Neural Persistence Dynamics
Authors: Sebastian Zeng
Florian Graf
Martin Uray
Stefan Huber
Roland Kwitt
Venue: NeurIPS 2024 poster (2024)
Keywords: Persistent Homology
Multi-Time Attention
Latent ODE
Collective Behavior
Physical Sciences
Primary area: machine_learning_for_physical_sciences
TL;DR: We consider the problem of learning the dynamics in the topology of time-evolving point clouds, observed in systems that exhibit collective behavior such as swarms of birds, insects, or fish.

Abstract: We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i.e., a requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from a _topological perspective_, but instead of summarizing entire observation sequences (as done previously), we propose learning a latent dynamical model from topological features _per time point_. The latter is then used to formulate a downstream regression task to predict the parametrization of some a priori specified governing equation. We implement this idea based on a latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this modeling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model -- _Neural Persistence Dynamics_ -- substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

Neural Persistence Dynamics
Sebastian Zengi/ Florian Graft, Martin Urayt/1, Stefan Huber Roland Kwitt
University of Salzburg, Austria
'Josef Ressel Centre for Intelligent and Secure Industrial Automation, University of Applied Sciences, Salzburg, Austria (sebastian, zeng, florian, graf, roland. kwitt)@plus. ac. (martin.ural stefan. stefan.huberr)ffree www.rheffh-aallbbbueee ac.a.g...c.a... ac. at
Abstract
We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self ceff-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i.e., requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from topological perspective, but instead of sum- marizing entire observation sequences (as done previously), we propose learning latent dynamical model from topological features per time point. The latter is then used to formulate a downstream regression task to predict the parametrization of some priori specified governing equation. We implement this idea based latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this mod- eling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model - Neural Persistence Dynamics substantially outperforms the state-of-the the-thh-ae across diverse set of parameter regression tasks.
1 Introduction
Understanding emerging behavioral patterns of a collective through the interaction of individual entities is key elucidate many phenomena in nature Prominent examples are coherently moving flocks of birds, the swarming behavior of insects and fish or the development of cancerous cells, all understood as 2D/3D point clouds that evolve over time. Importantly, several widely-accepted governing equations for collective behavior exist [18, 40, 56] which, when appropriately parameterized, can reproduce different incarnations of typically observed patterns. Even more importantly, these equations are tied to physically interpretable parameters and can provide detailed insights into the intrinsic mechanisms that control various behavioral regimes. However, while it is fairly straightforward to simulate collective behavior from governing equations (see [21]). the inverse problem, i.c.. identifying the model parameters from the data, turns out be inherently difficult. Confounding factors include the often large number of observed entities and the difficulty of reliably identifying individual trajectories across point clouds possibly non-equidistant observation times.
macroscopic and microscopic scale.
However, several works [4, 23, 52] have recently demonstrated, may be necessary rely on individual trajectories for parameter identification. In fact, collective behavior is characterized by global patterns that emerge from local interactions, and observe the emergence of these patterns through changes to the "shape" of point clouds over time. For instance, birds may form flock, split
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

into groups, and then merge again. This perspective has prompted the idea of summarizing such topo- logical events over time and then phrasing model identification as a downstream prediction/regresionn task. The key challenge here lies in the transition from topological summaries of point clouds at specific observation times, typically obtained via persistent homology (PH) [3, 9, 22], to the dynamic regime where the temporal dimension plays crucial role.
Yet, despite the often remarkable perfor- mance of topological approaches in terms of parameter identification for models of col- lective behavior, it remains unclear in which situations they are preferable over more tra- ditional (learning) methods that can handle point cloud data, as, e-g., used in computer vision problems [24, 46]. In the spirit of [53], we highlight this point by previewing snapshot of one ablation experiment from Sec. 4. In particular, Tbl. compares the parameter regression performance of our proposed approach, using PH, to a variant where instead use PointNet++ [45] representations. Referring Fig. 1, this means that the v,, computed by a PointNet++ model. As can be seen in Tbl. I, relying on representations computed via PH works well on both datasets, while PointNet++ representations fail one dataset (vicsek-10k), but indeed perform reasonably well on the other (dorsogna-1k)
Table 1: PointNet++ [45] persistent homology (PH) representations: means higher and 4 means lower better.
VE
SMAPE
vicsek-10k
Ours (PointNet++, v2) Ours (PH, v1)
0.274=00.085 0.579=00034
0.199:00.014
0.146:+0.006
dorsogna-11 1k
Ours (PointNet++, v2)
0.816:10.031
0.1
Ours (PH, v1)
0.851=00.088
0.096:10.004
In light of this observation, it is worth emphasizing that there is clear distinction in terms of the source of point cloud dynamics when comparing collective behavior to problems in computer vision where moving objects are of primary interest and PointNet variants (see [24]) have shown stellar performance: particular, point clouds in vision primarily evolve due to a change in pose or camera motion, whereas collective behavior is driven by (local) intrinsic point interactions. The latter induces changes to the "shape" of the point clouds, i.e., a phenomenon that is typically not observed in vision.
Contribution(s). Our key idea is to learn generative model that can reproduce topological sum- maries at each point in time. This contrasts prior work, which primarily aims to extract one summary representation of the entire timeline. In detail, we advocate for modeling the dynamics of vectorized persistence diagrams via a continuous latent variable model (e.g., a latent ODE), see Fig. 1, and to use the resulting latent paths as input downstream parameter regression task. Recent stability results for vectorizations of persistence diagrams- - relating distances among the latter to the Wasserstein distance between point clouds - justify this modeling choice. Aside from state-of-the-art performance on various parameter identification tasks for established models of collective behavior, our approach scales favorably with the number of observed sequences, accounts for non-equidistant observation times, and is easily combinable with other sources of information.
latent path
Zero
Zr,
Zr
(vectorication) vec
i
Vr
VaN
time
(persistence diagram)
Prx
dgma (Rips (Pro))
Figure 1: Conceptual overview of Neural Persistence Dynamics. Given is sequence of observed point clouds Pro...., Pry- First, we summarize each P.,, via (Vietoris-Ring Rips)) persistent homology into persistence diagrams (zero-, one- and two-dimensional; only the zero-dimensional diagrams are shown) which are then vectorized into V-; via existing techniques. Second, we model the dynamics in the sequence Vro..... Vry via a continuous latent variable model (in our case, latent ODE) and then use summary of the latent path predict the parameters of specific governing cquation(s) of collective behavior. Precomputed steps are highlighted in red.

---

2
Related work
Our work is partially related to the literature on learning with dynamic point clouds in vision problems, but primarily connects work on summarizing topological features over time and inferring interaction laws of collective behavior from data.
Summarizing topological features over time. A common denominator in the literature on encoding topological changes in dynamic point clouds is the use of persistent homology, extended to accom- modate the temporal dimension in various ways. One of the early works along this direction are persistence vineyards [16], introduced as a means to study folding trajectories of proteins. Based on the idea of tracking points in persistence diagrams over time, vineyards come with stability properties similar to persistent homology [41]. but are expensive to compute and compare on a large scale. In cases where one would know the "right" scale at which to compute homology. one may also use zigzag persistence [8], as done in [17, 54], to track homological changes over time. Nevertheless, for problems with large number of observation sequences, scale selection per sequence is nontrivial and highly impractical.
Alternatively, instead of thinking about the evolution of individual points in persistence diagrams over time, one may discard the matching between points and instead focus on sequences of summary representations of said diagrams. In [25], for instance, the authors work directly with persistence diagrams (per time point) to identify changes in the topology of time-varying graphs. In terms of temporal summary representations, [52] introduce crocker plots to encode the evolution of topological features by stacking discretized Betti curves over time. Crocker stacks [57], an extension of this concept, adds a smoothing step that gradually reduces the impact of points of low persistence and, upon discretization, yields a third dimension to crocker plots. In our context, both crocker plots & stacks have been successfully used input regression methods to estimate the parametrization of models of collective behavior [4]. By drawing on prior work on kernels for sequentially ordered data [33], [23] follow conceptually similar strategy as [52, 57], introducing a path signature kernel (PSK) for sequences of summary representations of persistence diagrams.
Along different line of research, [28, 29] propose formigrams summaries of dynamic metric data, encoding the evolution of connected components. In subsequent work, [30] construct mul- tidimensional (i.c., spatiotemporal) persistent homology modules from dynamic metric data and compare invariants of these modules. While these works provide important theoretical stability results for direeiiienional homological features, i.e., connected components, it is unclear how their construction extends to homological features of higher dimension in a tractable manner. However, it is worth pointing out that recent progress along the lines of vectorizations of multiparameter persis- tent homology [36] in combination with [30] (who essentially construct multiparameter persistence modules) might, in the future, constitute tractable approach to study dynamically changing point clouds with learning methods.
Notably, computational challenges also arise in the context of crocker plots/stacks and PSKs. Despite their remarkable performance in distinguishing different configurations of models for collective behavior, both approaches suffer scalability issues: either (i) in terms of unfavorable scalability with respect to the dimensionality of vectorized persistence diagrams (as with the PSK approach of [23]), or (ii) in terms of unfavorable scalability with the number of observation sequences (as is the case for crocker plots/stacks, due to the need for extensive cross- validation of the discretization parameters). As we show in Sec. 4, our method not only outperforms these techniques by large margin, but also scales to large amounts of training sequences and requires little hyperparameter tuning.
In addition to the closely related works discussed above, we highlight that there is large body of literature on the topological analysis of time-varying signals, such as studying fMRI data via cubical persistence [47], or the persistent homology of delay embeddings [43, 44] of time series. In our context, however, these works are only partially related/relevant, as they do assume precise knowledge about individual trajectories over time (e.g., voxel IDs in IMRI data), which is unrealistic when seeking to infer parametrizations of models for collective behavior from data.
Inferring interaction laws for models of collective behavior. In the context of studying character- istics of collective behavior, there is second line of closely related work on inferring interaction laws (see, e.g., [2. 5, 6, 27, 39, 61]), ranging from metric-disstane-basee models and topological interaction models to con-paraettric estimators of interaction kernels. While a thorough survey of this literature is beyond the scope of this paper, we highlight that one common denominator in
3

---

these works is their reliance on correspondences between points across time, e.g., to infer individual point velocities or trajectories. To give an example, in recent work [39, 61]. the authors derive non-parametric estimators for interaction kernels (certain functions of pairwise distances) between observed points which crucially hinges on the traceability of each particle over time. While such approaches are conceptually appealing and even allow for reconstructing or extrapolating trajectories, they operate in the observation space. Even under a moderate number of points m, computing these estimators becomes prohibitively expensive (as, e.g., in 3D, the state space is ())). In contrast, our approach only requires positional information and can handle larger point clouds. Furthermore, although only present results predicting parameters for a class of priori specified governing equations, by formulating an auxiliary regression task, our underlying model may also be used to predict other quantities of interest.
Finally, taking a slightly broader perspective on prior art, we want to highlight recent progress on learning-based approaches that study inverse problems in the context of classic partial differential equations (PDEs). It might be possible to apply such approaches [37, 55, 60] to specific models of collective behavior such volume exclusion [40], for which the asymptotics of infinitely many particles are solutions to parametric PDEs [42]. However, for other models, the number of particles strongly influences the dynamics. For instance, in the D' D'Orsogna model [18], particle distances can collapse to zero as the number of particles tends to infinity.
3 Method
Below, we present method with focus on its application to modeling collective behavior in point clouds. Importantly, the latter represents only one of many conceivable use cases. In fact, the core ideas be applied in exactly the same way whenever one can extract topological features from time-dependent data (e.g., graphs or images).
Notation. In the following, P (xI...., XM3 C R denotes point cloud with M points and y) = ||x - y denotes the Euclidean metric. Point clouds may be indexed by 4 (or 73) to highlight the dependence on time 44- If necessary, we clearly distinguish between T; as time point with an available observation, and t; as a general placeholder for time.
Problem statement. Given a sequence of point clouds Pro...., PAN. observed at possibly non- equidistant time points Tú, we (I) seek to model their topological evolution over time and then (2) use this model to predict the parametrization of an a priori defined governing equation of collective behavior. The latter is typically specified by small number of parameters B1..... ... .pp that control the motions dx,/ ddt of individual points x; and specify (local) interactions among neighboring points.
As preparation for our model description, we first briefly establish how one may extract topological features from point cloud P. using persistent homology [3, 9, 22] - the arguably most prominent and computationally most feasible approach.
Persistent homology of point clouds. Persistent homology seeks to uncover and concisely summarize topological features of P. To this end, one constructs a topological space from P in the form of simplicial complex, and studies its homology across multiple scales. The most relevant construction for our purposes is the VittorisRRpp complex Rips(P)s. with vertex set P. This complex includes an m-simplex .x...., xm,] iff d(x,, x,) I 5 for all S j, S m at given threshold 8. The "shape" of this complex can then be studied using homology, tool from algebraic topology, with zero- dimensional homology (Ho) encoding information about connected components, one-dimensional homology (H)) ) encoding information about loops and two-dimensional homology (H2) encoding information about voids; we refer to this information as homologica//ooplooicc features. Importantly, if 6, < 6g. then Rips(P) S Riips())s_ inducing a sequence of inclusions when varying 6, called filtration. The latter in turn induces a sequence of vector spaces Hę (Rips(P)), ),) + H4 (Rips( (P)s)) basis at the homology level. Throughout this sequence, homological features (corresponding vectors of the different vector spaces) may appear and disappear; we say they are born at some &, and die at da > 6g. For instance, and disappear at later dz: in other words, the hole persists from ò, to dg. hence the name persistent homology. As different features may be born and die at the same time, the collection of (birth, death) tuples is multiset of points, often represented in the form of a persistence diagram, which we denote as dgmm (Rips(P)). Here, the notation (ips(P) refers to the full filtration and dgmy indicates that we have tracked k-dimensional homological features throughout this filtration.
mimeimmesio hole might appear at a particular value of &

