# Stochastic Deep Networks

arXiv: https://arxiv.org/abs/1811.07429 · HTML: https://mlsanity.com/paper/1811.07429/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2018.

## Built on by (most cited first, 13 of 13)

- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [How Smooth Is Attention?](https://mlsanity.com/paper/2312.14820/) (2023)
- [Transformers are Universal In-context Learners](https://mlsanity.com/paper/2408.01367/) (2024)
- [On the Universality of Graph Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2105.13099/) (2021)
- [Understanding Catastrophic Forgetting In LoRA via Mean-Field Attention Dynamics](https://mlsanity.com/paper/2402.15415/) (2024)
- [Transformers through the lens of support-preserving maps between measures](https://mlsanity.com/paper/2509.25611/) (2025)
- [Coin Flipping Neural Networks](https://mlsanity.com/paper/2206.09182/) (2022)
- [Learning Theory of Distribution Regression with Neural Networks](https://mlsanity.com/paper/2307.03487/) (2023)
- [Token Sample Complexity of Attention](https://mlsanity.com/paper/2512.10656/) (2025)
- [Softmax as Linear Attention in the Large-Prompt Regime: a Measure-based Perspective](https://mlsanity.com/paper/2512.11784/) (2025)
- [Concentration inequalities and optimal number of layers for stochastic deep neural networks](https://mlsanity.com/paper/2206.11241/) (2022)

## Abstract (arXiv)

Machine learning is increasingly targeting areas where input data cannot be accurately described by a single vector, but can be modeled instead using the more flexible concept of random vectors, namely probability measures or more simply point clouds of varying cardinality. Using deep architectures on measures poses, however, many challenging issues. Indeed, deep architectures are originally designed to handle fixedlength vectors, or, using recursive mechanisms, ordered sequences thereof. In sharp contrast, measures describe a varying number of weighted observations with no particular order. We propose in this work a deep framework designed to handle crucial aspects of measures, namely permutation invariances, variations in weights and cardinality. Architectures derived from this pipeline can (i) map measures to measures - using the concept of push-forward operators; (ii) bridge the gap between measures and Euclidean spaces - through integration steps. This allows to design discriminative networks (to classify or reduce the dimensionality of input measures), generative architectures (to synthesize measures) and recurrent pipelines (to predict measure dynamics). We provide a theoretical analysis of these building blocks, review our architectures' approximation abilities and robustness w.r.t. perturbation, and try them on various discriminative and generative tasks.