VisFlow: Interpretable Flow Matching for Visibility Reconstruction, Characterization, and Recognition

Methods for learning continuous representations, such as latent manifolds, are fundamental and ubiquitous in modern machine learning. Yet many geometric objects combine continuous shape information with discrete combinatorial structure. This combination presents a fundamental challenge for representation learning because small geometric perturbations can induce abrupt changes in the underlying discrete representation. As a result, the smoothness assumptions that make continuous representations effective no longer hold. Polygons provide a canonical example. Their vertex locations vary continuously, but infinitesimal changes in geometry can dramatically alter the associated visibility graph. Existing methods address the inverse mapping from visibility graphs to polygon geometry using vertex-, triangulation-, or signed-distance-function (SDF)-based representations. Vertex- and triangulation-based approaches construct polygons through explicit geometric operations. However, they are less effective at modeling the relationship between geometry and visibility. SDF-based approaches capture this relationship more naturally. Their intermediate states, however, do not correspond to explicit stages of polygon construction. We present VisFlow, a novel, interpretable flow formulation that combines continuous SDF representations with explicit polygon evolution. Our method models polygon generation as rectified flow in the latent space of continuous SDFs. Standard rectified flow linearly interpolates between noise and the target SDF, retaining the key limitation of SDF-based generation. Its intermediate states lack an interpretation of polygon formation. In contrast, VisFlow learns approximately linear latent paths to represent explicit stages of polygon growth along SDF normals. We evaluate VisFlow on Visibility Reconstruction and Visibility Characterization. Visibility Reconstruction generates one polygon consistent with a visibility graph \(G\), while Visibility Characterization characterizes all such polygons. VisFlow maintains 20% more visibility relations on Visibility Reconstruction and improves coverage by 30% on Visibility Characterization compared with prior methods. We further show that VisFlow generalizes to Visibility Recognition, where the input graph is not guaranteed to be valid. We also demonstrate that VisFlow extends to a real-world protein reconstruction problem by applying its trajectory-supervision formulation to generate protein structures from contact graphs.