Steve Oudot
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View article: Counts and end-curves in two-parameter persistence
Counts and end-curves in two-parameter persistence Open
Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of ba…
View article: Estimating the persistent homology of $\mathbb{R}^n$-valued functions using function-geometric multifiltrations
Estimating the persistent homology of $\mathbb{R}^n$-valued functions using function-geometric multifiltrations Open
Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered i…
View article: SING: Stability-Incorporated Neighborhood Graph
SING: Stability-Incorporated Neighborhood Graph Open
International audience
View article: Differential Calculus and Optimization in Persistence Module Categories
Differential Calculus and Optimization in Persistence Module Categories Open
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other ar…
View article: Computation of $γ$-linear projected barcodes for multiparameter persistence
Computation of $γ$-linear projected barcodes for multiparameter persistence Open
The $γ$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along li…
View article: ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities
ADAPT: Multimodal Learning for Detecting Physiological Changes under Missing Modalities Open
Multimodality has recently gained attention in the medical domain, where imaging or video modalities may be integrated with biomedical signals or health records. Yet, two challenges remain: balancing the contributions of modalities, especi…
View article: Multimodal Learning for Detecting Stress under Missing Modalities
Multimodal Learning for Detecting Stress under Missing Modalities Open
International audience
View article: D-GRIL: End-to-End Topological Learning with 2-parameter Persistence
D-GRIL: End-to-End Topological Learning with 2-parameter Persistence Open
End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique c…
View article: Differentiability and Optimization of Multiparameter Persistent Homology
Differentiability and Optimization of Multiparameter Persistent Homology Open
Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single…
View article: Efficient computation of topological integral transforms
Efficient computation of topological integral transforms Open
Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric …
View article: Local characterization of block-decomposability for multiparameter persistence modules
Local characterization of block-decomposability for multiparameter persistence modules Open
Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a pr…
View article: On the Stability of Multigraded Betti Numbers and Hilbert Functions
On the Stability of Multigraded Betti Numbers and Hilbert Functions Open
International audience
View article: Efficient Computation of Topological Integral Transforms
Efficient Computation of Topological Integral Transforms Open
Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric …
View article: Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures Open
Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parame…
View article: On the bottleneck stability of rank decompositions of multi-parameter persistence modules
On the bottleneck stability of rank decompositions of multi-parameter persistence modules Open
A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal r…
View article: Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions
Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions Open
In this paper we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart i…
View article: On the stability of multigraded Betti numbers and Hilbert functions
On the stability of multigraded Betti numbers and Hilbert functions Open
Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modu…
View article: Seabed prediction from airborne topo-bathymetric lidar point cloud using machine learning approaches
Seabed prediction from airborne topo-bathymetric lidar point cloud using machine learning approaches Open
International audience
View article: A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis
A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis Open
We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regula…
View article: Barcode embeddings for metric graphs
Barcode embeddings for metric graphs Open
Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, …
View article: Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams
Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams Open
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…
View article: Exact computation of the matching distance on 2-parameter persistence modules
Exact computation of the matching distance on 2-parameter persistence modules Open
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonabl…
View article: Local characterizations for decomposability of 2-parameter persistence modules
Local characterizations for decomposability of 2-parameter persistence modules Open
We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corre…