Related papers: From Universal Approximation Theorem to Tropical G…

Motivated by interpretability and reliability, we investigate whether large language models (LLMs) deploy universal geometric structures to encode discrete, graph-structured knowledge. To this end, we present two complementary experimental…

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of…

In this paper we fully describe all tropical linear mappings in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by an order 3 matrix. An erratum has been added fixing two…

Global pooling is one of the most significant operations in many machine learning models and tasks, whose implementation, however, is often empirical in practice. In this study, we develop a novel and solid global pooling framework through…

Neural networks can accurately forecast complex dynamical systems, yet how they internally represent underlying latent geometry remains poorly understood. We study neural forecasters through the lens of representational alignment,…

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…

This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial,…

The recurrent geometric network (RGN), the first end-to-end differentiable neural architecture for protein structure prediction, is a competitive alternative to existing models. However, the RGN's use of recurrent neural networks (RNNs) as…

The paper proposes representation functionals in a dual paradigm where learning jointly concerns both linear convolutional weights and parametric forms of nonlinear activation functions. The nonlinear forms proposed for performing the…

In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our…

The construction of a multilayer perceptron (MLP) as a piecewise low-order polynomial approximator using a signal processing approach is presented in this work. The constructed MLP contains one input, one intermediate and one output layers.…

In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural…

We investigate the universal approximation property (UAP) of transformer-type architectures, providing a unified theoretical framework that extends prior results on residual networks to models incorporating attention mechanisms. Our work…

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…

We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…

Meta-learning has emerged as a powerful approach to train neural networks to learn new tasks quickly from limited data. Broad exposure to different tasks leads to versatile representations enabling general problem solving. But, what are the…

This paper is a continuation of [arXiv:1603.02204]. Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove…

Printed Electronics (PE) feature distinct and remarkable characteristics that make them a prominent technology for achieving true ubiquitous computing. This is particularly relevant in application domains that require conformal and…

We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes…

Large language models (LLMs) such as ChatGPT, Gemini, LlaMa, and Claude are trained on massive quantities of text parsed from the internet and have shown a remarkable ability to respond to complex prompts in a manner often indistinguishable…