Wasserstein Dictionaries of Persistence Diagrams
Pages 1638 - 1651
Abstract
This article presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters Turner et al. (2014), Vidal et al. (2020) of a dictionary of <italic>atom diagrams</italic>. 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 <italic>atom diagrams</italic>. 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 <italic>data reduction</italic> and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a <italic>dimensionality reduction</italic> 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.
References
[1]
ISO/IEC Guide 98–3:2008, “Uncertainty of measurement - part 3: Guide to the expression of uncertainty in measurement (GUM),” 2008.
[2]
A. Acharya and V. Natarajan, “A parallel and memory efficient algorithm for constructing the contour tree,” in Proc. IEEE Pacific Visualization Symp., 2015, pp. 271–278.
[3]
K. Anderson, J. Anderson, S. Palande, and B. Wang, “Topological data analysis of functional MRI connectivity in time and space domains,” in Proc. Connectomics NeuroImaging: 2nd Int. Workshop, 2018, pp. 67–77.
[4]
T. Athawale and A. Entezari, “Uncertainty quantification in linear interpolation for isosurface extraction,” IEEE Trans. Vis. Comput. Graph., vol. 19, no. 12, pp. 2723–2732, Dec. 2013.
[5]
T. Athawale, E. Sakhaee, and A. Entezari, “Isosurface visualization of data with nonparametric models for uncertainty,” IEEE Trans. Vis. Comput. Graph., vol. 22, no. 1, pp. 777–786, Jan. 2016.
[6]
T. M. Athawale, D. Maljovec, C. R. Johnson, V. Pascucci, and B. Wang, “Uncertainty visualization of 2D morse complex ensembles using statistical summary maps,” 2019,.
[7]
U. Ayachit et al., “ParaView catalyst: Enabling in situ data analysis and visualization,” in Proc. 1st Workshop Situ Infrastructures Enabling Extreme-Scale Anal. Visualization, 2015, pp. 25–29.
[8]
D. P. Bertsekas, “A new algorithm for the assignment problem,” Math. Program., vol. 21, pp. 152–171, 1981.
[9]
H. Bhatia, A. G. Gyulassy, V. Lordi, J. E. Pask, V. Pascucci, and P.-T. Bremer, “TopoMS: Comprehensive topological exploration for molecular and condensed-matter systems,” J. Comput. Chem., vol. 39, pp. 936–952, 2018.
[10]
H. Bhatia et al., “Flow visualization with quantified spatial and temporal errors using edge maps,” IEEE Trans. Vis. Comput. Graph., vol. 18, no. 9, pp. 1383–1396, Sep. 2012.
[11]
S. Biasotti, D. Giorgio, M. Spagnuolo, and B. Falcidieno, “Reeb graphs for shape analysis and applications,” Theor. Comput. Sci., vol. 392, pp. 2–22, 2008.
[12]
T. Bin Masood et al., “An overview of the topology ToolKit,” in Proc. Topological Methods Data Anal. Visualization VI, 2019, pp. 327–342.
[13]
A. Bock, H. Doraiswamy, A. Summers, and C. T. Silva, “TopoAngler: Interactive topology-based extraction of fishes,” IEEE Trans. Vis. Comput. Graph., vol. 24, no. 1, pp. 812–821, Jan. 2018.
[14]
G. Bonneau et al., “Overview and state-of-the-art of uncertainty visualization,” Math. Visual., vol. 37, pp. 3–27, 2014.
[15]
P. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci, “A multi-resolution data structure for 2-Dimensional morse functions,” in Proc. IEEE VIS, 2003, pp. 139–146.
[16]
P. Bremer, G. Weber, J. Tierny, V. Pascucci, M. Day, and J. Bell, “Interactive exploration and analysis of large scale simulations using topology-based data segmentation,” IEEE Trans. Vis. Comput. Graph., vol. 17, no. 9, pp. 1307–1324, Sep. 2011.
