arXiv subjects

Julien Tierny

Publications and source records attributed to Julien Tierny.

At least 19 recordsLinked to original sources

arXiv2026

Sierpi\'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation

This paper introduces the Sierpi\'nski-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpi\'nski-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in $O(N\log N)$ steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical $2$-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted $W_\Gamma$, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of $d_{\mathrm{SK}}$ over state-of-the-art approximations of $W_2$ is $626\times$, while the aggregate speedup over the full benchmark is $2100\times$. Average-linkage partitions obtained from $d_{\mathrm{SK}}$ and $W_\Gamma$ each exactly match the corresponding $W_2$ partition on 8 of the 12 collections. Hilbert $k$-means and Gaussian spectral clustering, both based on $d_{\mathrm{SK}}$, achieve mean adjusted Rand indices (ARI) of $0.756$ and $0.800$, respectively, with respect to the benchmark reference partitions, compared to $0.750$ obtained by average linkage on $W_2$. The Gaussian $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

cs.CG↗

arXiv2026

DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging

This application paper presents DebrisTracer, a framework for the reliable tracking of debris in hypervelocity impact fast imaging. These noisy and highly specific datasets capture the ejection of a large number of debris fragments after the impact of a projectile launched at hypervelocity into a target material. The reliable estimation of debris mass and speed distributions is of major importance in aerospace applications. We document how to extend an off-the-shelf topology tracking framework based on critical point extraction and matching, in order to incorporate domain knowledge and physical assumptions. Our approach automatically produces an accurate and reliable debris tracking, enabling an interpretable visual analysis of this complex space-time phenomenon. Extensive experiments demonstrate the accuracy improvements provided by our approach over established tools used by domain experts in terms of physical validation, specifically via the prediction of the experimental ejected mass and crater depth profiles. We illustrate the utility of our approach across several use cases (with varying impact angles and physics). We show that our statistical summaries enable the visual identification of distinct regimes within the debris population, corroborating and refining prior expectations of domain experts. Our database and our C++ implementation are available at this address: https://github.com/tloloum/DebrisTracer.

cs.LG↗

arXiv2025

Delaunay-Rips filtration: a study and an algorithm

The Delaunay-Rips filtration is a lighter and faster alternative to the well-known Rips filtration for low-dimensional Euclidean point clouds. Despite these advantages, it has seldom been studied. In this paper, we aim to bridge this gap by providing a thorough theoretical and empirical analysis of this construction. From a theoretical perspective, we show how the persistence diagrams associated with the Delaunay-Rips filtration approximate those obtained with the Rips filtration. Additionally, we describe the instabilities of the Delaunay-Rips persistence diagrams when the input point cloud is perturbed. Finally, we introduce an algorithm that computes persistence diagrams of Delaunay-Rips filtrations in any dimension. We show that our method is faster and has a lower memory footprint than traditional approaches in low dimensions. Our C++ implementation, which comes with Python bindings, is available at https://github.com/MClemot/GeoPH.

cs.CG↗

arXiv2025

Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams

We introduce the Continuous Edit Distance (CED), a geodesic and elastic distance for time-varying persistence diagrams (TVPDs). The CED extends edit-distance ideas to TVPDs by combining local substitution costs with penalized deletions/insertions, controlled by two parameters: $\alpha$ (trade-off between temporal misalignment and diagram discrepancy) and $\beta$ (gap penalty). We also provide an explicit construction of CED-geodesics. Building on these ingredients, we present two practical barycenter solvers, one stochastic and one greedy, that monotonically decrease the CED Frechet energy. Empirically, the CED is robust to additive perturbations (both temporal and spatial), recovers temporal shifts, and supports temporal pattern search. On real-life datasets, the CED achieves clustering performance comparable or better than standard elastic dissimilarities, while our clustering based on CED-barycenters yields superior classification results. Overall, the CED equips TVPD analysis with a principled distance, interpretable geodesics, and practical barycenters, enabling alignment, comparison, averaging, and clustering directly in the space of TVPDs. A C++ implementation is provided for reproducibility at the following address https://github.com/sebastien-tchitchek/ContinuousEditDistance.