---

Clearly this construction does not account for any temporal changes to point cloud, but reveals features that are present at a specific time point. Furthermore, due the inconvenient multiset structure of persistence diagrams for learning problems, one typically resorts to appropriate vec- torizations (see, e.g.. [1, 7, 10. 26]), all accounting for the fact that points close to the diagonal A = [(x,,)) x)) x E R contribute less (0 at A) to the vectorization. The latter is important to preserve stability (see paragraph below) with respect to perturbations of points in the diagram. In the following, we refer to a vectorized persistence diagram of a point cloud Pt,
Vt,,k == veo(dgm (Ripss(s)
where Vt,,& E TRd and d is controlled by hyperparameter(s) of the chosen vectorization technique. For brevity, we omit the subscript when referring to vectorizations unless necessary.
Remark 1. As our goal is model the dynamics of vectorized persistence diagrams using continuous latent variable model, it is important to discuss the dependence of the vectorizations on the input data. In particular, for our modeling choice to be sound, vectorizations should vary (Lipschitz) continuously with changes in the point clouds over time. We discuss this aspect next.
Stability/Cnntinuity aspects. First, we point out that persistence diagrams can be equipped with different notions of distance, most prominently the bottleneck distance and Wasserstein distances, both based upon the cost of an optimal matching between two diagrams, allowing for matches to the diagonal A. We refer the reader to [12, 51] for a detailed review. Stability in the context of persistent homology is understood as persistence diagrams varying (Lipschitz) continuously with the input data. While seminal stability results exist for the Wasserstein distances [15] and the bottleneck distance [12, 14], most results from the literature focus on the latter.
In the case of vectorization techniques for persistence diagrams (e.g., [1, 26]), it turns out that, most of the time, vectorizations are only Lipschitz continuous with respect to Wasserstein distances, and existing results are typically of the form
d(vec(F), vec(G)) S (FF,))
where W1 denotes the (1, 22-sasssrnttinn distance as defined in [51, Def. 2.7], F, G are two persis- tence diagrams and K > 0 is Lipschitz constant. In the context of Rem. 1, one may be tempted to combine Eq. (2) with the seminal Wasserstein stability theorem from [15] to infer stability of
vectorizations with respect to the input data. Yet, the conditions imposed in [15] actually eliminate
a direct application of this result in most practical cases, as discussed in [51]. However, the latter work also provides alternative: in our particular case of Vietoris-Rips persistent homology, one
can upper bound W1 in terms of the standard point set Wasserstein distance Wi. Specifically, upon relying on a stable vectorization technique, we obtain the inequality chain
[26, Thm. 12]
d( d(veo(Fs), veo(Gx))
K (iFk; Gk)
[5L. Thm. 5.91
M - 1 1)
2K
1:
where Fj- dgm (Rips(P)), G2 dgm (Rips( Q)) denote the k-dimensional persistence dia- grams of point clouds P and Q of equal cardinality M. In particular, we realize vec via the approach outlined in [26, Def. 10] using exponential structure elements [26, Def. 19].
Overall, the continuity property in Eq. (3) guarantees that small changes in dynamic point clouds over time only induce small changes in their vectorized persistence diagrams, and therefore provides a solid justification for the model discussed next.
Latent variable model for persistence diagram vectorizations. As we presume that the dynamic point clouds under consideration are produced (or can be described sufficiently well) by equations of motions with only a few parameters, cf. Fig. 3, it is reasonable to assume that the dynamics of the (vectorized) persistence diagrams are equally governed by somewhat simpler unobserved/latent dynamic process in IR' with 2 < d. We model this latent dynamic process via a neural ODE [13, 49], Iearned in variational Bayes regime' [32]. In this setting, one chooses recoonitiion/encoder network (Encg) ) to parametrize an approximate variational posterior go (2,,) [vv,, h²), an ODE solver
'Other choices, e.g.. latent SDE in [35, 58] are possible, but did not explore this here.
5

---

Ours
Ours
Ours
Baseline (Pl only)
Baseline (PoimiNett)
(PI) only)
(PeiniNet++)
(join))
(V1)
(12)
Phy Pry
(v3)
(H4)
(V5)
Dynanics
Preseor Prw
PointNet+*
Rto m gelte (v
1 COE solver
Ency Go CofKial ((r,)))
Vagecse Ves
Frg Tig.t.ryrp Fay
Painthitte
Frg Frw
cat
Yeg Ves
Vageces Vea
Pryy
PointNot
Ency
Prus Maule..
Prrpe Meyl
| Rega
Fegror..Po Fry Rege
Broacer Bey
Dynantes Beger
Te (Pt (Pt,
| Decy
Pp....
Dynamics
Dynonies
ReRa
Rega
Figure 2: Schematic illustration of different model variants. The first three variants (lefi to right) explicitly model latent dynamics (later denoted as w/ dynamics), the baseline variants not (later denoted as w/o dynamics), but still incorporate the attention mechanism of the encoder from [50], which we use throughout.
to yield latent states (zr, observed time points т; E [0, 7] and suitable generative/decoder network (Dec,) to implement the likelihood (v [.,)). Upon choosing a suitable prior p(zt,), ). one can then train the model via ELBO maximization, i.e.,
0, Y arg max Ezi, 40 0,y
Σ log by
- (ge Kv hill p(zra))
(4)
Different to [49], we do not implement the recognition/encoder network via another neural ODE, but rather choose an attention- manner as strong baseline (see Sec. 4). In our implementation, the recognition network yields the parametrization (μt, Σ) of a multivariate Gaussian I' with diagonal covariance, and the prior is a standard Gaussian (0, I,). Furthermore, the ODE solver (c.g.. Euler) can yield 2r, at any desired t, however, we can only evaluate the ELBO at observed time points T₁.
approach (m (mTAN) [50] which can even be used a standalone
Regression objective. To realize our downstream regression task, i.e., predicting parameters 3......pp Bp of an underlying governing equation for collective behavior (see Fig. 3) from given observation sequence, we have multiple choices. By our assumption of a latent dynamic process that carries information about the dynamic nature of the topological changes, is reasonable to tie the simulation parameter estimates 3,,..., 3P to the latent path (Zt, J: via regression network Reg.. that accepts (2:,) J: as input. In particular, we re-use the iitentioo-bbsed encoder architecture Enco to allow attending different parts of this path. However, different to Ence. which parametrizes the approximate posterior from observations at time points T,, the regression network Rega. (with its own set of parameters a) receives latent states žt, at equidistant t; E [0, 27. then summarizes this sequence into a vector and linearly maps the latter to 32.....9. 3p,, cf. Fig. 2. As in [49], for training, extend the ELBO objective of Eq. (4) by additive auxiliary regression loss, such the mean-squared error (MSE), between the predictions B, and the ground truth 3p. implicitly making a Gaussian noise assumption.
A schematic overview of our Neural Persistence Dynamics approach is shown in Fig. Additional details, including different model variants, are illustrated in Fig. 2.
Remark 2. Clearly, our presented framework allows for many architectural choices. While some components affect downstream (regression) performance only marginally, others have more pro- found impact, and we have already identified some recommended choices above (based on our experiments in Sec. 4). This includes (I) our choice of a stable persistent homology vectorization vec, where we rely on [26] due to consistently reliable performance without much hyperparameter tuning, and (2) our choice of recognition network, where choose attention- approach (mTAN) [50] which has proven to be very effective in practice [58]. In Sec. 4.2, we will provide additional configuration details for our experimental study.
4 Experiments
4.1 Datasets
Similar to previous work [4, 23, 52, 57], we evaluate and compare our approach on simulation data that is generated from parametric models of collective behavior. Specifically, consider

---

D'Orsogna (dorsoggaa [18]
dx;
m dvi (a-3lvil)) V.- M 5=
1
Vx; Σ
M
U[]x,,xx1)), Ce -C (5)
dt dt
Viesek (vicsek) [20, 56]
dx; cV., dv. dv. (I- (vv.dt+v2DDD) v.=
dt
|
a,
Σ
V
(6)
-x;
ISR
Volume exclusion [40] (volex) variant from [21]
dx.
Σ
M
|xg-X.1
(xx-xx), with <(r)
1 - 0<rsi
dt
R
172
0.
else
j=1.453
+ cell division & cell death at constant rate(s) Aa, da E [0.
Figure 3: Models of collective behavior. Parameters that varied to obtain different behavior are highlighted red; the range of each parameter is listed Appendix In the Viesek model, B, denotes Brownian motion.
three different models in R D' D'Orsogna [18], Vicsek [20] and volume exclusion [40], using the publicly-availabe implementations in the SiSyPHE library [21]?. The corresponding equations of motion (and interaction laws) are summarized in Fig. 3. The parameters that are varied to generate the datasets, i.e., the response variables of the regression tasks, highlighted in red. We sample these parameters, as specified in Appendix A, to cover a wide range of macroscopic behavioral regimes. For each sampled parameter configuration, 10,000 sequences per model. All simulations are run for 1,000 time steps (with step size 0.01, starting t = 0) on point clouds of size M 200. We take every 10th time step an observation, yielding observation sequences of length 100. At t = O, points are drawn independently and uniformly in [-0.5, 0.5j and initial velocities are uniformly distributed on the unit sphere. For direct comparison to [23], we also simulated their parameter configuration of the D' D'Orsogna model. This setup yields four datasets: dorsogna- 1 (from [23]), dorsogna- 10k, vicsek- 10k and volex-10k.
simulate one sequence of point clouds,
total of
4.2 Implementation, training & evaluation metrics
Implementation. The model variants from Fig. can be realized in many ways. Below, we specify the configuration that was used to run the experiments. For the encoder Ence, as well as the regression network Rega. we use the attention- based mTTAN architecture [50]. As decoder network Dec., we choose a two-layer MLP with ReLU activations. and as ODE solver, we select the Euler method. Each model is trained for 150 epochs using ADAM [31] (with weight decay of 0.001), starting at learning rate of 0.001 (decaying according to a cosine annealing schedule) and MSE as reconstruction (i.e., to evaluate the first term in Eq. (4)) and regression loss. case a model uses topological features, we use Ripser++ [59] to compute (prior training) persistent homology of dimension up to two, i.e., Ho, H,, and H2. Vectorizations of each persistence diagram are then obtained using exponential structure elements from [26, Def. 19]. In particular, we use 20 structure elements per diagram, which yields a = 3.20 dimensional representation per point cloud and time point. The location of each structure element is set to of 20 k-means++ cluster centers, obtained by running the latter on a random subset of 50,000 points (per dimension) selected from all persistence diagrams available during training; the scale parameter of each structure element is set according to [48, Eq. (2)) To model the dynamics, we fix the latent space dimensionality to 2 20 and scale the ODE integration time [0, 1]. While other settings are undoubtedly possible, we did not observe any noticeable benefits from increasing the dimensionality of the vectorizations or the latent space. Our publicly available reference implementation can be found at https://gi com/ wwwpsuuwwiitttteuuua.pprrrre
cs.
Evaluation metrics. We randomly partition each dataset into five training/testing splits of size 80/20. To obtain a robust performance estimate for different regimes missing and unevenly spaced observations, train three models (per split) using only fraction (i.c., 20%, 50,, and 80%) of
7

