CORDS: Continuous Representations of Discrete Structures

TL;DR

CORDS tackles the problem of predicting variable-sized sets by introducing a bijective field-based representation that maps discrete objects to continuous density and feature fields over a domain, with the total density mass encoding the cardinality. By ensuring exact invertibility, models can learn entirely in field space while guaranteeing precise recovery of discrete sets, regardless of unknown set size. Across molecular generation, object detection, SBI for FRBs, and a synthetic maxima task, CORDS demonstrates competitive performance with the added ability to encode non-categorical object features directly in the field space. The approach offers a unified, scalable framework for handling variable cardinality across diverse domains, reducing reliance on padding or fixed-slot budgets and enabling principled, end-to-end field-space learning with exact decoding when needed.

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.

Figures (8)

• Figure 1: An image with $N$ MNIST digits (top) is encoded with CORDS into a density field $\rho(\mathbf r)$ (middle) and per-class feature fields $h_k(\mathbf r)$ (bottom). The number of objects is encoded directly in the density mass, $N = \int\rho(\mathbf r)\, d\mathbf r$.
• Figure 2: Sampling strategies for evaluating fields. Left: Importance sampling draws coordinates in proportion to the density $\rho$, concentrating samples where signal is present. Right: Uniform sampling evaluates fields on a fixed grid, covering the domain evenly. In both panels, the curves are isocontours of $\rho(\mathbf{r})$, and the colors show the values of the feature fields $\mathbf{h}_k(\mathbf{r})$.
• Figure 3: Left: Conditional generation on QM9. Histogram of the predicted atom count distribution $p(N | c)$ when conditioning on property ranges unseen during training. Right: Unconditional generation results on QM9, evaluated using OpenBabel postprocessing, following VoxMol. Baseline results are adapted from funcmol.
• Figure 4: Simulation-based inference on light curves. (a) Observed light curve $\ell$ (blue) with reconstructions from posterior samples $\theta \sim p(\theta \mid \ell)$. Each reconstruction is obtained by decoding sampled fields into component parameters $\theta$ and simulating the resulting light curve. (b) Posterior over the number of components $p(N \mid \ell)$.
• Figure 5: A molecular graph (top) is encoded with Cords into a density field $\rho(\mathbf r)$ (middle) and feature fields $h_k(\mathbf r)$ (bottom), which correspond to atom types here. The number of atoms is encoded directly in the density mass, $K = \int\rho(\mathbf r)\, d\mathbf r$.

Theorems & Definitions (12)

• Proposition A.1: Cardinality
• proof
• Lemma A.2: Gram matrix is SPD
• proof
• Proposition A.3: Feature inversion with correct $\alpha$
• proof
• Proposition A.4: Position recovery
• proof
• Definition 1: Set--field dual pair
• Proposition B.1: Continuous convolution $\;\longrightarrow\;$ message passing