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arxiv: 2606.28974 · v1 · pith:ROYP2SSZnew · submitted 2026-06-27 · 🧮 math.OC · cs.NA· math.NA· math.ST· stat.TH

Faster than Fast-LTS: Robust Regression and Outlier Detection with DC Programming

Marah-Lisanne Thormann , Phan Tu Vuong , Alain B. Zemkoho , Tri-Dung Nguyen This is my paper

Pith reviewed 2026-06-30 08:31 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NAmath.STstat.TH
keywords robust regressionleast trimmed squaresDC programmingoutlier detectionconcave minimizationdifference of convex functionspreconditioningoptimization algorithms
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The pith

Reformulating Least Trimmed Squares as a DC program enables the sBDCA algorithm to solve robust regression faster and more accurately than Fast-LTS.

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

The paper shows that the combinatorial Least Trimmed Squares problem can be recast exactly as a concave minimization over a capped simplex. It proposes the successive Boosted Difference of Convex Functions Algorithm (sBDCA) to solve this formulation and proves linear convergence to local solutions using the Lojasiewicz property. A custom preconditioning matrix is derived to achieve robust results from a single starting point. Experiments on synthetic and real datasets demonstrate that this method runs up to 3.25 times faster than the standard Fast-LTS heuristic while producing objective values up to 90 percent lower, especially when the number of dimensions is large. The work includes open Python code to make the approach practical.

Core claim

The LTS problem can be exactly recast as a concave minimization subject to a capped simplex constraint. The sBDCA algorithm solves this reformulation and, when combined with a derived preconditioning matrix, converges to a local solution with linear rate in the fastest case while delivering robust performance from a single initialization.

What carries the argument

The successive Boosted Difference of Convex Functions Algorithm (sBDCA) applied to the DC reformulation of the LTS estimator under a capped simplex constraint, augmented by a problem-specific preconditioning matrix.

If this is right

  • sBDCA converges linearly to local solutions via the Lojasiewicz property.
  • The preconditioning matrix enables robustness from one initialization without loss of solution quality.
  • The method runs up to 3.25 times faster than Fast-LTS on tested instances.
  • Objective function values are up to 90% lower than those from Fast-LTS, especially in high dimensions.
  • Open Python code is provided for practical use in robust regression.

Where Pith is reading between the lines

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

  • The DC reformulation approach could be tested on other robust estimators that involve combinatorial subset selection.
  • Preconditioning strategies derived here might improve convergence for similar DC programs in statistics.
  • High-dimensional performance gains suggest potential for scaling robust methods to large datasets where Fast-LTS struggles.

Load-bearing premise

The combinatorial LTS problem admits an exact reformulation as a concave minimization over a capped simplex whose solutions match the original problem, and the preconditioning matrix ensures single-start robustness.

What would settle it

Running the method on a high-dimensional dataset where the preconditioned sBDCA returns a worse objective value or requires multiple starts to match Fast-LTS performance would falsify the practical claims.

Figures

Figures reproduced from arXiv: 2606.28974 by Alain B. Zemkoho, Marah-Lisanne Thormann, Phan Tu Vuong, Tri-Dung Nguyen.

Figure 1
Figure 1. Figure 1: Masking, Swamping, and Types of Outliers in Regression Analysis. In regression analysis, different types of outliers can be distinguished. If an observation de￾viates significantly from the true underlying regression line, it is classified as a regression outlier [cf. Rousseeuw and van Zomeren (1990, p. 636)]. In contrast, an instance is considered a leverage 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Influence of ρk on Average Trimmed Distance (ATD) and Computation Time. The results for the ten toy examples are summarized in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Influence of α on the PDF of the Dirichlet Distribution. Since the inspection of the gradient ∇f at the beginning of this subsection suggested that starting points should ideally not be too far from the global solution – which corresponds to a vertex – we next consider an initialization strategy based on an extreme point of the feasible set. Specifically, we denote by v0 ∈ {0, 1} n a vertex satisfying v ⊤ … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of different Initialization Strategies for sBDCA. Notably, Fast-LTS does not suffer from the drawback of the p-subset initialization. As explained in Subsection 1.4, the heuristic internally generates 500 random p-subsets during each initialization, which are processed separately through the internal sorting procedure. After a few iterations, only the most promising results are retained and run … view at source ↗
Figure 5
Figure 5. Figure 5: Influence of Preconditioner on ATD and Computation Time. 4. Applications to Robust Regression In the theoretical part of this paper, preliminary numerical results were already presented in Subsection 3.2 and Subsection 3.3 to showcase the influence of the successive DC decompositions, the choice of starting points, and the proposed preconditioning matrix. To provide a more compre￾hensive evaluation of the … view at source ↗
Figure 6
Figure 6. Figure 6: Simple Linear Regression Examples. The regression lines in [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: compares the fraction of infeasible solutions (left y-axis, bar plots) and the ATD of all feasible outcomes (right y-axis, box plots) across the selected settings. Overall, the subplots confirm that sBDCA with preconditioning achieves superior performance in terms of both solu￾tion quality and algorithmic reliability. Compared to Fast-LTS, the results of the proposed DC programming algorithm are less sensi… view at source ↗
Figure 8
Figure 8. Figure 8: Performance comparison between sBDCA with preconditioning and Fast-LTS using the OSCM and focusing on algorithmic efficiency. Each subplot displays the median number of iterations / c-steps (left y-axis, bar plots) and the computation time of all feasible solutions (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initialization stra… view at source ↗
Figure 9
Figure 9. Figure 9: Performance comparison between sBDCA with preconditioning and Fast-LTS using the DMLP and focusing on algorithmic reliability and output quality. Each subplot displays the fraction of infeasible solutions (left y-axis, bar plots) and the ATD of all feasible outcomes (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initialization str… view at source ↗
Figure 10
Figure 10. Figure 10: Optimization Paths of Fast-LTS and sBDCA with Preconditioning. To provide the final part of the performance comparison using the DMLP, [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance comparison between sBDCA with preconditioning and Fast-LTS using the DMLP and focusing on algorithmic efficiency. Each subplot displays the median number of iterations / c-steps (left y-axis, bar plots) and the computation time of all feasible solutions (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initialization str… view at source ↗
Figure 12
Figure 12. Figure 12: Performance comparison between sBDCA with preconditioning and Fast-LTS using the MSD / SCD and focusing on algorithmic reliability and output quality. Each subplot displays the fraction of infeasible solutions (left y-axis, bar plots) and the ATD of all feasible outcomes (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initializati… view at source ↗
Figure 13
Figure 13. Figure 13: Performance comparison between sBDCA with preconditioning and Fast-LTS using the MSD / SCD and focusing on algorithmic efficiency. Each subplot displays the median number of iterations / c-steps (left y-axis, bar plots) and the computation time of all feasible solutions (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initializatio… view at source ↗
Figure 14
Figure 14. Figure 14: Specifically, the three subplots are based on three variables z1, z2, z3 ∈ [0, 1], which satisfy either z1 + z2 + z3 = 2 (green triangle), or z1 + z2 + z3 = 1 (blue triangle). The latter triangle is also known as the probability or unit simplex. From these representations, it can be observed that the feasible set is closed, bounded, and convex, as it corresponds to a hypercube (due to the box constraints)… view at source ↗
Figure 15
Figure 15. Figure 15: Performance comparison between sBDCA with preconditioning and Fast-LTS using the DMLP and focusing on algorithmic reliability and output quality. Each subplot displays the fraction of infeasible solutions (left y-axis, bar plots) and the ATD of all feasible outcomes (right y-axis, box plots) across different numbers of input variables on the x-axis. Results are grouped by solver type and initialization st… view at source ↗
read the original abstract