---

randomly chosen time points per sequence. Similarly, all testing splits undergo the same sampling procedure. All scores (see below) are reported as an average (# one standard deviation) over the five splits and the three time point sampling percentages. Specifically, we report the variance explained (VE) [34] (in [o, 1]: higher is better 1) and the symmetric mean absolute percentage error (SMAPE) (in [o. 1]; lower is better 4). For each testing sequence in each split, the scores are computed from the true simulation parameters (see variables marked red in Fig. 3) and the corresponding predictions (denoted by 3, in Fig. 2). Finally, when reporting results on dataset, we mark the best score in bold, as well as all other scores that do not show a statistically significant difference in mean (assessed via Mann-Whitney Whitney test at 5% significance and correcting for multiple comparisons).
4.3 Ablation
In our ablation study, assess (1) the relevance of different point cloud representations (PH vs. PointNet+++, (2) any potential benefits of modeling latent dynamics and (3) the impact of varying the observation timeframe. Additional ablation experiments can be found in Appendix B.3.
Are representations complementary? We first investigate the impact of different point cloud representations (from PH and PointNet++, resp.), by comparing variants v1 v2 and v3 from Fig. 2.
Tbl. 2 shows an extension of Tbl. 1, listing re- sults for the vicsek-1 0k and dorsogna-1K data. While using PointNet++ representa- tions on vicsek-10k yields rather poor per- formance, combining them with representa- tions computed from PH is at least nor detri- mental. On the other hand, on dorsogna- -k, where PointNet++ yields decent SMAPE and VE scores, the combination of both sources substantially outperforms each single source in isolation. Although, the complementary nature of a topological perspective on data has been pointed out many times in the lit- erature, it is rarely as pronounced as in this particular experiment. Hence, we will stick to the combination of PH+PPinnNer++ representations
Table 2: Ablation study the relevance of different point cloud representations.
Source
VE
SMAPE
vicsek-10k
PointNet++ (v2)
0.274+000085
0.199++0.014
PH (v1)
0.579+10.034
0.146++0.006
PH++PintNer++ (v3)
0.576+00000
0.144
dorsogna-1k 1k
PointNet++ (v2)
0,816110.031
0.132.10.018
PH (v1)
0.851110.008
0.097
PH+PointNet++ (v3)
0.931#0.004
0.067#0.002
(i.e., v3 in Fig. 2) in any subsequent experiments.
Are explicit latent dynamics beneficial? To assess the impact of explicitly modeling the dynamics of persistence diagram vectorizations, we ablate the latent ODE part of our model. In detail, we compare v3 from Fig. against using Reg.. operating directly on concatenated point cloud representations from PH and PointNet++ (i.e., combination of variants v4 & v5 from Fig. 2). The latter approach constitutes an already strong buseline (cf. [58]) as Reg.. incorporates an attention mechanism that allows attending to relevant parts of cach sequence. For a fair comparison, we increase the size of Rega to approximately match the number of parameters to our latent dynamic model.
As in Tbl. 2, we list results for the vicsek- and dorsogna-1k 1k data in Tbl. First, we see that ex- plicitly modeling the latent dynamics is beneficial, with considerable improvements in the reported scores across both datasets. While differences are quite prominent on the dorsogna- k data, they are less pronounced in the SMAPE score on vicsek- 10k. but still statistically significant. Second, it is important to point out that even without any explicit latent dynamics, the regression performance (see Tbl. 4) is already above the current state-of-the-art (i.e., compared to PSK [23]), highlighting the necessity for strong baselines.
Table 3: Results on the relevance of latent dynamies.
VE
SMAPE
vicsek-10k
w/ dynamics
0.576:10.030 0.512-40.020
0.144:10.006
w/o dynamics
0.1554
dorsogna-1k
w/ dynamics
0.9311100004 0.067
w/o dynamics
0.850110.006
0.09810.004
Impact of the observation timeframe. So far, we have presented results (as in prior work) where observation sequences are extracted from the beginning of a simulation (i.e., starting from 60 = 0), which corresponds to observing the emergence of patterns out of random yet qualitatively similar, initial configuration of points. the following experiment, we investigate the impact of extracting observation sequences with starting points randomly spread across much longer simulation time.
8

---

This is relevant as it is unlikely to observe uniform initial positions in any data obtained from real- world experiments. To this end, we create variations of the dorsogn-100k data by progressively increasing the simulation time 7 from 1,000 to 20,000 and then randomly extracting sub-sequences of length 1,000 (again, as before, taking each 10th step as an observation). The extension of the simulation timeframe has the effect that the difficulty of the learning task considerably increases with T. We argue that this is due to the increased variation across the observed sequences while the amount of training data remains the same. For comparison, of [23], however, only for the optimal case of observing all time points within a sequence (due to the massive PSK computation time; 3 days). As the PSK relies persistent homology only, we limit our model to the v1 variant from Fig. 2. As seen from Fig. 4, we observe an increase/drop in SMAPE and VE, resp., for both approaches, yet our approach degrades much slower with T.
also list results for the PSK approach
0.6
0.14
SMAPE (L)
VE (
0.13
0.5
0.12
Ours,
Ours, v1
PSK [23]
0.11
PSK [23]
0.4
2,000
5,000
10,000
20,000
2,000
5,000
10,000
20,000
Simulation time T
Simulation time T
Figure 4: Impact of the maximal simulation time T for extracting training/testing sequences starting at
To E [0, T' 1000]. assessed the dorsogna- 10k dataset.
4.4 Comparison to the state-of-the-art
Finally, we present parameter regression results on the full battery of datasets and compare against two of the most common approaches from the literature: the path signature kernel (PSK) of [23] and the crocker stacks approach from [57]- For both competitors, we report results when all time points are observed to establish baseline of what could be achieved in the best case. Note that evaluate the PSK approach [23] on exactly the same vectorizations as our approach, and we replicate their experimental setting of choosing the best hyperparameters via cross-validation. Similarly, for crocker stacks, we cross- cross-validart validate the discretization parameters, see Appendices B. and B.2. Tbl. lists the corresponding results. Aside from the observation that approach largely outperforms the state-of-th of-the-arr art across all parameter regression tasks, we also highlight that our model is trained with exactly the same hyperparameters all datasets.
Remark 3. closer look at the volex-10k 10k dataset, in particular its governing equations in Fig. 3,
shows that the cardinality of the point clouds may change over time due to cell division death. While this is no practical limitation for our approach, our stability arguments from Eq. (3) no longer apply. We conjecture that one may be able to extend the result of [51] account for such cases, but this might require a case-by-case analysis and possibly include additional assumptions.
5 Discussion
We introduced framework capture the dynamics observed in topological summary representations over time and presented one incarnation using a latent ODE on top of vectorized Vietoris-Rips persistence diagrams, addressing the problem of predicting parametrizations for models of collective behavior. Conceptually, Neural Persistence Dynamics embodies the idea of learning the dynamics in temporal sequence of vectorized topological summaries instead of, e.g., trying to track individual homology classes over time (as with persistence vineyards, see Sec. 2). Our approach successfully scales to a large number of observation sequences, requires little no parameter tuning, and vastly outperforms the current ot sfate-of-the-are as demonstrated by several experiments on dynamic point cloud datasets with varying characteristics. Finally, we emphasize that the fundamental ideas of Neural Persistence Dynamics may also be applied to other types of time-varying data (e.g., graphs or images) and downstream objectives simply by adjusting the filtration choice. We hope that our work will stimulate further research in this direction.

---

four diverse datasets of collective behavior. Ours (joint) refers to variant v3 of Fig. 2, Ours (PH-only) refers to variant v1. The latter setting is directly comparable to the PSK and crocker stacks approach. Best results are marked bold. Multiple bold entries indicate that there is no significant difference in mean. Note that dorsogna-1k 1k is simulated exactly as in [23] varying only two parameters, whereas for dorsogna- 10k vary four parameters (as with vicsek- 10k and volex-10k)
Table 4: Comparison the state-of-the-art
Z VE
SMAPE
Ours (joint, v3)
0.931=000004
0.067=0.002
Ours (PH-only, v1)
0.851-10.008
0.097
dorsogna-1k 1
PSK [23]
0.828=0.016
0.096=0.006
Crocker Stacks [57]
0.746=:0.023
0.150=00.005
Ours (joint, v3)
0.689+10.021
0.088=00004
Ours (PH-only, v1)
0.68010.025
0.090=0.005
Morsaana-100
PSK [23]
0.647+00.00
0.100=00003
Crocker Stacks [57]
0.343-+00016
0.145=00001
Ours (joint, v3)
0.576:10.030
0.144-000006
Ours (PH-only, v1)
0.579-10.034
0.146=0.006
vicsek-10k
PSK [23]
0.466110.009
0.173
Crocker Stacks [57]
0.345.100055
0.190
Ours (joint, v3)
0,871=0.019
0.081=0.006
Ours (PH-only, v1)
0.869-10.018
0.082
volex-10k
PSK [23]
0.509=+0.005
0.190=0.003
Crocker Stacks [57]
0.076=0.0099
0.292=0.004
Limitation(s). One obvious limitation in the presented implementation the reliance on Vietoris- Rips persistent homology of point clouds. In fact, the underlying simplicial complex will become prohibitively large (especially for H2) once we scale up the number of points by, e.g., an order of magnitude. Using Alpha [22] Witness complex [19] might be viable alternative to mitigate this issue. Similarly, one may explore subsampling strategies for persistent homology, as in [11]. and learn latent ODE from combination of estimates.
Societal impact. The present work mainly deals with an approach to capture the dynamics observed in data representing collective behavior. We assume that this has no direct societal impact, but point out that any application of such a model (aside from synthetic data), e.g., in the context of behavioral studies on living beings or health sciences (e.g., to study the development of cancer cells), should carefully reviewed. This is especially advisable when drawing conclusions from missing measurements, as these are based on imputations and may be biased.
Acknowledgments
This work was supported by the Land Salzburg within the EXDIGIT project 20204- 20204-WISS/26636-62222 and projects 0102-F1901 166- KZP, 20204- 200002-WS///2//197-200 M. Uray and S. Huber are supported by the Christian Doppler Research Association (JRC ISIA). We also thank all reviewers for the valuable feedback during the review process.
10

---

References
[1] H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, Shipman, Chepushtanova
E. Hanson, F. Motta, and L. Ziegelmeier. "Persistence Images: A stable vector representation of persistent homology". In: JMLR 18.8 (2017), pp- 1-35.
[2] M. Ballerini, N. Cabibbo, R. Candelier, V. D. F. D., E. Marinari, O. Pellizzola, G. Parisi, A. Viale, and V. Zdravkovic. "Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study". In: PNAS 105.4 (2008), pp. 1232- 1237.
[3] S. Barannikov. "Framed Morse complex and its invariants". In: Adv. Sovier Math. 21 (1994),
Pp. 93-115.
[4] D. Bhaskar, A. Manhart, J. Milzman, J. T. Nardini, K. M. Storey, C. M. Topaz, and L. Ziegelmeier. "Analyzing collective motion with machine learning and topology". In: Chaos 29.12 (2019), 123125.
[5] W. Bialek, A. Cavanga, I. Giardina, and A. Walezak. "Statistical mechanies for natural flocks birds". In: PNAS 109.13 (2012), pp. 4786-4791.
[6] S. L. Brunton, J. L. Proctor, and J. N. Kutz. "Discovering govering equations from data by sparse identification of nonlinear dynamical systems". PNAS 113.15 (2016). pp. 3932- 3937.
[7] P. Bubenik. "Statistical Topological Data Analysis using Persistence Landscapes". In: JMLR
16.1 (2015), pp. 77-102.
[8] G. Carlsson and V. de Silva. "Zigzag Persistence". In: Found. Comput. Math. 10.4 (2010),
pp. 367-405.
[9] Carlsson and M. Vejdemo-Johansson. Topological Data Analysis with Applications. Cam- bridge University Press, 2021.
[10] M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer, and Y. Umeda. "PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures". In: AISTATS.
2020.
[11] F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo, and I.. Wasserman. "Subsampling Methods
for Persistent Homology". In: ICML.. 2015.
[12] F. Chazal, V. de Silva, and S. Y. Oudot. "Persistence Stability for Geometric Complexes". In: Geometriae Dedicata 173.1 (2014), pp. 193-214.
[13] R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud. "Neural Ordinary Differential Equations". In: NeurIPS. 2018
[14] Cohen-Steiner, H. Edelsbrunner, and J. Harer. "Stability of persistence diagrams". In: Discrete Comput. Geom. 37.1 (2007), pp. 103-120.
[15] D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and Y. Mileyko. "Lipschitz Functions Have Sta-lt Persistence". In: Found. Comput. Math. 10.2 (2010), pp. 127-139.
[16] D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. "Vines and vineyards by updating persistence linear time". In: SCG. 2006.
[17] P. Corcoran and C. B. Jones. "Modelling Topological Features of Swarm Behaviour Space and Time With Persistence Landscapes". In: /EEE Access 5 (2017), pp. 18534- 18544. [18] M. R. DOrsogna, Y. L. Chuang, A. L. Bertozzi, and L. Chayes. "Self- Propelledd Particles with Soft-Core Interactions: Patterns, Stability, and Collapse". In: Phys. Rev. Lett. 96.10 (2006),
p. 104302.
[19]
Silva and Carlsson. "Topological estimation using witness complexes", In: Sympos. Point-Based Graphics. 2004.
[20] P. Degond and S. Motsch. "Continuum limit of self-driven particles with orientation interac-
tion". In: Math. Models Methods Appl. Sci. 18. (2008), pp. 1193-1215.
[21] A. Diez. "SiSyPHE: A Python package for the Simulation of Systems of interacting mean-field
Particles with High Efficiency". In: J. Open Source Softw. 6.65 (2021), p. 3653.
[22] H. Edelsbrunner and J. Harer. Computational Topology. An Introduction. AMS, 2010.
[23] C. Giusti and D. Lee. "Signatures, Lipschitz-Free Spaces, and Paths of Persistence Diagrams".
In: SIAM Appl. Algebra Geom. 7.4 (2023), pp. 828-866.

