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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.63
URN: urn:nbn:de:0030-drops-146443
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14644/
Lokshtanov, Daniel ;
Suri, Subhash ;
Xue, Jie
Efficient Algorithms for Least Square Piecewise Polynomial Regression
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 (?₁, y₁),… , (?_n, y_n) ∈ ℝ^d × ℝ where d ∈ {1,2}, the goal is to segment ?_i’s into some (arbitrary) number of disjoint pieces P₁, … , P_k, where each piece P_j is associated with a fixed-degree polynomial f_j: ℝ^d → ℝ, to minimize the total loss function λ k + ∑_{i = 1}ⁿ (y_i - f(?_i))², where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f: ⨆_{j = 1}^k P_j → ℝ is the piecewise polynomial function defined as f|_{P_j} = f_j. The pieces P₁, … , P_k are disjoint intervals of ℝ 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 + ε)-approximation algorithm with time complexity O(n/(ε) log 1/(ε)), assuming that data is presented in sorted order of x_i’s. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(√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.
BibTeX - Entry
@InProceedings{lokshtanov_et_al:LIPIcs.ESA.2021.63,
author = {Lokshtanov, Daniel and Suri, Subhash and Xue, Jie},
title = {{Efficient Algorithms for Least Square Piecewise Polynomial Regression}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {63:1--63:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14644},
URN = {urn:nbn:de:0030-drops-146443},
doi = {10.4230/LIPIcs.ESA.2021.63},
annote = {Keywords: regression analysis, piecewise polynomial, least square error}
}
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Keywords: |
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regression analysis, piecewise polynomial, least square error |
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Collection: |
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29th Annual European Symposium on Algorithms (ESA 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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31.08.2021 |
