# Nonnegative spatial factorization

arXiv: https://arxiv.org/abs/2110.06122 · HTML: https://mlsanity.com/paper/2110.06122/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 82 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [A Framework for Interdomain and Multioutput Gaussian Processes](https://mlsanity.com/paper/2003.01115/) (2020)
- [A Tutorial on Sparse Gaussian Processes and Variational Inference](https://mlsanity.com/paper/2012.13962/) (2020)
- [Variational Latent Gaussian Process for Recovering Single-Trial Dynamics from Population Spike Trains](https://mlsanity.com/paper/1604.03053/) (2016)
- [PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference](https://mlsanity.com/paper/1709.09216/) (2017)
- [A joint model of unpaired data from scRNA-seq and spatial transcriptomics for imputing missing gene expression measurements](https://mlsanity.com/paper/1905.02269/) (2019)
- [Contrastive latent variable modeling with application to case-control sequencing experiments](https://mlsanity.com/paper/2102.06731/) (2021)
- [Efficient non-conjugate Gaussian process factor models for spike count data using polynomial approximations](https://mlsanity.com/paper/1906.03318/) (2019)
- [Latent variable modeling with random features](https://mlsanity.com/paper/2006.11145/) (2020)
- [Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition](https://mlsanity.com/paper/2106.05992/) (2021)

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

- [Spatial Transcriptomics Dimensionality Reduction using Wavelet Bases](https://mlsanity.com/paper/2205.11243/) (2022)
- [hSNMF: Hybrid Spatially Regularized NMF for Image-Derived Spatial Transcriptomics](https://mlsanity.com/paper/2602.02638/) (2026)

## Abstract (arXiv)

Gaussian processes are widely used for the analysis of spatial data due to their nonparametric flexibility and ability to quantify uncertainty, and recently developed scalable approximations have facilitated application to massive datasets. For multivariate outcomes, linear models of coregionalization combine dimension reduction with spatial correlation. However, their real-valued latent factors and loadings are difficult to interpret because, unlike nonnegative models, they do not recover a parts-based representation. We present nonnegative spatial factorization (NSF), a spatially-aware probabilistic dimension reduction model that naturally encourages sparsity. We compare NSF to real-valued spatial factorizations such as MEFISTO and nonspatial dimension reduction methods using simulations and high-dimensional spatial transcriptomics data. NSF identifies generalizable spatial patterns of gene expression. Since not all patterns of gene expression are spatial, we also propose a hybrid extension of NSF that combines spatial and nonspatial components, enabling quantification of spatial importance for both observations and features. A TensorFlow implementation of NSF is available from https://github.com/willtownes/nsf-paper .