  Learning representations that capture rich semantic relationships and
accommodate propositional calculus poses a significant challenge. Existing
approaches are either contrastive, lacking theoretical guarantees, or fall
short in effectively representing the partial orders inherent to rich
visual-semantic hierarchies. In this paper, we propose a novel approach for
learning visual representations that not only conform to a specified semantic
structure but also facilitate probabilistic propositional reasoning. Our
approach is based on a new nuclear norm-based loss. We show that its minimum
encodes the spectral geometry of the semantics in a subspace lattice, where
logical propositions can be represented by projection operators.
