# CORDS: Continuous Representations of Discrete Structures

arXiv: https://arxiv.org/abs/2601.21583 · HTML: https://mlsanity.com/paper/2601.21583/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 52 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 30 of 31)

- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [Deformable DETR: Deformable Transformers for End-to-End Object Detection](https://mlsanity.com/paper/2010.04159/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [FCOS: Fully Convolutional One-Stage Object Detection](https://mlsanity.com/paper/1904.01355/) (2019)
- [EfficientDet: Scalable and Efficient Object Detection](https://mlsanity.com/paper/1911.09070/) (2019)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [CenterNet: Keypoint Triplets for Object Detection](https://mlsanity.com/paper/1904.08189/) (2019)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)
- [MoFlow: An Invertible Flow Model for Generating Molecular Graphs](https://mlsanity.com/paper/2006.10137/) (2020)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [Diffusion-based Molecule Generation with Informative Prior Bridges](https://mlsanity.com/paper/2209.00865/) (2022)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [COIN++: Neural Compression Across Modalities](https://mlsanity.com/paper/2201.12904/) (2022)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [3D molecule generation by denoising voxel grids](https://mlsanity.com/paper/2306.07473/) (2023)
- [Learn to Scale: Generating Multipolar Normalized Density Maps for Crowd Counting](https://mlsanity.com/paper/1907.12428/) (2019)
- [RT-DETRv3: Real-time End-to-End Object Detection with Hierarchical Dense Positive Supervision](https://mlsanity.com/paper/2409.08475/) (2024)
- [Learning a Continuous Representation of 3D Molecular Structures with Deep Generative Models](https://mlsanity.com/paper/2010.08687/) (2020)
- [Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systems](https://mlsanity.com/paper/2502.17019/) (2025)
- [Generative Neural Fields by Mixtures of Neural Implicit Functions](https://mlsanity.com/paper/2310.19464/) (2023)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [DeepSource: Point Source Detection using Deep Learning](https://mlsanity.com/paper/1807.02701/) (2018)
- [Redundant Queries in DETR-Based 3D Detection Methods: Unnecessary and Prunable](https://mlsanity.com/paper/2412.02054/) (2024)

## Built on by (most cited first, 1 of 1)

- [Generative Molecular Morphing for Flexible-Size Design via Unbalanced Optimal Transport](https://mlsanity.com/paper/2606.07239/) (2026)

## Abstract (arXiv)

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.