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Clustering of Incomplete Data via a Bipartite Graph Structure

Amirhossein Javaheri, Daniel P. Palomar

TL;DR

This paper proposes a clustering method based on a bipartite graph model that can infer clusters from incomplete data without requiring information about the center nodes and is designed to effectively handle heavy-tailed data.

Abstract

There are various approaches to graph learning for data clustering, incorporating different spectral and structural constraints through diverse graph structures. Some methods rely on bipartite graph models, where nodes are divided into two classes: centers and members. These models typically require access to data for the center nodes in addition to observations from the member nodes. However, such additional data may not always be available in many practical scenarios. Moreover, popular Gaussian models for graph learning have demonstrated limited effectiveness in modeling data with heavy-tailed distributions, which are common in financial markets. In this paper, we propose a clustering method based on a bipartite graph model that addresses these challenges. First, it can infer clusters from incomplete data without requiring information about the center nodes. Second, it is designed to effectively handle heavy-tailed data. Numerical experiments using real financial data validate the efficiency of the proposed method for data clustering.

Clustering of Incomplete Data via a Bipartite Graph Structure

TL;DR

This paper proposes a clustering method based on a bipartite graph model that can infer clusters from incomplete data without requiring information about the center nodes and is designed to effectively handle heavy-tailed data.

Abstract

There are various approaches to graph learning for data clustering, incorporating different spectral and structural constraints through diverse graph structures. Some methods rely on bipartite graph models, where nodes are divided into two classes: centers and members. These models typically require access to data for the center nodes in addition to observations from the member nodes. However, such additional data may not always be available in many practical scenarios. Moreover, popular Gaussian models for graph learning have demonstrated limited effectiveness in modeling data with heavy-tailed distributions, which are common in financial markets. In this paper, we propose a clustering method based on a bipartite graph model that addresses these challenges. First, it can infer clusters from incomplete data without requiring information about the center nodes. Second, it is designed to effectively handle heavy-tailed data. Numerical experiments using real financial data validate the efficiency of the proposed method for data clustering.
Paper Structure (13 sections, 2 theorems, 37 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 13 sections, 2 theorems, 37 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Problem Formulation
  4. Proposed Method
  5. $\mathbf{L}$ update step:
  6. $\mathbf{B}$ update step:
  7. $\mathbf{A}$ update step:
  8. Dual variable update step:
  9. Numerical Results
  10. Conclusion
  11. Proof of Lemma \ref{['lemma:1']}
  12. Simplex Projection Operator
  13. Proof of Lemma \ref{['lemma:2']}

Key Result

Lemma 1

The function $f_\mathbf{B}(\mathbf{B})$ in eq:B_subprob can be majorized as where

Figures (1)

  • Figure 1: The graphs learned from financial data corresponding to the log-returns of 100 stocks in S&P500 index (including $k=8$ sectors).

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof