Paul L Bendich
- Associate Research Professor of Mathematics
- Assistant Director of Curricular Engagement of the Information Initiative at Duke
Research Areas and Keywords
Computational Mathematics
topological data analysis, data science
Signals, Images & Data
topological data analysis, machine learning, applied topology, data science
Topology
topological data analysis, applied topology
I work in computational topology, which for me means adapting and using tools from algebraic topology in order to study noisy and high-dimensional datasets arising from a variety of scientific applications. My thesis research involved the analysis of datasets for which the number of degrees of freedom varies across the parameter space. The main tools are local homology and intersection homology, suitably redefined in this fuzzy multi-scale context. I am also working on building connections between computational topology and various statistical data analysis algorithms, such as clustering or manifold learning, as well as building connections between computational topology and diffusion geometry.
Selected Grants
Geometric and Topological Methods for Multi-Modal Data Analysis and Fusion awarded by Air Force Office of Scientific Research (Co-Principal Investigator). 2018 to 2021
BIGDATA: F: DKA: CSD: Topological Data Analysis and Machine-Learning with Community-Accepted Features awarded by National Science Foundation (Co-Principal Investigator). 2014 to 2019
Topological Signal Analysis for Multi-Modal Data Analysis awarded by Geometric Data Analytics, Inc. (Principal Investigator). 2016 to 2017
Bendich, Paul, et al. “Stabilizing the unstable output of persistent homology computations.” Journal of Applied and Computational Topology, Springer, Nov. 2019, pp. 1–30.
Bendich, P., et al. Scaffoldings and Spines: Organizing High-Dimensional Data Using Cover Trees, Local Principal Component Analysis, and Persistent Homology. Vol. 13, Jan. 2018, pp. 93–114. Scopus, doi:10.1007/978-3-319-89593-2_6. Full Text
Bendich, P., et al. “Topological and statistical behavior classifiers for tracking applications.” Ieee Transactions on Aerospace and Electronic Systems, vol. 52, no. 6, Dec. 2016, pp. 2644–61. Scopus, doi:10.1109/TAES.2016.160405. Full Text
Bendich, P., et al. “Persistent homology analysis of brain artery trees.” Annals of Applied Statistics, vol. 10, no. 1, 2016, pp. 198–218. Open Access Copy
Munch, E., et al. “Probabilistic Fréchet means for time varying persistence diagrams.” Electronic Journal of Statistics, vol. 9, Jan. 2015, pp. 1173–204. Scopus, doi:10.1214/15-EJS1030. Full Text Open Access Copy
Bendich, P., et al. “Homology and robustness of level and interlevel sets.” Homology, Homotopy and Applications, vol. 15, no. 1, Apr. 2013, pp. 51–72. Scopus, doi:10.4310/HHA.2013.v15.n1.a3. Full Text
Bendich, P., et al. “A point calculus for interlevel set homology.” Pattern Recognition Letters, vol. 33, no. 11, Aug. 2012, pp. 1436–44. Scopus, doi:10.1016/j.patrec.2011.10.007. Full Text
Bendich, P., et al. “Improving homology estimates with random walks.” Inverse Problems, vol. 27, no. 12, Dec. 2011. Scopus, doi:10.1088/0266-5611/27/12/124002. Full Text
Bendich, P., and J. Harer. “Persistent Intersection Homology.” Foundations of Computational Mathematics, vol. 11, no. 3, June 2011, pp. 305–36. Scopus, doi:10.1007/s10208-010-9081-1. Full Text
Bendich, P., et al. “The robustness of level sets.” Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6346 LNCS, no. PART 1, Nov. 2010, pp. 1–10. Scopus, doi:10.1007/978-3-642-15775-2_1. Full Text
Pages
Blasch, E., et al. “Machine learning in/with information fusion for infrastructure understanding, panel summary.” Proceedings of Spie the International Society for Optical Engineering, vol. 11423, 2020. Scopus, doi:10.1117/12.2559416. Full Text
Tralie, C. J., et al. “Multi-Scale Geometric Summaries for Similarity-Based Sensor Fusion.” Ieee Aerospace Conference Proceedings, vol. 2019-March, 2019. Scopus, doi:10.1109/AERO.2019.8741399. Full Text
Bendich, P. “Topology, geometry, and machine-learning for tracking and sensor fusion.” Proceedings of Spie the International Society for Optical Engineering, vol. 11017, 2019, pp. lxxxiii–cii.
Garagić, D., et al. “Upstream fusion of multiple sensing modalities using machine learning and topological analysis: An initial exploration.” Ieee Aerospace Conference Proceedings, vol. 2018-March, 2018, pp. 1–8. Scopus, doi:10.1109/AERO.2018.8396737. Full Text
Tralie, C. J., et al. “Geometric cross-modal comparison of heterogeneous sensor data.” Ieee Aerospace Conference Proceedings, vol. 2018-March, 2018, pp. 1–10. Scopus, doi:10.1109/AERO.2018.8396789. Full Text Open Access Copy
Bendich, P., et al. “Geometric models for musical audio data.” Leibniz International Proceedings in Informatics, Lipics, vol. 51, 2016, pp. 65.1-65.5. Scopus, doi:10.4230/LIPIcs.SoCG.2016.65. Full Text Open Access Copy
Bendich, P., et al. “Geometric Models for Musical Audio Data.” Proceedings of the 32st International Symposium on Computational Geometry (Socg), 2016.
Tralie, C. J., and P. Bendich. “Cover Song Identification with Timbral Shape Sequences.” 16th International Society for Music Information Retrieval (Ismir), 2015, pp. 38–44. Open Access Copy
Bendich, P., et al. “Multi-scale local shape analysis and feature selection in machine learning applications.” Proceedings of the International Joint Conference on Neural Networks, vol. 2015-September, 2015. Scopus, doi:10.1109/IJCNN.2015.7280428. Full Text Open Access Copy
Rouse, D., et al. “Feature-aided multiple hypothesis tracking using topological and statistical behavior classifiers.” Proceedings of Spie the International Society for Optical Engineering, vol. 9474, 2015. Scopus, doi:10.1117/12.2179555. Full Text
Pages
Bendich, P., et al. Stratification learning through homology inference. 1 Dec. 2010, pp. 10–17.
A key goal of Together Duke is to invest in faculty as scholars and leaders of the university’s intellectual communities. To foster collaboration around new and emerging areas of interest... read more »
For his excellent work in developing Data+ into a university model for undergraduate mentoring and research, Paul Bendich has been granted the Dean's Leadership Award. This award is given to a faculty or staff member who has made a distinctive... read more »
