Neural Persistence Dynamics

Authors: sebastian  zeng
florian  graf
martin  uray
stefan  huber
roland  kwitt
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 18:30:33

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.

Main paper: 
 introduction: 
 Understanding emerging behavioral patterns of a collective through the interaction of individual entities is key to elucidate many phenomena in nature on a macroscopic and microscopic scale. 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.e., identifying the model parameters from the data, turns out to be inherently difficult. Confounding factors include the often large number of observed entities and the difficulty of reliably identifying individual trajectories across point clouds at possibly non-equidistant observation times.However, as several works [4, 23, 52] have recently demonstrated, it may not be necessary to rely on individual trajectories for parameter identification. In fact, collective behavior is characterized by global patterns that emerge from local interactions, and we observe the emergence of these patterns through changes to the "shape" of point clouds over time. For instance, birds may form a flock, split 38th Conference on Neural Information Processing Systems (NeurIPS 2024). into groups, and then merge again. This perspective has prompted the idea of summarizing such topological events over time and then phrasing model identification as a downstream prediction/regression 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 a crucial role. Yet, despite the often remarkable performance of topological approaches in terms of parameter identification for models of collective behavior, it remains unclear in which situations they are preferable over more traditional (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 a snapshot of one ablation experiment from Sec. 4 . In particular, Tbl. 1 compares the parameter regression performance of our proposed approach, using PH, to a variant where we instead use PointNet++ [45] representations. Referring to Fig. 1 , this means that the v τi are computed by a PointNet++ model. As can be seen in Tbl. 1, relying on representations computed via PH works well on both datasets, while PointNet++ representations fail on one dataset (vicsek-10k), but indeed perform reasonably well on the other (dorsogna-1k).In light of this observation, it is worth emphasizing that there is a 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: in 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.Our key idea is to learn a generative model that can reproduce topological summaries 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 to a 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.  
related work: 
 Our work is partially related to the literature on learning with dynamic point clouds in vision problems, but primarily connects to 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 accommodate 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 a 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 as input to 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 a conceptually similar strategy as [52, 57] , introducing a path signature kernel (PSK) for sequences of summary representations of persistence diagrams.Along a different line of research, [28, 29] propose formigrams as summaries of dynamic metric data, encoding the evolution of connected components. In subsequent work, [30] construct multidimensional (i.e., spatiotemporal) persistent homology modules from dynamic metric data and compare invariants of these modules. While these works provide important theoretical stability results for zero-dimensional 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 persistent homology [36] in combination with [30] (who essentially construct multiparameter persistence modules) might, in the future, constitute a 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 a 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 a 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 fMRI 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 characteristics of collective behavior, there is a second line of closely related work on inferring interaction laws (see, e.g., [2, 5, 6, 27, 39, 61] ), ranging from metric-distance-based models and topological interaction models to non-parametric 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 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 R 3m ). In contrast, our approach only requires positional information and can handle larger point clouds. Furthermore, although we only present results on predicting parameters for a class of a 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 as 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'Orsogna model [18] , particle distances can collapse to zero as the number of particles tends to infinity.Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: Sec. 5 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 are encouraged to create a 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, e.g., if the approach was only tested on a few datasets or with a 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, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to 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 fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a 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 important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
method: 
