Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations

• URL: http://arxiv.org/abs/2608.24762v1
• Date: Tue, 25 Aug 2026 15:59:59 GMT
• Title: Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations
• Abstract summary: We introduce factor-space structure, combining factor domains, generator-induced identifications, and position-dependent scales.<n>We show that statistically independent factors need not be geometrically separable.<n>We propose the Factor-Space Topographic Map (FactoMap), which learns interpretable prototypes indexed by a factor-space lattice.
• Score: 5.350329706369602
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many disentanglement methods represent generative factors using Euclidean product coordinates, although the underlying factor spaces may wrap, collapse, or have position-dependent geometry. We introduce factor-space structure, combining factor domains, generator-induced identifications, and position-dependent scales to distinguish topologically equivalent spaces with different factor geometries. We show that statistically independent factors need not be geometrically separable: hue and scale produce effects that grow at different rates, yielding anisotropy that no fixed rescaling removes. We propose the Factor-Space Topographic Map (FactoMap), which learns interpretable prototypes indexed by a factor-space lattice. Topographic learning transfers the lattice's periodicity, collapses, and non-uniform extent to the representation. Experiments show that matching this structure preserves factor continuity and enables disentanglement of the underlying factors.

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