Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an arbitrary binary operation $\star$ that distributes over max, and outline applications to geometry, machine learning, and optimization. Further, we generalize tropical geometrical objects using weighted lattices. Finally, we provide the optimal solution of max-$\star$ equations using morphological adjunctions that are projections on weighted lattices, and apply it to optimal piecewise-linear regression for fitting max-$\star$ tropical curves and surfaces to arbitrary data that constitute polygonal or polyhedral shape approximations. This also includes an efficient algorithm for solving the convex regression problem of data fitting with max-affine functions.
Code (0)
등록된 구현이 없습니다.
Tasks
regressionSimilar Papers 제목 키워드 기반
A Tropical Approach to Neural Networks with Piecewise Linear Activations
We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations as tropical …
Toric geometry of ReLU neural networks
Given a continuous finitely piecewise linear function $f:\mathbb{R}^{n_0} \to \mathbb{R}$ and a fixed architecture $(n_0,\ldots,n_k;1)$ of feedforward ReLU neural networks, the exact function realization problem is to de…
Computing Linear Regions in Neural Networks with Skip Connections
Neural networks are important tools in machine learning. Representing piecewise linear activation functions with tropical arithmetic enables the application of tropical geometry. Algorithms are presented to compute regio…
On the Decision Boundaries of Neural Networks. A Tropical Geometry Perspective
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of alg…
Network PruningOn the Decision Boundaries of Neural Networks: A Tropical Geometry Perspective
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of alg…
Network Pruning