 Below, we present our method with a 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 can 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 = {x 1 , . . . , x M } ⊂ R 3 denotes a point cloud with M points and d(x, y) = xy denotes the Euclidean metric. Point clouds may be indexed by t i (or τ i ) to highlight the dependence on time t i . If necessary, we clearly distinguish between τ i as a time point with an available observation, and t i as a general placeholder for time.Problem statement. Given a sequence of point clouds P τ0 , . . . , P τ N , observed at possibly nonequidistant time points τ i , we (1) 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 a small number of parameters β 1 , . . . , β P that control the motions dx i /dt of individual points x i and specify (local) interactions among neighboring points.As preparation for our model description, we first briefly establish how one may extract topological features from a 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 a simplicial complex, and studies its homology across multiple scales. The most relevant construction for our purposes is the Vietoris-Rips complex Rips(P) δ , with vertex set P. This complex includes an m-simplex [x 0 , . . . , x m ] iff d(x i , x j ) ≤ δ for all 0 ≤ i, j ≤ m at a given threshold δ. The "shape" of this complex can then be studied using homology, a tool from algebraic topology, with zerodimensional homology (H 0 ) encoding information about connected components, one-dimensional homology (H As different features may be born and die at the same time, the collection of (birth, death) tuples is a multiset of points, often represented in the form of a persistence diagram, which we denote as dgm k (Rips(P)). Here, the notation Rips(P) refers to the full filtration and dgm k indicates that we have tracked k-dimensional homological features throughout this filtration.Clearly this construction does not account for any temporal changes to a point cloud, but reveals features that are present at a specific time point. Furthermore, due to the inconvenient multiset structure of persistence diagrams for learning problems, one typically resorts to appropriate vectorizations (see, e.g., [1, 7, 10, 26] ), all accounting for the fact that points close to the diagonal ∆ = {(x, x) : x ∈ R} contribute less (0 at ∆) 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 P ti asEQUATIONwhere v ti,k ∈ R d and d is controlled by hyperparameter(s) of the chosen vectorization technique.For brevity, we omit the subscript k when referring to vectorizations unless necessary.Remark 1. As our goal is to model the dynamics of vectorized persistence diagrams using a 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/Continuity 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 ∆. 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 formEQUATIONwhere W 1 denotes the (1, 2)-Wasserstein distance as defined in [51, Def. 2.7], F, G are two persistence diagrams and K > 0 is a 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 an alternative: in our particular case of Vietoris-Rips persistent homology, one can upper bound W 1 in terms of the standard point set Wasserstein distance W 1 . Specifically, upon relying on a stable vectorization technique, we obtain the inequality chainEQUATIONwhereF k = dgm k (Rips(P)), G k = dgm k (Rips(Q))denote the k-dimensional persistence diagrams 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 a somewhat simpler unobserved/latent dynamic process in R z with z d. We model this latent dynamic process via a neural ODE [13, 49] , learned in a variational Bayes regime1 [32] . In this setting, one chooses a recognition/encoder network (Enc θ ) to parametrize an approximate variational posterior q θ (z t0 |{v τi } i ), an ODE solver Ours (PH-only) Ours (PointNet++) Ours (joint) Baseline (PH-only) Baseline (PointNet++) (v1) (v2) (v3) (v4) (v5) β1, . . . , βP [rτ 0 , vτ 0 ], . . . , [rτ N , vτ N ] vτ 0 , . . . , vτ N cat Regα Pτ 0 , . . . , Pτ N rτ 0 , . . . , rτ N PointNet++ q θ (zt 0 |{[vτ i , rτ i ]}i) Enc θ Regα β1, . . . ,EQUATIONDifferent to [49] , we do not implement the recognition/encoder network via another neural ODE, but rather choose an attention-based approach (mTAN) [50] which can even be used in a standalone manner as a strong baseline (see Sec. 4). In our implementation, the recognition network yields the parametrization (µ, Σ) of a multivariate Gaussian in R z with diagonal covariance, and the prior is a standard Gaussian N (0, I z ). Furthermore, the ODE solver (e.g., Euler) can yield z ti at any desired t i , however, we can only evaluate the ELBO at observed time points τ i .Regression objective. To realize our downstream regression task, i.e., predicting parameters β 1 , . . . , β P of an underlying governing equation for collective behavior (see Fig. 3 ) from a 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, it is reasonable to tie the simulation parameter estimates β1 , . . . , βP to the latent path {z ti } i via a regression network Reg α that accepts {z ti } i as input. In particular, we re-use the attention-based encoder architecture Enc θ to allow attending to different parts of this path. However, different to Enc θ , which parametrizes the approximate posterior from observations at time points τ i , the regression network Reg α (with its own set of parameters α) receives latent states z ti at equidistant t i ∈ [0, T ], then summarizes this sequence into a vector and linearly maps the latter to β1 , . . . , βP , cf. Fig. 2 . As in [49] , for training, we extend the ELBO objective of Eq. ( 4) by an additive auxiliary regression loss, such as the mean-squared error (MSE), between the predictions βp and the ground truth β p , implicitly making a Gaussian noise assumption.A schematic overview of our Neural Persistence Dynamics approach is shown in Fig. 1 . 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 a more profound impact, and we have already identified some recommended choices above (based on our experiments in Sec. 4). This includes (1) 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 we choose an attention-based 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.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? 
