# The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators

arXiv: https://arxiv.org/abs/2310.06639 · HTML: https://mlsanity.com/paper/2310.06639/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 10 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [Learning Deep Morphological Networks with Neural Architecture Search](https://mlsanity.com/paper/2106.07714/) (2021)
- [Some open questions on morphological operators and representations in the deep learning era](https://mlsanity.com/paper/2105.01339/) (2021)
- [Discrete Morphological Neural Networks](https://mlsanity.com/paper/2309.00588/) (2023)
- [An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks](https://mlsanity.com/paper/2310.04584/) (2023)
- [Distribution-free Deviation Bounds and The Role of Domain Knowledge in Learning via Model Selection with Cross-validation Risk Estimation](https://mlsanity.com/paper/2303.08777/) (2023)

## Built on by (most cited first, 1 of 1)

- [On the representation of stack operators by mathematical morphology](https://mlsanity.com/paper/2504.09766/) (2025)

## Abstract (arXiv)

The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the sample size. From a computational perspective, there should be an efficient algorithm to minimize an empirical error over the class. From an understanding point of view, the properties of the learned operator need to be derived, so its behavior can be theoretically understood. The statistical bottleneck can be overcome due to the rich literature about the representation of lattice operators, but there is no general learning algorithm for them. In this paper, we discuss a learning paradigm in which, by overparametrizing a class via elements in a lattice, an algorithm for minimizing functions in a lattice is applied to learn. We present the stochastic lattice descent algorithm as a general algorithm to learn on constrained classes of operators as long as a lattice overparametrization of it is fixed, and we discuss previous works which are proves of concept. Moreover, if there are algorithms to compute the basis of an operator from its overparametrization, then its properties can be deduced and the understanding bottleneck is also overcome. This learning paradigm has three properties that modern methods based on neural networks lack: control, transparency and interpretability. Nowadays, there is an increasing demand for methods with these characteristics, and we believe that mathematical morphology is in a unique position to supply them. The lattice overparametrization paradigm could be a missing piece for it to achieve its full potential within modern machine learning.