cs.CG↗

arXiv2025

Robust Barycenters of Persistence Diagrams

This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the $q$-Wasserstein distance $W_q$ when $q=2$. We adapt an alternative fixed-point method to compute a barycenter diagram for generic transportation costs ($q > 1$), in particular those robust to outliers, $q \in (1,2)$. We show the utility of our work in two applications: \emph{(i)} the clustering of persistence diagrams on their metric space and \emph{(ii)} the dictionary encoding of persistence diagrams. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter .

cs.LG↗

arXiv2025

Topology Aware Neural Interpolation of Scalar Fields

This paper presents a neural scheme for the topology-aware interpolation of time-varying scalar fields. Given a time-varying sequence of persistence diagrams, along with a sparse temporal sampling of the corresponding scalar fields, denoted as keyframes, our interpolation approach aims at "inverting" the non-keyframe diagrams to produce plausible estimations of the corresponding, missing data. For this, we rely on a neural architecture which learns the relation from a time value to the corresponding scalar field, based on the keyframe examples, and reliably extends this relation to the non-keyframe time steps. We show how augmenting this architecture with specific topological losses exploiting the input diagrams both improves the geometrical and topological reconstruction of the non-keyframe time steps. At query time, given an input time value for which an interpolation is desired, our approach instantaneously produces an output, via a single propagation of the time input through the network. Experiments interpolating 2D and 3D time-varying datasets show our approach superiority, both in terms of data and topological fitting, with regard to reference interpolation schemes. Our implementation is available at this GitHub link : https://github.com/MohamedKISSI/Topology-Aware-Neural-Interpolation-of-Scalar-Fields.git.

cs.LG↗

arXiv2025

Distributed Discrete Morse Sandwich: Efficient Computation of Persistence Diagrams for Massive Scalar Data

The persistence diagram, which describes the topological features of a dataset, is a key descriptor in Topological Data Analysis. The "Discrete Morse Sandwich" (DMS) method has been reported to be the most efficient algorithm for computing persistence diagrams of 3D scalar fields on a single node, using shared-memory parallelism. In this work, we extend DMS to distributed-memory parallelism for the efficient and scalable computation of persistence diagrams for massive datasets across multiple compute nodes. On the one hand, we can leverage the embarrassingly parallel procedure of the first and most time-consuming step of DMS (namely the discrete gradient computation). On the other hand, the efficient distributed computations of the subsequent DMS steps are much more challenging. To address this, we have extensively revised the DMS routines by contributing a new self-correcting distributed pairing algorithm, redesigning key data structures and introducing computation tokens to coordinate distributed computations. We have also introduced a dedicated communication thread to overlap communication and computation. Detailed performance analyses show the scalability of our hybrid MPI+thread approach for strong and weak scaling using up to 16 nodes of 32 cores (512 cores total). Our algorithm outperforms DIPHA, a reference method for the distributed computation of persistence diagrams, with an average speedup of x8 on 512 cores. We show the practical capabilities of our approach by computing the persistence diagram of a public 3D scalar field of 6 billion vertices in 174 seconds on 512 cores. Finally, we provide a usage example of our open-source implementation at https://github.com/eve-le-guillou/DDMS-example.

cs.DC↗

arXiv2025

BondMatcher: H-Bond Stability Analysis in Molecular Systems

This application paper investigates the stability of hydrogen bonds (H-bonds), as characterized by the Quantum Theory of Atoms in Molecules (QTAIM). First, we contribute a database of 4544 electron densities associated to four isomers of water hexamers (the so-called Ring, Book, Cage and Prism), generated by distorting their equilibrium geometry under various structural perturbations, modeling the natural dynamic behavior of molecular systems. Second, we present a new stability measure, called bond occurrence rate, associating each bond path present at equilibrium with its rate of occurrence within the input ensemble. We also provide an algorithm, called BondMatcher, for its automatic computation, based on a tailored, geometry-aware partial isomorphism estimation between the extremum graphs of the considered electron densities. Our new stability measure allows for the automatic identification of densities lacking H-bond paths, enabling further visual inspections. Specifically, the topological analysis enabled by our framework corroborates experimental observations and provides refined geometrical criteria for characterizing the disappearance of H-bond paths. Our electron density database and our C++ implementation are available at this address: https://github.com/thom-dani/BondMatcher.

