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The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators

Diego Marcondes, Junior Barrera

TL;DR

This paper tackles learning lattice operators under principled constraints by introducing a lattice overparametrization framework. It formalizes algebraic representations, kernel-based bases, and sup-/inf-generating decompositions to enable interpretable parameterizations, and proposes the Stochastic Lattice Descent (SLDA) as a general, efficient optimizer over a fixed lattice parametrization. It also discusses hierarchical SLDA and model-selection-inspired strategies to cope with limited prior information, advocating for the three pillars of control, transparency, and interpretability that MM-based methods can offer over neural networks. The work situates mathematical morphology as a potent foundation for principled, interpretable operator learning and outlines proofs-of-concept through discrete DMNN, while outlining key open problems for extending to uncountable lattices.

Abstract

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.

The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators

TL;DR

This paper tackles learning lattice operators under principled constraints by introducing a lattice overparametrization framework. It formalizes algebraic representations, kernel-based bases, and sup-/inf-generating decompositions to enable interpretable parameterizations, and proposes the Stochastic Lattice Descent (SLDA) as a general, efficient optimizer over a fixed lattice parametrization. It also discusses hierarchical SLDA and model-selection-inspired strategies to cope with limited prior information, advocating for the three pillars of control, transparency, and interpretability that MM-based methods can offer over neural networks. The work situates mathematical morphology as a potent foundation for principled, interpretable operator learning and outlines proofs-of-concept through discrete DMNN, while outlining key open problems for extending to uncountable lattices.

Abstract

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.
Paper Structure (7 sections, 13 equations, 1 figure, 1 algorithm)

This paper contains 7 sections, 13 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Algebraic representations of operators
  2. Lattice overparametrization
  3. The machine learning of lattice operators
  4. Stochastic lattice descent as a general learning algorithm
  5. Degrees of prior information and hierarchical SLDA
  6. Control, transparency and interpretability
  7. Conclusion

Figures (1)

  • Figure 1: The lattice isomorphisms between representations of $(\Omega,\leq)$. The dashed lines represent an isomorphism that holds when the operators in $\Omega$ have a basis representation. The dotted lines represent isomorphisms that hold for t.i. and locally defined set operators when $\mathscr{L} = \mathcal{P}(E)$ and $(E,+)$ is an Abelian group. The orange arrows represent frameworks for the learning of lattice operators via the SLDA and via model selection based on data.