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Count Bridges enable Modeling and Deconvolving Transcriptomic Data

Nic Fishman, Gokul Gowri, Tanush Kumar, Jiaqi Lu, Valentin de Bortoli, Jonathan S. Gootenberg, Omar Abudayyeh

TL;DR

This work introduces Count Bridges, a stochastic bridge process on the integers that provides an exact, tractable analogue of diffusion-style models for count data, with closed-form conditionals for efficient training and sampling, and extends this framework to enable direct training from aggregated measurements via an Expectation-Maximization-style approach.

Abstract

Many modern biological assays, including RNA sequencing, yield integer-valued counts that reflect the number of molecules detected. These measurements are often not at the desired resolution: while the unit of interest is typically a single cell, many measurement technologies produce counts aggregated over sets of cells. Although recent generative frameworks such as diffusion and flow matching have been extended to non-Euclidean and discrete settings, it remains unclear how best to model integer-valued data or how to systematically deconvolve aggregated observations. We introduce Count Bridges, a stochastic bridge process on the integers that provides an exact, tractable analogue of diffusion-style models for count data, with closed-form conditionals for efficient training and sampling. We extend this framework to enable direct training from aggregated measurements via an Expectation-Maximization-style approach that treats unit-level counts as latent variables. We demonstrate state-of-the-art performance on integer distribution matching benchmarks, comparing against flow matching and discrete flow matching baselines across various metrics. We then apply Count Bridges to two large-scale problems in biology: modeling single-cell gene expression data at the nucleotide resolution, with applications to deconvolving bulk RNA-seq, and resolving multicellular spatial transcriptomic spots into single-cell count profiles. Our methods offer a principled foundation for generative modeling and deconvolution of biological count data across scales and modalities.

Count Bridges enable Modeling and Deconvolving Transcriptomic Data

TL;DR

This work introduces Count Bridges, a stochastic bridge process on the integers that provides an exact, tractable analogue of diffusion-style models for count data, with closed-form conditionals for efficient training and sampling, and extends this framework to enable direct training from aggregated measurements via an Expectation-Maximization-style approach.

Abstract

Many modern biological assays, including RNA sequencing, yield integer-valued counts that reflect the number of molecules detected. These measurements are often not at the desired resolution: while the unit of interest is typically a single cell, many measurement technologies produce counts aggregated over sets of cells. Although recent generative frameworks such as diffusion and flow matching have been extended to non-Euclidean and discrete settings, it remains unclear how best to model integer-valued data or how to systematically deconvolve aggregated observations. We introduce Count Bridges, a stochastic bridge process on the integers that provides an exact, tractable analogue of diffusion-style models for count data, with closed-form conditionals for efficient training and sampling. We extend this framework to enable direct training from aggregated measurements via an Expectation-Maximization-style approach that treats unit-level counts as latent variables. We demonstrate state-of-the-art performance on integer distribution matching benchmarks, comparing against flow matching and discrete flow matching baselines across various metrics. We then apply Count Bridges to two large-scale problems in biology: modeling single-cell gene expression data at the nucleotide resolution, with applications to deconvolving bulk RNA-seq, and resolving multicellular spatial transcriptomic spots into single-cell count profiles. Our methods offer a principled foundation for generative modeling and deconvolution of biological count data across scales and modalities.
Paper Structure (106 sections, 17 theorems, 141 equations, 10 figures, 14 tables, 7 algorithms)

This paper contains 106 sections, 17 theorems, 141 equations, 10 figures, 14 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Background on diffusion models
  3. Diffusion as a bridge between noise and data
  4. Sampling the Posterior
  5. Infinitesimal.
  6. Distributional.
  7. Count Bridges
  8. An integer bridge between distributions
  9. Distributional scoring loss for the denoiser
  10. Deconvolution with count bridges
  11. E-Step
  12. M-Step
  13. Approximate Sampling from the conditional distribution
  14. Related works
  15. Stochastic interpolants.
  16. ...and 91 more sections

Key Result

Proposition 2.1

Let $(X_t)_{t\in[0,1]}$ be given by equation eq:unconditional_forward_euclidean. For $0 < s < t \le 1$, consider $(X_s)_{s\in[0,t]}$ conditioned on $X_t = x_t$ and $X_0 = x_0$. Then the conditional law $K_{s\mid 0,t}(\cdot\mid x_0,x_t)$ is Gaussian and can be written where $Z \sim \mathcal{N}(0,\mathrm{Id})$ is independent of $(X_0,X_t)$ and $r({s,t}) = {\frac{\alpha(t)^2 \sigma(s)^2}{\alpha(s)^2

Figures (10)

  • Figure 1: Left: Sample paths for several endpoint gaps $d_1$ (top). Fixing the prefix $[0,t]$ resample $(t,1]$ by the recursive kernel (bottom). Middle: Bessel slack posteriors at initial and intermediate times. The slack $M_t$ concentrates near $0$ as $|d|$ grows. Right: ECDFs of $X_s$ from a one–step kernel $(1\!\to\!s)$ and a two–step kernel $(1\!\to\!t\!\to\!s)$ are indistinguishable, confirming composition.
  • Figure 1: Training Poisson–BD Bridge
  • Figure 2: A scaled and rounded variant of the classic 8 gaussian to two moons task. Here we compare the trajectories of continuous flow matching, discrete flow matching, and count bridges. CB achieves the lowest $W_2$, MMD, and EMD, see Table \ref{['tab:discrete_moons']}.
  • Figure 3: Guided Sampling to for $x_0^\approx$
  • Figure 3: CFM, DFM, and CB on our low-rank mixture of Gaussians transport experiment across dimensions and NFE. See App. \ref{['app:lowrank-gaussian']} for full details.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 4.1: First–order aggregate projection
  • Proposition A.1: Total jumps process and time-rescaling invariance
  • proof
  • Lemma A.2: Binomial--Hypergeometric structure of Count Bridges
  • Lemma A.3: Binomial composition
  • proof
  • Lemma A.4: Hypergeometric composition
  • proof
  • ...and 22 more