---

[24] Y. Guo, H. Wang. Q. Hu, H. Liu, and M. Bennamoun. "Deep Learning for 3D point clouds: A survey". In: TPAMI 43.12 (2021), pp. 4338 4364.
[25] M. Hajij, B. Wang. C. Scheidegger, and P. Rosen. "Visual Detection of Structural Changes in Time- Vame--aryinn Graphs Using Persistent Homology". In: Pacifiiciss. 2018.
[26] C. Hofer, R. Kwitt, and M. Niethammer. "Learning representations of persistence barcodes". 1-45.
In: JMLR 20. 10.12666 (2019), pp.
[27] Y. Katz, K. Tunstrøm, C. C. Ioannou, C. Huepe, and I. D. Couzin. "Inferring individual rules
from collective behavior". In: PNAS 108.46 (2010), pp. 18720-18725.
[28] W. Kim and F. Memoli, "Formigrams: Clustering Summaries of Dynamic Data". In: CCCG.
2018.
[29] W. Kim and F. Mémoli. "Extracting Persistent Clusters in Dynamic Data via Möbius inversion".
In: Discrete Comput. Geom. 71 (2024), pp. 1276-1342.
[30] W. Kim and F. Mémoli. "Spaaiotemporal Persistent Homology for Dynamic Metric Spaces".
In: DCG 66.4 (2021), pp. 831-875.
[31]
D. P. Kingma and Ba. "Adam: Method for Stochastic Optimization". In: ICLR. 2015.
[32]
D. P. Kingma and M. Welling. "Auto-Encoding Variational Bayes". In: ICLR. 2014.
[33]
F. J. Kiraly and H. Oberhauser. "Kernels for sequentially ordered data". In: JMLR 20.31 (2019),
pp. 1-45.
[34] J. Li. "Assessing the accuracy of predictive models for numerical data: Not r nor r², why not? Then what?" In: PLOS ONE 12.8 (2017), e01 83250.
[35] X. Li, T.-K. L. Wong, R. T. Q. Chen, and D. Duvenaud. "Scalable Gradients for Stochastic Differential Equations". ". In: AISTATS. 2020.
[36] D. Loiseaux, L. Scoccola, M. Carrière, M. B. Botnan, and S. Oudot. "Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures". In: NeurIPS. 2023. [37] D. Long and S. Zhe. Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems. 2024. arXiv: 2402. 11722 LG].
[38] M. Löning, A. Bagnall, S. Ganesh, V. Kazakov, J. Lines, and F. J. Király, "sktime: A Unified Interface for Machine Learning with Time Series". In: Workshop on Systems for ML at NeurIPS. 2019.
[39] F. Lu, M. Zhong, S. Tang, and M. Maggioni. "Nonparametric inference of interaction laws in systems of agents from trajectory data". In: PNAS 1 16.29 (2019), pp. 14424-14433.
[40] S. Motsch and D. Peurichard. "From short-range repulsion to Hele-Shaw problem in a model of tumor growth". In: Math. Biol. 76 (201 8), pp. 205-234.
[41] E. Munch. "Applications of Persistent Homology to Time Varying Systems". PhD thesis. Duke University, 2013.
[42] K. Oelschläger. "Larre systems of interacting particles and the porous medium equation". In:
J. Differ. Equ. 88.2 (1990), pp. 294-346.
[43] J. A. Perea, A. Deckard, S. B. Haase, and J. Harer. "SWIPerS: Sliding windows and 1- persistence scoring; discovering periodicity in gene expression time series data". In: BMC Bioinform 16.1 (2015), 257.
[44] J. A. Perea and J. Harer. "Sliding windows and persistence: An application of topological methods to signal analysis". In: Found. Comput. Math. 15.3 (2015), pp. 799-838.
[45] C. R. Qi, I. Yi, H. Su, and I. J. Guibas. "PointNet++: Deep Hierarchical Feature Learning on Point Sets in Metric Space". In: NeurIPS. 2017.
[46] D. Rempe, Birdal, Zhao, Z. Gojeic, S. Sridhar, and L. Guibas. "CaSPR: Learning Canonical Spatiotemporal Point Cloud Representations"" In: NeurlPS. 2020.
[47] B. Rieck, T. Yates, C. Bock, K. Borgwardt, G. Wolf, N. Bark-BBrowee, and S. Krishnaswany. "Uncovering the topology of time-varying iMRI data using cubical persistence". In: NeurIPS. 2020.
[48] M. Royer, F. Chazal, C. Levrard, Y. Umeda, and Y. Ike. "ATOL: Measure Vectorization for Automatic Yopologically-Orieen
I Learning". In: AISTATS. 2021.
[49] Y. Rubanova, R. T. Q. Chen, and D. K. Duvenaud. "Latent Ordinary Differential Equations for Time Series". In: NeurlPS. 2019.
Irregularl-SSample
[50] S. N. Shukla and M. Marlin. "Multi-Tine Maleii-iimeaa Attention Networks for Irregularly Sampled Time Series". In: ICLR. 2021.
12

---

[51] P. Skraba and K. Turner. Wasserstein Stability for Persistence Diagrams. 2023, arXiv: 2006.
16824 [math.AT].
[52] M. Topaz, L. Ziegelmeier, and T. Halverson. "Topological Data Analysis of Biological Aggregation Models". In: PLOS ONE 10.5 (2015), pp. 1-26.
[53] R. Turkes, G. Montufar, and N. Otter. "On the Effectiveness of Persistent Homology". In: NeurIPS. 2022.
[54] S. Tymochko, E. Munch, and F. A. Khasawneh. "Using Zigzag Persistent Homology to Detect Hopf Bifurcations Dynamical Systems". In: Algorithms 13.11 (2020), p. 278.
[55] A. Vadeboncoeur, Ó. D. Akyildiz, I. Kazlauskaite, M. Girolami, and F. Cirak. "Fully proba- bilistic deep models for forward and inverse problems in parametric PDEs". In: J. Comput. Phys. 491.C (2023).
[56] T. Viesek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet. "Novel type of phase transition a system of self-driven particles". In: Phys. Rev. Lett. 75.6 (1995), pp. 222-12299
[57] L. Xian, H. Adams, C. M. Topaz, and L. Ziegelmeier. "Capturing dynamics time-varying data via topology". In: Found. Data Sci. 4.1 (2022), pp. 1-36.
[58] S. Zeng, F. Graf, and R. Kwitt. "Latent SDEs Homogeneous Spaces". In: NeurIPS. 2023.
[59] S. Zhang, M. Xiao, and Wang. "GPU-Accelerated Computation of Vietoris-Rip Persistence Barcodes". In: SCG. 2020.
[60] Q. Zhao, D. B. Lindbell, and G. Wetzstein. "Learning to Solve PDE-constrained Inverse Problems with Graph Networks". In: ICML, 2022.
[61] M. Zhong, J. Miller, and M. Maggioni. "Data-driven discovery of emergent behavior in collective dynamics". In: Physica D 411 (2020), p. 132542.
13

---

Supplementary material
Simulation settings
B Comparison(s) to prior work & Additional ablation experiments
15
B.I Path Signature Kernel (PSK) [23]
15
B.2 Crocker stacks [57]
15
Additional ablation experiments
C Compatational resources
16
D Runtime analysis
16
A Simulation settings
As mentioned in Sec. 4.1, we create datasets of dynamic point clouds from simulations. All simula- tions with 200 points for 1,000 steps with a step size of 0.01, and every 10th step is taken as an observation. Only for the final ablation experiment timeframe). simulate 20,000 time steps (of the D' D'Orsogna model) and extract training/testing sequences from this extended timeframe. The model parameters for these simulations are randomly sampled as specified below, and for each sampled parameter tuple, we create one simulation.
Sec. 4.3 (i.e., Impact of the observation
To create the dor Sorganna-10k 10k dataset, we vary the following model parameters (see Fig. 3): overall, we have four parameters that need to be predicted. As macroscopic regimes mainly depend the ratios Cr C..Co, and l,/la. cf. [18, Fig. 1]. we fix C. = la = 1 and sample C, 2'c 4 = 2' with uniformly U[-1550.35 Similarly, 2 with fmm 2. The model from the SiSyPHE library used implement this simulation is Attractio ioooenulellsss sion. Note, that the SiSyPHE library implements the mass m in terms of the interaction radius parameter Rt i.e.,
distributed lo 1111,1] and
sample 2 with l, ~ 2-2222 and
Rt (as we simulate point clouds in 3D).
Remark 4. For comparability (and interpretability of the parameters) to the original D D'Orsogna model from [18], we adjusted the AttractionRepulsion implementation of SiSyPHE directly match [18, Eqs. (2) & (3)).
The dorsogna-1k 1 dataset is created by re-running the simulation provided as part of the public (Julia) implementation' of [23]. This dataset has two parameters to predict. Note that in [23], the authors simulated 500 sequences. For our work, we simulated 1,000 to have larger dataset, but still one magnitude smaller than dorsogna-10k, vicsek-10k and volex-10k. For this dataset, particle masses are 1, propulsion is = 1, Ca l = 1 and Cr, I, vary uniformly in [0.1, 2). and are selected the generated point clouds satisfy certain scale condition, leading to the parameter pairs illustrated in [23, Fig. 6].
For the vicsek-10k dataset, we sample R, c, uniformly from Mp.555 and D from U0. Overall, this gives four parameters that need to be predicted.
Finally, for the volex-10k 10k dataset, sample four parameters follows: Up.2). interaction radīi R ~ 24, and birth/death rates \b, Xa ~ Up. However, for the latter two parameters, we discard settings where da > >5 as, in this case, death rates almost impossible to distinguish; also, we discard settings where As > da as the resulting point cloud cardinalities exceed 2000 points during the simulation. The resulting (As, Aa) combinations are illustrated in 5.
14

---

Deuth rute Ad
06
60
Birth rate A
Figure 5: Birth rates As and death rates da used for generating the volex-10k dataset.
For reproducibility, we will release the simulation data publicly.
B Comparison(s) to prior work & Additional ablation experiments
B.1 Path Signature Kernel (PSK) [23]
For the PSK, we rely on the publicly available implementation in sktime" [38]. We compute the PSK (truncated signature depth 3) using our vectorized persistence diagrams per time point as input. For concatenated zero-, one- and two-dimensional persistence diagrams, we use 20-dimensional vectorizations (as specified in Sec. 4.2), yielding 60-dimensional input vectors per time point. The computed kernel is then input to kernel support vector regressor kermel-SVR)
Following the experimental protocol in [23, Sec. 7.3], cross-validate the sliding window bedding (across lags [1, 2..3) 3) and kernel-SVR hyperparameters on 20% validation portion of the training data, with hyperparameters selected based the average MSE (per governing equation parameter 3r to be predicted) across 5-folds. Due to the excessive runtime (approx. 8.3 hours per lag on the system specified in Appendix C), we only report results when all time points per observation sequence are considered. Hence, no subsampling of time points (as is done when evaluating our approach) performed, and the reported performance can be considered an optimistic estimate of what can be achieved with the PSK.
B.2 Crocker stacks [57]
As mentioned in Sec. 4.4, we compare against crocker stacks, introduced in [57], as one of the stae-ooftthe-ar approaches. Crocker stacks are an extension to crocker plots [52] and constitute topological summary for time-varying persistence diagrams. For the crocker stacks, we adapted the publicly available implementation of the crocker plots in the teaspoon" library.
The crocker plot computed as follows: for each time step, the persistence diagrams are computed up to scale parameter e. In discretized steps of the scale parameter, the Betti numbers are computed from the persistence diagrams, which results in 2D representation (i.c., vs. Betti number) for each homology dimension.
The extension to crocker stacks is achieved by adding a third dimension lt induced by the smoothing factor a. For given steps of a, a smoothing operation is applied, i.e., values within a specified distance (smoothing factor) from the diagonal of the persistence diagram ignored. Hence, for each homology dimension, a crocker stack is tensor in R with axes corresponding to discretizations of the (1) scale parameter E [0, xo), the (2) time E [0, T). and the (3) smoothing factor [0, oo).
We note that in [57], the authors set the scale parameter to 8 = 0.35, as they use data normalized to the range of to 1. We did not scale the data, however, we adjusted the scale parameter correspondingly:
Shttps: chttps://teaspoontde github...... oo/teaspoon
15

