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Documents authored by Carrière, Mathieu

Document
DOI: 10.4230/LIPIcs.SoCG.2020.11
Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study

Authors: Andrew Aukerman, Mathieu Carrière, Chao Chen, Kevin Gardner, Raúl Rabadán, and Rami Vanguri

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Persistent homology is a common tool of topological data analysis, whose main descriptor, the persistence diagram, aims at computing and encoding the geometry and topology of given datasets. In this article, we present a novel application of persistent homology to characterize the spatial arrangement of immune and epithelial (tumor) cells within the breast cancer immune microenvironment. More specifically, quantitative and robust characterizations are built by computing persistence diagrams out of a staining technique (quantitative multiplex immunofluorescence) which allows us to obtain spatial coordinates and stain intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers of cancer subtype and prognostic biomarker of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. This provides new insights and possibilities on the general problem of building (topology-based) biomarkers that are characteristic and predictive of cancer subtype, overall survival and response to therapy.
Cite as

Andrew Aukerman, Mathieu Carrière, Chao Chen, Kevin Gardner, Raúl Rabadán, and Rami Vanguri. Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aukerman_et_al:LIPIcs.SoCG.2020.11,
  author =	{Aukerman, Andrew and Carri\`{e}re, Mathieu and Chen, Chao and Gardner, Kevin and Rabad\'{a}n, Ra\'{u}l and Vanguri, Rami},
  title =	{{Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.11},
  URN =		{urn:nbn:de:0030-drops-121695},
  doi =		{10.4230/LIPIcs.SoCG.2020.11},
  annote =	{Keywords: Topological data analysis, persistence diagrams}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.21
On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces

Authors: Mathieu Carrière and Ulrich Bauer

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their diagram distances, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of persistence diagrams into Reproducing Kernel Hilbert Spaces, in which all computations can be performed easily. Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the metric properties of the feature map, i.e., the relationship between the Hilbert distance and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding. In this article, we study the possibility of embedding persistence diagrams into separable Hilbert spaces with bi-Lipschitz maps. In particular, we show that for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature, any lower bound must depend on the cardinalities of the persistence diagrams, and that when the Hilbert space is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.
Cite as

Mathieu Carrière and Ulrich Bauer. On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carriere_et_al:LIPIcs.SoCG.2019.21,
  author =	{Carri\`{e}re, Mathieu and Bauer, Ulrich},
  title =	{{On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.21},
  URN =		{urn:nbn:de:0030-drops-104259},
  doi =		{10.4230/LIPIcs.SoCG.2019.21},
  annote =	{Keywords: Topological Data Analysis, Persistence Diagrams, Hilbert space embedding}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.25
Local Equivalence and Intrinsic Metrics between Reeb Graphs

Authors: Mathieu Carrière and Steve Oudot

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.
Cite as

Mathieu Carrière and Steve Oudot. Local Equivalence and Intrinsic Metrics between Reeb Graphs. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carriere_et_al:LIPIcs.SoCG.2017.25,
  author =	{Carri\`{e}re, Mathieu and Oudot, Steve},
  title =	{{Local Equivalence and Intrinsic Metrics between Reeb Graphs}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.25},
  URN =		{urn:nbn:de:0030-drops-71794},
  doi =		{10.4230/LIPIcs.SoCG.2017.25},
  annote =	{Keywords: Reeb Graphs, Extended Persistence, Induced Metrics, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.25
Structure and Stability of the 1-Dimensional Mapper

Authors: Mathieu Carrière and Steve Oudot

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^{-1}(I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero.
Cite as

Mathieu Carrière and Steve Oudot. Structure and Stability of the 1-Dimensional Mapper. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{carriere_et_al:LIPIcs.SoCG.2016.25,
  author =	{Carri\`{e}re, Mathieu and Oudot, Steve},
  title =	{{Structure and Stability of the 1-Dimensional Mapper}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{25:1--25:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.25},
  URN =		{urn:nbn:de:0030-drops-59175},
  doi =		{10.4230/LIPIcs.SoCG.2016.25},
  annote =	{Keywords: Mapper, Reeb Graph, Extended Persistence, Topological Data Analysis}
}
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