Papers › Combinatorial optimization of the coefficient of determination
Combinatorial optimization of the coefficient of determination
Marc Harary
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 ( R² ). 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 ℝ⁵ 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 ℝ² by a conic section, which, in ℝ⁵, can be found by a topological sweep in Θ( n⁵ logn ) 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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