[17]
N. Brown et al., “Utilising urgent computing to tackle the spread of mosquito-borne diseases,” in Proc. IEEE/ACM HPC Urgent Decis. Mak., 2021, pp. 36–44.
[18]
H. Carr, J. Snoeyink, and U. Axen, “Computing contour trees in all dimensions,” in Proc. 11th Ann. ACM-SIAM Symp. Discrete Algorithms, 2000, pp. 918–926.
[19]
H. Carr, G. Weber, C. Sewell, and J. Ahrens, “Parallel peak pruning for scalable SMP contour tree computation,” in Proc. IEEE Symp. Large Data Anal. Vis., 2016, pp. 75–84.
[20]
H. A. Carr, J. Snoeyink, and M. van de Panne, “Simplifying flexible isosurfaces using local geometric measures,” in Proc. IEEE Visualization, 2004, pp. 497–504.
[21]
M. E. Celebi, H. A. Kingravi, and P. A. Vela, “A comparative study of efficient initialization methods for the K-means clustering algorithm,” Expert Syst. Appl., 2013.
[22]
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams,” in Proc. 21st ACM Symp. Comput. Geometry, 2005, pp. 263–271.
[23]
D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and Y. Mileyko, “Lipschitz functions have lp-stable persistence,” Found. Comput. Math., vol. 10, pp. 127–139, 2010.
[24]
M. Cuturi, “Sinkhorn distances: Lightspeed computation of optimal transport,” in Proc. Int. Conf. Neural Inf. Process. Syst., 2013, pp. 2292–2300.
[25]
M. Cuturi and A. Doucet, “Fast computation of wasserstein barycenters,” in Proc. Int. Conf. Mach. Learn., 2014, pp. II-685–II-693.
[26]
L. De Floriani, U. Fugacci, F. Iuricich, and P. Magillo, “Morse complexes for shape segmentation and homological analysis: Discrete models and algorithms,” Comput. Graph. Forum, vol. 34, pp. 761–785, 2015.
[27]
P. Diggle, P. Heagerty, K.-Y. Liang, and S. Zeger, The Analysis of Longitudinal Data. London, U.K.: Oxford Univ. Press, 2002.
[28]
H. Doraiswamy and V. Natarajan, “Computing reeb graphs as a union of contour trees,” IEEE Trans. Vis. Comput. Graph., vol. 19, no. 2, pp. 249–262, Feb. 2013.
[29]
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction. Providence, RI, USA: American Mathematical Society, 2009.
[30]
H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, “Morse-Smale complexes for piecewise linear 3-manifolds,” in Proc. 19th Annu. Symp. Comput. Geometry, 2003, pp. 361–370.
[31]
H. Edelsbrunner, J. Harer, and A. Zomorodian, “Hierarchical morse complexes for piecewise linear 2-manifolds,” in Proc. 17th Annu. Symp. Comput. Geometry, 2001, pp. 70–79.
[32]
H. Edelsbrunner and E. P. Mucke, “Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms,” ACM Trans. Graph., vol. 9, pp. 66–104, 1990.
[33]
G. Favelier, N. Faraj, B. Summa, and J. Tierny, “Persistence atlas for critical point variability in ensembles,” IEEE Trans. Vis. Comput. Graph., vol. 25, no. 1, pp. 1152–1162, Jan. 2018.
[34]
G. Favelier, C. Gueunet, and J. Tierny, “Visualizing ensembles of viscous fingers,” in Proc. IEEE SciVis Contest, 2016.
[35]
F. Ferstl, K. Bürger, and R. Westermann, “Streamline variability plots for characterizing the uncertainty in vector field ensembles,” IEEE Trans. Vis. Comput. Graph., vol. 22, no. 1, pp. 767–776, Jan. 2016.
[36]
F. Ferstl, M. Kanzler, M. Rautenhaus, and R. Westermann, “Visual analysis of spatial variability and global correlations in ensembles of iso-contours,” Comput. Graph. Forum, vol. 35, pp. 221–230, 2016.
