Kanno and Colleagues Propose Hodge Spectral Relaxations for Topology-Constrained Optimization

A new differentiable framework for topology-constrained optimisation tackles the challenge of directly controlling topological features such as Betti numbers and persistent homology. Satoshi Kanno and Yoshi-aki Shimada of the Quantum Technology Division propose using Hodge-spectral relaxations and low-pass spectral filters to create a system applicable to both soft graphs and point clouds. The approach represents homological information through zero and near-zero modes, enabling differentiable topological objectives and yielding improved gradient behaviour and smoother scale-normalisation compared to existing persistent-homology-based methods. Links to polynomial spectral methods and potential applications in quantum trace estimation are also revealed, offering a new pathway for optimising objects with prescribed topological structures. Topological Data Analysis (TDA) has become increasingly prevalent in fields ranging from materials science to data science, offering powerful tools for understanding the shape and structure of data. However, a significant hurdle has been the…