Embedding interpretable $\ell_1$-regression into neural networks for uncovering temporal structure in cell imaging

Fabian Kabus; Maren Hackenberg; Julia Hindel; Thibault Cholvin; Antje Kilias; Thomas Brox; Abhinav Valada; Marlene Bartos

cs.LGMar 3, 2026

Embedding interpretable $\ell_1$ -regression into neural networks for uncovering temporal structure in cell imaging

Authors: Fabian Kabus, Maren Hackenberg, Julia Hindel, Thibault Cholvin, Antje Kilias, Thomas Brox, Abhinav Valada, Marlene Bartos,

Organizations: Institute of Medical Biometry and Statistics (IMBI), Faculty of Medicine and Medical Center, University of Freiburg, Germany · Department of Computer Science, University of Freiburg, Germany · Institute of Physiology I, Faculty of Medicine, University of Freiburg, Germany · Freiburg Center for Data Analysis, Modeling and AI (FDMAI), University of Freiburg, Germany ·

Abstract

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 $\ell_1$ 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 $\ell_1$ -regularized VAR. $\ell_1$ -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.

Figures & tables

Figure 1: High-level overview of the sparse spatiotemporal dimension reduction in the end-to-end model. The frames

\bm{x}_{t}^{(i)}

for all time series are first aggregated over time to form a mean frame

\bar{\bm{x}}

, which captures the static structure. The dynamic component,

\bm{x}_{t}^{(i)}-\bar{\bm{x}}

, is encoded into a latent representation

\bm{z}_{t}^{(i)}

. The latent representation

\operatorname{vec}(\bm{z}_{t}^{(i)})

is modeled using a sparse vector autoregressive (VAR) model of order

p

, which forecasts

\hat{\bm{z}}_{t}^{(i)}

. The coefficient matrices

\bm{A}_{1}^{(i)},\ldots,\bm{A}_{p}^{(i)}

are fit using

\ell_{1}

-regression from scratch in each forward pass and they contain the learned spatiotemporal relationships. Finally, the decoder reconstructs the frame

\hat{\bm{x}}_{t}^{(i)}

from

\hat{\bm{z}}_{t}^{(i)}

and the static mean frame

\bar{\bm{x}}

, which is reintroduced via a skip connection.

Figure 2: Effect of the skip connection on the latent representation and reconstruction from one run at $t=263$ : Top row (a–c) : model without skip connection. Bottom row (d–f) : model with skip connection. (a, d) Latent representation $\bm{z}_{t}^{(i)}$ . (b, e) Decoder output $f_{\text{dec}}(\bm{z}_{t}^{(i)})$ , brightness enhanced for visual clarity. (c) Ground truth input $\bm{x}_{t}^{(i)}$ , contrast enhanced. (f) Reconstruction $\bm{\hat{x}}_{t}^{(i)}$ with skip connection, contrast enhanced.

$G$		$H$		p-value
$F_{1}$ to	$F_{10}$	$F_{11}$ to	$F_{20}$		1781
$N_{1}$ to	$N_{10}$	$N_{11}$ to	$N_{20}$		5136
$F_{1}$ to	$F_{10}$	$N_{1}$ to	$N_{10}$	0 .	0002
$F_{11}$ to	$F_{20}$	$N_{11}$ to	$N_{20}$	0 .	0021
$F_{1}$ to	$F_{10}$	$N_{11}$ to	$N_{20}$	0 .	0001
$F_{11}$ to	$F_{20}$	$N_{1}$ to	$N_{10}$	0 .	0002

Table 1: Group comparisons using VAR coefficients (

\lambda=0.005

), between runs F with familiar condition, between runs N with novel condition, and between runs F and N of different type. The similarity between sets of experimental runs based on their estimated dynamics is assessed by p-values. Significant p-values (after Bonferroni correction) are indicated by boldface.

Figure 3: Assessing the effect end-to-end learning via automatic differentiation (AD) with visual contribution maps, $\bm{\Omega}$ , which illustrate the spatial origins of estimated dynamics, averaged across all runs of an exemplary mouse: (a) Familiar condition without end-to-end training. (b) Familiar condition with end-to-end training. (c) Difference map ( $\bm{\Omega}^{F}-\bm{\Omega}^{N}$ ) without end-to-end training. (d) Difference map ( $\bm{\Omega}^{F}-\bm{\Omega}^{N}$ ) with end-to-end training.

