Fiber Bundle Morphisms as a Framework for Modeling Many-to-Many Maps

TL;DR

This paper introduces a fiber bundle morphism framework for modeling many-to-many relationships in data, capturing the distribution of outputs given inputs and vice versa, which is crucial for understanding complex real-world processes.

Contribution

It proposes using fiber bundle morphisms as a novel mathematical framework to model and analyze many-to-many mappings in machine learning.

Findings

Provides a theoretical foundation for modeling many-to-many processes.

Illustrates how fiber bundle morphisms capture input-output distributions.

Enhances understanding of variance and ambiguity in complex data mappings.

Abstract

While it is not generally reflected in the `nice' datasets used for benchmarking machine learning algorithms, the real-world is full of processes that would be best described as many-to-many. That is, a single input can potentially yield many different outputs (whether due to noise, imperfect measurement, or intrinsic stochasticity in the process) and many different inputs can yield the same output (that is, the map is not injective). For example, imagine a sentiment analysis task where, due to linguistic ambiguity, a single statement can have a range of different sentiment interpretations while at the same time many distinct statements can represent the same sentiment. When modeling such a multivalued function $f: X \rightarrow Y$, it is frequently useful to be able to model the distribution on $f(x)$ for specific input $x$ as well as the distribution on fiber $f^{-1}(y)$ for specific output $y$. Such an analysis helps the user (i) better understand the variance intrinsic to the process they are studying and (ii) understand the range of specific input $x$ that can be used to achieve output $y$. Following existing work which used a fiber bundle framework to better model many-to-one processes, we describe how morphisms of fiber bundles provide a template for building models which naturally capture the structure of many-to-many processes.