[37]
R. Forman, “A user's guide to discrete morse theory,” AM, 1998.
[38]
D. Guenther, R. Alvarez-Boto, J. Contreras-Garcia, J.-P. Piquemal, and J. Tierny, “Characterizing molecular interactions in chemical systems,” IEEE Trans. Vis. Comput. Graph., vol. 20, no. 12, pp. 2476–2485, Dec. 2014.
[39]
C. Gueunet, P. Fortin, J. Jomier, and J. Tierny, “Task-based augmented contour trees with fibonacci heaps,” IEEE Trans. Parallel Distrib. Syst., vol. 30, no. 8, pp. 1889–1905, Aug. 2019.
[40]
C. Gueunet, P. Fortin, J. Jomier, and J. Tierny, “Task-based augmented reeb graphs with dynamic ST-Trees,” in Proc. Eurographics Symp. Parallel Graph. Visualization, 2019.
[41]
P. Guillou, J. Vidal, and J. Tierny, “Discrete morse sandwich: Fast computation of persistence diagrams for scalar data–an algorithm and a benchmark,” IEEE Trans. Vis. Comput. Graph., 2023.
[42]
D. Günther, J. Salmon, and J. Tierny, “Mandatory critical points of 2D uncertain scalar fields,” Comput. Graph. Forum, vol. 33, pp. 31–40, 2014.
[43]
A. Gyulassy, P. Bremer, R. Grout, H. Kolla, J. Chen, and V. Pascucci, “Stability of dissipation elements: A case study in combustion,” Comput. Graph. Forum, vol. 33, no. 3, pp. 51–60, 2014.
[44]
A. Gyulassy, P. Bremer, and V. Pascucci, “Shared-memory parallel computation of morse-smale complexes with improved accuracy,” IEEE Trans. Vis. Comput. Graph., vol. 25, no. 1, pp. 1183–1192, Jan. 2019.
[45]
A. Gyulassy et al., “Topologically clean distance fields,” IEEE Trans. Vis. Comput. Graph., vol. 13, no. 6, pp. 1432–1439, Nov./Dec. 2007.
[46]
A. Gyulassy et al., “Interstitial and Interlayer Ion Diffusion Geometry Extraction in Graphitic Nanosphere Battery Materials,” IEEE Trans. Vis. Comput. Graph., vol. 22, no. 1, pp. 916–925, Jan. 2016.
[47]
C. Heine et al., “A survey of topology-based methods in visualization,” Comput. Graph. Forum, vol. 35, no. 3, pp. 643–667, 2016.
[48]
M. Hummel, H. Obermaier, C. Garth, and K. I. Joy, “Comparative visual analysis of lagrangian transport in CFD ensembles,” IEEE Trans. Vis. Comput. Graphics, vol. 19, no. 12, pp. 2743–2752, Dec. 2013.
[49]
C. R. Johnson and A. R. Sanderson, “A next step: Visualizing errors and uncertainty,” IEEE Comput. Graph. Appl., vol. 23, no. 5, pp. 6–10, Sep./Oct. 2003.
[50]
L. Kantorovich, “On the translocation of masses,” AS USSR, 1942.
[51]
J. Kasten, J. Reininghaus, I. Hotz, and H. Hege, “Two-dimensional time-dependent vortex regions based on the acceleration magnitude,” IEEE Trans. Vis. Comput. Graph., vol. 17, no. 12, pp. 2080–2087, Dec. 2011.
[52]
M. Kerber, D. Morozov, and A. Nigmetov, “Geometry helps to compare persistence diagrams,” ACM J. Exp. Algorithmics, vol. 22, 2017.
[53]
M. Kraus, “Visualization of uncertain contour trees,” in Proc. Int. Conf. Inf. Visualization Theory Appl., 2010, pp. 132–139.
