Title: CORDS - Continuous Representations of Discrete Structures
Primary area: learning on graphs and other geometries & topologies
Keywords: Continuous set representations, Neural fields, Variable-cardinality prediction, Invertible encoding/decoding, Diffusion and flow matching, Object detection, Molecular generation, Simulation-based inference

Abstract: Many learning problems require predicting sets of objects without knowing their
number in advance. Examples include object detection, molecular modeling, and
a variety of inference problems for scientific data, such as astrophysical source
detection. Existing methods often rely on padded representations, or must explicitly
infer the cardinality directly from data, which often poses challenges. We present
a novel strategy for addressing this challenge by casting prediction of variable
cardinality as a continuous inference problem, where the number of objects is
recovered directly from field mass. Our approach, CORDS (Continuous Representations
of Discrete Structures), provides a bijective representation that maps sets
of spatial objects with features to continuous density and feature fields. Because
the mapping is invertible, models can operate entirely in field space and still be
decoded back to discrete sets. We evaluate CORDS across molecular generation
and regression, object detection, simulation-based inference in astronomy, and a
mathematical task that recovers local maxima, demonstrating robust handling of
variable cardinality with competitive accuracy.

TL;DR: We turn discrete objects into continuous fields that implicitly encode their count, offering a simple way to handle variable cardinality across tasks and domains.