---

for all experiments, we set the scale parameter to 8 = 1/3 maxPers(dgnn (Riperr) where P denotes the point cloud, and maxPers(dgm, (Rips(P))) is the maximum persistence obtained from all point clouds in an observed sequence.
Hyperparameter choices. We made the following hyperparameter choices: (1) the Vietoris-Rips filtration is considered up to (see above), where the computations are discretized with 25 equally spaced values in [0, s]; (2) the smoothing values are discretized with 18 steps equally spaced in [0, 0.5 maxPers].
Eventually, the crocker stacks per homology dimension vectorized (i.e., the tensor is flattened) and concatenated into a single vector per observation sequence. These vectorizations are then input to linear support vector regressor (SVR). For each of the (varied) parameters in the governing equation of Fig. 3, a separate SVR is trained and evaluated. Hyperparameters of the SVR and the usage of preprocessing steps (feature scaling) tuned using Bayesian optimization. Each parameter configuration evaluated using a 5-fold cross- validation, and the best configuration is then used to train the final model on the full dataset.
B.3 Additional ablation experiments
To assess whether homology dimensions > 0 are beneficial to the downstream regression task, we experimented on dorsogna-1k. We find that when using our approach with Ho-features only, the regression quality drops. When additionally including Hz-features (i.e., Ho. H₃ and H2), the situation less clear, the results are not noticeably different. Quantitative results can be found in Tbl. below.
Table 5: Quantitative assessment the impact of including higher-dimensional homology features on down- stream regression performance, evaluated on dorsogna 1k.
VEt
C SMAPE
Ours (PH--nnl, v1; Ho)
0.819 1 0.015
0.101 1: 0,003
Ours (PHI-only, v1; Ho, Hi)
0.844 1 0.021
0.098 1 0.007
Ours (PH-only, v1; Ho, Hi, H2)
0.846
0.011
0.097 1 0.005
C
Computational resources
All experiments were run on an Ubuntu Linux system (22.04), running kernel 5.15.0- 1.150--000-eemeric. with 34 Intel@) TM i9- 10980XE CPU @ 3.00GHz cores, 128 GB of main memory, and two NVIDIA GeForce RTX 3090 GPUs.
D Runtime analysis
In the following, we present runtime breakdown of the pre- pre-processing steps (i.e., Vietoris-Ree diepsrs-Riiss persistent homology (PH) computation, and the vectorization of persistence diagrams), as well as runtime comparison prior work (PSK and crocker stacks) and our baseline approach. Runtime is measured on the system listed above, using our mublicly-availabee reference implementation.
At this point, it is also worth highlighting that our pre-processing step, i.e,, PH computation & vectorization, is trivially parallelizable and can easily be distributed across multiple CPUs/GPUs if needed.
Vietoris-Rips PH computation. In Tbl. 6, we list wall clock time measurements (using Ripser++ [59] on one GPU) per point cloud. In particular, the table lists runtime for computing PH of dimension zero and one (i.e., H, and H1), as well as PH of dimension zero, one and two (i.e., Hg, H, and Hz). All point clouds in this experiment of size 200 in R
Persistence diagram vectorization. Persistence diagram (PD) vectorization can essentially broken down into two steps: (1) parameter fitting for the structure elements of [26] and (2) mapping PDs to vector representations using those structure elements. Tbl. lists the runtime for both steps on the dorsogna- 1k data when vectorizing zero-, and onc-dimensional persistence diagrams. Throughout
16

---

Table 6: Runtime comparison for PfI computation for different homology dimensions. Reported is the average runtime per point cloud (in seconds), and overall runtime (in seconds) estimated from this average by multiplying by the number of processed point clouds on dorsogna- 1k.
Time per point cloud
Overall
PH computation (He, Ha) PH computation (Ho, H1, Hz)
0.018
1800
0.330
33000
all of our experiments, parameter fitting for the structure elements is done by first collecting all persistence diagrams for each homology dimension and then running k-means++ clustering 50,000 points uniformly sampled points from those diagrams.
Table 7:
Runtime breakdown of PD vectorization (per diagram and overall for Hn and Hi on sogna-kkk. split into (Step 1) time spent for fitting the parameters of the expemential structure elements from [26], and (Step 2) actually mapping PDs to vector representations.
Time per diagram
Overall
Step 1: Parameter fitting Step Mapping PDs vectors
n/a
17
9.4c-05
s
Comparison prior work. Next, we compare the runtime of our approach (here, variant from Fig. 2) to prior work that uses persistence diagrams as input. In particular, we compare against the PSK method from [23] and crocker stacks [57].
In Tbl. 8, we list the overall training time (on dorsonga-1k), where runtime for pre-processing (see above) is excluded from these measurements. Also, PSK and crocker stacks timings do include hyperparameter optimization, as suggested in the corresponding references. Importantly, this is not optional but required to obtain decent performance with respect to EV and SMAPE. Notably. for the PSK approach, kernel computation scales quadrati- cally with the number of sequences, and kernel-SVR training takes time somewhere between quadratic and cubic. Hence, scaling up the number of training sequences quickly becomes computationally pro- hibitive, especially light of the required hyperparameter tuning. Finding suitable hyperparameters is also the main bottleneck for crocker stacks (which rely linear SVR).
Table 8: Training time comparison to prior work (on dorsogna-1k).
Training time
Crocker Stacks PSK Ours (PH-only, v1)
24600
646
190
Comparison to baseline model. Finally, in Tbl. 9, we present a training time comparison to our baseline model which does not explicitly model any dynamics via a latent ODE (denoted as w/o dynamics). In this experiment, the training protocol remains unchanged, and we vary the type of input data, i.e., from using vectorized persistence diagrams only (Ho and H2) to using a combination of PointNet++ features and vectorized persistence diagrams. Importantly, as remarked in the main part of the manuscript, the baseline models (PH-only, v1) and PPH+-ointNet+++ v3) already yields strong performance across all parameter prediction problems.
Table 9: Training time comparison (on dorsogna-144 11k) our approach vs. the baseline model (w/o dynamics), using the same input data.
Training time
Ours (PII-only, v1)
190
Ours (PointNet+++, v2)
3780
Ours (PIL+PointNet++: v3)
4100
Baseline (w/o dynamics; PH-only, v1)
50 s
Baseline (w/o dynamics; PointNet++, v2)
525
Bascline (w/ dynamics; PH+PointNet++. v3)
600
17

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Claims made in the abstract are backed up by ablation experiments and comparisons to prior work baselines.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
Sec. of the manuscript specifically addresses limitations of the proposed
approach.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, c.g., if the approach was only tested few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-tex system might not be used reliably provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
18

---

Answer: [Yes]
Justification: Our justification for the modeling choice relies on two previous stability results for persistent homology. When discussing the stability claim in Sec. 3, we list exact references to theorems in the literature that lead to this result.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofss provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions the paper (regardless whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: We provide all experimental details in the main part of the paper and the appendix. Furthermore, source code (and data) publicly-availabl at https: thub. com/plus hom//pus-rkwiiee wwwus-rrwwitt/nuuallll.k...
Guidelines:
The answer NA means that the paper does not include experiments. If the paper includes experiments, No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way access this model for reproducing the results way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (c.g., to registered users), but it should be possible for other researchers to have some path to reproducing verifying the results.
19

---

reproducible, they 5. Open access to data and code
Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, described in supplemental material?
Question:
Answer: [Yes]
Justification: We provide source code with this submission and full details about the (sim- ulated) experimental data we use throughout. The code is released publicly and access to the already simulated data is provided to the community (via the code repository, see checklist item 4). Moreover, the data can easily be reproduced, publicly available simulation libraries, iee., https: /g thub. com/ antoinedie / and https: / http//ggtttbbbb com/ httmm//drricc///aa wttpe//pppatts----eessstteeeeeee diagranss......
it is generated from
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please the NeurlPS code and data submission guidelines (https://nips.c
htte//uuiee//eoeeiuueeeeee Sww.iiiiooooolicc for more details.
While we encourage the release of code and data, we understand that this might not be possible, so No an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https:
/ //nips ctppsscccpbblll htte//uiies//ooeoeeeeeeeee
www.llmisssoooooolloooooo onnooiccy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Question:
Answer: [Yes]
Justification: Sec. 4 lists all relevant hyperparameters for training. Exact architecture specifications are provided as part of the submitled source code.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: All reported results include error bars (#: 1 standard deviation); Sec. 4.2 clearly states how these were computed. Furthermore, we assessed the statistical significance of any results the listed Tables against the strongest obtained result the respective Table using Mann- Whitney test (correcting for multiple comparisons).
20

---

Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by enror bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
should be clear whether the error bar is the standard deviation the standard error of the mean.
is OK to report I-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).
If eπror bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information the puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: Computational resources and memory/compute requirements listed in the appendix.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (c.g., preliminary or failed experiments that didn't make it into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics http:::://eerrpsss.ccpptttttttte
Answer: [Yes]
Justification:
We follow the guidelines as specified.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make to preserve anonymity (e.g., if there is special consid- eration due to laws or regulations their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative socictal impacts of the work performed?
21

---

Answer: [Yes]
Justification: Sec. of the manuscript specifically addresses possible negative societal impact of the proposed approach. However, in view of the demonstrated improvements the current state of the art seems reasonable to expect a positive (societal) impact in related research areas.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g.. disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (c.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: Dutasets that are part of our experimental studies and that we release af- ter publication for reasons of reproducibility come exclusively from publicly accessible repositories described in Section P Pretrained models will not be part of the submission.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (c.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: The use of existing assets (i.e., code packages) was exclusively carried out with indication of the source in the manuscript.
22

---

Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paper swi thcode. thccode com com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re- ee-ackagedd, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification: All newly introduced assets have been carefully documented, i.e. details of the model have been described in Sec. and limitations have been described in Sec. of the manuscript. Furthermore, source code is attached the submission and will be released publicly in case of acceptance (including datasets used in this work as well as a documentation).
Guidelines:
The answer NA means that the paper does release assets.
Researchers should communicate the details the dataset/code/model as part their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Question:
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
23

---

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with
human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
24


