its consequences the LDDMM framework. We showcase the advantages of our representation compared to existing methods using synthetic data and real-world examples motivated by biomedical applications.
Introduction
Our goal is to analyze the inter-individual variability within a time series dataset, an approach of significant interest in physiological contexts (25 58 21]. Specifically, we aim develop an unsupervised feature representation method that encodes the specificities of individual time series in comparison a reference time series. In physiology, examining the various "shapes" in time series related to biological phenomena and their variations due to individual differences or pathological conditions is common. However, the term "shape" lacks precise definition and is more intuitively understood as the silhouette of pattern in time series. In this paper, we refer to the shape of a time series as the graph of this signal.
'Complete affiliation: Université Paris Saclay. Université Paris Cité, ENS Paris Saclay, CNRS, SSA. INSERM, Centre Borelli, F-91190, Gif--su-Yvettte, France. Corresponding author. Contact thibaut. beabaut.germaineerr.. wawteesss-aiiissssssaeee
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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space
space
time
time
cos (0) -sin(0) sin(O) cos(O)
Figure 1: A time series' graph G = {(t, s(t)) t I can lose its structure after applying general diffeomorphism G.G:: time value can be related to two values on the space axis.
Although community structures with representatives can be learned an unsupervised manner [55 [3] using contrastive loss (20 [4 39 similarity measures 2 21 45 62, the study of inter-
individual variability of shapes within a cluster [42 [1] remains an open problem in unsupervised
representation learning (URL). particularly for irregularly sampled time series with variable lengths.
Our work explicitly focuses on learning shape-based representation of time series. First, we propose to view the shape of time series not merely as its curve {s, E 1), but as its graph G(s)
{(t, : & E l]- Then, building on the shape analysis literature S 57], we adopt the Large Deformation Diffecmorphic Metric Mapping (LDDMM) framework [5 57 analyze these graphs. The core idea is to represent each element G(s²) of a dataset (s) jE[NN] as the transformation of reference graph G(so) by a diffeomorphism 6j. i.c. G(s)) G(so). The diffcomorphissm Φy learned by integrating ordinary differential equation parameterized by Reproducing Kernel Hilbert Space (RKHS). The parameters (05))E[N] encoding the diffemorphisms (dj))E[N) yield the representation features of the graphs (G(s)) ()) Finally, these shape-encoding features can be used inputs any statistical or machine- learnin model.
However, a time series graph transformation by a general diffeomorphism is not always a time series graph, c.g. Figure thus time series graph is more than simple curve (23] Our contributions arise from this observation: we specify the class of diffeomorphisms to consider and show how to learn them. This change is fruitful in representing transformations of time series graphs illustrated
in Figure [
Our contributions can be summarized as follows:
We propose an unsupervised method (TS-LDDMM) to analyze the inter-individual vari- ability of shapes time series dataset (Section particular, the method can handle multivariate time series irregularly sampled and with variable sizes.
We motivate extension of LDDMM time series by introducing theoretical framework with a representation theorem for time series graph (Theorem ! and kernels related to their structure (Lemma !
We demonstrate the identifiability of the model by estimating the true generating parameter of synthetic data, and we highlight the sensitivity of our method concerning its hyperparameters (Appendix G.I also providing guidelines for tuning (Appendix ())
We highlight the interpretability of TS-LDDMM for studying the inter-individual variability in a clinical dataset (Section
We illustrate the quantitative interest of such representation on classification tasks on real
shape-basee based datasets with regular and irregular sampling (Appendices H and
2 Notations
We denote by integer ranges by [k: : fk..... 1) CT (1, E) the set of m-times continously differentiable function defined on an open set to a normed vector space by ||ы] = supaeu [u(z)] for any bounded function u U E, and by Nso the set of positive integers.
P(Z) and [7] [1 with kt E N, by

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LDOMMM 754DDMM
Figure 2: LDDMM and TS-LDDMM are applied to ECG data. We observe that LDDMM, using general Gaussian kernel, does not learn the time translation of the first spike but changes the space values, i.e., one spike disappears before emerging at a translated position. At the same time, TS- LDDMM handles the time change the shape. This difference of deformations implies differences in features representations.
3 Background on LDDMM
In this part, we expose how to learn the diffeomorphisms (Øy)]E[N] using LDDMM, initially intro- duced in [5]. In nutshell, for any j [N]. Φj corresponds to a differential flow related to learnable velocity field belonging to a well-chosen Reproducing Kernel Hilbert Space (RKHS).
In the next section, time series are going to be represented by diffeomorphism parameters (az (aN)) That is why LDDMM is chosen since it offers parametrization for diffeomorphisms that sparse and interpretable, two features particularly relevant in the biomedical context.
The basic problem that consider in this section is the following. Given set of targets y (V)))e(TT)) in R set of starting points (27)) in Rd", we aim find diffeomorphism such that the finite set of points is similar in a certain sense to the set of finite sets of transformed points @ (0(x;))se(TT)) The function is occasionally referred to deformation. In general, these sets y, are meshes of continuous objects, e.g., surfaces, curves, images, and so on.
Representing diffeomorpshims deformations. Such deformations & are constructed via differ- ential flow equations, for any xp E IRd and 1]:
dX(()) = = X(0) xo 26 (2O) = X(T), 6² o
(1)
dr
where the velocity field is differentiable function on Rd If dulloo ||ull I ||ul|v for any u V and E [= 11, V) {u E C ( 1]. V): for || 1 dr < xo). by (22 Theorem 5] d² exists and belongs to D(RRd) where we denote by D(O) the set of diffeomorpshim defined on an open set O O. Therefore, for any choice of v, Φ² defines a valid deformation. This offers a general recipe construct diffeomorphism given a functional space V.
E [0, 1] o, E V and V is a Hilbert space of continuously
With this in mind, the velocity field v fitting the data can be estimated by minimizing v E L'( 1J, + LØd x. y), where L is an appropriate loss function. However, two computa- tional challenges arise. First, this optimization problem is ill-posed, and penalty term is needed to obtain a unique solution. In addition, parametric family Ve C LL V), parameterized by θ is sought to efficiently solve this minimization problem.
From deformations to geodesies. It has been proposed in 40 interpret V as a tangent space relative to the group of diffeomorphisms H = fØ"- v E ( 1J, V)J. Following this geometric point of view, geodesics be constructed H by using the following squared norm
inf vEL.( g=d" S.
g H
||e, 1/ dr
(2)
By deriving differential constraints related to the minimum of and using Cauchy-Lipschitz conditions, geodesics can be defined only by giving the starting point and the initial velocity vo e V 40). as straight lines in Euclidean space. Denoting by + Pvo (r) H the geodesic starting from
Note that denote by d' IN the ambient space
3

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the identity with inital velocity E V, the exponential map is defined as pp - Pvo (1). Using ( instead of Φ", the previous matching problem becomes a geodesic shooting problem:
inf No
(3)
Using ((o0)) instead of 6" for any e E ( 1), V) regularizes the problem and induces a sparse representation for the learning diffeomorphisms. Moreover, by setting V as an RKHS, the geodesic shooting problem has a unique solution and becomes tractable, as described in the next section.
this part, V is chosen as an RKHS 6 generated by smooth kernel K (e.g.. Gaussian). We follow [17] and define discrete parameterization of the velocity fields to perform geodesics shooting (3). The initial velocity field Up is chosen as a finite
Discrete parametrization of diffeomorpshim.
linear combination of the RKHS basis vector fields, control points Xo (ZI,0)kE(mo] E (R y and momentum vectors og (al,))kE(mol E (IRd) yuu are defined such that for any x Rd'
no
to (o):Xo) (I)
Σ k=1
K(I,
(4)
In our applications, the control points (IE)O))E(no) can be understood as the discretized graph (fa, so (tx) (E)))))())) of a starting time series So. With this parametrization of vo. [40 show that the velocity field v of the solution of (3) keeps the same structure along time, such that for any x e Rd and r E [0, 1].
no
(x Σ K(2, xx(T))au(T)
dax(T) (r)((r) 1y (Ix(
no
dr Σ
(1 ax(
dr
(5)
(xx(0) 0,k,0, (0) ok k E [no]
These equations are derived from the hamiltonian (axR, a&) xk))k + Bo -= 0f K (Ix, a))a
such that the velocity norm is preserved ||w~|lv = ||oolly for any T E [0. By (.. the velocity field related to geodesic v" is fully parametrized by its initial control points and momentum
(I an <eE Thus, given set of targets y (Bt))strss in Rd a of starting points
(TI,O))E(T)) in Rd", a RKHS's ' kernel K : Rdd x Rdd - Rd xd' a distance on sets L, a numerical integration scheme of ODE and a penalty factor À > the basic geodesic shooting step minimizes the following function using gradient descent method:
Fx. (OL)KE(TT) + L
L (1p(mo) x. + Allvblle.
where up is defined by (4) and (v0) x is the result of the numerical integration of (5) using control points x and initial momentums (ax)ke((s)
Relation to Continuous Normalizing Flows. One particular popular choice to address the problem
of learning a diffeomorphism or a velocity field is Normalizing Flows 47 32] (NF) or their continuous (CNF). However,
do not rely this class of learning algorithms for several reasons. Indeed, existing and simple normalizing flows are not suitable for the type of data that we interested in this paper [19 [6] In addition, they are primarily designed to have tractable Jacobian functions, while we do not require such property in our applications. Finally, the use of differential flow solution of an ODE (1) trick is also at the basis of CNF, which then consists of learning velocity field to address in fitting the data through a loss aiming to address the problem at hand. Nevertheless, the main difference between CNF and LDDMM lies in the parametrization of the velocity field. LDDMM uses kernels to derive closed form formula and enhance interpretability while NF and CNF take advantage of deep neural networks to scale with large dataset in high dimensions.
counterpart [13 24 48
4 Methodology
We consider in this paper observations which consist in population of N multivariate time series, for any E [V], sd (1, However, we can only access ny -y-mallees ÿ = =(3 = = st()))

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collected at timestamps (t) kejnl for any j E [M]. Note that the number of samples nj is not necessarily the same across individuals and the timestamps can be irregularly sampled. We assume the time series population is globally homogeneous regarding their "shapes" even if inter-individual variability exists. Intuitively speaking, the "shape" of time series & : - Rd is encoded in its graphs G(s) defined as the set {(t, s(t): & E I and not only in its values s(I) = {s(t) : t I) since the time axis is crucial. As a motivating use-case, s can be the time series of a heartbeat extracted from an individual's 's electrocardiogram (ECG), see Figure 2 The homogeneity in resulting dataset comes from the fact that humans have similar shapes of heartbeat (7]
The deformation problem. In this paper, we aim to study the inter-individual variability in the dataset by finding relevant representation of each time series. Inspired from the framework of shape analysis [57], addressing similar problems in morphology. we suggest to represent each time series' graph G( as the transformation of reference graph G(s,), related to time series So + Rd by [N].
a diffeomorphism by on Rd+1, for any
of G(sp) = f0y (t,))(t), tE
So. will be understood as the typical representative shape common to the collection of time series (s)) JE[]]- As So is supposed to be fixed, then the representation of the time series (s )JE[N] boils down to the one of the transformation (dy) We aim to learn G(so) and (d))zE(N)-
Optimization related to 7 Defining the discretized graphs of the time series (s JENN and a discretization of the reference graph G(so) as, for any j E [N]
yj = G(5)) (t), 3t )eIm,) (Io+,)" Go (t), 5 8.)iclmll E (I==1)
with no (ondia(((2))EE(T))) the representation problem given in 7 boils down solving:
Foo.N ((oE)
argmin
INI (o PO.
(8)
which is carried out by gradient descent on the control points G, and the momentums aj (a() (no)) mn)]
for any j E [AV] initialized by dataset's time series graph of size no and by (dd+1)n, respectively. The optimization hyperparameter details are given in Appendix IEII The result of the minimization Go is then considered as the ny samslee of common time series Sy and the momentums ay encoding Φj yields a feature vector in dRd of s for any j E [N]. Finally, the vectors (ai))EIM can be analyzed with any statistical or machine learning tools such as Principal Components Analysis (PCA), Latent Discriminant Analysis (LDA), longitudinal data analysis and so on.
Nevertheless, g asks to define a kernel and loss in order to perform geodesic shooting 6 which the purpose the following subsection.
4.1 Application of LDDMM to time series analysis: TS-LDDMM
This section presents our theoretical contribution: we tailor the LDDMM framework to handle time series data. The reason is that applying a general diffeomorphism e from R2++1 to time series' graph G(s) can result in set é. G(s) that does not correspond to the graph of any time series, illustrated in the Figure Thus, time series graphs have more structure than simple 1D curve 23 and deserve
their unique analysis, which will prove fruitful as demonstrated in Section 5
To address this challenge, we need to identify RKHS kernel K Rd+1 x Rd+1 + (d+1)² that generates deformations preserving the structure of the time series graph. This goal motivates us to clarify, in Theorem [ the specific representation of diffeomorphisms we require before presenting class of kernels that produce deformations with this representation.
Similarly, selecting a loss function on sets L that considers the temporal evolution in time series' graph is crucial for meaningful comparisons with time series data. Consequently, we introduce the oriented Varifold distance.
A representation separating space and time. We prove that two time series graphs can always be linked by a time transformation composed with space transformation. Moreover, a time series graph transformed by this kind of transformation is always time series graph. We define V,
5

