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

Conference · Mon Apr 25 04:00:00 EDT 2022

OSTI ID:2000666

Coda, Elizabeth D. ^[1]; Courts, Nicolas C. ^[1]; Wight, Colby L. ^[1]; Truong, Loc T. ^[1]; Choi, WoongJo ^[1]; Godfrey, Charles W. ^[1]; Emerson, Tegan H. ^[1]; Kappagantula, Keerti S. ^[1]; Kvinge, Henry J. ^[1]

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1. BATTELLE (PACIFIC NW LAB)

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.

• Research Organization: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: AC05-76RL01830
• OSTI ID: 2000666
• Report Number(s): PNNL-SA-170544
• Country of Publication: Netherlands
• Language: English