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, we considerD'Orsogna (dorsogna) [18] dxi dt = vi, m dvi dt = α-β|vi| 2 vi- 1 M ∇x i M j=1,j =i U ( xi-xj ), U (r) = Cre -r lr -Cae -r la (5)Vicsek (vicsek) [20, 56] EQUATIONVolume exclusion [40] (volex) variant from [21] three different models in R 3 : D'Orsogna [18] , Vicsek [20] and volume exclusion [40] , using the publicly-available implementations in the SiSyPHE library [21] 2 . 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, are highlighted in red. We sample these parameters, as specified in Appendix A, to cover a wide range of macroscopic behavioral regimes.EQUATIONFor each sampled parameter configuration, we simulate one sequence of point clouds, to a total of 10,000 sequences per model. All simulations are run for 1,000 time steps (with step size 0.01, starting at t = 0) on point clouds of size M = 200. We take every 10th time step as an observation, yielding observation sequences of length 100. At t = 0, points are drawn independently and uniformly in [-0.5, 0.5] 3 , and initial velocities are uniformly distributed on the unit sphere. For direct comparison to [23] , we also simulated their parameter configuration of the D'Orsogna model. This setup yields four datasets: dorsogna-1k (from [23] ), dorsogna-10k, vicsek-10k and volex-10k.Implementation. The model variants from Fig. 2 can be realized in many ways. Below, we specify the configuration that was used to run the experiments. For the encoder Enc θ , as well as the regression network Reg α , we use the attention-based mTAN architecture [50] . As decoder network Dec γ , 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 a weight decay of 0.001), starting at a learning rate of 0.001 (decaying according to a cosine annealing schedule) and MSE as a reconstruction (i.e., to evaluate the first term in Eq. ( 4)) and regression loss. In case a model uses topological features, we use Ripser++ [59] 2)]. To model the dynamics, we fix the latent space dimensionality to z = 20 and scale the ODE integration time to [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://github.com/plus-rkwitt/neural_persistence_dynamics.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 of missing and unevenly spaced observations, we train three models (per split) using only a fraction (i.e., 20%, 50%, and 80%) of 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 [0, 1]; higher is better ↑) and the symmetric mean absolute percentage error (SMAPE) (in [0, 1]; lower is better ↓). 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 βi in Fig. 2 ). Finally, when reporting results on a 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 a Mann-Whitney test at 5% significance and correcting for multiple comparisons).In our ablation study, we 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 results for the vicsek-10k and dorsogna-1k data. While using PointNet++ representations on vicsek-10k yields rather poor performance, combining them with representations computed from PH is at least not detrimental. On the other hand, on dorsogna-1k, 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 literature, it is rarely as pronounced as in this particular experiment. Hence, we will stick to the combination of PH+PointNet++ representations (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. 2 against using Reg α operating directly on concatenated point cloud representations from PH and PointNet++ (i.e., a combination of variants v4 & v5 from Fig. 2 ). The latter approach constitutes an already strong baseline (cf. [58] ) as Reg α incorporates an attention mechanism that allows attending to relevant parts of each sequence. For a fair comparison, we increase the size of Reg α to approximately match the number of parameters to our latent dynamic model. As in Tbl. 2, we list results for the vicsek-10k and dorsogna-1k data in Tbl. 3. First, we see that explicitly modeling the latent dynamics is beneficial, with considerable improvements in the reported scores across both datasets. While differences are quite prominent on the dorsogna-1k 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.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 t 0 = 0), which corresponds to observing the emergence of patterns out of a random yet qualitatively similar, initial configuration of points. In the following experiment, we investigate the impact of extracting observation sequences with starting points randomly spread across a much longer simulation time. This is relevant as it is unlikely to observe uniform initial positions in any data obtained from realworld experiments. To this end, we create variations of the dorsogna-10k data by progressively increasing the simulation time T 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, we also list results for the PSK approach 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 on 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 . 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 a baseline of what could be achieved in the best case. Note that we 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-validate the discretization parameters, see Appendices B.1 and B.2. Tbl. 4 lists the corresponding results. Aside from the observation that our approach largely outperforms the state-of-the-art across all parameter regression tasks, we also highlight that our model is trained with exactly the same hyperparameters on all datasets.Remark 3. A closer look at the volex-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 or 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] to account for such cases, but this might require a case-by-case analysis and possibly include additional assumptions. 
discussion: 
