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DOI: 10.4230/LIPIcs.ESA.2021.63
URN: urn:nbn:de:0030-drops-146443
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Lokshtanov, Daniel ; Suri, Subhash ; Xue, Jie

Efficient Algorithms for Least Square Piecewise Polynomial Regression

LIPIcs-ESA-2021-63.pdf (0.8 MB)


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 (š±ā‚, yā‚),ā€¦ , (š±_n, y_n) āˆˆ ā„^d Ɨ ā„ where d āˆˆ {1,2}, the goal is to segment š±_iā€™s into some (arbitrary) number of disjoint pieces Pā‚, ā€¦ , P_k, where each piece P_j is associated with a fixed-degree polynomial f_j: ā„^d ā†’ ā„, to minimize the total loss function Ī» k + āˆ‘_{i = 1}āæ (y_i - f(š±_i))Ā², where Ī» ā‰„ 0 is a regularization term that penalizes model complexity (number of pieces) and f: ā؆_{j = 1}^k P_j ā†’ ā„ is the piecewise polynomial function defined as f|_{P_j} = f_j. The pieces Pā‚, ā€¦ , P_k are disjoint intervals of ā„ 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 + Īµ)-approximation algorithm with time complexity O(n/(Īµ) log 1/(Īµ)), assuming that data is presented in sorted order of x_iā€™s. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(āˆšn)}; 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

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-146443},
  doi =		{10.4230/LIPIcs.ESA.2021.63},
  annote =	{Keywords: regression analysis, piecewise polynomial, least square error}

Keywords: regression analysis, piecewise polynomial, least square error
Collection: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue Date: 2021
Date of publication: 31.08.2021

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