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D( (t) E R + (7(t). for any E D(R) and fy (t,x) E Rd+1 + (t, f(t,
for any f C (R441 Rd). We have the following representation theorem. All proof's are given in Appendix B
Denote by G(s) - ((t)((t) E 1) the graph of time series - Rd and p.G(s) & {ō(t, s(t)) 1 EE!) the action of Φ E D(R++1) on G(s).
Rd and So + Ed be two continuously differentiable time seriess with I, two intervals of R. There exist f E C. (RR+1 Rd) and E D(R) such that r(1) = and 2 E D(Bd+1).
Theorem 1. Let
G(s) = (ly..-G(so)) 1h Vr o
Moreover, for any I C (IR4+1 1R and ((R), there exists a continously differentiable time series such that G(3) = I1 G(s))
Remark 2. Note that for any Y E ((K) and s (I,
((T(1), s()), 1 l = (11, LE (E)())
As a result, M, can be understood temporal reparametrization and 4 encodes the transformation about the space.
Choice for the kernel associated with the RKHS V As depicted Figure we not any kernel K to apply the previous methodology to learn deformations on time series' graphs. We describe and motivate
choice in this paragraph. Denote the mne-dimensional Gaussian kemnel by K(a) (ar, 3/ = exp(- - (o) for any (z, v) E (R*)², a E and a > To solve the geodesic shooting problem (6) on 18++1 we consider for the RKHS associated with the kernel defined for any (t, (f, (t',x') E (2:
Ka((t, 2), (f,(t,,,,t',z) = Kumeume
(9)
C1 CyKspace)
Kapace K! (t, (,,t)) (') (d) (x, x' la Kime (1) (t,t')
parametrized by the widths OT.0, TT,1, On >> O and the constants co, e1 > O. This choice for KG motivated by the representation Theorem [ and the following result.
Lemma 1. If we denote by V the RKHS associated with the kernel Kg. then for any vector field
generated by (5) with vo satisfying 4.. there exist y E D(R) and E C² (IR+1 RR) such that
6" = y
Instead of Gaussian kernels, other types of smooth kernels can be selected as long as the structure is respected.
(
Remark 3. With this choice of kernel, the features associated with the time transformation can be extracted from the momentums (okk,) kE(po] E (Rd+1)no in (4) by taking the coordinates related to time. However, the features related to the space transformation are not only in the space coordinates since the related kernel K Kpcce depends on time as well. The kernel's representation has been carefully designed to integrate both space and time, while ensuring that time remains independent of space. Initially, we considered separating the spatial and temporal components. However, post-hoc analysis of such a representation proved to be challenging. The separated spatial and temporal representations are correlated, and understanding this correlation essential for interpreting the data. As a result, concatenating the two representations becomes necessary, though there is no straighfforward method for doing so, as they are not commensurable. Consequently, we opted for representation that inherently integrates both space and time.
In Appendix we give guidelines for selecting the hyperparameters (OT,O, TT,1, Tz. Co, C1).
Loss This section specifies the distance function L introduced in the loss function defined (6).
In practice, we can only access discretized graphs of time series, (t/, x()se(n)) for any j E [N], that are potentially of different sizes n, and sampled at different timestamps (1 )se(n,) for any j E [N]. Usual metries, such as the Fuclidean distance, are not appealing as they make the underlying assumptions of equal size sets and the existence of a pairing between points. Distances between measures on sets

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(taking the empirical distribution), such as Maximum Mean Discaprency (MMD) 18 9]. alleviate those issues; however, MMD only accounts for positional information and lacks information about the time evolution between sampled points. A classical data fidelity metric from shape analysis corresponding to the distance between oriented varifolds associated with curves alleviates this last issue [30). Intuitively, an oriented varifold is measure that accounts for positional and tangential information about the underlying curves at sample points. More details and information about oriented varifolds can be found in Appendix C
More precisely, given two sets Gu = (g.)) (m()))) G1 (o)) E (R and kerne (IR-11 x ) + I verifying (30] Proposition 2 & 4], for any & E {0, 1] and i E [Tg - 1], denoting the center and length of the it segment [ s441 by of = (of + 9/41)/2, 1f - of ll. and
off ( /0, the varifold distance between Go and G₁ is defined as,
To-1 Σ
To
dy- (Go, G1)
Pk(lc, R. (
Σ
Σ
CK(C R.), (C)
b-=1
1=1
Σ ( (') w=
In practice, we set the kernel k the product of two anisotropic Gaussian kernels, kpos and kdirs such that for any (I,, )t), (v. 7) E (Re+1 g4)
A((2, 1)), (v. 2 = kpos y)kair v)
Note that the loss kernel & has nothing to do with the velocity field kernel denoted by KG or K specified in Section 4.1 Finally, we define the data fidelity loss function, L, as a sum of dz-- using different kernel's width parameters to incorporate multiscale information. L is indeed differentiable with respect to its first variable. The specific kernels kpos kair that we use in our experiments are given Appendix C.I For further readings on curves and surface representation as varifolds, readers can refer to 30 12
A pedagogical online application is available to inspect the effect of hyperprameters on geodesic shooting (5) and registration
5 Experiments
The source code is available on Githut For conciseness, several experiments are relegated in appendix:
1. TS-LDDMM representation identifiability, Appendix G On synthetic data, we evaluate the ability of our method to retrieve the parameter of that encodes the deformation p(v) acting on a time series graph G by solving the geodesic shooting problem (6) between G and (x()), GG. Results show that TS-LDDMM representations are identifiable or weakly identifiable depending on the velocity field kernel KG specification.
2. Robustness irregular sampling, Appendix Hi: We compare the robustness of TS- LDDMM representation with 9 URL methods handling irregularly sampled multivariate time series on 15 shape- based datasets (7 univariates & 8 multivariates). We assess methods' classification performances under regular sampling (0% missing rate) and three irregular sampling regimes (30%, 50% and 70% missing rates), according to the protocol depicted in 331. Results show that our method, TS-LDDMM, outperforms all methods for sampling regimes with missing rates: 0%, 30%, and 50%.
3. Classification benchmark on regularly sampled datasets, Appendix I We compare performances of a kernel support vector machine (SVC) algorithm based on TS-LDDMM representation with state-of-the-an classification methods from shape analysis 15 shape-based datasets (7 univariates & multivariates). Results show that the TS-LDDMM- based method outperforms other methods (best performances over 13 datasets), making TS-LDDMM representation relevant for time series shape analysis.
Ig = {i E Rd+1, |x|
[=|=1)
https://githut thibaut- gormain/TSLLDMM
7

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(a) TS-LDDMM shooting
(b) LDDMM shooting
dens tips
PC densities
-20c
+2
-20
+2
cole
Thorace
colo
202
102
200
Unn
Figure 3: Analysis of the two principal components (PC) related mice ventilation before exposure with TS-LDDMM representations (a), and LDDMM (b). In both cases and for all PC, the left plot displays PC densities according mice genotype and right plot displays deformations of the reference graph co along each PC.
so
Thoeact
at
Nasal
(a) ColQ cycle (b) PC1 vs PC2
(c) WT cycle
Figure 4: (a) ColQ respiratory cycle sample. (b) Referent respiratory cycle of individual mouse cj the TS-LDDMM PC1-PC2 coordinates system of Co. (c) a WT respiratory cycle sample.
4. Noise sensitivity for learning the reference graph, Appendix J
We evaluate the noise sensitivity of TS-LDDMM and Shape-FPCA [60] for learning the reference graph on synthetic dataset and for several levels of additive Gaussian noise. Results show that both methods are sensitive to noise. However, TS-LDDMM preserves the overall shape while shape-FIPC alters the shape depending on the noise level.
5.1 Interpretability: mice ventilation analysis
This experiment highlights the interpretability of TS-LDDMM representation for studying the inter- individual variability in biomedical applications. We consider a time series dataset monitoring the evolution of mice's nasal and thoracic airflow when exposed to a drug altering respiration [4T). The dataset includes recordings of 7 control mice (WT) and 7 mutant mice (ColQ) with enzyme deficiency. The enzyme involved in the respiration regulation, and the drug inhibits its activity. For each mouse, airflows were monitored for 15 to 20 minutes before the drug exposure and then for 35 40 minutes. A complete description of the dataset given the Appendix F.
Experimental protocol. We considered experimental scenarios; the first focuses mice ventilation before exposure to explore the inter-individual and genotype-specific variabilities. The second focuses whole recordings to analyze the evolution of mice's ventilation after drug exposure. In both cases, the baseline protocol consists of first extracting N respiratory cycles from the datasets with the procedure described in [2]]. Then, learning the referent respiratory cycle co and the representations of respiratory cycles (ag))E by solving (8) using TS-LDDMM. ag being the momentum of the initial velocity field of the geodesic encodings the diffeomorphismms mapping Co to the jth respiratory cycle. Finally, performing a Kernel-PCA on the initial velocity fields (4)
belonging V and encoded by the pairs (a) cu)j The first experiment includes N₁ 700 cycles collected before exposure. The second experiment includes N₂ = 1400 cycles with 25% (resp. 75%) before (resp. after) exposure. We also performed the first experimental scenario with LDDMM representation, and Appendix K describes the settings of both methods. Essentially, varifold losses are identical for both methods, and the velocity field kernels are set to encompass time and space scales. in addition, In addition, Appendix presents comparison between TS-LDDMM and Shape-FPCA on the second scenario.
Geodesic shooting along principal component directions. Any principal component (PC), noted vg from a kernel-PCA in V, itself an initial velocity field encoded by a pair (Co, ag"). PCs encode the principal axis of deformations, and it is possible to shoot along the geodesic they encode with the differential equations (5), enabling interpretation of the main sources of deformations.

