I will present Deep Neural Networks as Iterated Function Systems and a Generalization Bound at Curves and Surfaces 2026 in Saint-Malo. According to the conference program, the talk is scheduled for Friday, June 12, 2026, from 08:55 to 09:20 in the session Machine learning theory, Neural networks.

The presentation focuses on a stochastic IFS viewpoint on deep architectures and on the corresponding generalization questions for generative modeling. It is based on the following preprint.

This work extends our segmentation line to reaction times and natural images. We reconstruct subjective segmentation maps from human responses and use the temporal dynamics to test an iterative Bayesian inference account of visual segmentation.

1. Biswas, T. K., Vacher, J., Molholm, S., Mamassian, P. & Coen-Cagli, R. Natural scene segmentation dynamics reveal iterative Bayesian inference. bioRxiv 2026–01 (2026).

   @article{biswas2026natural,
     title = {Natural scene segmentation dynamics reveal iterative Bayesian inference},
     author = {Biswas, Tridib K and Vacher, Jonathan and Molholm, Sophie and Mamassian, Pascal and Coen-Cagli, Ruben},
     journal = {bioRxiv},
     pages = {2026--01},
     year = {2026},
     publisher = {Cold Spring Harbor Laboratory}
   }
   

Overview

When image evidence agrees with spatial priors, inference should settle faster. This prediction explains why reaction times can increase with distance for same-segment judgments yet decrease with distance for different-segment judgments, and it also captures individual differences in spatial biases. More broadly, the paper argues that the tempo of segmentation in natural scenes is shaped by recurrent probabilistic inference rather than by a purely feedforward grouping rule.

This preprint connects modern deep architectures to stochastic iterated function systems. The idea is to treat depth as a random dynamical system: once this is made explicit for architectures such as ResNets, Transformers, or mixture-of-experts layers, questions of stability become questions of contractivity, and generative generalization can be controlled in Wasserstein distance.

1. Vacher, J. Deep Neural Networks as Iterated Function Systems and a Generalization Bound. arXiv preprint arXiv:2601.19958 (2026).

   @article{vacher2026deep,
     title = {Deep Neural Networks as Iterated Function Systems and a Generalization Bound},
     author = {Vacher, Jonathan},
     journal = {arXiv preprint arXiv:2601.19958},
     year = {2026}
   }
   

Overview

The paper has three main pieces. First, it identifies conditions under which depth dynamics admit invariant measures and attractors. Second, it derives a generalization bound for generative modeling based on the gap between the data distribution and its image under the learned transfer operator. Third, it turns this bound into a collage-style training objective and tests it on a controlled 2D example together with latent-image experiments on MNIST, CelebA, and CIFAR-10.

This paper presents DynTex, a real-time toolbox for generating controlled naturalistic dynamic textures. It grows out of the Motion Clouds framework and is meant for experiments that need richer motion stimuli than simple gratings while preserving precise parameter control.

1. Meso, A. I. et al. DynTex: A real-time generative model of dynamic naturalistic scenes from luminance textures. Journal of Vision 25, 2–2 (2025).

   @article{meso2025dyntex,
     title = {DynTex: A real-time generative model of dynamic naturalistic scenes from luminance textures},
     author = {Meso, Andrew I. and Vacher, Jonathan and Gekas, Nikos and Mamassian, Pascal and Perrinet, Laurent U. and Masson, Guillaume},
     journal = {Journal of Vision},
     volume = {25},
     number = {11},
     pages = {2--2},
     year = {2025},
     publisher = {The Association for Research in Vision and Ophthalmology}
   }
   

Overview

The toolbox exposes orientation, speed, and spatiotemporal frequency structure in a form that can be explored interactively, integrated with Psychtoolbox or PsychoPy, and rendered on the GPU in real time. The broader goal is to make naturalistic motion stimuli practical for psychophysics, neurophysiology, and computational modeling.

This work studies formal control guarantees for a coupled ODE-PDE switched system. The main challenge is to certify stability for an infinite-dimensional system without relying on a direct discretization of the PDE state.

