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Hausdorff Moment Transforms and Their Performance

Xinyun Wang, Martin Haenggi

TL;DR

The paper investigates the truncated Hausdorff moment problem on $[0,1]$, seeking accurate reconstruction of a cdf from a finite moment sequence and providing a framework to compare diverse Hausdorff moment transforms. It analyzes classical approaches (BM and ME) and refinements (GP, FJ, FL) while introducing two new methods, FC and CM, plus a tweaking mechanism to enforce distribution properties. A key contribution is the explicit, offline-friendly framing of several transforms as linear mappings, enabling fast, scalable computation and enabling rigorous accuracy analyses. The study emphasizes that convergence and accuracy strongly depend on the decay type of the moment sequence, leading to a six-class taxonomy and practical guidance on method selection. Overall, the work offers a principled basis for selecting HMTs in applications like CT and meta-distribution reconstructions, balancing accuracy, complexity, and stability across diverse moment-decay regimes.

Abstract

Various methods have been proposed to approximate a solution to the truncated Hausdorff moment problem. In this paper, we establish a method of comparison for the performance of the approximations. Three ways of producing random moment sequences are discussed and applied. Also, some of the approximations have been rewritten as linear transforms, and detailed accuracy requirements are analyzed. Our finding shows that the performance of the approximations differs significantly in their convergence properties, accuracy, and numerical complexity and that the decay type of the moment sequence strongly affects the accuracy requirement.

Hausdorff Moment Transforms and Their Performance

TL;DR

The paper investigates the truncated Hausdorff moment problem on , seeking accurate reconstruction of a cdf from a finite moment sequence and providing a framework to compare diverse Hausdorff moment transforms. It analyzes classical approaches (BM and ME) and refinements (GP, FJ, FL) while introducing two new methods, FC and CM, plus a tweaking mechanism to enforce distribution properties. A key contribution is the explicit, offline-friendly framing of several transforms as linear mappings, enabling fast, scalable computation and enabling rigorous accuracy analyses. The study emphasizes that convergence and accuracy strongly depend on the decay type of the moment sequence, leading to a six-class taxonomy and practical guidance on method selection. Overall, the work offers a principled basis for selecting HMTs in applications like CT and meta-distribution reconstructions, balancing accuracy, complexity, and stability across diverse moment-decay regimes.

Abstract

Various methods have been proposed to approximate a solution to the truncated Hausdorff moment problem. In this paper, we establish a method of comparison for the performance of the approximations. Three ways of producing random moment sequences are discussed and applied. Also, some of the approximations have been rewritten as linear transforms, and detailed accuracy requirements are analyzed. Our finding shows that the performance of the approximations differs significantly in their convergence properties, accuracy, and numerical complexity and that the decay type of the moment sequence strongly affects the accuracy requirement.
Paper Structure (34 sections, 1 theorem, 52 equations, 13 figures, 2 algorithms)

This paper contains 34 sections, 1 theorem, 52 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and related work
  3. Contributions
  4. Preliminaries
  5. Existence and uniqueness of solutions
  6. Adjustment
  7. Distance metrics
  8. Solutions and approximations
  9. The CM inequalities
  10. Well-established methods
  11. BM method
  12. ME method
  13. Improvements and concretization of existing methods
  14. GP method
  15. FJ method
  16. ...and 19 more sections

Key Result

Lemma 4.4

For a uniformly distributed random vector $\left(m_k\right)_{k \in [n]}$ on the moment space $\mathcal{M}_n$, the corresponding canonical moments $p_k$, $k \in [n]$, are independent and beta distributed as

Figures (13)

  • Figure 1: The infima and suprema from the CM inequalities for $n = 4, 6, 8, 10$. $x$ is discretized to $\mathcal{U}_{50} = \{ i/50, \, i \in [50]_0\}$.
  • Figure 2: The infima and suprema from the CM inequalities for $m_n = 1/(n+1)$ and $F(x) = x^2$.
  • Figure 3: $F_{\mathrm{BM},10}$, $\hat{F}_{\mathrm{BM},10}$ and $F$, where $\bar{F} = \exp(-\frac{x}{1-x}) (1-x)/ (1-\log(1-x))$. The $\circ$ denote ${F}_{\mathrm{BM},10}|_{\mathcal{U}_{11}}$.
  • Figure 4: Average of the total and maximum distances between $F$ and polished $\hat{F}_{\rm{GP}(\Delta s,\upsilon )}$ which are averaged over $100$ randomly generated beta mixtures as described in \ref{['sec:bypdfs']}.
  • Figure 5: The upper two plots are for the FJ method guruacharya2018approximation with $n = 20$, and the lower two plots are for the FL series with $n = 20$. The cdf is given in \ref{['eq:accdf']}.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Example 2.1: Two sequences with $n = 2$
  • Definition 2.2: Tweaking mapping
  • Definition 2.3: Maximum and total distance
  • Definition 3.1: Approximation by the BM method
  • Definition 3.2: Solution by the ME method
  • Definition 3.3: Approximation by the Gil-Pelaez method
  • Definition 3.4: Approximation by the FJ method guruacharya2018approximation
  • Definition 3.5: Approximation by the FL method
  • Definition 3.6: Approximation by the FC method
  • Definition 3.7: Approximation by the CM method
  • ...and 6 more