When datasets contain outliers, robust regression is a well-established alternative to Ordinary Least Squares. A commonly employed robust estimator is Least Trimmed Squares (LTS), which computes the regression coefficients from a subset of observations. Determining the exact solution corresponds to a combinatorial problem with prohibitive computational costs, even for instances of moderate dimension. Thus, the most prevalent approach in practice remains a heuristic known as Fast-LTS. Although the heuristic often performs effectively, certain elements of the approach remain open to improvement. In particular, its core procedure provides robust results only when initialized with a large number of starting points. To address the heuristic's limitations, this paper reformulates the LTS problem as a concave minimization problem subject to a capped simplex constraint, and proposes the successive Boosted Difference of Convex Functions Algorithm (sBDCA) as a solution method. Theoretically, we establish via the \L ojasiewicz property that sBDCA converges to a local solution with a linear rate in the fastest case. To ensure robustness from a single initialization in practice, we derive and integrate a problem-specific preconditioning matrix into the algorithmic setup. Building on this theoretical foundation, we conduct numerical studies on various synthetic and real-world datasets to demonstrate the effectiveness of sBDCA with preconditioning. Specifically, we show that our approach is up to 3.25 times faster than Fast-LTS and achieves up to 90% lower objective function values, particularly in high-dimensional settings. As all code is openly available, this paper further provides a practical guide to robust regression in Python.

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

0 major / 2 minor

Summary. The manuscript reformulates the combinatorial Least Trimmed Squares (LTS) problem exactly as a concave minimization over a capped simplex constraint and introduces the successive Boosted Difference of Convex Functions Algorithm (sBDCA) together with a derived problem-specific preconditioning matrix. It establishes linear convergence to a local solution via the Łojasiewicz property and reports that sBDCA with preconditioning is up to 3.25 times faster than Fast-LTS while attaining up to 90% lower objective values on synthetic and real data, with all code released openly.

Significance. If the DC equivalence and preconditioner construction hold, the paper supplies a theoretically grounded, single-start robust alternative to Fast-LTS that is especially advantageous in high dimensions. The explicit reformulation, the Lojasiewicz-based rate, the open reproducible code, and the fair numerical comparisons (identical objective, same instances) are concrete strengths.

minor comments (2)
  1. The abstract states convergence 'in the fastest case'; a brief clarification of the precise Łojasiewicz exponent range that yields the linear rate would improve readability.
  2. Notation for the capped simplex and the preconditioning matrix could be introduced once in a dedicated preliminary section rather than inline in the algorithmic description.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the recognition of its theoretical and practical contributions, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation begins with an explicit combinatorial-to-DC reformulation of the LTS objective (concave minimization over capped simplex) and derives a preconditioning matrix from the problem geometry; both steps are presented as direct mathematical constructions rather than fits. Convergence follows from the external Łojasiewicz inequality with a stated linear rate. Numerical claims compare runtime and attained objective values on identical instances against Fast-LTS; these are empirical measurements, not quantities forced by internal fitting or self-citation. No load-bearing step reduces a reported result to its own inputs by construction, and the chain is self-contained against external benchmarks and open code.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central modeling step is the assumption that LTS admits an exact DC reformulation; convergence relies on the Lojasiewicz property (standard in nonsmooth optimization). No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The LTS objective admits an exact reformulation as concave minimization subject to a capped simplex constraint.
    This modeling choice is the load-bearing step that enables the DC algorithm.

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