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(a) PC1 densities
(b) TS-LDDMM PC1 shooting
(c) Scatter PC1 PC3
-1.54m
+1.50M
50
colo
Nasal
Thotatic
40
-000 -500 300
500 1800
500
1000
0
500
1000
200
time (ms)
Figure 5: Analysis of the first Principal Component (PC1) related to mice ventilation before and after exposure with IS-LDDMM representations. (a) displays PC densities per mice genotype, (b) illustrates deformations of the reference respiratory cycle co along PCI, and (c) displays all respiratory cycles with respect to time in PC1 and PC3 coordinates
Mice ventilation before exposure. We focus on the amalysis of the two first Principal Components (PC) for TS-LDDMM (Figure 31) and LDDMM (Figure 33)) Looking at the geodesic shooting along PCs, Figure 3 shows that principal components learned with TS-LDMM lead deformations that remain respiratory cycles. In contrast, deformations learned with LDDMM are challenging to interpret respiratory cycles. The LDDMM velocity field kernel is Gaussian anisotropic kernel that accounts for time and space scales; however, the entanglement of time and space dimensions in the kernel does not guarantee the graph structure, and it makes the convergence of the method complex (relative varifold loss error: TS-LDDMM: 0.06, LDDMM: 0.11).
Regarding TS-LDDMM Figure its PCs refer deformations directions carrying different physio- logical meanings. Indeed, the geodesic shooting along these directions indicates that PCI accounts for variations of the total duration of a respiratory cycle, while PC2 expresses the trade-off between inspiration and expiration duration. In addition, the distribution of ColQ respiratory cycles along PC1 is wider than in WT mice, indicating that the adaptation of mutant mice their enzyme deficiency variable. This observation can also be seen in Figure 4b where a referent respiratory cycle cf is learned by solving (8) for each mouse and is encoded in the (PC1,PC2) coordinate system of co by registration (3). Indeed, the average respiratory cycles of ColQ mice more spread out than WT mice's. s. Going back to the densities of PCI, ColQ mice distribution has a heavier tail toward negative values compared to WT mice. When shooting the opposite direction of PCI, we can observe that the inspiration is divided into two steps. Congruently with [2T. such inspirations indicate motor control difficulties due to enzyme deficiency. Figure 4a an example of ColQ respiratory cycle with negative PC1 coordinate.
Mice ventilation evolution after drug exposure. This experiment focuses the first principal components learned from TS-LDDDM representations of respiratory cycles randomly sampled before and after drug exposure. Figure 5h illustrates the geodesic shootings along PCI. Again, PC1 accounts for variations in respiratory cycle duration, but more importantly, it can be observed on the deformation at -15 drc the apparition of long pause after inspiration. Congruently, Figure Sp indicates that pauses appear after drug exposure as cycles with negative PC1 values mainly occur after 20 minutes and present more variability along PC3. In addition, Figure 5p shows a bimodal distribution for WT mice with one of the peaks in the negative values. This peak was not observed in the previous experiment Figure hh. It indicates that pauses after inspiration are prevalent in WT mice after drug exposure. On the other hand, the distributions of ColQ mice's 's respiratory cycles along PC1 in both experiments are similar and account for the same deformation, suggesting that ColQ mice weakly react to the drug exposure as they already adapt their enzyme deficiency.
Experiment Conclusion. Analyzing mice ventilation with TS-LDDMM representation highlights the method's s ability to create meaningful interaction between experts and the data. Indeed, combining statistical and visual results shows that main deformations carry physiological meaning, enabling the characterization of some mice genotypes and the effects of drug exposure.
6 Related Works
Shape analysis focuses on statistical analysis of mathematical objects invariant under some deforma- tions like rotations, dilations, or time parameterization. The main idea is to represent these objects in a complete Riemannian manifold (M,g) with a metric g adapted to the geometry of the problem 40 Then, any set of points in M can be represented as points in the tangent space of their Frechet
44 33 by considering their logarithms. The goal is to find well-suited Riemannian structure according to the nature of the studied object.
mean mg

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LDDMM framework is a relevant shape analysis tool to represent curves as depicted in 233 However, graphs of time series are a well-structuree type of curve due to the inclusion of the temporal dimension that requires specific care (Figure T. In similar vein, Qiu ef al [46] proposes method for tracking anatomical shape changes serial images using LDDMM. They include temporal evolution, but not for the same purpose: the aim is to perform longitudinal modeling of brain images.
Leaving the LDDMM representation, the results of [53 26 address the representation of curves with the Square- Root Velocity (SRV) representation. However, the SRV representation is applied after reparametrization of the temporal dimension of the unit length segment. Consequently, the graph structure of the time series is not respected, and the original time evolution of the time series is not encoded in the final representation. Very recently, functional data analysis (FDA) framework, a improved by representing the original time evolution. However, the space and time representations remain correlated, complicating post-hoc analysis, as discussed in Remark Additionally, this method is tailored for continuous objects and applies only to time series of the same length, making the estimation more sensitive to noise. This issue can be addressed through interpolation, but this approach is not always reliable in sparse and irregular sampling scenarios.
paper [60] (Shape-FPCC)
Most FDA approaches, as seen [50 63 [9]. address this challenge using interpolation or basis function expansion. In summary, FDA methods typically separate space and time representations for continuous objects, whereas DS-MM algorithm maintain a discrete-to-diserete analysis, inherently integrating both space and time representations.
Balancing between discrete and continuous elements is a challenging task. In the deep learning 29 36 I1 Neural Ordinary Differential Equations (Neural ODEs) 13 learn continuous latent representations using vector field parameterized by neural network, serving as a continuous analog to Residual Networks 644 This approach was further enhanced by Neural Controlled Differential Equations (Neural CDEs) (31] for handling irregular time series, functioning as continuous-time analogs of RNNs [49. Extending Neural ODEs, Neural Stochastic Differen- tial Equations (Neural SDEs) introduce regularization effects (36), although optimization remains challenging. Leveraging techniques from continuous-discres
literature 13
31
56
filtering theory, Ansari et al. II] applied successfully Neural SDEs to irregular time series. Oh et al. (43) improved these results by incorporating the concept of controlled paths into the drift term, similar to how Neural CDEs outperform Neural ODEs. With TS-LDDMM, the representation is also derived from an ODE, but the velocity field is parameterized with kernels and optimized to have minimal norm, which enhances interpretability.
All these state-of-the-art methods previously mentionned LDDMM in Appendix and Appendix
23
43
60
26
are compared to TS-
Compared to the Metamorphosis framework 77). LDDMM framework has weaker assumptions. The 3DMM framework requires that each mesh be parammetrized into a consistent form where the number of vertices, triangulation, and the anatomical meaning of each vertex are consistent across all meshes, as stated in the introduction of [8]. In our context, we do not need such procesccs,,,g; the time series graph can have different sizes.
Limitations and conclusion
This paper proposes a feature representation method, TS-LDDMM, designed for shape comparison on homogeneous time series datasets. We show on a real dataset its ability to study, with high interpretability, the inter-individual shape variability. As an unsupervised approach, it is user-friendly and enables knowledge transfer for different supervised tasks such as classification.
Although TS-LDDMM is already competitive for classification, its performances can be leveraged on more heterogeneous datasets using a hierarchical clustering extension, which is relegated for future work.
TS-LDDMM employs kernel computations, which require specific libraries (e.g., KeOps (II) to be efficient and scalable. However, in our experiments, the time complexity of TS-LDDMM is comparable to that of competitors. It is clear that TS-LDDMM needs to be extended to handle very large datasets with high- nigh-dimensional time series (such as videos).
Additionally, TS-LDDMM requires tuning several hyperparameters, though this is common re- quirement among competitors (3 43
60) 26]. In future work, adaptive methods are expected to be developed to provide a more user-friendly interface.
10

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Acknowledgments and Disclosure of Funding
This work was supported by grants from Région Ile-de-Frand Frence (DIM MathInnov). Charles Truong is funded by the PhLAMES chair of ENS Paris-Saclay.
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A
Societal impact
We believe that the paper has positive societal impact for the following reasons:
TS-LDDMM is an interpretable method for understanding inter-individual variability in biomedical datasets, potentially offering new insights in medicine.
TS-LDDMM bridges the gap between the shape analysis community and the unsupervised representation learning (URL) community, fostering potential future collaborations between these fields.
However, the computational cost the method may raise environmental concerns similar to those associated with deep learning 43 Additionally, while TS-LDDMM has promising biomedical applications, could also be misused for creating poison.
B
Proofs
Denote by G(s) - (1)) E 1) the graph of time series s - TRd and di. db. - {o(1, s())) t E l} the action of Φ D(R++1) on G(s).
Theorem 4. Let s : J + Rd and So - Rd be two continuously differentiable time seriess with I, J two intervals of R. There exist f C¹ [R Rd) and E ((R) such that y(I) and 4 E D(R4+1),
G(s) = G(So)), 1hy = Mr b
Moreover, for any f E C [Rd+1 I and 7 D(R), there exists a continously differentiable time series such that G(3) = II, f G(s))
Proof. Let 8 J + d and so - Rd be two continuously differentiable time seriess with I= (a, b), (a, B) two intervals of R. By setting 7 : + (8 a) aa/(b---/ae--- a) a R, have 7()
and 7 D(R). By defining f (t, Rd41 + so(t) 5 7t), the map $ D(R(+1), indeed, its inverse is 4 (1, E Rd+1 - (1, so(() - s(I) and is continuously differentiable. Moreover, we have G(s,) (()(),
E = G(s).
Let f (Rd+1 (,2+H,, E D(R) and so E C' (I, with an interval of R. We have
G.s) [(<((), f(t,s, tel} [tt, (t), (()) t t (
(10)
By defining s: t (1) + f (t), su ( (f))), we have ø C² Rd) by composition of C² functions and =[ y G(so) by 10 which concludes the proof.
Lemma we denote by V the RKHS associated with the kernel KG, then for any vector field generated by (5) with vo satisfying there exist 7 E D(R) and f C (RR++ RR) such that d² V, o 6j.
Proof. Let be a vector field generated by (5) with un satisfying 4 We remark that the first coordinate of the velocity field e, denoted by ultiee only depends the time variable for any Thus, when computing the first coordinate of the deformation Φ", denoted by y, we integrate (1) with v, replaced by ufime, thus y is independant of the variable æ.. Moreover, 7 D(R) since Gaussian kernel induced an Hilbert space V satisfying flv < If + | for any f e V by [22 Theorem 9]. For the same reason, we have o" E D ), and thus its last coordinates denoted by belongs to C2 (RRd+1 RRd), and by construction e V, o Dy.
T T [0,
C Oriented varifold
In this section, we introduce the oriented varifold associated with curves. For further readings on curves and surfaces representation varifolds, readers can refer to [30 12] We associate to
15