1. Le Coënt, A., Vacher, J. & Kergrene, K. Formal stabilization of a coupled ODE-PDE switched system. in 2024 IEEE 63rd Conference on Decision and Control (CDC) 6341–6348 (IEEE, 2024).

   @inproceedings{lecoent2024formalstab,
     title = {Formal stabilization of a coupled ODE-PDE switched system},
     author = {Le Co{\"e}nt, Adrien and Vacher, Jonathan and Kergrene, Kenan},
     booktitle = {2024 IEEE 63rd Conference on Decision and Control (CDC)},
     pages = {6341-6348},
     year = {2024},
     organization = {IEEE}
   }
   

Overview

The approach combines model reduction, superposition arguments, and tiling-based control synthesis to turn the problem into a finite-dimensional one that is numerically tractable while keeping the guarantee attached to the original infinite-dimensional dynamics. The concrete example is a switched ODE coupled to a heat equation.

We tested predictions of perceptual scales from Fisher information in a series of experiments. This will allow us to go beyond perceptual distances and get closer to perceptual geometry. .

1. Vacher, J. & Mamassian, P. Perceptual Scales Predicted by Fisher Information Metrics. in The Twelfth International Conference on Learning Representations (2024).

   @inproceedings{vacher2024perceptual,
     title = {Perceptual Scales Predicted by Fisher Information Metrics},
     author = {Vacher, Jonathan and Mamassian, Pascal},
     booktitle = {The Twelfth International Conference on Learning Representations},
     year = {2024}
   }
   

Abstract

Perception is often viewed as a process that transforms physical variables, external to an observer, into internal psychological variables. Such a process can be modeled by a function coined perceptual scale. The perceptual scale can be deduced from psychophysical measurements that consist in comparing the relative differences between stimuli (i.e. difference scaling experiments). However, this approach is often overlooked by the modeling and experimentation communities. Here, we demonstrate the value of measuring the perceptual scale of classical (spatial frequency, orientation) and less classical physical variables (interpolation between textures) by embedding it in recent probabilistic modeling of perception. First, we show that the assumption that an observer has an internal representation of univariate parameters such as spatial frequency or orientation while stimuli are high-dimensional does not lead to contradictory predictions when following the theoretical framework. Second, we show that the measured perceptual scale corresponds to the transduction function hypothesized in this framework. In particular, we demonstrate that it is related to the Fisher information of the generative model that underlies perception and we test the predictions given by the generative model of different stimuli in a set a of difference scaling experiments. Our main conclusion is that the perceptual scale is mostly driven by the stimulus power spectrum. Finally, we propose that this measure of perceptual scale is a way to push further the notion of perceptual distances by estimating the perceptual geometry of images i.e. the path between images instead of simply the distance between those.

(i) We introduce the first experimental method to measure perceptual segmentation on arbitrary images. (ii) We capture individual-level variability and relate it to perceptual uncertainty, which is necessary to understand human perception. (iii) We offer computational tools to fit any segmentation algorithm to the data, which will enable new benchmarks for computer vision algorithms, and testing computational theories of perceptual segmentation.

1. Vacher, J., Launay, C., Mamassian, P. & Coen-Cagli, R. Measuring uncertainty in human visual segmentation. PLOS Computational Biology 19, 1–24 (2023).

   @article{vacher2023measuring,
     author = {Vacher, Jonathan and Launay, Claire and Mamassian, Pascal and Coen-Cagli, Ruben},
     journal = {PLOS Computational Biology},
     publisher = {Public Library of Science},
     title = {Measuring uncertainty in human visual segmentation},
     year = {2023},
     month = sep,
     volume = {19},
     pages = {1-24},
     number = {9}
   }
   

Abstract

Segmenting visual stimuli into distinct groups of features and visual objects is central to visual function. Classical psychophysical methods have helped uncover many rules of human perceptual segmentation, and recent progress in machine learning has produced successful algorithms. Yet, the computational logic of human segmentation remains unclear, partially because we lack well-controlled paradigms to measure perceptual segmentation maps and compare models quantitatively. Here we propose a new, integrated approach: given an image, we measure multiple pixel-based same–different judgments and perform model–based reconstruction of the underlying segmentation map. The reconstruction is robust to several experimental manipulations and captures the variability of individual participants. We demonstrate the validity of the approach on human segmentation of natural images and composite textures. We show that image uncertainty affects measured human variability, and it influences how participants weigh different visual features. Because any putative segmentation algorithm can be inserted to perform the reconstruction, our paradigm affords quantitative tests of theories of perception as well as new benchmarks for segmentation algorithms.

