Quantum Generalized Linear Models

Abstract

Generalized linear models (GLM) are link function based statistical models. Many supervised learning algorithms are extensions of GLMs and have link functions built into the algorithm to model different outcome distributions. There are two major drawbacks when using this approach in applications using real world datasets. One is that none of the link functions available in the popular packages is a good fit for the data. Second, it is computationally inefficient and impractical to test all the possible distributions to find the optimum one. In addition, many GLMs and their machine learning extensions struggle on problems of overdispersion in Tweedie distributions. In this paper we propose a quantum extension to GLM that overcomes these drawbacks. A quantum gate with non-Gaussian transformation can be used to continuously deform the outcome distribution from known results. In doing so, we eliminate the need for a link function. Further, by using an algorithm that superposes all possible distributions to collapse to fit a dataset, we optimize the model in a computationally efficient way. We provide an initial proof-of-concept by testing this approach on both a simulation of overdispersed data and then on a benchmark dataset, which is quite overdispersed, and achieved state of the art results. This is a game changer in several applied fields, such as part failure modeling, medical research, actuarial science, finance and many other fields where Tweedie regression and overdispersion are ubiquitous.