eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-31
63:1
63:15
10.4230/LIPIcs.ESA.2021.63
article
Efficient Algorithms for Least Square Piecewise Polynomial Regression
Lokshtanov, Daniel
1
Suri, Subhash
1
Xue, Jie
1
Department of Computer Science, University of California, Santa Barbara, CA, USA
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±_iβs 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}βΏ (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_iβs. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(βn)}; 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol204-esa2021/LIPIcs.ESA.2021.63/LIPIcs.ESA.2021.63.pdf
regression analysis
piecewise polynomial
least square error