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define distance sets of triplets as C' ((a, R an oriented varifold p4y. i... distribution on the space Rd+1 x S defined as follows, for any smooth test function w : Rd+1 x Sd - R,
By-p, (v)) u (w( w (1(0), 7(() 19((0) (4(t))tt
Denoting by W the space of smooth test function, we have that H4y belongs to its dual W Thus, distance on W is sufficient to set distance on oriented varifolds associated to curve and thus on ((a, b), Id+1) by the identification 7 - Hy- Remark that (TS-LDDMM), y should be the parametrization of time series' graph G(s), i... 7 y (t, (t)) R denoting by Rd the time series. However, in practice, we work with discrete objects. That why, set W as an RKHS to use its representation theorem. More specifically [30, Proposition 2 & 4] encourages consider kernel & (IR+1 8d + E such that there exist two positive and continuously differentiable kernels Kpos and kdirs such that for any (x, d (v. 7) E (R4+1 S4)²
A((2, T.). F) Kios y)kaiee ( L d
with moreover kair > O and kipos which admits an RKHS W Wpas dense in the space of continous function on Rd+1 vanishing at infinite (10].
Given such kernel l: (Rd+1 8d)² - R verifying 30 Proposition 2 & 4], we have that for any
(z, v) E 12+1 Sd, 7) belongs W as a distribution and that the dual metric (--)ws satisfies for any (21, v1), (22, U2) (12+11 x x
(6. Pi) i (s,, v,), k((21, 11), (12, 72))
Thus, given two sets of triplets X (4 v (e) (1, yo, w. ()et))))) x Rd+1 x sdyTi-1 and denoting by
(R& x Rd+1 BITTo-1,
#x Σ
Σ 1=1
4 ())
(11)
we have,
1xx- HY -yl--
Σ Eis (k((2,, v)), (2,, 0y ") + (Kl( wi), (yi. w())''
-2 Io-1 -Ti-1 LK((2s, wt). (1;, wiI''
Then, using the identification X px, Y fy, dw- 3XX, Y)= |ax- HY ly-.
Now, we aim to discretize the oriented varifold μıg related to time series' graph G(s) by using set of triplets. This is carried out by using discretized version of G(s), i.e. G = (g. (4,, ((:)))))(r)) (Rd-11T, in the following way: For any i [T 1), denoting the center and length of the tth segment [9s 9:+1 by C, = (g: + 9:+1 /, 4 = || gill. and the unit norm vector of direction 9:9:+1 by Jे = = 9.)/4. we define the set of triplets X x() (4, ct, V v)) and its related oriented varifold ( Σ 1,0,, as in (II). This is a valid discretization of the oriented varifold μG according to 30 Proposition 1]: (G) converges towards PaG as the size of the descretization mesh suPre[T-1] |+++1 4 converges to O.
Finally, we define a distance on discretized time series' graphs Gi G2 as dw- (Gi, G2) = dw- (G))
C.1 Varifold kernels
Denote the one-dimensional Gaussian kernel by K:(") (z, y) exp(-Ix- (r) for any (z, v) ( a E N and a > In the implementation. we use the following kernels, for any (1),,21, (t2, x2)) E (12+112, ((w1, v1), (w2, v2)) E (Sd)2
kpoo (I, y) (!)) (ti, 1g (dd) opon. (2), Kpos (z, y) k((1) (wi, W2) K (V1, v2)
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where po ppoe,z ddir,t, ddi.. > 0 are hyperparameters. In practice, we select pos, 22 dir, R I when the times series are centered and normalized. Otherwise we select pos,,r 2 dir. o, with o, the average standard deviation of the time series. We choose postt R d mfe with fe the sampling frequency of the time series and m E [5] an integer depending on the time change between the starting and the target time series graph. The more significant the time change, the higher m should be. The intuition comes from the fact that the width pos,t, dir, rules the time windows used to perform the comparison, and pos,a d affects the space window. The size of the windows should be selected depending on the variations the data.
D Tuning the hyperparameters of the TS-LDDMM velocity field kernel
The parameter dr,o should be chosen large compared the sampling frequency fe and compared to average standard deviation 0, of the time series, e.g ØT.0 100 as 0, fe 1. It makes the time transformation smoother. If T is too small, for instance, σT,0 = fe. the effect of the time deformation is too localized, and there are not enough samples to make it visible.
The parameter σT,1 should be of the same order as fe: two different points in time can have various space transformations. σ₂ should be of the same order of ō. two points with a big difference regarding space compared to o2 can have very different space transformations.
We take 2 10c1, we want to encourage time transformation before space transformation. We take (co, C1) = (1, 0.1) all experiments.
E Experimental settings
All experiments were performed on a Debian 6. 1.699 server with NVIDIA RTX A2000 12GB GPU, Intel(R) Xeon(R) Gold 5220R CPU @ 2.20GHz, and 250 GB of RAM. The source code is available on Githut
E.1 Optimization details of TS-LDDMM LDDMM
We implemented TS-LDDMM in Python with the JAX library
Initialization. As initialization of 88 all momentum parameters of reference is picked from the dataset such that its length is equal to the median length observed in the dataset.
and the initial graph
Gradient descent. The chosen gradient descent method is "adabelief" [65] implemented in the OPTAX library 7 The gradient descent has two main parameters: the number of steps (nb_steps) and the maximum stepsize value (7M)). The stepsize has a scheduling scheme:
Warmup period 0.1x nb steps steps: the stepsize increases linearly from 0 to 7)M- The goal is learn progressively the parameters. If the step size is too large at the start, smaller steps the end cannot make up for the mistakes made at the beginning.
Fine tuning periode on 0.9x nb steps the stepsize decreases from Mas to with a cosine decay implemented in the OPTAX scheduler, i.c. the decreasing factor as the form 0.5(1 cos(nt/TT).
By default, set nb steps to 400 and 7M to 0.
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A
B
Flow (milsec))
Nasal chamber
Thorax chamber
Oime (sec)
Volumel(m)
Inspiration
Expiration (sec)
Figure 6: A: Illustration of double-chamber plethysmograph. The term dpr stands for differential pressure transducer which measures the pressure in each compartment, the pressure then being converted flow. B: Nasal airflow (top) and lung volume (bottom). During inspiration, airflow is positive (grey) and during expiration, airflow is negative (pink).
F
Datasets
F1 Mouse respiratory cycle dataset
Ventilation is simple physiological function that ensures vital supply of oxygen and the elimination of CO2. Acetylcholine (Ach) is a neurotransmitter that plays an important role in muscular activity, notably for breathing. Indeed, muscle contraction information passes from the brain to the muscle through the nervous system. Achs are located in synapses of the nervous system (central and periph- eral) and skeletal muscles. They ensure the information transmission from nerve nerve. However, the transmission cannot end without the hydrolysis of Ach by the enzyme Acetylcholinesterase (AchE), allowing nerves return to their resting state. Inhibition of (AchE) with, for instance, nerve gas, pesticide, or drug intoxication leads to respiratory arrests.
The dataset comes from the experiment [1), where they studied the consequences of partial deficits in AChE and AChE inhibition on mice respiration. AchE inhibition was induced with an irritant molecule called physostigmine (an AchE inhibitor). Mice nasal airflows were sampled at 2000Hz with a Double Chamber plethysmograph (28). as depicted Figure (). The flow is expressed in ml.s-!; it has positive value during inspiration and negative value expiration Figure B). Among the mice population, we selected control mice (wt) and ColQ mice (colg), which do not have AChE anchoring in muscles and some tissues. As described in 41), mice experiments were follows:
1. The mouse is placed in DCP for 15 20 min to
internal control.
2. The mouse is removed from the DCP and injected with physostigmine.
3. The mouse is placed back into the DCP, and its nasal flow is recorded for 35 or 40 min.
Respiratory cycles were extracted following procedure 21 We removed respiratory cycles whose duration exceeds second; the average respiratory cycle duration 300 ms. We randomly sampled 10 respiratory cycles per minute and mouse. It leads to dataset of 12,732 (time, respiratory cycles.
htttp:://githuu..co/bbbbbbab www.habbbaattggrraaaaii/DDDDDD wwwp::/wwwwwweeeeddeeeeeoeee/eee
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F.2 Shape-based UCR/UEA time series classification datasets
selected 15 shape-based datasets (7 univariates and 8 multivariates) from the from University of East Anglia (UEA) and the University of California Riverside (UCR) Time Series Classification
Repository 8 [3 3]. All datasets were downloaded with the python package acon" Essential datasets
information summarized in Table I and further can be found in [15 3].
Table 1: UCR/UEA shape-
shape-based based time series datasets for classification.
Dataset
Size
Lengh
Number classes
Number dimensions
Type
ArrowHead
251
3
IMAGE
BMK
180
128
3
SIMULATES
ECG200
200
96
2
ECG
Univariate
FacesUCR
2250
IMAGE
GunPoint
200
150
2
MOTION
PhalaagesOutlinssCorree
2658
80
2
IMAGE
Trace
200
SENSOR
Articulary Rendneoognition
575
144
SENSOR
Cricket
180
1197
12
MOTION
ERing
65
6
SENSOR
Multivariate
Handwriting
1000
152
MOTION
Libras
3601
45
15
VIDEO
NATOPS
51
6
MOTION
RacketSports
303
30
SENSOR
U UWaveCiestureL.bbrry
240
315
8
SENSOR
G Appendix for experiment: TS-LDDMM representation identifiability
In this experiment, we evaluate the ability of TS-LDDMM to retrieve the parameter vo that encodes the deformation 1 acting on time series graph G by solving the geodesic shooting problem (6) between G and f Parameter identifiability is an important property for subsequent statistical analysis. Results show that TS-LDDMM representations are identifiable weakly identifiable depending on the velocity field kernel Ke specification.
G.1 Settings
This experiment only involves the TS-LDDMM method two different settings:
The velocity field kernel KG well-specified: The velocity field kernel KG is set to
(Co, C1, az)
01,0.1, 100, 1), the varifold loss kernels (kpoa: kair)) are set to (apos,ts pos,ts dii,,t d ) = 2,,1,2,0.6), and the optimizer has 400 steps with
maximum stepsize 7JM of 0.05.
The velocity field kernel KG is missspecified: The velocity field kernel Ke is set with (Cn, C1 TT,1) = (1, 0.1,1), σT,o ranging in (1,5, 5, 10, 50, 100, 200, 300), and ax ranging in (0.1, 10, 100). The varifold loss kernels (kipas kair) set to (Tpos,t. pos,t, Adie. a = 1, 0.6)) and the optimizer has 400 steps with maxi- mum stepsize Mas of 0.05.
provided that the hyperparameters and the reference graph are wisely selected, i.c., the parameter vp generating deformation (v6) of time series graph G can be estimated from the data G, ((6)), by solving the geodesic shooting problem 66
The velocity field kernel Kg is well specified. First, we show the model identifiability when the kernel KG is well specified: the estimated parameter is good approximation of the generating parameter when the generation and the estimation procedure use the same hyperparameters for the RKHS kernel Kg. All the hyperparameter values for generation and estimation given in Appendix G.I
httt:://tmesseressssssscciiooono
19

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15-
releerrce
10-
14020 5,120 4,100-my1000 fa15 5220
05
my.30
0.0
-05
100
-100
100 200
300
Figure 7: Plots of (/oo(a",))) x)} X for different values of a* according to its sampling parameter laz Sa, M2 taking X = G(8o) with : [300] + sin(2mk/300).
Table 2: Values of (lypteala x)) X, ( X) a is sampled according to Gen( 10, 10,50) and no
estimated using KG with varying parameters 0T,1, Ow.
10
50
100
200
300
0.1
2e+0
3e-4
le-5
4e-6
7e-4
4e-3
4e-2
le-4
le-5
4e-6
7e-4
4e-3
100
4e-2
2e-4
le-5
4e-6
7e-4
4e-3
We fix the initial control points as X = (xx = (ki (inn2k/3300)) xe(30)) Given m, E N and Lay So O, randomly generate initial momentums a* (a )e|ool with the following sam- pling. called Gen(m,, ta, s,): For any & [no], a is sampled according to Gaussian normal distribution (00+1, 1a+1). Then, (o<))e[mo) is regularized by a rolling average of size m,s we get á' = (a)o) Finally, we normalize â' to derive a* such that ([(a))))))))) (e = tamp and [[lo])))<((oo] Samp for any & [ng]. denoting by [ag]e, [oğ]s the time and space coordinates of a respectively. Note that the regularizing step (a() (e))ss ino] + ā' is necessary to obtain realistic deformations which take into account the regularity induced by the RKHS V.
Then, using (a X) defined in (4) with initial momentums a" and control points X, apply solve 66 to recover
the induced deformation (m0) by
X and obtain (+o) X. Finally,
////000(aa (a" X) on 50 repetitions. This procedure is performed for any ms: las 8a (10,55 50, 100) x (5, 10, 15, 20) Mean, standard deviation, and maximum of the ARE on all these hyperparameters choices are respectively 0.10, 0.03, 0.17. Therefore, the estimation procedure (6) offers good approximation of the true parameter when the kernel KG is well specified. We observe that the estimation difficult when sa because the time series can be very noisy illustrated in Figure 7 this impacts the Varifold loss which is sensitive to tangents.
estimation a and report the average relative error (ARE) [uo à X) - Uy x) X) h
The velocity field kernel KG is misspecitied. We demonstrate a weak identifiability when the kernel KG is misspecified: we can reconstruct the graph time series' after deformations even the hyperparameters of KG are different during the generation and the estimation. The hyper- parameters of KG during generation are (Cu, C1, Fr (T)
(1, 0.1, 100, and we fix 01.1, Co. C1 = (1, 1, 0.1) 1) for Kg during estimation. We aim to understand the impact of or.1, on the reconstruction since they are encoding the smoothness of the transformation according to time and space.
For any choice of the hyperparameters TT.1, og {1, 10, 50, 100, 200, 3000 x (0.1, 1,00) related to (x)(a"))),
Ke in the estimation, we average ( (vo(a x)) x)),x, XX (6)) X) on 50 repetitions when a* is sampled according to Gen (10, 10, 50) and 0n vo X) denoting by à the result of the minimization (6) We observe Table that the reconstruction almost perfect except in the case when ot.0 = during estimation, while T..0 100 during generation. Compared to OT,0. On has nearly no impact on the
reconstruction. Appendix C we propose guidelines to drive future hyperparameters tuning and further discussions related to ØT,1, Co, C1.
20

