# Avoiding subtraction and division of stochastic signals using normalizing flows: NFdeconvolve

arXiv: https://arxiv.org/abs/2501.08288 · HTML: https://mlsanity.com/paper/2501.08288/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 14 Jan 2025.

## Builds on (most cited first, 8 of 8)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [Density Deconvolution with Normalizing Flows](https://mlsanity.com/paper/2006.09396/) (2020)

## Built on by (most cited first, 2 of 2)

- [Mamba time series forecasting with uncertainty quantification](https://mlsanity.com/paper/2503.10873/) (2025)
- [Inherited or produced? Inferring protein production kinetics when protein counts are shaped by a cell's division history](https://mlsanity.com/paper/2506.09374/) (2025)

## Abstract (arXiv)

Across the scientific realm, we find ourselves subtracting or dividing stochastic signals. For instance, consider a stochastic realization, x, generated from the addition or multiplication of two stochastic signals a and b, namely x=a+b or x = ab. For the x=a+b example, a can be fluorescence background and b the signal of interest whose statistics are to be learned from the measured x. Similarly, when writing x=ab, a can be thought of as the illumination intensity and b the density of fluorescent molecules of interest. Yet dividing or subtracting stochastic signals amplifies noise, and we ask instead whether, using the statistics of a and the measurement of x as input, we can recover the statistics of b. Here, we show how normalizing flows can generate an approximation of the probability distribution over b, thereby avoiding subtraction or division altogether. This method is implemented in our software package, NFdeconvolve, available on GitHub with a tutorial linked in the main text.