cs.LG↗

arXiv2025

Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the $0$-dimensional persistent homology (PH$^0$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH$^d$ for $d\ge 1$. Based on this observation, we introduce a novel generalization of TopoAE to $1$-dimensional persistent homology (PH$^1$), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the $2$-chains filling persistent $1$-cycles, hence resulting in more faithful geometrical reconstructions of the $1$-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

cs.CG↗

arXiv2025

A User's Guide to Sampling Strategies for Sliced Optimal Transport

This paper serves as a user's guide to sampling strategies for sliced optimal transport. We provide reminders and additional regularity results on the Sliced Wasserstein distance. We detail the construction methods, generation time complexity, theoretical guarantees, and conditions for each strategy. Additionally, we provide insights into their suitability for sliced optimal transport in theory. Extensive experiments on both simulated and real-world data offer a representative comparison of the strategies, culminating in practical recommendations for their best usage.

cs.LG↗

arXiv2024

Identifying Locally Turbulent Vortices within Instabilities

This work presents an approach for the automatic detection of locally turbulent vortices within turbulent 2D flows such as instabilites. First, given a time step of the flow, methods from Topological Data Analysis (TDA) are leveraged to extract the geometry of the vortices. Specifically, the enstrophy of the flow is simplified by topological persistence, and the vortices are extracted by collecting the basins of the simplified enstrophy's Morse complex. Next, the local kinetic energy power spectrum is computed for each vortex. We introduce a set of indicators based on the kinetic energy power spectrum to estimate the correlation between the vortex's behavior and that of an idealized turbulent vortex. Our preliminary experiments show the relevance of these indicators for distinguishing vortices which are turbulent from those which have not yet reached a turbulent state and thus known as laminar.

physics.flu-dyn↗

arXiv2024

Distributed Path Compression for Piecewise Linear Morse-Smale Segmentations and Connected Components

This paper describes the adaptation of a well-scaling parallel algorithm for computing Morse-Smale segmentations based on path compression to a distributed computational setting. Additionally, we extend the algorithm to efficiently compute connected components in distributed structured and unstructured grids, based either on the connectivity of the underlying mesh or a feature mask. Our implementation is seamlessly integrated with the distributed extension of the Topology ToolKit (TTK), ensuring robust performance and scalability. To demonstrate the practicality and efficiency of our algorithms, we conducted a series of scaling experiments on large-scale datasets, with sizes of up to 4096^3 vertices on up to 64 nodes and 768 cores.

cs.DC↗

arXiv2024

Localized Evaluation for Constructing Discrete Vector Fields

Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

cs.GR↗

arXiv2024

A Practical Solver for Scalar Data Topological Simplification

This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of "signal" persistence pairs to maintain, our approach produces an output field g that is close to f and which optimizes (i) the cancellation of "non-signal" pairs, while (ii) preserving the "signal" pairs. In contrast to pre-existing simplification algorithms, our approach is not restricted to persistence pairs involving extrema and can thus address a larger class of topological features, in particular saddle pairs in three-dimensional scalar data. Our approach leverages recent generic persistence optimization frameworks and extends them with tailored accelerations specific to the problem of topological simplification. Extensive experiments report substantial accelerations over these frameworks, thereby making topological simplification optimization practical for real-life datasets. Our approach enables a direct visualization and analysis of the topologically simplified data, e.g., via isosurfaces of simplified topology (fewer components and handles). We apply our approach to the extraction of prominent filament structures in three-dimensional data. Specifically, we show that our pre-simplification of the data leads to practical improvements over standard topological techniques for removing filament loops. We also show how our approach can be used to repair genus defects in surface processing. Finally, we provide a C++ implementation for reproducibility purposes.

