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Monotone Curve Estimation via Convex Duality

Tongseok Lim, Kyeongsik Nam, Jinwon Sohn

TL;DR

This work develops a monotone principal-curve estimator built on convex duality and optimal transport, enforcing monotonicity through a dual convex-pair exposure and a reconstruction penalty. A rigorous theory establishes existence of the estimator and provides finite-sample bounds on the empirical and generalization mean-squared error, including a rate of $O(n^{-1/3})$ for the generalization gap under compact support. The authors implement a neural-network based numerical algorithm that parameterizes the convex components and uses an orthogonal transformation for data alignment, with early stopping justified by validation error guarantees. Empirical studies on synthetic data and real-world datasets (commodity prices and avocado demand) demonstrate improved accuracy and robust performance under variable transformations, highlighting the method’s potential for monotone-structure problems in noisy, high-dimensional settings.

Abstract

A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve framework constrained by monotonicity with rigorous theoretical justifications. We establish statistical guarantees for our monotone curve estimate, including expected empirical and generalized mean squared errors, while proving the existence of such estimates. These statistical foundations justify adopting the popular early stopping procedure in machine learning to implement our numeric algorithm with neural networks. Comprehensive simulation studies reveal that the proposed monotone curve estimate outperforms competing methods in terms of accuracy when the data exhibits a monotonic structure. Moreover, through two real-world applications on future prices of copper, gold, and silver, and avocado prices and sales volume, we underline the robustness of our curve estimate against variable transformation, further confirming its effective applicability for noisy and complex data sets. We believe that this monotone curve-fitting framework offers significant potential for numerous applications where monotonic relationships are intrinsic or need to be imposed.

Monotone Curve Estimation via Convex Duality

TL;DR

This work develops a monotone principal-curve estimator built on convex duality and optimal transport, enforcing monotonicity through a dual convex-pair exposure and a reconstruction penalty. A rigorous theory establishes existence of the estimator and provides finite-sample bounds on the empirical and generalization mean-squared error, including a rate of for the generalization gap under compact support. The authors implement a neural-network based numerical algorithm that parameterizes the convex components and uses an orthogonal transformation for data alignment, with early stopping justified by validation error guarantees. Empirical studies on synthetic data and real-world datasets (commodity prices and avocado demand) demonstrate improved accuracy and robust performance under variable transformations, highlighting the method’s potential for monotone-structure problems in noisy, high-dimensional settings.

Abstract

A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve framework constrained by monotonicity with rigorous theoretical justifications. We establish statistical guarantees for our monotone curve estimate, including expected empirical and generalized mean squared errors, while proving the existence of such estimates. These statistical foundations justify adopting the popular early stopping procedure in machine learning to implement our numeric algorithm with neural networks. Comprehensive simulation studies reveal that the proposed monotone curve estimate outperforms competing methods in terms of accuracy when the data exhibits a monotonic structure. Moreover, through two real-world applications on future prices of copper, gold, and silver, and avocado prices and sales volume, we underline the robustness of our curve estimate against variable transformation, further confirming its effective applicability for noisy and complex data sets. We believe that this monotone curve-fitting framework offers significant potential for numerous applications where monotonic relationships are intrinsic or need to be imposed.
Paper Structure (34 sections, 7 theorems, 47 equations, 6 figures, 8 tables, 1 algorithm)

This paper contains 34 sections, 7 theorems, 47 equations, 6 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Formulation of monotone curve-fitting task
  4. Monotone set and diagonal parametrization
  5. Exposure of a monotone set through convex functions
  6. Loss function to extract a monotone set
  7. Exposure of a monotone curve in higher dimensions
  8. Monotone curve-fitting task
  9. Extension of Task \ref{['algo']} via orthogonal transformation
  10. Theoretical analysis
  11. Existence
  12. Statistical error analysis
  13. Numerical algorithm
  14. Simulation
  15. Experiment data
  16. ...and 19 more sections

Key Result

Proposition 2.1

Let $f, g \in {\cal A}({\cal H})$ satisfy $f(x) + g(y) \ge \langle x, y \rangle$ for all $x,y \in {\cal H}$. Then the contact set $\Gamma_{f,g} := \{(x,y) \in {\cal H}^2 \ | \ f(x) + g(y) = \langle x, y \rangle \}$ is monotone. Moreover, $f, g$ are mutually conjugate if and only if $\Gamma_{f,g}$

Figures (6)

  • Figure 1: Contour of $H({\bf x};f,f^*)$: $f(x)=|x|^p/p$ and $f^*(x)=|x|^q/q$, the convex conjugate of $f$, with $1/p+1/q=1$. The star points ($\filledstar$) represent the zero set where $H({\bf x};f,f^*)=0$.
  • Figure 2: Task \ref{['algo2']} for the toy example: $(p_1,p_2)$ is the first principal component, $(1,1)$ indicates the diagonal direction, and $\gamma_{{\bf X}}$ is the true curve associated to ${\bf X}$.
  • Figure 3: Comparison of three methods on 2 and 3 dimensions: the ground truth curves (black points), Ours (blue stars), SCMS (orange rhombuses), and HS (green rectangles).
  • Figure 4: Comparison of commodity prices: the prices of copper, silver and gold. Gray points are observed data points. Ours and HS curves are colored blue and red respectively.
  • Figure 5: Comparison of commodity prices in $\mathbb{R}^3$: (Left) the prices are standardized after logarithmic transformation, and (right) the prices are just standardized before estimation.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Proposition 2.1
  • Remark 2.1: Construction of $f,f^*$ given $\Gamma$
  • Definition 2.2
  • Remark 2.2
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 7.1
  • ...and 3 more