Invariant Signal Representations with Wavelets and Deep Learning

This project is one of a number that are in competition for funding from the University of Bath for entry in October 2027, supervised by Dr Georgios Exarchakis, Department of Computer Science, and Dr Carolin Villforth, Department of Physics.

Overview of the Research:

Many signals carry the same information regardless of how they are rotated or shifted. A galaxy image means the same thing in any orientation, and the same holds for a molecule or a crystal structure. Representations that remove the effect of such transformations make learning easier and are particularly useful when data is scarce. This project will develop new representations of this kind and study how invariance is encoded in them. Deep networks often learn features that are approximately invariant, but we lack a clear account of how and when this happens. A central question for the project is whether designed representations can explain this behaviour, or even replace it.

The work builds on Mallat’s wavelet scattering transform and on recent models that are simple yet capture invariance well. Brochard, Zhang and Mallat showed that wavelet covariance models can match deep networks in texture synthesis, and Sanborn and colleagues showed that neural networks can learn group-invariant bispectra directly from data. We have previously developed the solid harmonic wavelet scattering transform and, more recently, the solid harmonic wavelet bispectrum. The bispectrum is invariant to rotations and translations and keeps the relative phase between wavelet coefficients at different scales. In a recent paper in Advanced Science it matched or outperformed deep networks in texture classification, medical imaging and the prediction of galaxy properties, with particularly strong results on small datasets.

The project will start with the development of a novel family of wavelets that we are designing to study roto-translation invariance further. We pursue the design of a multi-scale signal representation with invariance properties that linearise decision boundaries in physics tasks. While our focus is on signals such as images, we can also consider structured data through graph-based representations. For example, we can identify features of the graph Laplacian that reflect graph structure and are invariant to actions on the graph. We also draw on the structure of biological and artificial neural networks when designing new algorithms. Symmetry is central to many physical systems, and we expect applications in astrophysics, in collaboration with Dr Villforth, as well as in materials science and chemistry. We expect the student to reproduce some of our preliminary work in the first year and to aim for original results by the end of it. From the second year onwards the student will take part in designing the exact research direction. Multiscale operators that encode symmetries typically use the same symmetries at every scale. An interesting avenue is to develop operators that encode different symmetries at different scales and the relationships between them.

The student will be co-supervised across Computer Science and Physics and will have access to Hex, the GPU cloud of the Department of Computer Science. We aim to publish the work at leading machine learning conferences and in journals.

Research Environment:

The successful applicant will experience a unique and intellectually stimulating research environment within the Department of Computer Science and the Department of Physics, undertake a specific programme of specialist research skill development, and contribute to advancing a strategic research agenda. The successful applicant will be a part of the Artificial Intelligence and Machine Learning group of the Computer Science department. It is an expanding group with a wide variety of research interests that will enhance the applicant’s knowledge. The successful applicant will also be welcome to join relevant seminars and research discussions within the Physics department. The successful applicant will enjoy a collaborative research environment within the group and the departments. They will have full access to University of Bath resources, mentoring, and all offered in-house training.