Reviews:
[{"review_id": "gz3xYov9df", "reviewer": "Reviewer_bRy6", "summary": "This work considers the problem of learning the latent, continuous-time dynamics underlying time-evolving point clouds. To solve it, it leverages previous work on the persistent homology of point clouds and their vectorization, as well as the PointNet++ network, to obtain static representations of the point clouds at a set of observation times. It then makes use of the LatentODE framework of Rubanova et al. (2019) to infer a continuous-time representation encoding the observed representations. \n\nThe authors employ the learned continuous-time representations to tackle the inverse problem of regressing the ground-truth parameters of a set of governing equations from which the point cloud observations were simulated. They show their model outperforms two recent baselines. They also empirically demonstrate that (i) the deep representations obtained via PointNet++ are complementary to those obtained via persistent homology; that (ii) actually modelling the continuous-time dynamics helps with the regression task; and that (iii) the regression tasks increases its complexity with that of the initial condition in the simulation.\n\n*References:*\n- Latent ODEs for Irregularly-Sampled Time Series. Rubanova et al. (2019)", "questions": "1. Neural ODEs are well known to be very difficult to train (see e.g. Dupont et al. (2019), Finlay et al. (2020), Choromanski et al. (2020) or Pal et al. (2021), just to cite a few). Can you comment on how difficult (or easy) was to train your Latent ODE network on the synthetic data you studied? What about the training time?\n\n2. How did you parametrize the decoder network? I don’t find it in the manuscript.\n\n3. Did you consider adding some noise to your observations in point cloud space? As presented, your network should easily be able to handle noisy observations. Similarly, how does the persistent homology representation deal with noise? These questions are of course relevant if one wants to apply your methods to empirical data.\n\n4. How many points from the latent path are used as input to the regressor model? Does the model work with a single point, as e.g. the last point along the latent trajectory? It’d have been nice to understand what information of the latent path is important for the regression task, specially given that completely dispensing from the latent dynamics still gives compelling results (i.e. Table 3).\n\n*References:*\n- Augmented neural odes.  Dupont et al. (2019)\n- How to train your neural ode: the world of jacobian and kinetic regularization. Finlay et al. (2020)\n- Ode to an ode. Choromanski et al. (2020)\n- Opening the blackbox: Accelerating neural differential equations by regularizing internal solver heuristics. Pal et al. (2021)", "limitations": "Yes, they did address the limitations of their method.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is very well written, and provides enough information for a reader not versed in the use of persistent homologies for point cloud summarization. The authors justify their methodology, viz. the encoding of vectorized persistent diagrams into a continuous-time latent process, by using stability arguments of previous work. Their argumentation reads well and is convincing. \n\nAnother strenght is that, besides empirically demonstrating that their method outperforms two recent baselines, the authors additionally perform reasonable ablation studies that further justify their methodology.", "weaknesses": "The main contribution of this work could be read as a direct application of the Latent ODE framework of Rubanova et al. (2019) to the problem of time-evolving point clouds. It can be seen as incremental too, for it leverages stablished methods for point cloud representations. One could therefore argue that, despite its merits, the paper would better fit a conference or journal with a more applied character. \n\nIt is also not clear how the method would perform on real-world regression tasks with empirical point cloud data. Can one, for example, use the proposed methodology to study the recorded data presented in the seminal work of Bialek et al. (2012)? \n\nIt’d be nice if the authors could comment on the applicability of their method to empirical data. \n\n*References:*\n- Statistical mechanics for natural flocks of birds. Bialek et al. (2012)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "For better readability, we restate your comments/questions in *italic*, our response(s) are marked by &#9654;\n\n**Ad contribution:** The reviewer is correct in that we use the latent ODE framework of (Rubanova et al., 2019). However, the latter is only one particular variant (of many; other options are, e.g., latent SDEs or Continuous Recurrent Units (CRUs)) for modeling latent dynamics. Our main contribution is the idea of modeling the point cloud dynamics through the lens of vectorized topological summaries, i.e., capturing topological changes on a *persistence diagram level*, as opposed to capturing changes on the level of individual points in such topological summaries. Even this arguably \"simple\" approach yields remarkable predictive performance for the parameters of governing equations of collective behavior. Furthermore, our approach avoids the need to track particles over time (and infer velocities) or to rely on computationally expensive summaries of time-dependent persistence.\n\n**Ad empirical data:** Thank you for that comment. Yes, our approach would work on the data from (Bialek et al., 2012). In fact, one motivation for our work was to alleviate the challenge of having to compute/estimate per-particle velocities (this is required in the work of Bialek et al. as they compute correlations from normalized velocities). In many real-world settings, one might not even be able to unambiguously track particles (let alone when the cardinality of the point clouds may vary as well). Instead, our approach only hinges on the position of the particles (and can even naturally deal with point clouds of varying size, as demonstrated with the `volex-10k` experiment).\n\n**Ad training time**: Please see our runtime analysis in the *General Response* section.\n\n*Neural ODEs are well known to be very difficult to train (see e.g. Dupont et al. (2019), Finlay et al. (2020), Choromanski et al. (2020) or Pal et al. (2021), just to cite a few). Can you comment on how difficult (or easy) was to train your Latent ODE network on the synthetic data you studied? What about the training time?*\n\n&#9654; First, we did not experience any difficulty training the full model. However, we did only experiment with the arguably simplest ODE model (an autonomous ODE). We did preliminary experiments also with a non-autonomous variant, but did not observe any noticeable improvements. Nevertheless, our experiments support the conclusion that even the \"simplest\" choice of latent ODE already suffices to largely outperform existing techniques.\n\n*How did you parametrize the decoder network? I don’t find it in the manuscript.*\n\n&#9654; We apologize for not being clear enough on that point. The decoder (aka reconstruction network, denoted as $Dec_\\\\gamma$ in Fig. 2 of the manuscript) is a simple 2-layer MLP with ReLU activation that maps latent states (from the latent ODE) to reconstructed persistence diagram vectorizations. We will include this information in the revised manuscript's appendix (and point to the locations in our source code).\n\n*Did you consider adding some noise to your observations in point cloud space? As presented, your network should easily be able to handle noisy observations. Similarly, how does the persistent homology representation deal with noise? These questions are of course relevant if one wants to apply your methods to empirical data.*\n\n&#9654; Currently, only the governing equations of the Vicsek model (see Fig. 3) include noise through the Brownian motion. In general, we point out that due to the stability of the persistence diagrams wrt. perturbations of the input (see the *Stability/Continuity aspects* section of the manuscript), our method is suitable in case of observation noise (unless the noise is excessively large). E.g., on `dorsogna-1k` where points are within $[-1,1]^3$, adding Gaussian noise $\\\\mathcal{N}(0,\\\\sigma)$ to all point coordinates and all time points, ($R^2$,SMAPE) drops from (0.846 $\\\\pm$ 0.011, 0.097 $\\\\pm$ 0.005) to\n(0.822 $\\\\pm$ 0.016, 0.098 $\\\\pm$ 0.002) at $\\\\sigma=0.01$, and to (0.730 $\\\\pm$ 0.013, 0.150 $\\\\pm$ 0.001) at $\\\\sigma=0.1$.\n\n*How many points from the latent path are used as input to the regressor model? Does the model work with a single point, as e.g. the last point along the latent trajectory? It’d have been nice to understand what information of the latent path is important for the regression task, specially given that completely dispensing from the latent dynamics still gives compelling results (i.e. Table 3).*\n\n&#9654; Again, we apologize for not being precise enough. In fact, we experimented with multiple variants: you are correct that one could use the “last” point along the latent trajectory. Another variant would be summarizing the latent trajectory via a signature approach (although this summary can become quite high-dimensional). The most conceptually elegant approach, from our perspective, was to re-use the encoder architecture (i.e., a duplicate of the mTAN encoder module with its own set of parameters, denoted as $Enc_{\\alpha}$ in Fig. 2). However, instead of receiving an unequally-spaced sequence of vectorized persistence diagrams, $Enc_{\\alpha}$ receives ALL points along the latent trajectory. In our case, “all” means 100 latent states at equally spaced time points in $[0,1]$, which we obtain by integrating the latent ODE forward in time.\n\nAlso, our baseline (w/o dynamics) could be considered quite strong since it uses an mTAN encoder (i.e., an attention-based approach) that directly maps sequences of vectorized persistence diagrams to parameter predictions. While mTANs can attend to relevant information in a sequence, this baseline still consistently falls short of our approach, which explicitly models the dynamics.", "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": "O6M06NoV9o", "reviewer": "Reviewer_dbj5", "summary": "This paper considers the problem of learning some parameters $\\theta$ in---roughly speaking---a dynamical system of the form $\\dot X = \\phi(X, \\theta)$, where $X \\in \\mathbb{R}^{n \\times d}$ (with typically $d=3$), from an observed time-discrete trajectory $X(\\tau_0),\\dots,X(\\tau_N)$. \n\nThe idea conveyed by this paper is that instead of tracking the dynamic of the whole point cloud ($n$ can be quite large) to infer the parameter $\\theta$, it may be somewhat sufficient to look at a \"featurization\" of it---here, of a topological nature, yielding a sequence of vectors $v_{\\tau_0},\\dots,v_{\\tau_n}$ that somewhat summarize the dynamic while being (hopefully) sufficient to infer the parameters $\\theta$. \n\nThe parameters of the system are then learned by existing techniques, essentially by using a low-dimensional latent dynamic $(z_t)_t$ modeled by a neural ODE and then connected through the observed $(v_t)_t$ by encoder-decoder; training being performed using ELBO maximization. \n\nThe experiments are conducted on synthetic data (from rather sophisticated dynamical systems), containing an extensive ablation study, and showcasing the usefulness of the proposed method on the considered models.", "questions": "1. Related to my comments above, could you make explicit the way the parameters of the systems are encoded and learn? Say that I consider the Volume exclusion model with parameters $(\\alpha, R, \\lambda_b, \\lambda_d)$. How are these parameters actually estimated once $\\texttt{Enc}_\\alpha, \\texttt{Enc}_\\theta, \\texttt{Dec}_\\gamma$ are trained with the ELBO+regression rule? How are the $\\hat{\\beta}_t$ (showcased in Figure 3) used afterward?\n\n2. (more genuine question out of curiosity) In some sense, the seminal point cloud $X$ contains implicitly all the possible topological information one could extract using persistent homology. It also seems that PointNet-like architectures can extract this topological information---see for instance the RipsNet model proposed in [1]. So in some sense, this suggests that turning the observed $X(\\tau)$ into diagrams is \"the way to go\" but PointNet fails to learn that. Do you agree with this statement? If so, do you have any intuition on why? \n\n3. Still related to the above, the paper mentions that it uses PD with homology dimension 0, 1 and 2. Did you run the experiments using $H_0$ alone? If the performance significantly decreases, that would be a nice example where non-trivial topology is useful (as frustrating as it can be, I often observed that $H_0$ alone, i.e. looking mostly at the distribution of pairwise distances, was sufficient to reach good performances). Note: please do not feel obliged to run new experiments during the rebuttal period! This is just a genuine question.\n\n4. In the conclusion, you mention that computing $H_2$-PD with Rips filtration is quickly computationally expensive (I completely agree), and propose to use alpha-filtration instead. Is there any difficulty to do so right now? (Alpha filtrations are readily implemented in several Tda-libraries according to https://cat-list.github.io/ ).\n\n\nRef : \n\n[1] \tRipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds, de Surrel et al.", "limitations": "NA", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The proposed use-case of Topological Data Analysis in the context of dynamical systems feels quite convincing and refreshing. \n\nThe methodology, at a global picture level, is fairly clear. \n\nThe performance showcased by the experimental results, supported by an extensive ablation study, seems to robustly support the usefulness of the method.", "weaknesses": "My main concern is about the clarity of the paper in the following sense: at the global picture level, the paper feels very clear and convincing. However, I think that it is not clear at the low-level scale, in that I feel completely unable to implement the method based on what is described in the (main) paper.\nFor, the PD vectorizations are never described precisely, the final loss function (ELBO + regression) is only informally discussed in lines 226-234, I actually do not exactly understand how the parameters of the systems are encoded in the models, between which variables the $R^2$ coefficient (line 268) is computed (I infer that it is the parameters of the models wrt the one inferred at training time), etc. \n\nI understand that code and so on will be delivered and that the method will be usable by others if needed, but I believe that the impact of the paper would be strengthen if one could understand what is done _at the technical level_ (i.e., be able to roughly implement it) after reading it.\n\nAdding a minimalist proof-of-concept experiment, with a trivial dynamical system like $\\dot x = \\theta x$ or something like that, may possibly be helpful (or not).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "For better readability, we restate your comments/questions in *italic*, our response(s) are marked by &#9654;\n\n**ad Clarity:** We tried to balance the presentation of our conceptual idea vs. the technical detail of its particular realization. We can provide more detail on the ELBO objective and the added regression loss in the appendix. The same holds for the PD vectorization technique we use. As both parts appeared separately in prior work, we sacrificed technical detail for those parts in favor of a more comprehensive conceptual perspective. We will add these details to the appendix to facilitate reproducibility, as you suggested, including pointers to the respective parts in the source code.\n\nFurther, for **training**, the regression loss is the MSE between the parameter estimates and the ground truth simulation parameters. One may also use $R^2$ (or any other reasonable objective) during training, but we found the MSE to already yield good results. During **inference/evaluation** (on held-out testing data), we report the $R^2$ and SMAPE computed from predictions vs. ground truth values, just as you correctly inferred.\n\n*Related to my comments above, could you make explicit the way the parameters of the systems are encoded and learned? Say that I consider the Volume exclusion model with parameters. How are these parameters actually estimated once $Enc_\\\\alpha, Enc_\\\\theta, Dec_\\\\gamma$ are trained with the ELBO + regression rule. How are the $\\hat{\\beta}_t$ (showcased in Figure 3) used afterward?*\n\n&#9654; Inferring the simulation model parameters (e.g., for the Volume Exclusion model) is done as follows: For a given sequence of point clouds (at time points $\\tau_0, \\\\dots,\\\\tau_n$), one first pre-computes Vietoris-Rips persistent homology (e.g., for $H_0, H_1, H_2$) per available time point and vectorizes the diagrams. This yields a sequence of vectors, i.e., one vector per available time point. This sequence is then fed to the encoder $Enc_\\theta$ which outputs a parametrization for the approximate posterior $q_\\\\theta(z_{t_0}|\\\\{ v_{\\tau_i} \\\\})$. Upon sampling from $q_\\\\theta$, we get an initial latent state $z_{t_0}$ and integrate the latent ODE forward in time to $t_n$. $Enc_\\\\alpha$ then summarizes this sequence of latent states and linearly maps it to a vector of *simulation parameter estimates*\n$\\beta_1, \\\\ldots, \\beta_P$. (For Volume Exclusion, we have four simulation parameters, so $P=4$).\n\n*(more genuine question out of curiosity) In some sense, the seminal point cloud contains implicitly all the possible topological information one could extract using persistent homology. It also seems that PointNet-like architectures ...*\n\n&#9654; An important point in this context is that RipsNet is *trained* (in a supervised manner) to predict precomputed *vectorizations* of persistence diagrams, not the persistence diagrams themselves. This means the desired vectorized diagram is known in advance and can be used in a loss function. Clearly, this strategy is designed to guide the network towards capturing topological information. Instead, in our experiments, the PointNet++ *directly* predicts the parameterization of the simulation; hence, it would have to learn topological information implicitly (as there is no external guidance towards this goal). The only guidance given is a loss wrt. the predicted simulation parameters. It might, however, be possible to replace our full pre-computation step (for Vietoris-Rips PH) with a pre-trained RipsNet (i.e., pre-trained on point clouds from such simulation experiments). We did not explore this direction here, but we greatly appreciate the comment.\n\n*Still related to the above, the paper mentions that it uses PD with homology dimension 0, 1 and 2. Did you run the experiments using $H_0$ alone? If the performance significantly decreases, that would be a nice example where non-trivial topology is useful (as frustrating as it can be, I often observed that $H_0$ alone, i.e. looking mostly at the distribution ...*\n\n&#9654; Yes, initially, we experimented with $H_0$-only and found that prediction performance drops in that case. The situation is less clear when also including $H_2$ features, where the results are more mixed and often not noticeably different. From a more quantitative perspective, below you can find a table for the `dorsogna-1k` experiment comparing $H_0$ vs. $(H_0, H_1)$ vs. $(H_0, H_1, H_2)$ for our approach (i.e., the `v1` variant from Fig. 2):\n\n|   | $R^2$ $\\uparrow$  | SMAPE $\\downarrow$ |\n|---|---|---|\n| Ours (PH-only, v1, $H_0$)               | 81.9 $\\pm$ 0.015  |  0.101 $\\pm$ 0.003 |\n| Ours (PH-only, v1, $H_0, H_1$)          | 84.4 $\\pm$ 0.021  |  0.098 $\\pm$ 0.007 |\n| Ours (PH-only, v1, $H_0, H_1, H_2$)     | 84.6 $\\pm$ 0.011  |  0.097 $\\pm$ 0.005 |\n\n*In the conclusion, you mention that computing $H_2$-PD with Rips filtration is quickly computationally expensive (I completely agree), and propose to use alpha-filtration instead. Is there any difficulty to do so right now? (Alpha filtrations are readily implemented in several Tda-libraries according to https://cat-list.github.io/ ).