### Primoz Skraba

Reader of Applied & Computational Topology
School of Mathematical Sciences
Queen Mary University of London
initial dot lastname @qmul.ac.uk

I do applied topology: mostly geometry, algebra, and recently stochastic topology -- with a bunch of other things such as algorithms, computational geometry, and machine learning. A long time ago I used to do signal processing. Most of my recent papers are on arxiv (see paper airplane) or my Google Scholor page. Otherwise, see publications.

If you are interested in doing a PhD, post-doc with me - I will advertise positions here as they become available.

#### Persistent Homology

Much of my research is on understanding persistent homology and its various aspects. The list are just some bits and pieces.

Stability: Persistence diagrams are the main invariant that is studied. We often try to understand spaces from finite samples, in which case, stability is important as it allows us to get quantitative control. I am very interested in stability statements and how they relate to classical objects such as exact sequences. In general there is a mixture of combinatorial and algebraic aspects to these type of questions.

Variants:There are several variants to persistence, such as zig-zag, robustness, and multiparameter. There are many open and interesting questions in this area. I have worked on some of these and have recently started working on multiparameter questions more concretely.

Algorithms: A key factor in persistent homology is that it can be computed quite efficiently (thanks to the hard work of many other people). I have worked on various complexity questions and there are still interesting questions here.

#### Stochastic Topology

This is, roughly speaking, asking about the topology of a space which comes from a random process. In particular, I am interested in the homology group (and ultimately homotopy groups) of these spaces. The random models I am interested in are usually geometric (Poisson, Boolean, or some regular tiling). My interest began with the work on maximal classes (the persistent classes which have the largest $$\frac{death}{birth}$$ - which for a Boolean model, we found was $$O\left(\frac{\log n}{\log\log n}\right)^{1/i}$$, where $$i$$ is the homological dimension.

Percolation: There is an active area of study around the sudden appearance of an infinite component in many random models - often called a sharp threshold. I am interested in these for the appearance of homology, e.g. the essential classes of a manifold. We know these appear at $$O(1/n^d)$$ but there are many open questions here.

Universality: If one considers$$\frac{death}{birth}$$ as a distribution, amazingly it seems that this distribution only depends on homological and ambient dimension (and what complex we build) but not the underlying generating distribution. Empirically, even this dependence can be removed with the appropriate normalization, which we are investigating.

Spanning Acycles: Minimal spanning trees have been well studied in both the probability and computer science literature. My interest is in minimum spanning acycles. Introduced by Kalal, they are also called generalized trees and are higher dimensional analogues.

#### Machine Learning

As part of the Department for Artificial Intelligence at the Jozef Stefan Institute, I worked on several aspects of machine learning.

Topological Information in Neural Networks: Deep networks have revolutionized machine learning with an immense amount of research going into them. I have focused on where and how topological (and geoemtric) information can be incorporated into networks to improve/control learning or understand the result.

Other: I am also still interested in other techniques from machine learning, e.g. clustering, functional data analysis, both related to topology/geometry/algebra as well as more classically machine learning techniques.

#### Applications

While I do more theory these days, I am still interested in applications to data. This includes applications as well as interesting theoretical questions which arise.

Euler Surfaces/Transforms: Rather then consider homology, we can consider the Euler characteristic. This is very efficient to compute and captures a lot of interesting information - although it is not well understood what information or how to best use it. Additionally, there are connections with constructible functions (and (co)sheaves) which are interesting directions to explore.

Visualization: Topology can be very useful for visualizations. In addition to work on vector fields, there was alos work on capturing states in time series. Topological visualization is an active area of research with many interesting problems.