[54]
J. B. Kruskal and M. Wish, “Multidimensional scaling,” SUPS, 1978.
[55]
T. Lacombe, M. Cuturi, and S. Oudot, “Large scale computation of means and clusters for persistence diagrams using optimal transport,” in Proc. Int. Conf. Neural Inf. Process. Syst., 2018, pp. 9792–9802.
[56]
D. E. Laney, P. Bremer, A. Mascarenhas, P. Miller, and V. Pascucci, “Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities,” IEEE Trans. Vis. Comput. Graph., vol. 12, no. 5, pp. 1053–1060, Sep./Oct. 2006.
[57]
M. Li, S. Palande, L. Yan, and B. Wang, “Sketching merge trees for scientific visualization,” IEEE TopoInVis, 2023.
[58]
T. Liebmann and G. Scheuermann, “Critical points of gaussian-distributed scalar fields on simplicial grids,” Comput. Graph. Forum, vol. 35, pp. 361–370, 2016.
[59]
S. Maadasamy, H. Doraiswamy, and V. Natarajan, “A hybrid parallel algorithm for computing and tracking level set topology,” in Proc. 19th Int. Conf. High Perform. Comput., 2012, pp. 1–10.
[60]
A. Maceachren et al., “Visualizing geospatial information uncertainty: What we know and what we need to know,” Cartography Geographic Inf. Sci., vol. 32, pp. 139–160, 2005.
[61]
D. Maljovec et al., “Topology-inspired partition-based sensitivity analysis and visualization of nuclear simulations,” in Proc. IEEE Pacific Visualization Symp., 2016, pp. 64–71.
[62]
M. Mirzargar, R. Whitaker, and R. Kirby, “Curve boxplot: Generalization of boxplot for ensembles of curves,” IEEE Trans. Vis. Comput. Graph., vol. 20, no. 12, pp. 2654–2663, Dec. 2014.
[63]
G. Monge, “Mémoire sur la théorie des déblais et des remblais,” Académie Royale des Sci. de Paris, 1781.
[64]
J. Munkres, “Algorithms for the assignment and transportation problems,” J. Soc. Ind. Appl. Math., vol. 5, pp. 32–38, 1957.
[65]
F. Nauleau, F. Vivodtzev, T. Bridel-Bertomeu, H. Beaugendre, and J. Tierny, “Topological analysis of ensembles of hydrodynamic turbulent flows–an experimental study,” in Proc. IEEE Symp. Large Data Anal. Vis., 2022, pp. 1–11.
[66]
M. Olejniczak, A. S. P. Gomes, and J. Tierny, “A topological data analysis perspective on non-covalent interactions in relativistic calculations,” Int. J. Quantum Chem., vol. 120, no. 8, 2019, Art. no.
[67]
M. Olejniczak and J. Tierny, “Topological data analysis of vortices in the magnetically-induced current density in LiH molecule,” Phys. Chem. Chem. Phys., vol. 25, pp. 5942–5947, 2023.
[68]
Organizers, The IEEE SciVis Contest. 2004. [Online]. Available: http://sciviscontest.ieeevis.org/
[69]
M. Otto, T. Germer, H.-C. Hege, and H. Theisel, “Uncertain 2D vector Field Topology,” CGF, vol. 31, no. 3pt2, pp. 1045–1054, 2010.
[70]
M. Otto, T. Germer, and H. Theisel, “Uncertain topology of 3D vector fields,” in Proc. IEEE Pacific Visualization Symp., 2011, pp. 67–74.
[71]
A. T. Pang, C. M. Wittenbrink, and S. K. Lodha, “Approaches to uncertainty visualization,” Vis. Comput., vol. 13, no. 8, pp. 370–390, 1997.
[72]
S. Parsa, “A deterministic o(m log m) time algorithm for the reeb graph,” in Proc. 28th Ann. Symp. Comput. Geometry, 2012, pp. 269–276.
[73]
V. Pascucci, G. Scorzelli, P. T. Bremer, and A. Mascarenhas, “Robust on-line computation of Reeb graphs: Simplicity and speed,” ACM Trans. Graph., vol. 26, pp. 58–es, 2007.
