A logifold structure on measure space

TL;DR

This work develops a measure-theoretic, local-to-global framework for understanding datasets by modeling them as fuzzy measure spaces equipped with charts, and assembling these charts into a linear logifold. It introduces linear logical functions, their fuzzy analogs, and a corresponding zero-locus/semilinear interpretation, establishing a universal approximation theorem for measurable functions via these graphs. The paper then extends the framework to measure spaces, showing the equivalence with semilinear and, more broadly, o-minimal structures, and defines linear logifolds as global structures built from chart graphs with Hausdorff-measured domains. Building on this, it formalizes ensemble learning as logifold assembly, with fuzzy model domains, specialization, and a detailed voting system across target graphs and refined targets, including path-based aggregation and validation-history guidance. The approach aims to improve classification by exploiting fuzzy domains, specialized ensembles, and graph-based aggregation, offering a mathematically grounded, interpretable alternative to conventional neural ensembles with empirically notable accuracy gains when employing fuzzy domains and refined voting.

Abstract

In this paper,we develop a local-to-global and measure-theoretical approach to understand datasets. The idea is to take network models with restricted domains as local charts of datasets. We develop the mathematical foundations for these structures, and show in experiments how it can be used to find fuzzy domains and to improve accuracy in data classification problems.

Figures (5)

• Figure 1: An example of a logifold. The graph jumps over values $0$ and $1$ infinitely in left-approaching to the point marked by a star (and the length of each interval is halved). This is covered by infinitely many charts of linear logical functions, each of which has only finitely many jumps. Moreover, the base is a measurable subset of $\mathbb{R}$ (which is hard to depict and not shown in the picture).
• Figure 2: The left hand side shows a simple example of a logifold. It is the graph of the step function $[-1,1] \to \{0,1\}$. The figure in the middle shows a fuzzy deformation of it, which is a fuzzy subset in $[-1,1]\times\{0,1\}$. The right hand side shows the graph of probability distribution of a quantum observation, which consists of the maps $\frac{|z_0|^2}{|z_0|^2 + |z_1|^2}$ and $\frac{|z_1|^2}{|z_0|^2 + |z_1|^2}$ from the state space $\mathbb{P}^1$ to $[0,1]$.
• Figure 3: The left side shows a partial directed graph at vertex $v$, with five outgoing arrows. On the right, chambers are formed in $\mathbb{R}^2$ by the affine maps $L_v = (l_1, l_2, l_3)$ defined on $\mathbb{R}^2$. A point $x$ is marked in the chamber defined by $\{l_1 \leq 0, l_2 \geq 0, l_3 \leq 0\}$. One of the arrows corresponding to the shaded chamber containing $x$ is highlighted in the left diagram.
• Figure 4:
• Figure 5:

Theorems & Definitions (57)

• Theorem 1.1: Universal approximation theorem by linear logical functions
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• Proposition 2.6