*\n\n&#9654; There is no inherent limitation to switching out Vietoris-Rips complexes for Alpha complexes. However, we did find in a preliminary experiment that different implementations of Alpha complexes (some of which are in the link you provided) surprisingly yield different diagrams, and without any further in-depth investigation refrained from using the latter in our experiments. Nevertheless, switching from VR to Alpha complexes would, as you suggested, significantly improve the performance of the preprocessing step in which we compute PH for different dimensions.", "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": "z9Nrrhdm9w", "reviewer": "Reviewer_83FG", "summary": "The paper addresses the challenge of predicting the specific parameters of models yielding point cloud dynamical systems known only partially from a set of observations in different time steps. This is achieved by leveraging information about the evolution of persistent homology vectorizations of the observed point clouds at different time steps. Particularly, the paper uses a specific, previously published, vectorization that is Lipschitz with respect to the standard point set Wasserstein distance. The evolution of the topological vectorization for the different point cloud dynamical systems is assumed to be governed by latent dynamic processes, which are approximated via a neural ODE and used to infer, given a set of observations of an evolving point cloud, the model parameters that produce these observations. The proposed baseline method is tested and benchmarked across three different scenarios, demonstrating the utility of its components and showing a significant superiority over the leading state-of-the-art methods.", "questions": "- Why do you use $R^2$? I thought that it was a misleading metric [4]. This is my main concern, and the core reason why the score is a borderline accept.\n- Can your approach be used for discrete dynamical systems (where time is discrete and not continuous?)\n- Since the method is very general and can work with any kind of vectorizations, not just topological ones, is there a way to regularize or add explicit bias during training to inform the training process/neural ODE that the representations are topological?\n\n[4] Li J (2017) Assessing the accuracy of predictive models for numerical data: Not r nor r2, why not? Then what?. PLOS ONE 12(8): e0183250.", "limitations": "Some limitations have been addressed, although I would appreciate a better discussion, for example, adding that the evolution of individual topological features cannot be studied.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "**Originality**: To the best of my knowledge, the idea of modeling the dynamics of vectorized persistence diagrams via a continuous latent variable model is novel and powerful.\n\n**Significance**: As stated and demonstrated in the paper, neural persistence dynamics, and the evolution of point clouds over time seen as a group rather than individual points, provide significant insights into the dynamical systems governing the behavior of the individual points. This approach has great potential in studying real-world problems, especially in \"natural systems\" where the interaction of individuals as a group is key to understanding their behavior (e.g., flocks of birds, insects, fish, cells in an organism). \nSpecifically related to neural networks, the method proposed in this paper has the potential to improve the analysis of the dynamics of data going through the different layers of a neural network, as done in a more simplistic way in [1] and [2], where contradictions were found that this project might help resolve. Additionally, the method's significant superiority compared to other approaches for predicting the parameters of dynamical systems from evolving point clouds suggests that this is indeed a promising direction for further research in the study of evolving topology and in the aforementioned areas.\n\n**Clarity and Quality**: The paper is generally well-written, with a few exceptions that I will address in the weaknesses section. The comprehensive literature review significantly enhances the quality of the manuscript. Figure 1 and 2 are particularly clarifying when reading the text.\n\n[1] Naitzat, Gregory, Andrey Zhitnikov, and Lek-Heng Lim. \"Topology of deep neural networks.\" Journal of Machine Learning Research 21.184 (2020): 1-40.\n\n[2] Wheeler, Matthew, Jose Bouza, and Peter Bubenik. \"Activation landscapes as a topological summary of neural network performance.\" 2021 IEEE International Conference on Big Data (Big Data). IEEE, 2021.", "weaknesses": "- I think that the sentence in line 32 \"... due to missing correspondences between individuals...\" is hard to read. Which missing correspondences?\n\n-Table 1 lacks context and is duplicated in Table 2. The experiments of this table are not explained and metrics are not justified. I think a more qualitative explanation may be better than the quantitative explanation given in the introduction. Alternatively, providing more information about the experiments would help, but it could lead to information duplicity, which might be undesirable.\n\n- The proposed approach does not track of the evolution of individual topological features of the persistence diagram. This could be important in scenarios requiring fine-grained topological information.\n\n- The authors claim that their method is more efficient than other methods, but computation times are not reported in the main text.\n\n- In the persistent homology paragraph (starting at line 154), the authors state that there is a decomposition in persistence diagrams (births and deaths) when using abelian groups in the chain complexes. As far as I know, this is not a trivial property and does not happen always [Theorem 2, reference 3] for abelian groups. The usual theorem applies to vector spaces. \n\n- Only one vectorization method is used, without significant justification. I would have appreciated a more detailed ablation on this point. \n\n- Maybe I'm wrong, but when I access the arxiv version of reference [23] in your paper, Table 4 contains execution times, and not SOTA results.\n\n[3] Jiajie Luo, , Gregory Henselman-Petrusek. \"Interval Decomposition of Persistence Modules over a Principal Ideal Domain.\" (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "For better readability, we restate your comments/questions in *italic*, our response(s) are marked by &#9654;\n\n*I think that the sentence in line 32 \"... due to missing correspondences between individuals...\" is hard to read. Which missing correspondences?*\n\n&#9654; By \"correspondences\" we mean that many previous works rely on being able to unambiguously identify particles across point clouds over time. We will clarify this unfortunate formulation.\n\n*The proposed approach does not track of the evolution of individual topological features of the persistence diagram. This could be important in scenarios requiring fine-grained topological information.*\n\n&#9654; Indeed, our method does not retain information about how individual homology classes evolve. We model the evolution of features *on the level of persistence diagram vectorizations*, not on the level of individual points in the persistence diagram(s). Persistence vineyards would allow us to do this, but they are (i) hard to vectorize for ML tasks, (ii) computationally expensive to compute, and (iii) often infeasible to obtain without particle correspondences over time. Following your suggestion, we will extend our discussion to include these points.\n\n*The authors claim that their method is more efficient than other methods, but computation times are not reported in the main text.*\n\n&#9654; Please see our runtime analysis in the *General Response* section.\n\n*In the persistent homology paragraph (starting at line 154), the authors state that there is a decomposition in persistence diagrams (births and deaths) when using abelian groups ...*\n\n&#9654; When sketching persistent homology in Sec. 3, our writing unintentionally and wrongly suggests that birth/death times (interval compositions) are well-defined for homology with coefficients in arbitrary Abelian groups, e.g., in the presence of torsion. We will clarify this part; thank you for pointing it out. We use coefficients in the field $\\\\mathbb{Z}/2\\\\mathbb{Z}$ (as is usual).\n\n*Only one vectorization method is used, without significant justification. I would have appreciated a more detailed ablation on this point.*\n\n&#9654; Our contribution is not within the particular realization of our core idea, but in the surprising insight that one can obtain remarkable predictive performance from capturing the dynamics of vectorized topological summaries per time point, without tracking particles, or without computationally-expensive summaries of time-varying persistence. On could, however, use other vectorization techniques; we experimented with ATOL vectorizations (which are conceptually very close to ours) and persistence images, but only observed minor changes in predictive performance. Nevertheless, ATOL vectorizations are not stable, and persistence images can be hard to parametrize (and discretize) without obtaining (very) high-dimensional descriptors.\n\n*Maybe I'm wrong, but when I access the arxiv version of reference [23] in your paper, Table 4 contains execution times, and not SOTA results.*\n\n&#9654; Thank you for pointing this out! Table 4 is meant to refer to the table in the submitted manuscript, not in the referenced work. We will rephrase our formulation.\n\n*Why do you use 𝑅2 ? I thought that it was a misleading metric [4]. This is my main concern, and the core reason why the score is a borderline accept.*\n\n&#9654; We agree, $R^2$ has known flaws, much like other metrics to assess regression performance, yet it is easy to interpret. As a summary impression across *all* predicted parameters, MSE and RMSE cannot be applied due to differing ranges of simulation parameters. We used $R^2$ and SMAPE to have two different evaluation scores. Our *joint* variant (v3) outperforms the state-of-the-art in both of these scores across all datasets, and even our PH-only variant (v1) does so on 3 out of 4 datasets.\n\nTo further reduce evaluation bias, we will add the explained variance (EV) as suggested in your ref [4] to all tables, and also list the RMSE per parameter in our appendix. Below is an example for the `dorsogna-1k` data (parameters $C,l$ are as in (Guisti et al., 2023)):\n\n| | $R^2$ $\\uparrow$  | SMAPE $\\downarrow$ | RMSE ($C$) $\\downarrow$ | RMSE ($l$) $\\downarrow$ | EV $\\uparrow$ |\n| ---|---|---|---|---|---|\n| Ours (joint, v3) | 0.930 $\\pm$ 0.003 | 0.068 $\\\\pm$ 0.003 | 0.080 $\\\\pm$ 0.005 | 0.190 $\\\\pm$ 0.004 | 0.931 $\\\\pm$ 0.004 |\n| Ours (PointNet++, v2) | 0.814 $\\\\pm$ 0.032 | 0.132 $\\\\pm$ 0.018 | 0.178 $\\\\pm$ 0.017 | 0.242 $\\\\pm$ 0.020 | 0.816 $\\\\pm$ 0.003 |\n| Ours (PH-only, v1)| 0.846 $\\\\pm$ 0.011 | 0.097 $\\\\pm$ 0.005 | 0.116 $\\\\pm$ 0.004 | 0.275 $\\\\pm$ 0.006 | 0.851 $\\\\pm$ 0.008 |\n| PSK | 0.816 $\\pm$ 0.015 | 0.096 $\\\\pm$ 0.006  | 0.112 $\\\\pm$ 0.010 | 0.305 $\\\\pm$ 0.013 | 0.819 $\\\\pm$ 0.016 |\n| Crocker Stacks | 0.743 $\\\\pm$ 0.083 | 0.150 $\\\\pm$ 0.005 | 0.156 $\\\\pm$ 0.007 | 0.331 $\\\\pm$ 0.011 | 0.746 $\\\\pm$ 0.023|\n\n*Can your approach be used for discrete dynamical systems (where time is discrete and not continuous?)*\n\n&#9654; We are confident that our method can be used with discrete systems. However, some components of our architecture may need to be adjusted; e.g., switching from neural ODEs for modeling latent dynamics to, say, LSTMs or GRUs. Thank you for this comment, we will add a remark!\n\n*Since the method is very general and can work with any kind of vectorizations, not just topological ones, is there a way to regularize or add explicit bias during training ....?*\n\n&#9654; If we understand your question correctly, you are asking whether some form of topological prior could be used to inform the neural ODE training process. This is an interesting question, and while we have not experimented with topological regularization so far, it might be possible to enforce specific topological/geometrical properties through an appropriate loss (on persistence diagrams) and differentiating through the PH computation (as done in prior works).", "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": "Ihw8jWskBN", "reviewer": "Reviewer_HnKN", "summary": "This work presents a novel approach to infer the parameters of governing equations describing the collective behavior of systems like point clouds. It leverages persistent homology to capture the topological features of the system's state. These features are then modeled using a Latent ODE system, capturing the temporal evolution of the system.  By analyzing the latent dynamics, the method can identify and regress the parameters of the underlying governing equations (e.g., PDE) that govern the system's behavior.", "questions": "- Does the model output a single point estimate for the parameters of interest or a corresponding distribution (mean and std)?\n- Can authors describe the regression objective w.r.t to underlying parameters of interest, as it is unclear from the paper, maybe including the correct equations they use?\n- Since PH does not account for the temporal changes in the point cloud, \n- How is the work different from this https://arxiv.org/abs/2406.03164? It seems like it can be subsumed as one can define a graph out of point clouds and operate on them.\n- Does the fitting of the ODE system deal with stiffness as you are fitting the topological features via an ODE, which can be non-smooth in between, leading to inefficient modeling?", "limitations": "See questions and weaknesses.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- This work proposes a novel method based on persistent homology (PH) to infer the parameters of governing equations describing the collective behavior of systems like point clouds.\n- A novel application in combining persistent homology with inverse problems.\n- The paper discusses various aspects of combining PH with modeling temporal correlations, giving an in-depth insight into the field.", "weaknesses": "- This work resonates closely with inverse problems for dynamical systems, where several works have been conducted to infer the initial parameters or parameters of interest based on observed data. Some of them are:\n\n           1. Learning to Solve PDE-constrained Inverse Problems with Graph Networks (https://arxiv.org/pdf/2206.00711)\n           2. Fully probabilistic deep models for forward and inverse problems in parametric PDEs \n           3. Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems (https://arxiv.org/pdf/2402.11722)\n\nA comparison with these methods will strengthen the aspect of this work, as these methods do not explicitly model the topological features. \n\n- I think the paper should include some additional discussion about the computational limitations of the method. While these are briefly discussed at a high level, a more quantitative analysis (e.g., comparing the effective run times and computational complexity) seems essential to give the reader a better idea of the overhead incurred when using this method.\n- Despite the capability of their model to incorporate persistent homology, enabling the use of topological features, they only utilize classical persistent homology via Rips filtrations. The potential advantages of this feature are not adequately explored in the model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "For better readability, we restate your comments/questions in *italic*, our response(s) are marked by &#9654;\n\n*This work resonates closely with inverse problems for dynamical systems, where several works have been conducted to infer the initial parameters ...*\n\n▶ Currently, we discuss related work on the inverse problem, but only in the context of models of collective behavior. We thank the reviewer for suggesting a broader perspective and comparing with prior learning-based works on inverse problems for dynamical systems in general, including the three listed ones. This will certainly be a good addition to the manuscript.\n\nRegarding refs 1-3, we note that the underlying differential equations (i.e., classic PDEs) differ from the multi-particle systems we consider. In our case, forces on each particle depend on its relative position towards all other (or its neighboring) particles, resulting in strong couplings between individuals. For specific systems (Volume exclusion), it is possible to derive parametric PDEs that describe the asymptotics of infinitely many particles (Oelschläger, 1990), and ideas from references 1-3 might be applicable. However, in general, the number $N$ of particles influences the dynamics. E.g., for the D'Orsogna model, particle distances can converge to 0 as $N\\to \\infty$.\n\n*I think the paper should include some additional discussion about the computational limitations of the method ...*\n\n&#9654; We agree; please see our *General Response* section.\n\n*Despite the capability of their model to incorporate persistent homology, enabling the use of topological features, they only utilize classical persistent homology ...*\n\n&#9654; Vietoris-Rips persistent homology is the de-facto workhorse for point cloud data. While one could use other geometric complexes (e.g., Alpha complexes as mentioned by reviewer **dbj5**) or different vectorization strategies (ATOL, persistence images, etc.), we decided to work with the most prevalent tools, but acknowledge that other choices are possible. Also, using Vietoris-Rips PH facilitates straightforward comparisons to previous topology-related works on the same problem, such as Crocker Stacks or the PSK approach.\n\nImportantly, our goal was not to assess the impact of different design choices but to introduce a latent dynamical model based on topological per-time-point descriptors for the inverse problem at hand. We are convinced that future work will explore different variants of our approach that may be more suitable in certain situations (e.g., using LSTMs in case of discrete dynamical systems, see **83FG**).\n\n*Does the model output a single point estimate for the parameters of interest or a corresponding distribution (mean and std)?*\n\n&#9654; At the moment, all models output *point estimates* of the sought-for parameters. However, one could easily sample $q_{\\\\theta}$, integrate forward, and obtain a distribution of estimates. We will add tables to the appendix that report an estimated mean & std. dev. from this sampling.\n\n*Can authors describe the regression objective w.r.t to underlying parameters of interest, ...?*\n\n&#9654; The objective is to predict the simulation parameters that led to a particular realization of a point cloud sequence. Take the PH-only variant `v1` of Fig. 1, for instance: input is a sequence of vectorized persistence diagrams $v_{\\\\tau_0}, \\\\ldots, v_{\\\\tau_n}$. The encoder $Enc_\\\\theta$ yields parameters of the approx. posterior $q_\\\\theta$ from which we sample and integrate the latent ODE forward in time to get latent states along the trajectory; this latent state sequence is then fed through $Enc_\\\\alpha$ which summarizes the sequences into a vector and linearly maps the latter to simulation parameter estimates \n$\\\\hat\\\\beta = (\\hat{\\beta}\\_1, \\ldots, \\hat\\beta\\_P)$. During training, we minimize the mean-squared-error (MSE) between the predictions and the ground-truth simulation parameters, implicitly making a Gaussian noise assumption. One could write the regression model as\n$$\\\\hat\\\\beta = \\\\phi(v_{\\\\tau_0}, \\\\ldots, v_{\\\\tau_0}) + \\epsilon, \\epsilon \\sim \\\\mathcal N(0,\\sigma I)$$\nwhere $\\phi$ subsumes all steps above. \n\n*Since PH does not account for the temporal changes in the point cloud, how is the work different from this https://arxiv.org/abs/2406.03164? ...*\n\n&#9654; As correctly pointed out, we do not track topological features on the level of *individual points* in a persistence diagram but on the level of *vectorized persistence diagrams*. The crucial difference to the arXiv paper is that the integration of neural ODEs is on the *message passing level* of a GNN for a *fixed* graph. In our context, this means that the graph object needs to be such that each vertex corresponds to the same particle at all times. However, such correspondences are unknown or hard to obtain in practice. In fact, in our work, we deliberately avoid this \"tracking\" step.\n\n*Does the fitting of the ODE system deal with stiffness as you are fitting the topological features via an ODE, which can be non-smooth in between, leading to inefficient modeling?*\n\n&#9654; This is an interesting point! However, we *do not* model the dynamics directly in the space of persistence diagrams but rather in a latent space learned from diagram vectorizations. This modeling could be inefficient to some extent, yes, but the latent ODE approach seeks to learn the most suitable latent space for the task at hand, which is to minimize prediction error for the model parameters (via the MSE) and to minimize the reconstruction error for the vectorized persistence diagrams (as part of the ELBO). This strategy is common in the literature, as, e.g., seen in the PhysioNet 2012 experiments of (Rubanova et al., 2019), where variables include binary indicators (e.g., of whether mechanical ventilation is used at available time points).", "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:
# General Response

