DataSig are looking for students to undertake summer research projects as part of a collaboration with GCHQ. Funding is available for 5 projects of around 4-8 weeks long. To apply please email Viktoriia Davletshyna by the 18th June.
Project 1: Measuring when learned classifiers stretch reality.
How much does a trained model stretch a data-defined function? Lipschitz norms, robustness, and adversarial images.
Suppose we know a function only through data: for example, images x_i and labels or scores y_i. If we train a function f to extend this data to unseen inputs, does it extend the geometry of the data in a trustworthy way? In particular, does it create regions where tiny changes in input cause large changes in output? The project studies this through empirical Lipschitz constants and adversarial image perturbations.
Project 2: Anomalies at scale with signatures, nearest neighbours, and reproducible workflows.
From signature anomaly scores to a production-style detector for millions of sensor windows.
Can the nearest-neighbour/signature approach to anomaly detection be turned from a research notebook into a small professional system that ingests many streams, computes robust path features, builds a nearest-neighbour index, and produces ranked “worrisome behaviour” reports?
Project 3: Messages in the Roughness: Signature Kernels and Rough Path Geometry
Many data streams contain weak but highly structured signals hidden within overwhelming noise. In this project we will study a controlled “needle in a haystack” problem in which a finite collection of messages is encoded as smooth rough paths and inserted into Brownian motion (including its Lévy area). The task is to determine which message was embedded in a given observation. While the signal may be invisible to methods based only on first-order information, it can remain detectable through the higher-order geometric information captured by path signatures. Of particular interest are messages encoded through area and other genuinely rough-path effects, providing a mathematically natural setting in which higher-order features become essential. The project will combine mathematical modelling with computational experimentation, using Python and JAX to simulate data, implement kernel methods, and investigate the performance of different approaches in practice.
The project will focus on signature kernels and their partial differential equation formulation. Signature kernels can be computed via a hyperbolic Goursat PDE, while recent work of Lemercier et al. extends this framework by incorporating higher-order log-signature information directly into the PDE system, allowing rough-path features such as area to be incorporated without explicitly resolving fine-scale oscillations. This leads to a range of mathematical and computational questions: which levels of the signature are genuinely informative, how much discrimination is gained by incorporating higher-order log-signatures, and under what conditions can a hidden message be recovered reliably from noise? Further directions include efficient numerical methods for the associated PDEs, the statistical limits of message recovery, and connections with attention-based constructions arising from hyperbolic developments and linear controlled differential equations.
Project 4: Principled loss functions for continuous-time learning
Most time series models are sequence-to-sequence and trained with pointwise losses such as mean squared error (MSE). Neural differential equations enable path-to-path learning, where inputs and outputs are continuous-time trajectories. An open question is which loss functions are best suited to path-to-path learning, as previous work has shown that pointwise objectives can overfit local noise and fail to capture temporal structure (Ramsay and Silverman, 2005; Kudrat et al., 2025).
Project 5: Low-rank and sparse structured approximations for large linear maps
Modern machine learning models rely on many large linear maps, usually implemented as matrix-vector or matrix-matrix multiplications. These operations often dominate memory use and computational cost, especially in neural networks with large hidden layers, attention projections, or embedding maps. A classical way to simplify such operators is to replace a large matrix by a low-rank approximation, for example using the singular value decomposition, which gives an optimal approximation in standard matrix norms (Eckart and Young, 1936). Randomised numerical linear algebra extends these ideas by using sampling and sketching to obtain cheaper approximations for large matrices (Halko et al., 2011; Woodruff, 2014).
An alternative approach is to seek sparse or structured sparse approximations. Rather than replacing a matrix by a dense low-dimensional factorisation, these methods try to select, prune, or recombine a small number of informative components, such as rows, columns, blocks, or entries. Such approximations can be more interpretable and hardware friendly, but may be harder to analyse and optimise. This project will compare these two perspectives on simple examples motivated by machine learning, with the goal of understanding the trade-offs between efficiency, accuracy, and interpretability in model compression.