[74]
C. Petz, K. Pöthkow, and H.-C. Hege, “Probabilistic local features in uncertain vector fields with spatial correlation,” Comput. Graph. Forum, vol. 31, pp. 1045–1054, 2012.
[75]
T. Pfaffelmoser, M. Mihai, and R. Westermann, “Visualizing the variability of gradients in uncertain 2D scalar fields,” IEEE Trans. Vis. Comput. Graph., vol. 19, no. 11, pp. 1948–1961, Nov. 2013.
[76]
T. Pfaffelmoser, M. Reitinger, and R. Westermann, “Visualizing the positional and geometrical variability of isosurfaces in uncertain scalar fields,” Comput. Graph. Forum, vol. 30, no. 3, pp. 951–960, 2011.
[77]
T. Pfaffelmoser and R. Westermann, “Visualization of global correlation structures in uncertain 2D scalar fields,” Comput. Graph. Forum, 2012.
[78]
M. Pont, J. Vidal, J. Delon, and J. Tierny, “Wasserstein distances, geodesics and barycenters of merge trees–ensemble benchmark,” Vis, vol. 28, no. 1, 2021, pp. 291–301. [Online]. Available: https://github.com/MatPont/WassersteinMergeTreesData
[79]
M. Pont, J. Vidal, J. Delon, and J. Tierny, “Wasserstein distances, geodesics and barycenters of merge trees,” IEEE Trans. Vis. Comput. Graph., vol. 28, no. 1, pp. 291–301, Jan. 2022.
[80]
M. Pont, J. Vidal, and J. Tierny, “Principal geodesic analysis of merge trees (and persistence diagrams),” IEEE Trans. Vis. Comput. Graph., vol. 29, no. 2, pp. 1573–1589, Feb. 2023.
[81]
K. Pöthkow and H.-C. Hege, “Positional uncertainty of isocontours: Condition analysis and probabilistic measures,” IEEE Trans. Vis. Comput. Graph., vol. 17, no. 10, pp. 1393–1406, Oct. 2011.
[82]
K. Pöthkow and H.-C. Hege, “Nonparametric models for uncertainty visualization,” Comput. Graph. Forum, vol. 32, no. 3, pp. 131–140, 2013.
[83]
K. Pöthkow, B. Weber, and H.-C. Hege, “Probabilistic marching cubes,” Comput. Graph. Forum, vol. 30, no. 3, pp. 931–940, 2011.
[84]
K. Potter, S. Gerber, and E. W. Anderson, “Visualization of uncertainty without a mean,” IEEE Comput. Graph. Appl., vol. 33, no. 1, pp. 75–79, Jan./Feb. 2013.
[85]
K. Potter, P. Rosen, and C. R. Johnson, “From quantification to visualization: A taxonomy of uncertainty visualization approaches,” in Proc. IFIP Work. Conf. Uncertainty Quantification, 2012, pp. 226–249.
[86]
K. Potter et al., “Ensemble-vis: A framework for the statistical visualization of ensemble data,” in Proc. IEEE Int. Conf. Data Mining Workshops, 2009, pp. 233–240.
[87]
V. Robins, P. J. Wood, and A. P. Sheppard, “Theory and algorithms for constructing discrete morse complexes from grayscale digital images,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, no. 8, pp. 1646–1658, Aug. 2011.
[88]
J. Sanyal, S. Zhang, J. Dyer, A. Mercer, P. Amburn, and R. Moorhead, “Noodles: A tool for visualization of numerical weather model ensemble uncertainty,” IEEE Trans. Vis. Comput. Graph., vol. 16, no. 6, pp. 1421–1430, Nov./Dec. 2010.
[89]
S. Schlegel, N. Korn, and G. Scheuermann, “On the interpolation of data with normally distributed uncertainty for visualization,” IEEE Trans. Vis. Comput. Graph., vol. 18, no. 12, pp. 2305–2314, Dec. 2012.