 We introduced a framework to 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 a 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 to no parameter tuning, and vastly outperforms the current state-of-the-art, 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. Limitation(s). One obvious limitation in the presented implementation is the reliance on Vietoris-Rips persistent homology of point clouds. In fact, the underlying simplicial complex will become prohibitively large (especially for H 2 ) once we scale up the number of points by, e.g., an order of magnitude. Using an Alpha [22] or a Witness complex [19] might be a viable alternative to mitigate this issue. Similarly, one may explore subsampling strategies for persistent homology, as in [11] , and learn a latent ODE from a 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 be carefully reviewed. This is especially advisable when drawing conclusions from missing measurements, as these are based on imputations and may be biased. To create the dorsogna-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 on the ratios C r /C a and l r /l a , cf. The dorsogna-1k dataset is created by re-running the simulation provided as part of the public (Julia) implementation3 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 a larger dataset, but still one magnitude smaller than dorsogna-10k, vicsek-10k and volex-10k. For this dataset, particle masses are m = 1, propulsion is α = 1, C a = l a = 1 and C r , l r vary uniformly in [0.1, 2], and are selected if the generated point clouds satisfy a certain scale condition, leading to the parameter pairs illustrated in [23, Fig. 6 ].For the vicsek-10k dataset, we sample R, c, ν uniformly from U [0.5,5] and D from U [0,2] . Overall, this gives four parameters that need to be predicted. For reproducibility, we will release the simulation data publicly.B.1 Path Signature Kernel (PSK) [23] For the PSK, we rely on the publicly available implementation in sktime4 [38] . We compute the PSK (truncated at 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 a kernel support vector regressor (kernel-SVR).Following the experimental protocol in [23, Sec. 7 .3], we cross-validate the sliding window embedding (across lags [1, 2, 3] ) and kernel-SVR hyperparameters on a 20% validation portion of the training data, with hyperparameters selected based on the average MSE (per governing equation parameter β i 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) is performed, and the reported performance can be considered an optimistic estimate of what can be achieved with the PSK.As mentioned in Sec. 4.4, we compare against crocker stacks, introduced in [57] , as one of the state-of-the-art approaches. Crocker stacks are an extension to crocker plots [52] and constitute a topological summary for time-varying persistence diagrams. For the crocker stacks, we adapted the publicly available implementation of the crocker plots in the teaspoon5 library.The crocker plot is computed as follows: for each time step, the persistence diagrams are computed up to a scale parameter ε. In discretized steps of the scale parameter, the Betti numbers are computed from the persistence diagrams, which results in a 2D representation (i.e., vs. Betti number) for each homology dimension.The extension to crocker stacks is achieved by adding a third dimension k, induced by the smoothing factor α. For given steps of α, a smoothing operation is applied, i.e., values within a specified distance (smoothing factor) from the diagonal of the persistence diagram are ignored. Hence, for each homology dimension, a crocker stack is a tensor in R 3 , with axes corresponding to discretizations of the (1) scale parameter ε ∈ [0, ∞), the (2) time t ∈ [0, T ], and the (3) smoothing factor α ∈ [0, ∞).We note that in [57] , the authors set the scale parameter to δ = 0.35, as they use data normalized to the range of 0 to 1. We did not scale the data, however, we adjusted the scale parameter correspondingly:for all experiments, we set the scale parameter to δ = 1 /3 • maxPers(dgm k (Rips(P))), where P denotes the point cloud, and maxPers(dgm k (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 δ (see above), where the computations are discretized with 25 equally spaced values in [0, ε];(2) the smoothing values are discretized with 18 steps equally spaced in [0, 0.5 • maxPers].Eventually, the crocker stacks per homology dimension are vectorized (i.e., the tensor is flattened) and concatenated into a single vector per observation sequence. These vectorizations are then input to a 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) are tuned using Bayesian optimization. Each parameter configuration is evaluated using a 5-fold cross-validation, and the best configuration is then used to train the final model on the full dataset.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 H 0 -features only, the regression quality drops. When additionally including H 2 -features (i.e., H 0 , H 1 and H 2 ), the situation is less clear, as the results are not noticeably different. Quantitative results can be found in Tbl. 5 below.Table 5 : Quantitative assessment of the impact of including higher-dimensional homology features on downstream regression performance, evaluated on dorsogna-1k.Ours (PH-only, v1; H0) 0.819 ± 0.015 0.101 ± 0.003 Ours (PH-only, v1; H0, H1) 0.844 ± 0.021 0.098 ± 0.007 Ours (PH-only, v1; H0, H1, H2) 0.846 ± 0.011 0.097 ± 0.005All experiments were run on an Ubuntu Linux system (22.04), running kernel 5.15.0-100-generic, with 34 Intel® Core™ i9-10980XE CPU @ 3.00GHz cores, 128 GB of main memory, and two NVIDIA GeForce RTX 3090 GPUs.Table 6 : Runtime comparison for PH computation for different homology dimensions. Reported is the average runtime per point cloud (in seconds), and the overall runtime (in seconds) estimated from this average by multiplying by the number of processed point clouds on dorsogna-1k.PH computation (H0, H1) 0.018 s 1800 s PH computation (H0, H1, H2) 0.330 s 33000 s 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 on 50,000 points uniformly sampled points from those diagrams. Comparison to prior work. Next, we compare the runtime of our approach (here, variant v1 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 quadratically 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 prohibitive, 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).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 (H 0 and H 1 ) 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 (PH+PointNet++, v3) already yields strong performance across all parameter prediction problems. 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 in the paper and important assumptions and limitations. 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 Appendix:  