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H Appendix for experiment: Robustness to irregular sampling
This experiment is inspired by 4 where the authors perform extensive comparison of Neural Or- dinary Differential Equations (Neural ODEs) methods [31]. We assess the classification performances of several methods under regular sampling (0% missing rate) and three irregular sampling regimes 15 shape-based datasets (7 univariate & 8 multivariate). Methods and training strategy are taken from its associated Githut and described in what follows. We conclude with the results, which show that our method, TS-LDDMM, outperforms all methods for sampling regimes with missing rates; 0%, 30%, and 50%.
H.1 Benchmark methods
In related work, we give an overview of Neurals ODEs methods and their relation with TS-LDDMM.
RNN-based methods: Baseline reccurent neural networks including RNN 38
LSTM 27
and GRU [4].
Attention-based methods: Multi-Time Attention Networks (MTAN) 52 and Multi-
Integration Attention Module (MIAM) [35]. Both handle multivariate time series irregularly sampled with attention mechanisms.
Neural ODEs: ODE-LSTM representations.
34 form of Neural-ODEs used to learn continuous latent
Neural SDEs: Neural SDE 36 and Neural LNSDE 43 have been proposed to model randomness time-series using drift and diffusion terms as an extension of Neural-ODEs.
methods: TS-LDDMM (ours) and LDDMM (23], From shape analysis, both methods learn representations by solving ODEs parametrized with Kernels. While both methods handle multivariate signals irregularly sampled, TS-LDDMM is specifically designed for time series.
H.2 Model architecture
Neural ODEs methods As depicted in 43 any Neural ODEs layer in Appendix H.1 is followed by an MIP with two fully connected layers with ReLU activations. The risk of overfitting and the model regularization are handled with a dropout rate of 10% and an early-stopping mechanism, ceasing the training when the validation loss does not improve for 10 successive epochs.
For each method and dataset, the learning rate, the hidden vector dimensions, and the number of layers are optimized to minimize the CrossEntropy loss validation set using the Ray Python library. The learning rate varies from 10 to 10 using log uniform search, the hidden vector dimension ranges from 16, 32, 64, 128 using grid search, and the number of layers ranges from 1, 2, 3, 4 using grid search. The batch size was selected from 16, 32, 64, 128 according to the size of the dataset. All methods were trained for 100 epochs, and the best method was selected based on the lowest validation loss.
TS-LDDMM and LDDMM Representations learned with TS-LDDMM or LDDMM are fed to a Support Vector Classifier (SVC) from scikit-learn All SVC's hyperparameters are set to default except the regularization term C, which is set through grid search on a validation set with the macro fl-score 13
To learn TS-LDDMM (resp. LDDMM) representations, the velocity field kernel KG is set to (co, C1, TT,0. TT,1, ox) (1, 0.1, 0.33I, na). (resp. (ØT, σ²) (0.337, n2)) where is the average time series length and na the number of dimensions. For both methods and all datasets, the varifold loss kernels (kpas kdtr) are identical and set (opos.t: possts o dir, ddi,,)) (2, n. 2, na). For TS-LDDMM (resp. LDDMM), the optimizer is set with 400 epochs (resp. 400) and maximum learning rate 7/M = 0.1 (resp. A) = 0.01). In all cases, the initial reference graph is selected in the dataset as time series with the median length.
21

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H.3 Protocol
In this experiment, we investigate the robustness to missing samples and the classification performance of TS-LDDMM compared to Neural ODEs 15 datasets described in Appendix [F2 For fairness between methods of different architectures, the evaluation protocol on each dataset and method is as follows:
1. Spilt the dataset in train 75%, validation 15%, and test 15%.
2. Tune hyperparameters with train and validation sets and missing rate of 0%.
3. For each missing rate in 096,30%,50%%,705
Remove samples time series in the train and test sets according the missing rate and the drop procedure described in [31.
Train the model on the train set
Evaluate the macro fl -c-rcore on the test set
H.4 Results
In this experiment, we investigate the robstuness to missing samples and the classification performance of TS-LDDMM representations. We compare TS-LDDMM with LDDMM and 8 neural ODEs networks. Performances are evaluated in terms average macro fl fl-score and rank on four different regimes of missing rate 0%,30%,50%, and 70%. Results are aggregated in Table 3
three out of four regimes (0%,30%, 50%) TS-LDDMM classifier is the best performer in terms of fl-score and rank. For missing rates of 0% and 30%, the score increases by 10% compared to the second-best performer, LDDMM. However, LDDMM is not the second-best performer in rank (Neural LNSDE), showing its sensitivity parameterization, unlike TS-LDDMM, which remains consistent. Performances of Neural LNSDE remain constant with the increase the missing rate as observed in 43 and it becomes the best performer for missing rate 70%. The decrease in TS-LDDMM performances with the increasing missing rate is due to the varifold loss, which poorly approximates the time series shape. Other losses might be more relevant for high missing rates.
Overall, TS-LDDMM a relevant and consistent chape-based representation for irregularly sampled multivariate time series for missing rates up to 50%.
Table 3: Comparison of average macro fl fl-score and rank the sample dropping rate increases. First
second best performers. TS DS-LDDMM is the best performer on three out of four regimes.
Regular -ccore
30 % dropped
50 % dropped
70 dropped
Methods
Rank
FI-score
Rank
FI-score
Rank
FI-score
Rank
RNN (1999)
0.64 + 0.21
6.2
0.53 = 0.23
6.6
0.48 + 0.21
7.2
0.44 + 0.21
6.07
LSTM (1997)
0.61 0.29
6.0
0.57
0.29
6.27
0.53 0.25
6.07
0.51 + 0.29
5.27
GRU (2014)
0.266
4.2
0.68
4.27
0.66 0.28
0.28
3.67
MTAN (2021)
0.22
7.13
0.58 = 0.28
5.8
0.54:0.29
5.33
0.51 0.28
5.0
MIAM (2022)
0.33
6.93
0.42 0.33
8.27
0.47
6.93
0.35 +.31
7.6
ODE-L (2020)
0.63
6.0
0.57 +.25
6.53
0.51 + 0.24
7.27
0.45+0.22
6.73
Neural SDE (2019)
0.28
7.67
0.27
7.47
0.45
7.13
0.225
6.0
Neural LNSDE (2024)
0.27
3,87
0.68 0.68
4.0
0.67
3.53
0.22
2.47
LDDMM (2008)
4.53
0.21
4.2
0.57 0.25
0.4
7.13
TS-LDDMM (ours)
0.83 0.18
2.93
0.8 0.18
2.07
0.7 0.26
3.33
0.51
5.67
I Appendix for experiment: Classification benchmark on regularly sampled datasets
In this section, we compare the classification performances of TS-LDDMM with other methods from shape analysis on 15 shape-based datasets of time series regularly sampled. TS-LDDMM outperforms other methods on 12 out of 15, highlighting its relevance for shape analysis when dealing with time series.
22

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1.1 Benchmark methods
logistic regression the tangent space of the Frechet mean with Square Root Velocity (SRV representation). We also include Shape-FPCA [60] that encodes both the time series and its time parameterization.
SRV-based method: include TCLR 26
LDDMM-Based TS-LDDMM (ours) and LDDMM 23 Both methods learn representa- tions by solving ODEs parametrized with Kernels. While both methods handle multivariate signals, TS-LDDMM is specifically designed for time series.
1.2
Model settings
TCLR & Shape-FPCA Shape-FiPCA available in the Python library FDASRSF Once the shape- FPCA representations are learned, they are fed to an SVC from scikit-leart FDASRSF provides SRV representation methods that we combined with a logistic regression from scikit- implement TCLR. For both methods, the number of steps to learn the Frechet mean is set to 50, and the regularization hyperparameter C is set through grid search on validation set with the macro fl-score. Other parameters are set to default.
to
TS-LDDMM & LDDMM
Representations learned with TS-LDDMM or LDDMM fed an SVC from scikit-leese All SVC's hyperparameters are set to default except the regularization term C, which through grid search on validation with the macro fl-score.
To learn TS-LDDMM (resp. LDDMM) representations, the velocity field kernel KG is set to (Co; C1, dr,0; 0T,,, 0x) (1, 0.1, 0.33I, na), (resp. (ØT, az) (0.337, na)) where I is the average time series length and ny the number of dimensions. For both methods and all datasets, the varifold loss kernels (kipss ) are identical and set posos:: 0 pos dir,r, ddr,,)) (2, na, 2, na). For TS-LDDMM (resp. LDDMM), the optimizer is set with 400 epochs (resp. 400) and maximum learning rate 7/M 0.1 (resp. 7/M 0.01). In all cases, the initial reference graph is selected in the dataset as time series with the median length.
I.3 Protocol
For each dataset and method, the evaluation protocol is simple validatio test with hyperpa- rameter tuning:
1. Split The dataset in train 75%, validation 15%, and test 15%.
2. Training and hyperparameters tuning with train and validation sets
3. Evaluate the macro fl-score on the test set
1.4 Results
In this experiment, we investigate the classification performances of several methods from shape analysis on 15 shape-based time series datasets (7 univariate and 8 multivariate). The performances are evaluated in terms of macro fl-score. Results are aggregated in Table 4
The TS- TTS-LDDM-bbase classifier outperforms other methods 12 out of 15 datasets. TCLR is the second-ber best performer on univariate datasets; however, its current implementation with FDASRSF does not extend the multivariate case, which limits usage. LDDMM performances are lower than TCLR, and Shape-FPCA the worst performer.
Overall, TS-LDDMM representations are well suited for shape- based time series classification, and its extension to multivariate irregularly sampled time series makes a relevant option for time series shape analysis.
J Appendix for the experiment: Noise sensitivity
This experiment evaluates the influence noise on the learning of the reference sequence for TS-LDDMM and SRVF Kacher-mean, subroutine of Shape-FPCA 66).
23

