# Embedding interpretable ℓ1-regression into neural networks for uncovering temporal structure in cell imaging

arXiv: https://arxiv.org/abs/2603.02899 · HTML: https://mlsanity.com/paper/2603.02899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 46 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
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- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
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- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images](https://mlsanity.com/paper/1506.07365/) (2015)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
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- [Model-Based Deep Learning](https://mlsanity.com/paper/2012.08405/) (2020)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [The Elements of Differentiable Programming](https://mlsanity.com/paper/2403.14606/) (2024)
- [Implicit differentiation of Lasso-type models for hyperparameter optimization](https://mlsanity.com/paper/2002.08943/) (2020)
- [Longitudinal Variational Autoencoder](https://mlsanity.com/paper/2006.09763/) (2020)
- [ICE-NODE: Integration of Clinical Embeddings with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2207.01873/) (2022)
- [A Differentiable Alternative to the Lasso Penalty](https://mlsanity.com/paper/1609.04985/) (2016)

## Abstract (arXiv)

While artificial neural networks excel in unsupervised learning of non-sparse structure, classical statistical regression techniques offer better interpretability, in particular when sparseness is enforced by ℓ1 regularization, enabling identification of which factors drive observed dynamics. We investigate how these two types of approaches can be optimally combined, exemplarily considering two-photon calcium imaging data where sparse autoregressive dynamics are to be extracted. We propose embedding a vector autoregressive (VAR) model as an interpretable regression technique into a convolutional autoencoder, which provides dimension reduction for tractable temporal modeling. A skip connection separately addresses non-sparse static spatial information, selectively channeling sparse structure into the ℓ1-regularized VAR. ℓ1-estimation of regression parameters is enabled by differentiating through the piecewise linear solution path. This is contrasted with approaches where the autoencoder does not adapt to the VAR model. Having an embedded statistical model also enables a testing approach for comparing temporal sequences from the same observational unit. Additionally, contribution maps visualize which spatial regions drive the learned dynamics.