Topics Financial Mathematics and Artificial Intelligence
Hi, my name is Mattia Villani.
I am a researcher in AI with applications in mathematics and finance. I did my PhD at King’s under supervision of Peter McBurney and Dr. Frederik Mallmann-Trenn. Then I worked at Symbolica for a brief while, when they had just secured their series A. During and after my PhD I worked as an Applied Research Scientist at JP Morgan Chase, in the AI Research team first and then in the Quantum-Inspired Algorithms team, in Global Technology.
I am interested in using mathematics to understand how AI works and using AI to solve mathematical problems. I am also interested in financial applications. I am collating here some of my research interests and sharing guides to how I understand topics in mathematics.
I’d like this to be an introduction in topics that interest me. I’d like to write in a style that comes from intuitions, but also maintains a good level of formal argumentations.
These are some of the topics I will cover:
Mathematics for Quantitative Finance,
Artificial Intelligence,
Category Theory and Automated Theorem Proving.
This is a work in progress. Content will be added over time. If you have any suggestions, please reach out to me on LinkedIn (Mattia Jacopo Villani).
Recent Research.
MetaTT: A global tensor-train Adapter for parameter-efficient fine-tuning (2025): you might have used LoRA before. Here we propose a new method for parameter-efficient fine-tuning. High parameter compression compared to LoRA (~10x) and comparable performance using tensor train decomposition. I helped with some engineering here.
A Unified Framework for Provably Efficient Algorithms to Estimate Shapley Values (2025 - NeurIPS): there are many ways to estimate Shapley values (an important feature importance technique). Two of the main approaches are matrix-vector and least-squares estimation. We look at unifying these approaches and provide a general framework for understanding Shapley estimation. I developed the algorithmic implementation, which now works for high dimensional problems, and performed experiments.
Any Deep ReLU Network is Shallow (2025 - ECAI): Nandi and I showed that you can write a network with layers as a network with 3 layers, if you allow your weights to be in the extended reals. The proof is constructive and has some interesting implications for interpretability and pruning.
Relating Piecewise Linear Kolmogorov Arnold Networks to ReLU Networks (2025 - AISTATS): Nandi, Niels and I had a look at how to relate KANs and ReLU Networks. Turns out we can write any ReLU network as a KAN and vice versa. We all developed the theory together.
PICE: Polyhedral Complex Informed Counterfactual Explanations (2025 - AIES in AAAI): we can get exactly minimal counterfactuals for ReLU networks, with sufficient compute. This was my internship project.
Trading-off accuracy and communication cost in federated learning (2025 - AAMAS): using training by sampling, we can get incredibly cheap communication costs in federated learning. There’s also a nice connection to Zonotope geometry.
Graph Convolutional Neural Networks as Parametric CoKleisli morphisms (2022 - SYCO-10 workshop): Bruno and I sat down to look at GCNs via category theory. This paper was the result of a weekend together in London. The paper just describes weight sharing in GCNs. Bruno presented it nicely here.
Feature Importance for Time Series Data: Improving KernelSHAP(2022 - ICAIF workshop): using Shapley value as a feature importance techniques relies on the assumption that every feature is a player in a game. If our model is a time series model, this isn’t quite right. We propose a method for computing Shapley for time series. This was my internship project.
Past Projects.
United Italian Societies (2017-2024): I helped Umberto set up this non-profit organisation, back in 2022. Before that, I managed a precursor project called Italian Societies Student Network. Within the UIS I founded the Research Centre. We helped Italian students in the UK connect, orgnaise educational events and provided support.
The Red Flower Factory (2020): I helped set up this spin-off start-up from Sevendots - a CPG marketing consultancy.