Claire Launay applied our algorithm based on mixture models to the task of video segmentation by propagating segmentation information across frames !

1. Launay, C., Vacher, J. & Coen-Cagli, R. Unsupervised Video Segmentation Algorithms Based On Flexibly Regularized Mixture Models. in 2022 IEEE International Conference on Image Processing (ICIP) 4073–4077 (IEEE, 2022).

   @inproceedings{launay2022unsupervised,
     title = {Unsupervised Video Segmentation Algorithms Based On Flexibly Regularized Mixture Models},
     author = {Launay, Claire and Vacher, Jonathan and Coen-Cagli, Ruben},
     booktitle = {2022 IEEE International Conference on Image Processing (ICIP)},
     pages = {4073--4077},
     year = {2022},
     organization = {IEEE}
   }
   

Abstract

We propose a family of probabilistic segmentation algorithms for videos that rely on a generative model capturing static and dynamic natural image statistics. Our framework adopts flexibly regularized mixture models (FlexMM) [1], an efficient method to combine mixture distributions across different data sources. FlexMMs of Student-t distributions successfully segment static natural images, through uncertainty-based information sharing between hidden layers of CNNs. We further extend this approach to videos and exploit FlexMM to propagate segment labels across space and time. We show that temporal propagation improves temporal consistency of segmentation, reproducing qualitatively a key aspect of human perceptual grouping. Besides, Student-t distributions can capture statistics of optical flows of natural movies, which represent apparent motion in the video. Integrating these motion cues in our temporal FlexMM further enhances the segmentation of each frame of natural movies. Our probabilistic dynamic segmentation algorithms thus provide a new framework to study uncertainty in human dynamic perceptual segmentation.

We propose a new method to regularize mixture models using the data topology. We demonstrate multiple advantages of our models by apply them to the taks of image segmentation :

• flexible update rule in the EM algorithm
• binding together mixture models (mutual-supervision ?)

1. Vacher, J., Launay, C. & Coen-Cagli, R. Flexibly Regularized Mixture Models and Application to Image Segmentation. Neural Networks 149, 107–123 (2022).

   @article{vacher2022flexibly,
     title = {Flexibly Regularized Mixture Models and Application to Image Segmentation},
     journal = {Neural Networks},
     volume = {149},
     pages = {107-123},
     year = {2022},
     author = {Vacher, Jonathan and Launay, Claire and Coen-Cagli, Ruben},
     issn = {0893-6080}
   }
   

Abstract

Probabilistic finite mixture models are widely used for unsupervised clustering. These models can often be improved by adapting them to the topology of the data. For instance, in order to classify spatially adjacent data points similarly, it is common to introduce a Laplacian constraint on the posterior probability that each data point belongs to a class. Alternatively, the mixing probabilities can be treated as free parameters, while assuming Gauss–Markov or more complex priors to regularize those mixing probabilities. However, these approaches are constrained by the shape of the prior and often lead to complicated or intractable inference. Here, we propose a new parametrization of the Dirichlet distribution to flexibly regularize the mixing probabilities of over-parametrized mixture distributions. Using the Expectation-Maximization algorithm, we show that our approach allows us to define any linear update rule for the mixing probabilities, including spatial smoothing regularization as a special case. We then show that this flexible design can be extended to share class information between multiple mixture models. We apply our algorithm to artificial and natural image segmentation tasks, and we provide quantitative and qualitative comparison of the performance of Gaussian and Student-t mixtures on the Berkeley Segmentation Dataset. We also demonstrate how to propagate class information across the layers of deep convolutional neural networks in a probabilistically optimal way, suggesting a new interpretation for feedback signals in biological visual systems. Our flexible approach can be easily generalized to adapt probabilistic mixture models to arbitrary data topologies.