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Machine Learning Model for Sparse PCM Completion

Selcuk Koyuncu, Ronak Nouri, Stephen Providence

TL;DR

This work addresses completing sparse pairwise comparison matrices (PCMs) and extracting robust rankings by blending classical PCM methods with graph-based learning. It introduces a graph neural approach that embeds items, propagates information over observed comparisons, and enforces multiplicative transitivity via a triangle-consistency loss, while producing reciprocal PCMs through projection. The method achieves competitive accuracy to log-least-squares on synthetic data, with the key advantage of scalable training on sparse graphs, enabling near-linear per-epoch complexity in the number of observed edges. Extensions emphasize sparse computation, mini-batch training, and large-scale applicability, with substantial speedups demonstrated on sizable PCMs. Overall, the paper provides a practical, scalable framework for sparse PCM completion that preserves multiplicative consistency and yields coherent rankings in large-scale settings.

Abstract

In this paper, we propose a machine learning model for sparse pairwise comparison matrices (PCMs), combining classical PCM approaches with graph-based learning techniques. Numerical results are provided to demonstrate the effectiveness and scalability of the proposed method.

Machine Learning Model for Sparse PCM Completion

TL;DR

This work addresses completing sparse pairwise comparison matrices (PCMs) and extracting robust rankings by blending classical PCM methods with graph-based learning. It introduces a graph neural approach that embeds items, propagates information over observed comparisons, and enforces multiplicative transitivity via a triangle-consistency loss, while producing reciprocal PCMs through projection. The method achieves competitive accuracy to log-least-squares on synthetic data, with the key advantage of scalable training on sparse graphs, enabling near-linear per-epoch complexity in the number of observed edges. Extensions emphasize sparse computation, mini-batch training, and large-scale applicability, with substantial speedups demonstrated on sizable PCMs. Overall, the paper provides a practical, scalable framework for sparse PCM completion that preserves multiplicative consistency and yields coherent rankings in large-scale settings.

Abstract

In this paper, we propose a machine learning model for sparse pairwise comparison matrices (PCMs), combining classical PCM approaches with graph-based learning techniques. Numerical results are provided to demonstrate the effectiveness and scalability of the proposed method.
Paper Structure (30 sections, 1 theorem, 57 equations, 2 figures, 1 table)

This paper contains 30 sections, 1 theorem, 57 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Step 1: Priority Vector.
  3. Step 2: Maximum Eigenvalue.
  4. Step 3: Consistency Index (CI).
  5. Step 4: Consistency Ratio (CR)
  6. The Bradley--Terry--Luce (BTL) Model
  7. The Log--Least--Squares (LLS) Method
  8. A Machine Learning Model for Sparse PCM Completion
  9. Step 1:
  10. Step 2:
  11. Step 3:
  12. Step 4:
  13. Step 5:
  14. Step 6:
  15. Step 7:
  16. ...and 15 more sections

Key Result

Proposition 6.1

Under sparse connectivity with average node degree bounded by a constant, and assuming mini-batch sizes $B_e, B_t = O(|\Omega| / \log n)$, the proposed training procedure achieves an expected per-epoch computational cost of $O(|\Omega|\log n)$.

Figures (2)

  • Figure 1: (Left) RMSE on held-out log-ratios vs edge density $p$. (Right) Kendall’s $\tau$ rank correlation vs edge density.
  • Figure 2: Wall-clock time vs number of observed edges $|\Omega|$ on synthetic PCMs. LLS is significantly faster than ML, but both scale roughly linearly.

Theorems & Definitions (5)

  • Example 4.1
  • Example 4.2
  • Proposition 6.1
  • proof
  • Example 6.2