Figure 4: Effect of $\ell_{1}$ regularization parameter $\lambda$ on time series forecast and reconstruction. (a) Selected dimensions of $\bm{z}_{t}$ in black (reference) and their forecasts $\hat{\bm{z}}_{t}$ for varying $\lambda$ (in colors) over time. (b–d) Reconstructions at $t=263$ : (b) Ground truth $\bm{x}_{t}$ . (c) Reconstruction $\hat{\bm{x}}_{t}$ for $\lambda=0.003$ . (d) Reconstruction $\hat{\bm{x}}_{t}$ for $\lambda=0.032$ .

		Sequential	Embedded	End-to-End
$\lambda=0.005$	$\mathcal{L}_{\text{rec}}$ [ $10^{-3}$ ]	0.087 $\pm$ 0.030	0.101 $\pm$ 0.035	0.103 $\pm$ 0.038
$\lambda=0.005$	$\mathcal{R}_{\text{var}}$	0.472 $\pm$ 0.040	0.421 $\pm$ 0.054	0.398 $\pm$ 0.068
$\lambda=0.01$	$\mathcal{L}_{\text{rec}}$ [ $10^{-3}$ ]	0.087 $\pm$ 0.030	0.106 $\pm$ 0.037	0.112 $\pm$ 0.042
$\lambda=0.01$	$\mathcal{R}_{\text{var}}$	0.682 $\pm$ 0.063	0.611 $\pm$ 0.074	0.530 $\pm$ 0.081
$\lambda=0.02$	$\mathcal{L}_{\text{rec}}$ [ $10^{-3}$ ]	0.087 $\pm$ 0.030	0.113 $\pm$ 0.040	0.118 $\pm$ 0.045
$\lambda=0.02$	$\mathcal{R}_{\text{var}}$	0.853 $\pm$ 0.061	0.791 $\pm$ 0.086	0.699 $\pm$ 0.072

Table 2: Ablation study comparing different methods of combining the autoencoder and the VAR model (sequential training, embedded VAR without end-to-end gradients, and our proposed end-to-end approach) via reconstruction error (

\mathcal{L}_{\text{rec}}

) and prediction error

\mathcal{R}_{\text{var}}

at three different levels of

\lambda

Autoregressive Modeling

Autoregressive modeling predicts sequences step by step from prior context, spanning language, time series, images, audio, and scientific simulations.

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Mar 18, 2026J/K move · Enter open · S save

Fundamental Limits of Neural Network Sparsification: Evidence from Catastrophic Interpretability Collapse

Dip Roy, Rajiv Misra, Sanjay Kumar Singh

Department of Computer Science and Engineering, Indian Institute of Technology, Patna, India · Department of Computer Science, Rajarshi School of Management Technology, Varanasi, India

Activation Sparsity Neural Network Interpretability Sparse Autoencoders Neural Network Pruning

Open paper AskPDF arXiv

Extreme neural network sparsification (90% activation reduction) presents a critical challenge for mechanistic interpretability: understanding whether interpretable features survive aggressive compression. This work investigates feature survival under severe capacity constraints in hybrid Variational Autoencoder--Sparse Autoencoder (VAE-SAE) architectures. We introduce an adaptive sparsity scheduling framework that progressively reduces active neurons from 500 to 50 over 50 training epochs, and provide empirical evidence for fundamental limits of the sparsification-interpretability relationship. Testing across two benchmark datasets -- dSprites and Shapes3D -- with both Top-k and L1 sparsification methods, our key finding reveals a pervasive paradox: while global representation quality (measured by Mutual Information Gap) remains stable, local feature interpretability collapses systematically. Under Top-k sparsification, dead neuron rates reach

34.4\pm0.9\%

on dSprites and

62.7\pm1.3\%

on Shapes3D at k=50. L1 regularization -- a fundamentally different "soft constraint" paradigm -- produces equal or worse collapse:

41.7\pm4.4\%

on dSprites and

90.6\pm0.5\%

on Shapes3D. Extended training for 100 additional epochs fails to recover dead neurons, and the collapse pattern is robust across all tested threshold definitions. Critically, the collapse scales with dataset complexity: Shapes3D (RGB, 6 factors) shows

1.8\times

more dead neurons than dSprites (grayscale, 5 factors) under Top-k and

2.2\times

under L1. These findings establish that interpretability collapse under sparsification is intrinsic to the compression process rather than an artifact of any particular algorithm, training duration, or threshold choice.

Abstract

Autoregressive Modeling

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