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Table 4: Fi-score comparison between methods from shape analysis on 15 datasets. First and second
best performers.
Dataset
Shape-FPCA (2024)
TCLR (2024)
LDDDM (2008)
TS-LDDMM (ours)
AnowHead
0.18
0.75
0.84
0.91
0.82
ECG200
0.40
0.67
0.81
0.79
Univariate
FacesUCR
0.08
0.73
0.69
0.86
GunPoint
0,93
0.97
0.83
1,00
PhulangesOuttinessConett
0.39
0.63
0.53
0.52
Trace
0.55
1.00
0.46
1,00
Asticulary WordReccogniion
0.98
1,00
Cricket
0.77
0.93
ERing
0.95
0.98
Multivariate
Handwriting
0.22
0.44
Libras
0.56
0.60
NATOPS
0.82
RacketSports
0.83
0.79
UWaveGestureL.bvaary
0.72
0.81
J.1 Protocol
The dataset includes 100 sine waves with randomly generated time parametrization by following the procedure described in Appendix with
with Gen(50, 1, 0) and uniformly resampled. The dataset has been altered under four scenarios with an additive Gaussian noise centered and with standard deviation o, {0, 0.05, 0.1, 0.2}. The referent sequence is learned for each scenario, and the L2- error between the exact and the learned barycenter is computed.
J.2 Method settings
For SRVF, the number of steps to learn the Kacher-mean set to 20. Regarding TS-LDDMM,
the velocity field kernel Kg is set to (Cu, C1, 02, (1, 0.1, 65, 1, 1), and the varifold loss kernels (Kpos: kair)) are set to pooss, pos,t, ddir, dir..=) = (5, 1, For TS-LDDMM, the optimizer is set with 400 epochs and maximum learning rate 7Jм 0.1. In all cases, the initial reference graph is selected in the dataset as time series with the median length.
J.3 Results
Figure 8 illustrates the results. Noise level affects the learning reference graph in both cases, as depicted by the increasing error and the illustrations. However, the overall sine wave shape is better preserved by TS-LDDMM compared to SRVF Kacher-mean, for which the sine wave amplitude decreases as the noise increases. addition, for TS-LDDMM, the regularity the reference graph can be controlled by penalizing the norm of the velocity fields in the loss function. Further work on penalization will be conducted to handle noisy data better.
K
Appendix for experiment: Analysis of respiratory behavior in mice
K.1
Settings
This experiment involves TS-LDDMM. LDDMM 23 and Shape-FPCA (60) methods. Two scenarios are investigated: before drug exposure and before/after drug exposure. All methods are investigated on both scenarios.
TS-LDDMM parameters.
Before exposure: The velocity field kernel Ka is set to (Io TT,0, OT,1, Ga.)
0.1, 150, 2). The varifold loss is the sum of three varifolds capture shapes variations at different scales with parameters: (Varifold 1, Varifold 2, Varifold 3): ((5, 1), (2, 1, 0.6), (1, 0,6, and the mapper (apos,ts poa.ts ddir, ddn..)). The optimizer has 800 steps with maximum stepsize aa of 0.3.
Before/after exposure: The velocity field kernel K set (Cn, C1, or aTr.O.TT..... OT,1, oz) ) (1, 220, 1,2). The varifold loss is the sum of four varifolds capture shapes

---

(a) TS-LDDMM barycenter
emonn: 2.22
0.05, emrn 2.28
0.1, error: 2.42
0.2. error: 2.6
Icamed
10
05-
-15
120
200
100
100
200
100
100
200
300
100
200
300
(b) Shape-FPCA barycenter
D, error: 0.51
0.05, error: 2
0 1. error: 2 76
0.2. error: 2.95
leameed
1.0
05
-10
115
120
200 300 120
200
300
120
200
300
120
200
300
Figure 8: Illustration of the learned barycenter (red) compared to the exact barycenter (green) for both TS-LDDMM (a) and Shape-FPCA (b). The computation has been done for different level of noise (5 o.) with o. E {0, 0.05, 0.2).
variations at different scales with parameters: (Varifold 1, Varifold 2, Variflld 3,
Varifold 4): ((30, 2, 30, 1), (5, 2, 5, 1), (2, 1, 2, 0.6), (1, 0.1, 0.)) and the mapper (a pon,t pos,t, Odir,t n. The optimizer has 800 steps with a maximum stepsize yoar of 0.3.
LDDMM parameters. Note that varifold losses are unchanged between TS-LDDMM and LD- DMM. Compared TS-LDDMM, the convergence LDDMM is more sensitive the maximum stepsize 7m, which must remain small for LDDMM to guarantee the convergence.
Before exposure: The velocity field kernel KG is anysotropic Gaussian kernel with parameters dr = 150 for the time dimension and a₂ 2 for space dimensions. The varifold loss is the sum of three varifolds capture shapes variations at different scales with parameters: (Varifold I, Varifold 2, Varifold 3): ((5, 2, 5, 1), (2, 1, 2, 0.6), (1, 0.6, 1, 0.6) and the mapper (a pooott pos,t Ad a. The optimizer has 800 steps with a maximum stepsize 7/M of 0.01.
Before/after exposure: The velocity field kernel KG is an anysotropic Gaus- sian kernel with parameters or 220 for the time dimension and = 2 for space dimensions. The varifold loss the sum of four varifolds to capture shapes variations at different scales with parameters: (Varifold 1, Varifold 2, Varifold 3, Varifold 4): ((30, 2, 30, 1), (5, 2, 1), (2, 1, 2, 0.6), (1, 0.1, 0.1)) and the mapper (apos,t: pos,t, Odir.t, .... The optimizer has 800 steps with maximum stepsize nas of 0.01.
Shape-FPCA parameters. For both scenarios, respiratory cycles are linearly interpolated and resampled to 200 points, and the length the original time interval kept. The computation of the Kacher-mean is done in maximum of 50 iterations, and srv representations of the realigned time series and time parametrization are concatenated with cycle durations. When concatenating these vectors, the choice of amplitude factors is made minimize the reconstruction error from the
25