cs.LG↗

arXiv2023

TTK is Getting MPI-Ready

This system paper documents the technical foundations for the extension of the Topology ToolKit (TTK) to distributed-memory parallelism with the Message Passing Interface (MPI). While several recent papers introduced topology-based approaches for distributed-memory environments, these were reporting experiments obtained with tailored, mono-algorithm implementations. In contrast, we describe in this paper a versatile approach (supporting both triangulated domains and regular grids) for the support of topological analysis pipelines, i.e. a sequence of topological algorithms interacting together. While developing this extension, we faced several algorithmic and software engineering challenges, which we document in this paper. We describe an MPI extension of TTK's data structure for triangulation representation and traversal, a central component to the global performance and generality of TTK's topological implementations. We also introduce an intermediate interface between TTK and MPI, both at the global pipeline level, and at the fine-grain algorithmic level. We provide a taxonomy for the distributed-memory topological algorithms supported by TTK, depending on their communication needs and provide examples of hybrid MPI+thread parallelizations. Performance analyses show that parallel efficiencies range from 20% to 80% (depending on the algorithms), and that the MPI-specific preconditioning introduced by our framework induces a negligible computation time overhead. We illustrate the new distributed-memory capabilities of TTK with an example of advanced analysis pipeline, combining multiple algorithms, run on the largest publicly available dataset we have found (120 billion vertices) on a cluster with 64 nodes (for a total of 1536 cores). Finally, we provide a roadmap for the completion of TTK's MPI extension, along with generic recommendations for each algorithm communication category.

cs.DC↗

arXiv2023

Merge Tree Geodesics and Barycenters with Path Mappings

Comparative visualization of scalar fields is often facilitated using similarity measures such as edit distances. In this paper, we describe a novel approach for similarity analysis of scalar fields that combines two recently introduced techniques: Wasserstein geodesics/barycenters as well as path mappings, a branch decomposition-independent edit distance. Effectively, we are able to leverage the reduced susceptibility of path mappings to small perturbations in the data when compared with the original Wasserstein distance. Our approach therefore exhibits superior performance and quality in typical tasks such as ensemble summarization, ensemble clustering, and temporal reduction of time series, while retaining practically feasible runtimes. Beyond studying theoretical properties of our approach and discussing implementation aspects, we describe a number of case studies that provide empirical insights into its utility for comparative visualization, and demonstrate the advantages of our method in both synthetic and real-world scenarios. We supply a C++ implementation that can be used to reproduce our results.

cs.GR↗

arXiv2023

Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)

This paper presents a computational framework for the Wasserstein auto-encoding of merge trees (MT-WAE), a novel extension of the classical auto-encoder neural network architecture to the Wasserstein metric space of merge trees. In contrast to traditional auto-encoders which operate on vectorized data, our formulation explicitly manipulates merge trees on their associated metric space at each layer of the network, resulting in superior accuracy and interpretability. Our novel neural network approach can be interpreted as a non-linear generalization of previous linear attempts [79] at merge tree encoding. It also trivially extends to persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our algorithms, with MT-WAE computations in the orders of minutes on average. We show the utility of our contributions in two applications adapted from previous work on merge tree encoding [79]. First, we apply MT-WAE to merge tree compression, by concisely representing them with their coordinates in the final layer of our auto-encoder. Second, we document an application to dimensionality reduction, by exploiting the latent space of our auto-encoder, for the visual analysis of ensemble data. We illustrate the versatility of our framework by introducing two penalty terms, to help preserve in the latent space both the Wasserstein distances between merge trees, as well as their clusters. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used for reproducibility.

cs.LG↗

arXiv2023

Wasserstein Dictionaries of Persistence Diagrams

This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [100], [102] of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

cs.LG↗