CORDS: Continuous Representations of Discrete Structures

2026-01-29

AI Summary

This work addresses the challenge of predicting variable-sized discrete sets with unknown cardinality—such as in object detection and molecular modeling—where existing approaches often rely on fixed-size assumptions or padding strategies, leading to inherent modeling limitations. The authors propose the first invertible continuous field representation framework, which leverages a bijective mapping to losslessly transform discrete sets into continuous density and feature fields. This formulation enables modeling in a continuous space while allowing exact reconstruction of the original discrete structures. The method requires no predefined set size or padding, supports end-to-end training, and achieves competitive accuracy across diverse tasks including molecular generation and property regression, object detection, simulation-based inference, and recovery of local extrema, demonstrating robustness to unknown set cardinalities.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Representation Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Many learning problems require predicting sets of objects when the number of objects is not known beforehand. Examples include object detection, molecular modeling, and scientific inference tasks such as astrophysical source detection. Existing methods often rely on padded representations or must explicitly infer the set size, which often poses challenges. We present a novel strategy for addressing this challenge by casting prediction of variable-sized sets as a continuous inference problem. Our approach, CORDS (Continuous Representations of Discrete Structures), provides an invertible mapping that transforms a set of spatial objects into continuous fields: a density field that encodes object locations and count, and a feature field that carries their attributes over the same support. Because the mapping is invertible, models operate entirely in field space while remaining exactly decodable to discrete sets. We evaluate CORDS across molecular generation and regression, object detection, simulation-based inference, and a mathematical task involving recovery of local maxima, demonstrating robust handling of unknown set sizes with competitive accuracy.

Problem

• variable-sized sets
• unknown set size
• discrete structures
• set prediction
• object detection

Innovation

• continuous representation
• invertible mapping
• set prediction
• density field
• discrete-continuous transformation