Relative Topology as a Supervisory Signal for Long-Range Dependency

Topological data analysis provides principled descriptors of global structure beyond local visual appearance. Standard topological invariants, however, characterise an entire space and are therefore insensitive to which elements are selected by a query, limiting their use for query-dependent relational tasks. We address this mismatch using _relative homology_, which characterises a space with respect to a designated subspace. We introduce _relative topological supervision_, a training-only auxiliary objective in which the query determines the reference subspace and relative homology provides a target encoding the resulting query-dependent structural relation. The target is computed directly from each training image and query, requires no additional annotation, and the auxiliary prediction head is discarded at inference. On the long-range same-path task in Pathfinder, relative topological supervision enables a plain feedforward convolutional network trained from scratch, without residual connections, recurrence, or attention, to solve the task, whereas matched controls using query-independent global topology or permuted topological targets fail. The method remains effective with compact models and limited training data, and we further demonstrate its applicability to retinal vessel segmentation. These results show that relative topology provides a principled mechanism for aligning topological supervision with query-dependent visual reasoning, allowing simple neural architectures to learn global relational structure that is otherwise difficult to capture.