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arxiv: 2408.13751 · v3 · submitted 2024-08-25 · 📊 stat.ML · cs.LG· math.OC

Improved identification of breakpoints in piecewise regression and its applications

Taehyeong Kim , Hyungu Lee , Myungjin Kim , Hayoung Choi This is my paper

Pith reviewed 2026-05-23 22:19 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OC
keywords piecewise regressionbreakpointsgreedy algorithmchange point detectionpolynomial regressionoptimizationdata fittingmodel interpretability
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The pith

A greedy neighborhood exploration algorithm locates optimal breakpoints in piecewise polynomial regression more accurately than existing methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents new greedy algorithms for identifying breakpoints where the data switches between different polynomial regimes. These algorithms repeatedly check points near each current breakpoint and move it to the spot that most reduces the overall fitting error. The approach converges quickly, stays stable across runs, and can automatically select how many breakpoints to use. Experiments on both synthetic data with known structure and real datasets show higher accuracy than previous techniques. The identified breakpoints make the fitted models easier to interpret and reveal useful structure in the data.

Core claim

The proposed greedy algorithm updates breakpoints by exploring their local neighborhoods to minimize the regression error, achieving fast convergence and stability while also determining the optimal number of breakpoints, outperforming existing methods on synthetic and real-world data.

What carries the argument

Greedy local neighborhood search that iteratively updates each breakpoint position to reduce total piecewise polynomial error.

If this is right

  • Piecewise models fitted with these breakpoints achieve lower error on data containing abrupt changes.
  • The number of segments can be chosen automatically rather than by manual search or cross-validation alone.
  • Breakpoints recovered this way yield more stable and interpretable segment boundaries in applications.
  • The same procedure applies directly to both polynomial and other basis expansions in regression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The local-search idea might transfer to change-point problems in time-series classification or survival models.
  • Combining the method with global techniques such as dynamic programming could handle very high-dimensional inputs.
  • Real-time monitoring systems could use the fast convergence to update breakpoint locations as new observations arrive.

Load-bearing premise

Repeated small local moves around each breakpoint will reliably reach the global best locations and the correct total number without becoming trapped in worse configurations.

What would settle it

A synthetic dataset whose true breakpoints are known in advance, on which the algorithm returns a different set whose total squared error is higher than the known optimum.

Figures

Figures reproduced from arXiv: 2408.13751 by Hayoung Choi, Hyungu Lee, Myungjin Kim, Taehyeong Kim.

Figure 1
Figure 1. Figure 1: Update breakpoints among ξ − j , ξ j , ξ+ j . where pj : Ij → R is a dth degree polynomial for all j = 1, . . . , k, given by pj(x) = θ0 j + θ1 jx + θ2 jx 2 + · · · + θd jx d . Since this optimization problem is nonconvex [17], it is not easy to solve the problem. Due to the number of too many breakpoint candidates, numerical algorithms may need an ex￾tensive computational cost. To overcome this situation,… view at source ↗
Figure 2
Figure 2. Figure 2: Comparing MSE as the number of breakpoints is reduced from 7 to 4. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Algorithm 4 can avoid local minimum well. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Result of each regression models. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of ℓ1 trend filter, APLR, PELT and proposed method for S&P 500 data. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of ℓ1 trend filter, APLR, PELT and proposed method for COVID-19 cases dataset. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

Identifying breakpoints in piecewise regression is critical in enhancing the reliability and interpretability of data fitting. In this paper, we propose novel algorithms based on the greedy algorithm to accurately and efficiently identify breakpoints in piecewise polynomial regression. The algorithm updates the breakpoints to minimize the error by exploring the neighborhood of each breakpoint. It has a fast convergence rate and stability to find optimal breakpoints. Moreover, it can determine the optimal number of breakpoints. The computational results for real and synthetic data show that its accuracy is better than any existing methods. The real-world datasets demonstrate that breakpoints through the proposed algorithm provide valuable data information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes novel greedy algorithms for breakpoint identification in piecewise polynomial regression. Breakpoints are updated by exploring the local neighborhood of each to minimize fitting error; the method is claimed to have fast convergence and stability, to automatically determine the optimal number of breakpoints, and to achieve higher accuracy than existing methods on both synthetic and real data, with the resulting breakpoints providing useful information on real-world datasets.

Significance. If the empirical claims are substantiated with reproducible experiments and the local-search procedure is shown to be reliable, the work could supply a practical, automatic tool for piecewise regression that improves interpretability without manual tuning of breakpoint count. The emphasis on real-data applications is a positive aspect.

major comments (2)
  1. [Abstract] Abstract: the central claims (fast convergence to optimal breakpoints, automatic determination of the optimal number, and accuracy superior to all existing methods) rest entirely on computational experiments whose methodology, baselines, error bars, and data details are not supplied, so the assertions cannot be checked against the evidence.
  2. [Abstract] Abstract: the algorithm is described as a greedy procedure that repeatedly updates each breakpoint by exploring its immediate neighborhood; no argument is given that this local operator is sufficient to escape the combinatorial local minima known to exist in breakpoint placement, leaving the headline optimality claims without theoretical support.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims (fast convergence to optimal breakpoints, automatic determination of the optimal number, and accuracy superior to all existing methods) rest entirely on computational experiments whose methodology, baselines, error bars, and data details are not supplied, so the assertions cannot be checked against the evidence.

    Authors: The experimental methodology, baselines, error bars, and data details are described in Section 4 (Experimental Results) and the appendix of the full manuscript. The abstract is intended as a high-level summary. We will revise the abstract to add a sentence directing readers to Section 4 for the supporting experimental details, methodology, and data descriptions. revision: yes

  2. Referee: [Abstract] Abstract: the algorithm is described as a greedy procedure that repeatedly updates each breakpoint by exploring its immediate neighborhood; no argument is given that this local operator is sufficient to escape the combinatorial local minima known to exist in breakpoint placement, leaving the headline optimality claims without theoretical support.

    Authors: We agree that the manuscript provides no theoretical argument or proof that the local neighborhood search escapes combinatorial local minima. The claims of finding optimal breakpoints are supported solely by empirical results on synthetic data (where ground truth is known) and real data (where the method outperforms baselines). We will revise the abstract and introduction to replace references to 'optimal breakpoints' with 'breakpoints that minimize fitting error in practice' and add a paragraph in the discussion section acknowledging the lack of theoretical guarantees. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical validation of greedy algorithm

full rationale

The paper introduces a greedy neighborhood-exploration algorithm for breakpoint detection in piecewise polynomial regression and supports its claims of convergence, stability, and superior accuracy solely through computational experiments on synthetic and real datasets. No derivation chain exists that reduces a claimed result to its own inputs by definition, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked. The central assertions rest on external empirical benchmarks rather than any self-referential construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5631 in / 1153 out tokens · 25395 ms · 2026-05-23T22:19:04.337088+00:00 · methodology

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