---

(a) PC1 densities
(b) TS-LDDMM PC1 shooting
(c) Scatter PC1
PC3
-1.54
+1.50M
colo
Nasal
Thoratic
40
-C0 -500 380
500 1800
1000
500
1000
time (ms)
Figure 9: Analysis of the first Principal Component (PC1) related to mice ventilation before and after exposure with TS-LDDMM representations. (a) displays PC densities per mice genotype, (b) illustrates deformations of the reference respiratory cycle co along PCI, and (c) displays all respiratory cycles with respect to time in PC1 and PC3 coordinates
(a) PC1 densities
Shape-PAAA PC1 shooting
Scatter PC1 PC3
-1.5
+1. +1.50mc
Kosal
PC1
Figure 10: Analysis of the first Principal Component (PC1) related to mice ventilation before and after exposure with shape-FPCA representations. (a) displays PC densities per mice genotype, (b) illustrates deformations of the reference respiratory cycle Co along PCI, and (c) displays all respiratory cycles with respect to time PCI and PC2 coordinates
principal components analysis by following the procedure described in [60]. Shape-FPCA does not handle multivariate data, and we only kept the nasal airflow for this method.
K.2 Addiotinal results
Figure 9 presents results for TS-LDDMM and Figure 10 presents results for Shape-FPCA, The main components look similar. However, a subtle difference, yet important, can be noticed. With Shape-FPCA, the deformation tends to be uniform time scaling, whereas, with TS-LDDMM, the time dilatation mainly occurs during the pause between inspiration and expiration. Qualitatively, this last deformation fits the physiological phenomenon: Mice's muscles cannot relax after exposure the irritant molecule, leading to pauses between inspiration and expiration [41]. Qualitatively, contrary to Shape-FPCA, which manages to represent the main phenomena in the data, the deformations of TS-LDDMM capture subtle physiological behaviors essential for understanding the phenomenon at hand.
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Reviews:
[{"review_id": "6EWmhlfyWR", "reviewer": "Reviewer_xHE6", "summary": "This paper introduces an unsupervised method based on LDDMM to highlight inter-sample shape variability in time series, with the model being able to work on irregular multivariate time series. Extensive studies are conducted to theoretically and experimentally justify the authors' choices. The interpretability and usefulness of the proposed method are demonstrated through experiments on clinical datasets.", "questions": "- How long does a full training process take on the mentioned hardware?", "limitations": "The authors address both the technical limitations (specific libraries, trouble handling high-dimensional time series...) and the societal impacts (positive and negative) : energy consumption and clinical misuse.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- Originality : The proposed model innovates by bringing LDDMM to the time series analysis field, taking advantage of the research done in the representation learning domain (with, for example, their distance between oriented manifolds loss function).\n- Quality : The claims made in the paper are all supported with thorough proofs or evidence (with varied datasets). Efforts were also focused on interpretability, which is paramount in clinical settings.\n- Clarity : Thanks to the appendix containing the bulk of the technical details, the main body of the paper is easy to follow and well-explained.\n- Significance : Inter-individual variability is a very useful tool for clinicians, as referring to standard practice and examples is common in medical settings - hence, this model could greatly help practitioners.", "weaknesses": "- It seems that the TS-LDDMM model is computationally expensive, which may be a tall hurdle to clinical usability, but this point is merely mentioned/addressed by the authors. \n- For a work intended for adoption by medical professionals, this paper is technical and relies on mathematical concepts and equations that might hinder its reach as is.\n- As far as I am aware, code was not made available for this review.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and valuable comments. Here are the responses to the weaknesses and questions you raised:\n\n- As depicted in Table 1 of the PDF and thanks to its minimal architecture, the training time of TS-LDDMM is below that of large neural networks and beyond that of classic statistical tools. Note that the TS-LDDMM computation time and memory usage can be further optimized with the dedicated package named KeOps. \n     \n- We deeply apologize for the codes; we thought the deadline was one day after the paper. We send the link to an anonymized Github to the Area Chair in the hope that it will be made available to you soon.    \n    \n- Though the method is technical, its outcomes are easy for medical professionals to interpret. One possibility consists of visually representing a PCA's principal components as shape deformation (see Figures 3 \\& 5), enabling the physicians to interpret the principal sources of variability within a population. For example, in the mice experiment and from the deformations shown in Figure 5 we can see that the primary source of variability corresponds to the pause duration between inspiration and expiration. We are also working on creating an online demo with a user-friendly interface to enhance the impact of this paper on medical professionals.\n\nWe greatly appreciate your interest in the application and usability of the method. We hope to make this work available soon. Thank you again.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sSBukE0yck", "reviewer": "Reviewer_HVwJ", "summary": "This paper extends the large deformation diffeomorphic metric mapping (LDDMM) framework to the case of univariate and multivariate functional data (time series). In particular, the focus is on understanding sample variation in the shape of irregularly sampled time series. The proposed framework leverages a graph representation, which is subsequently deformed via a space-time diffeomorphism. The main methodological contribution of the paper appears to be an appropriate parameterization for the tangent space of the diffeomorphism group, which results in time series deformations that preserve the graph structure. The main computational task is to learn a reference graph and a set of shooting vectors (vector fields), which subsequently define the deformation of the reference graph to each time series in the sample via a flow on the diffeomorphism group. For this purpose, the oriented varifold representation is used. The main experiment presented in the manuscript is aimed at showcasing the interpretability of the proposed framework. The dataset under consideration is composed of respiratory cycles for mice (7 controls and 7 deficient in a particular enzyme). The appendices describe proofs a theorem and a lemma, various settings (including hyperparameter and experimental/computational) for the presented method. Additionally, the authors provide two sets of classification experiments where they compare their approach to competitors in the literature.", "questions": "I've included a number of questions that require clarification in the Weaknesses above. The authors should carefully justify the novel contributions of the presented method. As written, the approach appears to be fairly straightforward extension of the LDDMM framework. Elaboration of various claims made in the real data analysis section is needed. Some of the claims do not appear well-supported by the presented results. Finally, the paper needs substantial editing to correct existing typos and to streamline notation throughout.", "limitations": "The authors have adequately addressed limitations of their work.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. The paper appears to be technically sound. The authors encode the shape of a time series as its graph. Given a sample of such graphs, the main task is to learn a reference graph as well as a set of space-time diffeomorphisms that deform the reference graph to each graph in the sample. The entire framework leverages the LDDMM paradigm, which has proven extremely useful in the context of shape analysis of point clouds, landmarks, curves and surfaces.\n2. The authors carefully consider an appropriate structure on the (tangent space of the) group of diffeomorphisms to ensure that learned deformations preserve the time series graph structure. The framework is supported by a representation theorem that ensures existence of a space-time transformation that is able to warp one time series to another.\n3. The proposed method is able to handle time series that are irregularly sampled and with different numbers of observations of in time.\n4. The appendices support the authors' claims that the proposed framework performs better than competitors on two classification tasks, one that involves irregularly sampled time series and one that involves regularly sampled time series.\n5. The presented application to understanding variation in respiration cycles of mice is interesting and sufficient details to understand the main motivation behind the analysis are presented.", "weaknesses": "1. In view of existing literature on shape analysis and functional data analysis, the presented framework essentially alters the popular LDDMM framework for the purposes of analyzing variation in univariate and multivariate time series. As such, the novelty of the proposed methods is not very high. The main contribution appears to be an appropriate parameterization of the diffeomorphism group tangent space, which allows the graph structure of time series to be preserved during deformation. This is done through a restriction of general diffeomorphisms of $\\mathbb{R}^2$ via a kernel that splits the time and space deformations (nonetheless, the space deformation does depend on the time deformation).\n2. I'm not sure I would consider sensitivity analyses and experiments as contributions. I appreciate the work the authors did to compare their approach to some of the existing approaches in the literature via classification tasks. I additionally like the real data example presented in the main manuscript. However, I'm not quite sure this example shows the benefits of the proposed approach or its interpretability. Figure 3 does perhaps show better principal directions of variation for the proposed TS-LDDMM vs standard LDDMM. I don't quite understand the following sentence though: \"Compared to wt mice, the distribution of colq mice feature along the PC1 axis has a heavy left tail, and the associated deformation (-2 σPC) shows an inspiration with two peaks.\" Why is the left tail in the density for PC1 score significant here? Also, I don't quite see inspiration with two peaks at -2 σPC or why this is related to the two types of mice since I assume PCA was carried out using the pooled data. If I understand correctly, based on the densities of the PC scores, it appears that wt and colq mice differ in terms of the time warping of their respiration cycles since the two PC1 densities are quite different. At the same time, it appears that PC2 scores are fairly similar across the two mouse groups. Figure 4 is also not explained sufficiently. Are the presented respiration cycles examples from the data or reference cycles computed using the proposed method? In the PC1 vs. PC2 plot of the PC scores for the 14 learned reference time series graphs, each corresponding to one mouse, three of the seven colq mice appear more similar to wt mice. Why is this the case? This is not addressed in the description of the figure. Also, it is unclear to me how PCA was carried out in this case, i.e., was PCA carried out with respect to the overall reference graph? In Figure 5, the change in PC1 after exposure to the irritant is interesting and the exposure does appear to affect wt mice more than colq mice.\n4. Related to the real data analysis, I'm not sure this example fully showcases the usefulness of the presented methodology. Based on my understanding of the dataset, it appears that the respiration cycles are densely (and prehaps regularly) sampled. Each cycle starts at t=0 and the cycles progress for different amounts of time. As such one could alternatively perform the following analysis using existing methods: (1) resample all cycles to the same number of time points (using some form of interpolation), (2) standardize the domain of each cycle to some interval (perhaps [0,1]) and store the time dilation factor for each cycle, (3) apply the SRV framework, for example, to separate phase (x-axis or time) and amplitude (y-axis) variation, (4) apply separate PCA to time variation and amplitude variation to understand overall variation in the data. Could authors elaborate on the benefits of their approach over the one described in points (1)-(4)? I understand that redoing the analysis in the way I described is not realistic during the rebuttal period.\n5. Overall, the paper is not written well with many typos and grammatical error throughout. Some typos make the presented material difficult to understand. Notation throughout the paper is very dense and I'm not sure that some of the described concepts are necessary. In particular, I fee that the general description of LDDMM could be condensed quite a bit and most concepts related to oriented varifolds could be relegated to one of the appendices that already exists. These concepts are not described very clearly in the main manuscript anyway with the introduced additional notation.\n6. I appreciate the authors' efforts in assessing sensitivity of their method to different hyperparameter values. How does noise in the observed time series affect the learned reference graph and observation specific deformations? Is the approach fairly robust to noise or does one have to be quite careful about selecting the necessary hyperparameters? The presented sensitivity analysis only consider a sine function without noise.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed comments and your time, which will help us improve the quality and clarity of this paper. We appreciate your accurate understanding of the paper. We are pleased to say that we have been able to carry out the experiment you suggested during the rebuttal period.\n\nHere are our answers to your concerns:\n\n- First, we sincerely apologize for the typos and are thankful for your advice. The article has been reviewed and corrected.\n\n- **Regarding the novelty.**\n    Representing multivariate and irregularly sampled time series of different lengths for analyzing inter-individual variability is a complex problem.\n    \n    Our proposed solution might seem simple for someone knowing LDDMM, but the problem we tackled was never appropriately addressed with LDDMM. For instance, in Figure 1, we show that some deformations in the set of deformations considered by a classical LDDMM do not preserve the time series structure. Additionally, Figure 3 shows that the deformations learned on time series with LDDMM do not carry physiological meaning in the Mice experiment. \n    \n    The given representation of the kernel has been carefully designed to integrate space and time while keeping time independent of space. Initially, we considered separating space and time, as suggested in Weakness comment 3. However, post-hoc analysis of this representation is not straightforward. The separated space and time representations correlate, and understanding this correlation is crucial to interpreting the data. Consequently, you must concatenate the space and time representations, but there is no single way to do this because the two representations are not commensurable. Therefore, we decided to have a representation that includes both space and time by design.\n\n    Moreover, we compare favorably to the state-of-the-art in both deep learning and Functional Data Analysis (FDA) literature addressing the same problem. In summary, we have capitalized on a gap in the LDDMM literature and shared our results with other communities working on similar topics.\n\n\n- **The experiment you suggested.** In the author rebuttal, you will find the experiment you suggested in Weakness comment 3. In brief, TS-LDDMM compares favorably to Shape-FPCA in this example, even though Shape-FPCA already provides impressive results. Moreover, Shape-FPCA requires good interpolation of time series, which is not feasible in scenarios with few or missing samples. Additionally, Shape-FPCA has a very low classification score compared to TS-LDDMM (0.38 for Shape-FPCA against 0.83 for TS-LDDMM). This low performance stems not from the package (which we verified with the owner) but from the method itself. This observation opens an avenue for future improvements to the Shape-FPCA method.\n    \n- **Mice breathing behavior experiment.** To clarify the computation of the PCA, we performed a Kernel-PCA in the RKHS encoding the velocity fields. Each respiratory cycle is represented by its initial velocity field, which is the velocity field that encodes the geodesic from the reference graph to the respiratory cycle at time t=0. \n    \n    When describing Figure 3, we stated: \"Compared to wt mice, the distribution of colq mice feature along the PC1 axis has a heavy left tail, and the associated deformation (-2$\\sigma$) shows an inspiration with two peaks\". Indeed, inspirations in two peaks are representative of colq mice as they suffer from a motor control impairment [1]. Figure 4.a shows an extreme real example of colq respiratory cycle; it is the furthest on the left of PC1. The principal component preserves physiological meaning, and it can be used to differentiate colq mice from wt mice. \n    \n    Figure 4.b shows the coordinates of individual learned reference graphs in the PCA coordinates associated with the overall learned reference graph. Figure 4.c shows an example of an individual learned reference graph. As you mentioned, some colq mice are close to wt mice. Indeed, colq mice suffer from a genetic mutation that affects them throughout their growth. The impact on motor control is variable, and some colq mice may appear closer to WT mice. \n    \n    As you mentioned: \"In Figure 5, the change in PC1 after exposure to the irritant is interesting, and the exposure does appear to affect wt mice more than colq mice\". This remark is highly relevant. The irritant molecule prohibits the action of a neurotransmitter involved in motor control. Due to their genetic mutation, colq mice have a deficiency in this neurotransmitter, which leads to motor control impairments. When exposed to the irritant molecule, colq mice are already partially accustomed to the neurotransmitter deficiency, whereas WT mice, the control group, highly suffer from the prohibition.\n\n    In light of your remarks, we will improve the soundness of this part. \n    \n\n- **Noise sensitivity.** TS-LDDMM focuses on fine-shape registration to encompass small sources of variability. It, therefore, has some sensitivity to noise with pros and cons. As depicted in Appendix G, TS-LDDMM performs good registrations for numerous kernel settings, even for noisy data, meaning that we register the exact time-series shape. On the downside, like other shape registration methods, the learning of the reference graph is affected by noise, as illustrated in the second experiment in the attached pdf. However, the regularity of the reference graph can be controlled by penalizing the velocity fields' norm in the loss. Further work on penalization will be conducted to leverage noisy data.\n\n\nWe thank you for your investment in reviewing our paper and the suggested experiments that we conducted, allowing us to demonstrate our method's significance further. \n\n[1] Aurélie Nervo, André-Guilhem Calas, Florian Nachon, and Eric Krejci. Respiratory failure triggered by cholinesterase inhibitors may involve activation of a reflex sensory pathway by acetylcholine spillover. Toxicology, 424:152232, 2019.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ODL6or34L0", "reviewer": "Reviewer_LvUA", "summary": "The paper presents and extension of the LDDMM shape analysis framework to time series data. The concepts of using deformations (diffeomorphisms) to transform data is extended to the graph of time series data. The authors define a kernel that extends to the augmented space of time+data by treating the time and data components individually, thus decoupling the regularization and ensuring the graph stays a graph. Techniques from the LDDMM world, sparse representations and varifolds, are used in the setup. The method is tested on synthetic and real-world data.", "questions": "- how does the method compare to methods like the metamorphosis framework where there is also to variable (space and image) similarly to the presented case (time and space) With the specific kernel, I could imagine that the underlying structure is very similar", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- well-written and clearly presented paper\n- interesting idea of deforming time and space together with the LDDMM framework\n- the non-linear flows generated by LDDMM may have real benefits compared to the mostly linear time-warping models used in e.g. functional data analysis\n- the method is evaluated on different datasets", "weaknesses": "- when reading the paper, I found the method to be a somewhat complicated setup for solving the problem, i.e. why apply the full LDDMM setup to graphs when in most applications the time and space warps could be treated separately at each step (each time-point of the flow). I believe the point is that this is handled in section 4 by the kernel. Perhaps the presentation would benefit if the method was presented with the new kernel from the start so that it was clear that the time-space separation happens\n- still, the use of LDDMM, sparse representations and varifolds is a quite complicated setup for solving a time-series analysis problem\n- there is extensive literature in the functional data analysis community on time-space warping and separation. I am not aware of methods using LDDMM for both time and space as the current paper does, but I think the background section should include relevant FDA papers", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and valuable comments. Here are the responses to the weaknesses and questions you raised:\n- **Regarding the Functional Data Analysis (FDA) literature.** We have compared TS-LDMMM to Shape-FPCA [1] in Figure 1 of the attached PDF to the common rebuttal and on the classification task in Appendix J.1 (Table 4). In both cases, TS-LDDMM compared favorably. Shape-FPCA employs the Square Root Velocity Field (SRVF) representation to separate space and time. However, this method is designed for continuous objects and only applies to time series of the same length. This issue can be addressed through interpolation, but this approach is not always reliable in sparse and irregular sampling scenarios. Most FDA papers [2,3,4] that we are aware of cope with the same issue using interpolation or basis function expansion. In a nutshell, FDA methods have to deal with continuous objects, while LDDMM algorithms can keep a discrete-to-discrete analysis. \nThanks to your comments, we plan to extend the discussion on FDA approaches in our background section. \n\n- **Time and space transformation analysis.** This topic is indeed addressed in Remark 3 of Section 4.1. Due to the TS-LDDMM's kernel parametrization, the time transformation is independent of the space transformation, but the reverse is not true. Our representation encodes both space and time by design which is a competitive advantage compared to method separating the time and space representation.\n    Indeed, post-hoc analysis of separated time and space representation are not straightforward. The separated space and time representations correlate, and understanding this correlation is crucial to interpreting the data. Consequently, you must concatenate the space and time representations, but there is no single way to do this because the two representations are not commensurable. \n    This fact might explain why TS-LDDMM compared favorably to Shape-FPCA, as depicted in Figure 1 of the PDF in attachment to the common rebuttal.\n    \n- **Difference with the metamorphosis framework introduced by 3DMM [5].** The primary difference lies in the pre-processing requirements. The 3DMM framework requires that each mesh be re-parametrized into a consistent form where the number of vertices, triangulation, and the anatomical meaning of each vertex are consistent across all meshes (as stated in the introduction of [6]). In our context, we do not need such pre-processing; the time series graph can have different sizes. However, applying TS-LDDMM to videos to analyze the inter-variability of facial movement is a promising idea.\n    \n- **Varifold loss complexity.** We acknowledge that the Varifold loss might seem quite complicated for solving a time-series analysis problem. However, in our framework, the loss must take as input two sets of possibly different sizes, which is not a common property for losses. During the design of TS-LDDMM, we also tried using the Maximum Mean Discrepancy (MMD), which is a more straightforward loss, but the performance was lower. \n    \n\nWe greatly appreciate your interest in connecting our paper to the broader literature. We believe these connections will help others situate our work within ongoing research. Thank you once again.\n\n\n[1] Yuexuan Wu, Chao Huang, Anuj Srivastava, Shape-based functional data analysis, 2023\n\n[2] John Warmenhoven , Norma Bargary , Dominik Liebl , Andrew Harrison , Mark A. Robinson , Edward Gunning , Giles Hooker PCA of waveforms and functional PCA: A primer for biomechanics, 2020\n\n[3] Han Lin Shang, A survey of functional principal component analysis, 2013\n\n[4] Yu, Q., Lu, X., Marron, J. S. Principal nested spheres for time-warped functional data analysis, 2017\n\n[5] Volker Blanz and Thomas Vetter, Face Recognition Based on Fitting a 3D Morphable Model, 1999\n\n[6] A 3D Morphable Model learnt from 10,000 face, 2016", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2rcQl37w8J", "reviewer": "Reviewer_xHtc", "summary": "This paper proposes a LDDMM method for time series data (TS-LDDMM), by representing time series as deformations of a reference time series. The TS-LDDMM can handle irregularly sampled multivariate time series of variable lengths, and provide shape-based representations of temporal data. They further show the advantages of the model using simulation and real-world examples motivated by biomedical applications, according to robustness to missing samples and classification performance.", "questions": "1. The base graph and diffeomorphism learned should be highly dependent on the model parametrization (otherwise they can be anything)? Maybe I miss some parts, but would you mind justifying the choice of your model?\n2. The results looks similar to naïve factor analysis models for time series (e.g. GP factor analysis model, or https://arxiv.org/abs/2307.02781), which can also handle irregular observations and downstream tasks easily. Would you mind providing some intuitions on benefits on your more complicated model?", "limitations": "The method may not be feasible for (super) high dimensional time series data. May consider do modeling in latent space for future research.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method can handle multivariate time series irregularly sampled and with variable sizes.\n2. The way adapting LDDMM to time series may be helpful for applications of other methods for time series.", "weaknesses": "1. The comparisons of methods mainly focus on robustness of missing and classification. It would be better to compare the “shape” detected for different methods, to show the interpretability advantage of TS-LDDMM.\n2. typos: line 58 (??) & line 106 (author?)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and valuable comments. Here are the responses to the weaknesses and questions you raised:\n\n- **Typos.** You have pointed out our typos accurately. We acknowledge these errors and will make the necessary corrections.\n\n- **Comparisons to other methods for the interpretability.** In the attached PDF of the common author rebuttal, you will find new figures comparing TS-LDDMM with Shape-FPCA [1] on the mouse dataset, as you requested, to assess the interpretability of these methods.\n    \n    The take-home message is the following: while Shape-FPCA managed to represent the main phenomena in the data, it missed the subtle behavior of the respiratory cycle after drug injection, which is captured by TS-LDDMM.\n\n- **impact of the model.** The goal is to encode time series by using the parametrization $\\mathbf{\\alpha}$ of the diffeomorphisms. In the LDDMM framework, the parametrization of the diffeomorphisms is highly dependent on the chosen RKHS's kernel. The kernel choice we give with TS-LDDMM offers a better representation $\\mathbf{\\alpha}$ compared to the classic LDDMM kernel, as depicted in Figures 2 and 3 in the paper. Remark that in both cases, TS-LDDMM and LDDMM, we can recover any time series graph (as you said) because of the great flexibility of LDDMM for learning diffeomorphisms, but the final representation $\\mathbf{\\alpha}$ is not the same depending on the chosen kernel. Although kernel hyperparameters inject prior information into the reference graph, the main source of dependency comes from the dataset.\n\n- **Factor Analysis.** Our method distinguishes of Factory analysis on two points.\n    First, TS-LDDMM's goal is to find a vector representation $\\alpha \\in \\mathbb{R}^d$ encoding an irregularly sampled time series. The dimension $d$ is not dependent on the size of the represented time series but depends on the size of the time series graph of reference $\\mathsf{G}_0$. It is particularly convenient to apply statistical methods to vectors of fixed dimension, such as PCA. In your suggested paper [2], the latent representation is related to each observation in such a way that if the dimension of the representation in $m$ and the size of the time series at hand is $T$, the dimension of the representation of the time series is $mT$. In the suggested paper, their goal is to perform a factor analysis on the gene observations *through time*, while in our case, we carried out a PCA/factor analysis on the vectors representing the whole time series (not the latent variable related to each observation). Therefore, the two problems are different: We apply a factor analysis (PCA) to the sequence of observations, and they apply a factor analysis to the observations during a sequence.\n\nWe greatly appreciate your interest in interpretability and modeling, which will help others be more receptive to our paper. Thank you once again.\n\n[1] Yuexuan Wu, Chao Huang, Anuj Srivastava, Shape-based functional data analysis, 2023\n\n[2] Cai, Jiachen, et al. \"Dynamic Factor Analysis with Dependent Gaussian Processes for High-Dimensional Gene Expression Trajectories.\" arXiv preprint arXiv:2307.02781 (2023).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Thank you for taking the time to review our manuscript. We are grateful for your constructive criticism and the effort you put into evaluating our work. Your careful analysis and suggestions have significantly enhanced the quality of our research.

