about |

Research · Teaching · Consulting

Kigali, Rwanda

I develop mathematical methods for understanding the shape and structure of data — and I make them available as research, teaching, and consulting. With a PhD in Topology from UCT and an MSc from AUST, my work sits at the frontier where topology, geometry, and machine learning meet. As Head of R&D at a technology startup, I lead applied research that bridges methodological innovation and operational deployment, with cross-industry experience in banking, energy, insurance, IT, and retail. I’m not a consultant who also publishes — I’m a researcher whose knowledge takes three forms.

Researcher

I develop and publish mathematical methods for understanding the shape and structure of data. My foundations are in asymmetric topology and fixed point theory in generalized metric spaces; my current work applies these ideas to machine learning, AI safety, and quantum-classical computation.

Topological foundations of AI & RL

Rigorous convergence, stability, and safety proofs for learning algorithms via quasi-metric and generalized metric spaces — building predictable, resilient autonomous agents with verifiable guarantees.

Quasi-Metrics Fixed Points AI Safety

TDA for robust machine learning

Persistent homology and sheaf theory for feature extraction and anomaly detection — with applications in cybersecurity, network intrusion detection, and interpretable, privacy-preserving AI.

Persistent Homology Anomaly Detection Cybersecurity

Asymmetric topology & complexity

Ordered structures and quasi-metrics to model non-commutative, time-irreversible computation — addressing algorithmic efficiency, quantum-resistant security, and quantum-inspired algorithms.

Ordered Spaces Quantum-Resistant Complexity

Quantum-topological AI for finance

Topological principles for quantum computational AI in finance — quantum-topological neural networks for market simulation, derivative pricing on hybrid hardware, and topology-based cryptographic schemes.

Quantum AI Derivatives Crypto Schemes

Co-author of The Shape of Data (No Starch Press). Currently at AIRINA Labs and ACAS. Previously at Quantum Leap Africa, North-West University, and IMSP.

Publications Google Scholar ORCID arXiv

Educator

I teach graduate courses, short courses, and workshops that make rigorous mathematical ideas accessible to scientists, engineers, and ML practitioners — from abstract topology to hands-on deep learning. - Pure Mathematics — Topology, Real Analysis, Abstract Algebra, Functional Analysis, Fixed Point Theory - Applied Mathematics — Probability, Stochastic Processes, Quantitative Finance, Optimization - Data Science & ML — TDA, Geometric Deep Learning, Deep RL, NLP, MLOps Organizer of the Workshop on Computational Topology & Quantum Computing (WoComToQC). Regular instructor for Data Science Africa and Data Science Makers. Committed to mentoring the next generation of African scientists.

Full course catalogue

Consultant

I advise research teams, startups, and organizations on how to apply topological and geometric methods to real problems — and serve as a fractional R&D partner for teams building at the frontier. - AI strategy & architecture — roadmaps, ML pipelines, platform evaluation - TDA-powered analytics — shape analysis, anomaly detection, persistent homology - Deep reinforcement learning — resource allocation, optimization, decision systems - ML workshops & training — custom programmes for universities, research centres, and companies

Consulting services Get in touch

news

Jun 15, 2025	New blog post: LLMs Meet Topology — exploring how topological data analysis can improve large language model interpretability. Read it here.
May 20, 2025	New preprint: Bellman Operator Convergence Enhancements in Reinforcement Learning Algorithms is now on arXiv, with David Krame Kadurha and Domini Jocema Leko Moutouo.
Jan 10, 2025	Joined AIRINA Labs as Head of R&D, leading research at the intersection of topology, geometry, and applied AI.
Oct 04, 2024	New preprint: Topological Foundations of Reinforcement Learning is now available on arXiv. We explore how algebraic topology illuminates the structure of RL state, action, and policy spaces.
Jan 15, 2024	The Shape of Data — our book on geometry-based machine learning and data analysis in R — is now available from No Starch Press! Co-authored with Colleen M. Farrelly.

latest posts

Aug 01, 2025	Persistence Landscapes as ML Features: A Complete Pipeline
Jul 01, 2025	Building AI Research Labs in Africa: Lessons from AIRINA Labs
Jun 15, 2025	LLMs Meet Topology: Can TDA Improve Language Model Interpretability?

selected publications

Book
The Shape of Data: Geometry-Based Machine Learning and Data Analysis in R

Colleen M. Farrelly and Yaé Ulrich Gaba

2024

Abs Bib HTML

A comprehensive guide to geometry-based machine learning and data analysis, covering topological data analysis, persistent homology, and geometric methods with practical implementations in R.
@book{farrelly2024shape, title = {The Shape of Data: Geometry-Based Machine Learning and Data Analysis in {R}}, author = {Farrelly, Colleen M. and Gaba, Ya{\'e} Ulrich}, year = {2024}, publisher = {No Starch Press}, isbn = {9781718503083}, }
arXiv
Bellman Operator Convergence Enhancements in Reinforcement Learning Algorithms

David Krame Kadurha, Domini Jocema Leko Moutouo, and Yaé Ulrich Gaba

2025

Abs arXiv Bib

This paper reviews the topological groundwork for the study of reinforcement learning by focusing on the structure of state, action, and policy spaces, and illustrates how the Banach fixed-point theorem explains the convergence of RL algorithms. Alternative Bellman operator formulations are explored and tested on standard environments including MountainCar, CartPole, and Acrobot.
@article{kadurha2025bellman, title = {Bellman Operator Convergence Enhancements in Reinforcement Learning Algorithms}, author = {Kadurha, David Krame and Moutouo, Domini Jocema Leko and Gaba, Ya{\'e} Ulrich}, year = {2025}, archiveprefix = {arXiv}, primaryclass = {cs.LG}, }
arXiv
Topological Foundations of Reinforcement Learning

David Krame Kadurha and Yaé Ulrich Gaba

2024

Abs arXiv Bib

This work serves as a foundation for deep studies of the topology of state, action, and policy spaces in reinforcement learning. It connects the Banach fixed point theorem to RL algorithm convergence and demonstrates how mathematical insights from topology can inform more efficient RL algorithm design.
@article{kadurha2024topological, title = {Topological Foundations of Reinforcement Learning}, author = {Kadurha, David Krame and Gaba, Ya{\'e} Ulrich}, year = {2024}, archiveprefix = {arXiv}, primaryclass = {cs.LG}, }
JMath
Startpoints and (α, γ)-Contractions in Quasi-Pseudometric Spaces

Yaé Ulrich Gaba

Journal of Mathematics, 2014

Abs DOI Bib HTML

Introduces the concept of startpoint and endpoint for multivalued maps defined on a quasi-pseudometric space and investigates the relation between these new concepts and the existence of fixed points.
@article{gaba2014startpoints, title = {Startpoints and ($\alpha$, $\gamma$)-Contractions in Quasi-Pseudometric Spaces}, author = {Gaba, Ya{\'e} Ulrich}, journal = {Journal of Mathematics}, year = {2014}, publisher = {Wiley/Hindawi}, doi = {10.1155/2014/709253}, }