[90]
M. A. Schmitz et al., “Wasserstein dictionary learning: Optimal transport-based unsupervised nonlinear dictionary learning,” SIAM J. Imag. Sci., vol. 11, no. 1, pp. 643–678, 2018.
[91]
N. Shivashankar and V. Natarajan, “Parallel computation of 3D morse-smale complexes,” CGF, vol. 31, no. 3, pp. 965–974, 2012.
[92]
N. Shivashankar, P. Pranav, V. Natarajan, R. van de Weygaert, E. P. Bos, and S. Rieder, “Felix: A topology based framework for visual exploration of cosmic filaments,” IEEE Trans. Vis. Comput. Graph., vol. 22, no. 6, pp. 1745–1759, Jun. 2016.
[93]
R. Sinkhorn, “Diagonal equivalence to matrices with prescribed row and column sums,” Amer. Math. Monthly, vol. 74, pp. 402–405, 1967.
[94]
M. Soler, M. Petitfrere, G. Darche, M. Plainchault, B. Conche, and J. Tierny, “Ranking viscous finger simulations to an acquired ground truth with topology-aware matchings,” in Proc. IEEE Symp. Large Data Anal. Visualization, 2019, pp. 62–72.
[95]
T. Sousbie, “The persistent cosmic web and its filamentary structure: Theory and implementations,” Monthly Notices Roy. Astronomical Soc., vol. 414, no. 1, pp. 350–383, 2011.
[96]
A. Szymczak, “Hierarchy of stable Morse decompositions,” IEEE Trans. Vis. Comput. Graph., vol. 19, no. 5, pp. 799–810, May 2013.
[97]
S. Tarasov and M. Vyali, “Construction of contour trees in 3D in O(n log n) steps,” in Proc. 28th Ann. Symp. Comput. Geometry, 1998, pp. 68–75.
[98]
J. Tierny, G. Favelier, J. A. Levine, C. Gueunet, and M. Michaux, “The topology ToolKit,” IEEE Trans. Vis. Comput. Graph., vol. 24, no. 1, pp. 832–842, Jan. 2018.
[99]
J. Tierny, A. Gyulassy, E. Simon, and V. Pascucci, “Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees,” IEEE Trans. Vis. Comput. Graphics, vol. 15, no. 6, pp. 1177–1184, Nov./Dec. 2009.
[100]
K. Turner, Y. Mileyko, S. Mukherjee, and J. Harer, “Fréchet means for distributions of persistence diagrams,” DCG, 2014.
[101]
L. P. van der Maaten and G. Hinton, “Visualizing data using t-SNE,” J. Mach. Learn. Res., vol. 9, pp. 2579–2605, 2008.
[102]
J. Vidal, J. Budin, and J. Tierny, “Progressive wasserstein barycenters of persistence diagrams,” IEEE Trans. Vis. Comput. Graph., vol. 26, no. 1, pp. 151–161, Jan. 2020.
[103]
R. T. Whitaker, M. Mirzargar, and R. M. Kirby, “Contour boxplots: A method for characterizing uncertainty in feature sets from simulation ensembles,” IEEE Trans. Vis. Comput. Graph., vol. 19, no. 12, pp. 2713–2722, Dec. 2013.
[104]
D. P. Woodruff, Sketching as a Tool for Numerical Linear Algebra. Boston, MA, USA: Now, 2014.
[105]
K. Wu and S. Zhang, “A contour tree based visualization for exploring data with uncertainty,” Int. J. Uncertainty Quantification, vol. 3, pp. 203–223, 2013.
[106]
L. Yan, Y. Wang, E. Munch, E. Gasparovic, and B. Wang, “A structural average of labeled merge trees for uncertainty visualization,” IEEE Trans. Vis. Comput. Graph., vol. 26, no. 1, pp. 832–842, Jan. 2019.
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Index terms have been assigned to the content through auto-classification.
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