We carried out all the experiments that you requested. Notably, most of you asked for additional experiments to compare the interpretability of TS-LDDMM with other methods. You will find our new experiments in the attached PDF: We have compared TS-LDDMM to Shape-FPCA [1] on the mouse dataset (Figure 1), we have investigated the sensitivity to noise of TS-LDDMM compared to Shape-FPCA (Figure 2), and we have performed a training time analysis (Table 1).

In what follows, we summarize the protocol and results of the additional experiments. 

1. We have compared TS-LDDMM with Shape-FPCA, a state-of-the-art method in Functional Data Analysis, on the mouse dataset.
    
    **Protocol.** As suggested, we carried out the following experiment: (1) we interpolate breathing cycles to an even number of samples and store the cycles' duration. (2) We aligned all cycles using the SRVF framework [1]. (3) We scale cycles' parametrization to their duration and perform a joint PCA of the parametrization and the aligned signals in their SRVF representation. We perform the analysis on instances after exposure to irritant molecules. We use the package fdasrsf, which only supports univariate signals. The results are presented in Figure 1 of the attached PDF.
    
    **Results.** The main components look similar. However, a subtle difference, yet important, can be noticed. With Shape-FPCA, the deformation tends to be a uniform time scaling, whereas, with TS-LDDMM, the time dilatation mainly occurs during the pause between inspiration and expiration. Qualitatively, this last deformation fits the physiological phenomenon: Mice's muscles cannot relax after exposure to the irritant molecule, leading to pauses between inspiration and expiration [2]. 
    
    The take-home message is the following: Qualitatively, contrary to Shape-FPCA, which manages to represent the main phenomena in the data, the deformations of TS-LDDMM capture subtle physiological behaviors essential to the understanding of the phenomenon at hand. 

2. As suggested, we have analyzed the sensitivity of TS-LDDMM and Shape-FPCA to noise.
    
    **Protocol.**
    Learning the reference graph with TS-LDDMM and SRVF Kacher-mean, a subroutine of Shape-FPCA, with different noise levels. The dataset includes 100 sine waves with randomly generated time parametrization. Four noise scenarios have standard deviations: 0, 0.05, 0.1, and 0.2. The results are presented in Figure 2 of the attached PDF.
    
    **Results.**
    Although the overall sine wave shape is preserved, the noise level affects the learned reference graph in both cases. However, the regularity of the reference graph can be controlled by penalizing the norm of the velocity fields in the loss function. Further work on penalization will be conducted to better handle noisy data.

3. We performed a training time comparison. 

    **Protocol.** 
    We compare the training time of TS-LDDMM with Shape-FPCA and Neural LNSDE [3] on 4 datasets: 2 univariate and 2 multivariate. We reported the time in Table 1 of the PDF. 

    **Results.**
    The training time of TS-LDDMM is below that of Neural LNSDE (2 to 3 times faster)and beyond that of Shape-FPCA (4 to 5 times slower). Given their respective classification performance, (f1score, Shape-FPCA: 0.38, Neural-LNSDE: 0.70, TS-LDDMM: 0.83), TS-LDDMM seems more relevant for shape analysis.

We hope that our personal responses will meet your expectations, as your feedback has been a great source of improvement for us.

[1] Yuexuan Wu, Chao Huang, Anuj Srivastava, Shape-based functional data analysis, 2023

 [2] Aurélie Nervo, André-Guilhem Calas, Florian Nachon, and Eric Krejci. Respiratory failure triggered by cholinesterase inhibitors may involve activation of a reflex sensory pathway by acetylcholine spillover. Toxicology, 424:152232, 2019.

[3] YongKyung Oh, Dongyoung Lim, and Sungil Kim. Stable neural stochastic differential equations in analyzing irregular time series data. In The Twelfth International Conference on Learning Representations, 2024.

Meta-review: The reviewers recognized the interest of this work extending the LDDMM paradigm to account for inter-sample shape variability in time series. The proposed Riemaniann approach to graph representation of time series was found theoretically founded and motivated, and demonstrated through a relevant application. Questions varied across reviewers and concerned several points, in particular related to the need for improving clarity and better acknowledging the state-of-the-art in functional data analysis, and the perceived incremental nature with respect to the LDDMM setting and relationship with respect to the metamorphosis framework. Questions about interpretability were also raised by several reviewers. 

The rebuttal focused on testing the proposed method with respect to alternatives from the state-of-the-art (Shape-FPCA), showcasing favorable results in terms of interpretability, sensitivity and computational complexity.  These aspects were further discussed with the reviewers, along with the positioning with respect to the state-of-the-art, and the general motivation for the work. The reviewers found the discussion positive, and reached a consensus towards acceptance for this work. 

Overall, the representation framework for analyzing time-series variability proposed in this paper is interesting and novel to the field. The results are convincing and potentially relevant to a broad community beyond the clinical/biostatistical domain. For these reasons, the paper is suitable for presentation to NeurIPS 2024.
Decision: Accept (poster)