Combinatorial optimization of the coefficient of determination

• URL: http://arxiv.org/abs/2410.09316v1
• Date: Sat, 12 Oct 2024 00:53:25 GMT
• Title: Combinatorial optimization of the coefficient of determination
• Authors: Marc Harary,
• Abstract summary: We develop an efficient algorithm for selecting the $k$- subset of $n$ points in the plane with the highest coefficient of determination. We experimentally demonstrate our method's optimality over several million trials up to $n=30$ without error.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Robust correlation analysis is among the most critical challenges in statistics. Herein, we develop an efficient algorithm for selecting the $k$- subset of $n$ points in the plane with the highest coefficient of determination $\left( R^2 \right)$. Drawing from combinatorial geometry, we propose a method called the \textit{quadratic sweep} that consists of two steps: (i) projectively lifting the data points into $\mathbb R^5$ and then (ii) iterating over each linearly separable $k$-subset. Its basis is that the optimal set of outliers is separable from its complement in $\mathbb R^2$ by a conic section, which, in $\mathbb R^5$, can be found by a topological sweep in $\Theta \left( n^5 \log n \right)$ time. Although key proofs of quadratic separability remain underway, we develop strong mathematical intuitions for our conjectures, then experimentally demonstrate our method's optimality over several million trials up to $n=30$ without error. Implementations in Julia and fully seeded, reproducible experiments are available at https://github.com/marc-harary/QuadraticSweep.

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