### Recent Paper (Topics I am working on)

O. Bobrowski, P. Skraba, On the Universality of Random Persistence Diagrams, Submitted

O. Bobrowski, P. Skraba, Wasserstein Stability for Persistence Diagrams, Submitted

P. Skraba, Yogeshwaran D., Central limit theorem for Euclidean minimal spanning acycles. Sumitted

O. Bobrowski, P. Skraba, Homological Percolation: The Formation of Giant k-Cycles, International Mathematics Research Notices, Volume 2022, Issue 8, p. 6186-6213

G. Beltramo, P. Skraba, Persistent Homology in the l-infinity metric, Computational Geometry: Theory and Applications, Volume 101

G. Beltramo, R. Andreeva, Y. Giarratano, M. Bernabeu, R. Sarkar, P. Skraba, Euler Characteristic Surfaces, Foundations of Data Science (special issue on Topological Methods), Vol. 4 No. 4: p.505-536 2020

O. Bobrowski, P. Skraba, Homological percolation and the Euler characteristic, Physical Review E, 101.3 (2020): 032304

P. Skraba, G. Thoppe, D. Yogeshwaran, Randomly Weighted d−Complexes: Minimal Spanning Acycles and Persistence Diagrams, Electronic Journal of Combinatorics 27(2) p. 11-47

R. Brüel-Gabrielsson, B.J. Nelson, A. Dwaraknath, P. Skraba, L.J. Guibas, G. Carlsson, A topology layer for machine learning, AISTATS 2020 (Corresponding issue in Proceeding of machine learning research)

R. Brüel-Gabrielsson, V. Ganapathi-Subramanian, P. Skraba, L.J. Guibas, Topology-aware surface reconstruction for point clouds, Computer graphics forum, vol. 39, no. 5, p 197-207 (appeared at the Symposium of Geometry Processing 2020)

#### Older publications

This is a complete list and contains some unpublished works.