We like to thank **all** reviewers for their overall positive feedback, their time, and their valuable comments and suggestions!

While we address all issues point by point per reviewer, we first comment on our approach's *computational aspects* and present a detailed runtime analysis as this issue has come up across multiple reviewers.

---
For reference, we refer to the following works in our rebuttal:

**(Rubanova et al., 2019)** Y. Rubanova, R.T.Q. Chen, and D. Duvenaud. Latent ODE for irregularly-sampled time series. In: NeurIPS 2019.

**(Hofer et al., 2019)** C. Hofer, R. Kwitt, and M. Niethammer. Learning representations of persistence barcodes. In: JMLR 20.126 (2019), pp. 1–45.

**(Bialek et al., 2012)** W. Bialek, A. Cavanga, I. Giardina, and A. Walczak. Statistical mechanics for natural flocks of birds. In: PNAS 109.13 (2012), pp. 4786–4791.

**(Carriere et al., 2021)** M. Carriere, F. Chazal, M. Glisse, Y. Ike, H. Kannan and Y. Umeda. Optimizing persistent homology based functions. In: ICML 2021.

**(Oelschläger, 1990)** K. Oelschläger. Large systems of interacting particles and the porous medium equation. In: Journal of Differential Equations, Volume 88, Issue 2.

---

## Runtime analysis (pre-processing)

We re-ran experiments on the `dorsogna-1k` dataset (1k sequences of length 100) and provide a breakdown of runtime below (measured on the same system as described in Appendix C of our manuscript):

First, we point out that computing Vietoris-Rips persistent homology (PH) as well as persistence diagram (PD) vectorization is done as a *pre-processing step* for all available point cloud sequences. *These steps are trivially parallelizable across multiple CPUs/GPUs.*

**Vietoris-Rips PH computation.** The following table contains wall clock time measurements (using Ripser++ on one GPU) per point cloud. We list runtime for computing $H_0$ and $H_1$ features, as well as for computing $H_0, H_1$ and $H_2$. Point clouds are of size 200, as in the manuscript:

|                              | Time per point cloud  | Overall          |
|---                           |---                    |---               |
| PH ($H_0$, $H_1$)            | 0.018 s               | $\approx$ 30 min |
| PH ($H_0$, $H_1$ and $H_2$ ) | 0.330 s               | $\approx$ 6 hrs  |

**PD vectorization.** Vectorization can essentially be broken down into two steps: (i) parameter fitting for the structure elements of (Hofer et al., 2019) and (ii) mapping PDs to vector representations using those structure elements. The following table lists the runtime for both steps on `dorsogna-1k` when vectorizing 0-, 1- and 2-dimensional persistence diagrams:

|                                | Time per diagram  | Overall        |
| ---                            | ---           | ---            |
| (i) Parameter fitting          | n/a           | $\approx$ 27 s |
| (ii) Mapping of PDs to vectors | 0.0052 s      | $\approx$ 52 s |

*PH computation & PD vectorization are both necessary steps for our approach, Crocker Stacks, as well as the PSK approach*.

## Runtime comparison to PRIOR WORK (using topological summaries)

Below, we list the overall **training times** for our approach, Crocker Stacks, and the PSK method. Runtime for pre-processing (see above) is *excluded* from these measurements. Also, PSK and Crocker Stack timings do include hyperparameter optimization, as suggested in the corresponding references. Importantly, this is *not* optional but required to obtain decent regression performance:

|                    | Time         |
| ---                | ---          |
| **Crocker stacks** | 24600 s (6 hrs 50 min) |
| **PSK**            | 646 s  |
| **Ours** (PH-only) | 190 s   |

Notably, PSK kernel computation scales quadratically with the number of sequences $n$, kernel-SVR training takes time somewhere between quadratic and cubic, hence scaling up the number of training sequences $n$ quickly becomes computationally prohibitive for the PSK method, especially in light of the required hyperparameter tuning. Finding suitable hyperparameters is also the main bottleneck for Crocker Stacks (which rely on a linear SVR).

## Runtime comparison to the BASELINE model (w/o dynamics)

We also compare the runtime of our approach to the *baseline* model which does not
explicitely model any dynamics via a latent ODE.

|                                                    | Time   |
| ---                                                | ---    |
| **Ours** (PH only)                                 | 190  s |
| **Ours** (PointNet++ only)                         | 3780 s |
| **Ours** (PH & PointNet++)                         | 4100 s |
|                                                    |        |
| **Baseline (w/o dynamics)** (PH only)              | 50 s   |
| **Baseline (w/o dynamics)** (PointNet++ only)      | 525 s  |
| **Baseline (w/o dynamics)** (PH & PointNet++)      | 600 s  |

We will include such a runtime study (across datasets) in our appendix.

Meta-review: The paper addresses the problem of modeling the dynamic evolution of point clouds based on partial observations at different time steps. This evolution is captured in a latent representation space using an encode-process-decode framework. The latent representation leverages concepts from persistent homology of point clouds and their vectorization. To achieve this, the paper employs an existing vectorization technique that is Lipschitz continuous with respect to the Wasserstein distance. The evolution of the point cloud's vectorized topological representations is modeled using a Neural-ODE-like formulation. This model is then applied to an inverse problem: inferring the parameters of a prior physical model, specifically the governing equations underlying the point cloud observations. The proposed model outperforms recent baselines, including the vision model PointNet, which operates directly in the observation space.
The main originality of the work lies in the combination of vectorized topological representations of point clouds and Neural ODEs to solve an inverse problem related to modeling collective behaviors. While the neural network architecture follows an encode-process-decode framework, which is now standard for modeling dynamical systems (e.g. PDEs), its application in the context of collective dynamics is novel.

All reviewers agree on the originality of the approach and the significance of the contribution. Modeling the evolution of point clouds using topological summaries, which capture global features, is considered a novel and interesting idea. The authors have thoroughly addressed the main concerns raised by the reviewers and added new experimental results that further reinforce the contribution. The paper would benefit from additional technical details and some rewriting to make it more accessible to readers who are not specialists in persistence diagrams and their vectorization. The authors are encouraged to consider these reviewer comments. All reviewers are in favor of acceptance.
Decision: Accept (poster)