Decision: 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.

Reviews: review: paper_summary: paper_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. 

The 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.

*References:*
- Latent ODEs for Irregularly-Sampled Time Series. Rubanova et al. (2019)
strength_weakness: strength_weakness: 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. 

It 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)? 

It’d be nice if the authors could comment on the applicability of their method to empirical data. 

*References:*
- Statistical mechanics for natural flocks of birds. Bialek et al. (2012)
questions: 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?

2. How did you parametrize the decoder network? I don’t find it in the manuscript.

3. 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.

4. 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).

*References:*
- Augmented neural odes.  Dupont et al. (2019)
- How to train your neural ode: the world of jacobian and kinetic regularization. Finlay et al. (2020)
- Ode to an ode. Choromanski et al. (2020)
- Opening the blackbox: Accelerating neural differential equations by regularizing internal solver heuristics. Pal et al. (2021)
limitations: limitations: Yes, they did address the limitations of their method.
paper_summary: paper_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 $X(\tau_0),\dots,X(\tau_N)$. 

The 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 $\theta$. 

The 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. 

The 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.
strength_weakness: strength_weakness: 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.
For, 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. 

I 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.

Adding 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).
questions: 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?

2. (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? 

3. 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.

4. 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/ ).


Ref : 

[1] 	RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds, de Surrel et al.
limitations: limitations: NA
paper_summary: paper_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.
strength_weakness: strength_weakness: - I think that the sentence in line 32 "... due to missing correspondences between individuals..." is hard to read. Which missing correspondences?

-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.

- 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.

- The authors claim that their method is more efficient than other methods, but computation times are not reported in the main text.

- 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. 

- Only one vectorization method is used, without significant justification. I would have appreciated a more detailed ablation on this point. 

- 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.

[3] Jiajie Luo, , Gregory Henselman-Petrusek. "Interval Decomposition of Persistence Modules over a Principal Ideal Domain." (2023).
questions: 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.
- Can your approach be used for discrete dynamical systems (where time is discrete and not continuous?)
- 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?

[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: 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.
paper_summary: paper_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.
strength_weakness: strength_weakness: - 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:

           1. Learning to Solve PDE-constrained Inverse Problems with Graph Networks (https://arxiv.org/pdf/2206.00711)
           2. Fully probabilistic deep models for forward and inverse problems in parametric PDEs 
           3. Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems (https://arxiv.org/pdf/2402.11722)

A comparison with these methods will strengthen the aspect of this work, as these methods do not explicitly model the topological features. 

- 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.
- 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.
questions: questions: - Does the model output a single point estimate for the parameters of interest or a corresponding distribution (mean and std)?
- 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?
- 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? It seems like it can be subsumed as one can define a graph out of point clouds and operate on them.
- 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: limitations: See questions and weaknesses.

Comments: title: Thanks
comment: We would like to kindly ask, at the end of this rebuttal phase, whether our response addressed your questions/concerns or whether we can provide any further clarifications.
We wanted to kindly ask, at the end of this rebuttal phase, whether the other reviewer's comments/answers to our response have helped in your deliberation or whether we can provide some further "short-term" answers to any remaining open issues.
I thank the authors for their detailed responses. After reading the reviews from other reviewers and your replies to them, I have decided to increase my score.

I believe that, if the updated manuscript addresses all the points raised during this discussion session, it will make a valuable contribution.
Really good answer, thank you very much. You addressed all my comments. Although I think it could have been really interesting to add individual tracking of persistent homology features (e.g., using vineyards, as you propose), I think the paper should be accepted for the conference. Also, I would love to see a continuation of this paper addressing discrete dynamical systems. I know several problems where this could be used successfully. In general, I'm really excited about this particular direction of applied TDA, so I'm increasing my score to 6. Congratulations and thank you very much again!
Thank you for taking time answering my review, giving experimental details and clarifying the content of the paper. 

I like that $H_1 + H_0 > H_0$, that's somewhat of a "good new" for TDA :-)

I still need some time to read other reviews / comments and discuss with other reviewers to make my mind clear but I feel more positive about this work.