• B. Wang, R. Bujack, P. Rosen, P. Skraba, H. Bhatia, H. Hagen, Interpreting Galilean Invariant Vector Field Analysis via Extended Robustness. Topological Methods in Data Analysis and Visualization V. (short version appeared at TopoinVis 2017)
• M. Senožetnik, L. Bradeško, T. Šubic, Z. Herga, J. Urbančič, P. Skraba, D. Mladenić, Estimating point-of- interest rating based on visitors geospatial behaviour. Computer science and information systems. Vol. 16, no. 1, pp 131-154
• A. Poulenard, P. Skraba, M. Ovsjanikov, Topological Function Optimization for Continuous Shape Matching, Computer Graphics Forum, Volume 37, Issue 5 p.13-25
• D. Govc, P. Skraba, An Approximate Nerve Theorem, Foundations of Computational Mathematics (FoCM), Vol. 18
• P. Skraba, Persistent homology and machine learning, Informatica Volume 42, Issue 2:253-258
• O. Bobrowski, M. Kahle, and P. Skraba, Maximally Persistent Cycles in Random Geometric Complexes, vol. 27, no. 4, Annals of Applied Probability https://arxiv.org/abs/1509.04347
• L. Stopar, P. Skraba, M. Grobelnik, D. Mladenic, StreamStory: Exploring Multivariate Time Series on Multiple Scales, IEEE Transactions on Computer Graphics and Visualization (http://streamstory.ijs.si), Volume 25, Issue 4, 1788-1802
• B. Kazic, J. Rupnik, P. Skraba, L. Bradesko, D. Mladenic, Predicting Users’ Mobility Using Monte Carlo Simulations, IEEE Access, Volume 5, p 27400-27420.
• G. Kudryavtseva and P. Skraba, The principal bundles over an inverse semigroup, Semigroup Forum, Vol. 94. No. 3 pp. 674-695 https://arxiv.org/abs/1503.08560
• M. Kerber, D. Sheehy, P. Skraba, Persistent Homology and Nested Dissection. Annual ACM-SIAM Symposium on Discrete Algorithms, Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2016, Arlington, Virginia, USA.
• P. Skraba, P. Rosen, B. Wang, G. Chen, V. Pasucci. Critical Point Cancellation in 3D Vector Fields: Robustness and Discussion, IEEE Pacific Visualization 2016 and in IEEE Transactions in Computer Graphics and Visualization vol. 22, no. 6, p. 1683-1693 (Best Paper Award)
• C. Fortuna, E. De Poorter, P. Skraba, I. Moerman, Data Driven Wireless Network Design: a Multi-level Modeling Approach, Wireless Personal Communications 88.1 (2016): 63-77.
• J. Rupnik, A. Muhic, G. Leban, P. Skraba, B. Fortuna, M. Grobelnik, News Across Languages - Cross-Lingual Document: Similarity and Event Tracking, JAIR: Special Track on Cross-language Algorithms and Applications
• P. Skraba and M. Vejdemo-Johansson, Topology, Big Data and Optimization, chapter in Big Data Optimization: Recent Developments and Challenges, Volume 18 of the series Studies in Big Data pp 147-176
• M. Vejdemo-Johansson, F. Pokorny, P. Skraba, D. Kragic, Cohomological learning of periodic motion, Applicable algebra in engineering, communication and computing, 2015, vol. 26, no. 1/2, p. 5-26. (https://www.youtube.com/watch?v=NGQ-M2gdibQ)
• P. Skraba, B. Wang, G. Chen, P. Rosen, Robustness-based simplification of 2D steady and unsteady vector fields. IEEE Trans. on visualization and computer graphics, 2015, vol. 21, issue 8, p. 930–944
• M. Mole, L. Wang, K. Bergant, W. Eichinger, S. Stanic, P. Skraba. Lidar measurements of Bora wind effects on aerosol loading. International Symposium on Atmospheric Light Scattering and Remote Sensing (ISALSaRS'15), June 1-5, 2015
• M. Vejdemo-Johansson, P. Skraba, Algebraic and Topological Perspectives on Semi-Supervised Clustering, European Conference on Complexity Science 2014
• J. Pita Costa, P. Skraba, A topological data analysis approach to epidemiology, European Conference on Complexity Science 2014
• P. Skraba, R. Adler, Topological Detection of Heavy Tailed Distributions, European Conference on Complexity Science 2014
• P. Skraba, M. Vejdemo-Johansson, Persistence modules: Algebra and algorithms, https://arxiv.org/abs/1302.2015
• J. Rupnik, P. Skraba, J. Shawe-Taylor, S. Guettes, A Comparison of Relaxations of Multiset Cannonical Correlation Analysis and Applications, https://arxiv.org/abs/1302.0974
• P. Skraba, B. Wang, G. Chen, P Rosen, 2D vector field simplification based on robustness, 7th Pacific Visualization Symposium, March 4-7, 2014, Yokohama, Japan. PacificVis 2014. (Best Paper)
• P. Skraba, B. Wang, Approximating local homology from samples, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, January 5-7, 2014, Portland, Oregon, USA, p. 174-192.
• P. Skraba, B. Wang, Interpreting feature tracking through the lens of robustness, Topological methods in data analysis and visualization III: theory, algorithms applications, (Mathematics and visualization), Springer, 2014, p. 19-37
• M .Mole, K. Bergant, L. Honzak, J. Rakovec, Joe, G. Skok, S. Stanic, R. Zabkar, P. Skraba. Analysis of measurements of the Bora wind in Vipava valley, European Geosciences Union, General Assembly 2014, Vienna, Austria, 27 April-02 May 2014.

#### Teaching

I was previously the Programme Director of the Data Analytics Msc. Some of the things I have taught (in various places):

• Computational Topology
• Discrete Mathematics
• Dynamic Systems
• Calculus/Analysis I
• Discrete Mathematics
• Introduction to Applied Topology
• Storing, Manipulating, and Visualizing Data

#### Previous Appointments

Previously, I was at the Jozef Stefan Institute, Slovenia (Deptartment for Artificial Intelligence - still hold a partial appointment there)

Also I was taught at the Faculty of Mathematics, Natural Sciences, and Information Technologies at the University of Primorska, Slovenia, and the Center for Atmospheric Research University of Nova Gorica, Slovenia.

Even before that I was at Geometrica (now DataShape), INRIA, France as a post-doctoral researchers.

#### Talks

I will put links to slides and talks here soon!