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# Efficient Algorithms for Least Square Piecewise Polynomial Regression

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## Cite As

Daniel Lokshtanov, Subhash Suri, and Jie Xue. Efficient Algorithms for Least Square Piecewise Polynomial Regression. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 63:1-63:15, Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.63

## Abstract

We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (π±β, yβ),β¦ , (π±_n, y_n) β β^d Γ β where d β {1,2}, the goal is to segment π±_iβs into some (arbitrary) number of disjoint pieces Pβ, β¦ , P_k, where each piece P_j is associated with a fixed-degree polynomial f_j: β^d β β, to minimize the total loss function Ξ» k + β_{i = 1}βΏ (y_i - f(π±_i))Β², where Ξ» β₯ 0 is a regularization term that penalizes model complexity (number of pieces) and f: β¨_{j = 1}^k P_j β β is the piecewise polynomial function defined as f|_{P_j} = f_j. The pieces Pβ, β¦ , P_k are disjoint intervals of β in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions. Our main results are the following. For univariate data, we present a (1 + Ξ΅)-approximation algorithm with time complexity O(n/(Ξ΅) log 1/(Ξ΅)), assuming that data is presented in sorted order of x_iβs. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(βn)}; a polynomial-time constant-approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness.

## Subject Classification

##### ACM Subject Classification
• Theory of computation β Computational geometry
##### Keywords
• regression analysis
• piecewise polynomial
• least square error

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