[Paper Review] CORDS: Continuous Representations of Discrete Structures

Tin Hadži Veljković, Erik J. Bekkers+2 more|arXiv (Cornell University)|Jan 29, 2026

Machine Learning in Materials Science0 citations

TL;DR

CORDS casts prediction of variable-sized sets as a continuous, invertible inference problem, encoding sets into density and feature fields that are exactly decodable back to discrete sets.

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.

Motivation & Objective

• Address prediction of variable-sized sets without pre-specifying set size or padding.
• Provide an invertible mapping from sets to continuous fields to enable learning entirely in field space.
• Demonstrate applicability across molecular generation, object detection, simulation-based inference, and mathematical recovery tasks.

Proposed method

• Define an invertible mapping that converts a discrete set of spatial objects into a density field (locations and count) and a feature field (attributes) over the same support.
• Operate models in field space rather than discrete sets, while guaranteeing exact decodability back to the original set.
• Evaluate the approach across multiple domains: molecular generation and regression, object detection, simulation-based inference, and a mathematical recovery task involving local maxima.

Experimental results

Research questions

• RQ1 Can variable-sized sets be accurately predicted without explicit set size or padding by using continuous field representations?
• RQ2 Does the invertible field-space mapping enable exact recovery of discrete sets across diverse tasks?
• RQ3 How does CORDS perform on tasks involving unknown set sizes compared to traditional padded representations?

Key findings

• CORDS provides an invertible mapping from discrete sets to continuous density and feature fields.
• Models trained in field space are exactly decodable to discrete sets.
• The approach is evaluated across molecular generation and regression, object detection, simulation-based inference, and a mathematical task, showing robust handling of unknown set sizes with competitive accuracy.