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Time-Varying Graph Learning for Data with Heavy-Tailed Distribution

Amirhossein Javaheri, Jiaxi Ying, Daniel P. Palomar, Farokh Marvasti

TL;DR

This work tackles learning time-varying graphs under heavy-tailed data and missing entries by introducing a probabilistic framework that combines a non-negative vector autoregressive model for graph weights with a Laplacian-based Student-t signal model. It employs a semi-online MAP approach updating graphs within individual frames and enforces spectral constraints to enable $k$-component graph structures for clustering. An ADMM-based algorithm with majorization-minimization surrogates provides a convergent solution, and the method demonstrates superior performance on synthetic data and real financial data for both clustering and portfolio design. The approach is robust to outliers and missing data and supports near real-time frame-wise updates, making it practically impactful for finance and other dynamic networks.

Abstract

Graph models provide efficient tools to capture the underlying structure of data defined over networks. Many real-world network topologies are subject to change over time. Learning to model the dynamic interactions between entities in such networks is known as time-varying graph learning. Current methodology for learning such models often lacks robustness to outliers in the data and fails to handle heavy-tailed distributions, a common feature in many real-world datasets (e.g., financial data). This paper addresses the problem of learning time-varying graph models capable of efficiently representing heavy-tailed data. Unlike traditional approaches, we incorporate graph structures with specific spectral properties to enhance data clustering in our model. Our proposed method, which can also deal with noise and missing values in the data, is based on a stochastic approach, where a non-negative vector auto-regressive (VAR) model captures the variations in the graph and a Student-t distribution models the signal originating from this underlying time-varying graph. We propose an iterative method to learn time-varying graph topologies within a semi-online framework where only a mini-batch of data is used to update the graph. Simulations with both synthetic and real datasets demonstrate the efficacy of our model in analyzing heavy-tailed data, particularly those found in financial markets.

Time-Varying Graph Learning for Data with Heavy-Tailed Distribution

TL;DR

This work tackles learning time-varying graphs under heavy-tailed data and missing entries by introducing a probabilistic framework that combines a non-negative vector autoregressive model for graph weights with a Laplacian-based Student-t signal model. It employs a semi-online MAP approach updating graphs within individual frames and enforces spectral constraints to enable -component graph structures for clustering. An ADMM-based algorithm with majorization-minimization surrogates provides a convergent solution, and the method demonstrates superior performance on synthetic data and real financial data for both clustering and portfolio design. The approach is robust to outliers and missing data and supports near real-time frame-wise updates, making it practically impactful for finance and other dynamic networks.

Abstract

Graph models provide efficient tools to capture the underlying structure of data defined over networks. Many real-world network topologies are subject to change over time. Learning to model the dynamic interactions between entities in such networks is known as time-varying graph learning. Current methodology for learning such models often lacks robustness to outliers in the data and fails to handle heavy-tailed distributions, a common feature in many real-world datasets (e.g., financial data). This paper addresses the problem of learning time-varying graph models capable of efficiently representing heavy-tailed data. Unlike traditional approaches, we incorporate graph structures with specific spectral properties to enhance data clustering in our model. Our proposed method, which can also deal with noise and missing values in the data, is based on a stochastic approach, where a non-negative vector auto-regressive (VAR) model captures the variations in the graph and a Student-t distribution models the signal originating from this underlying time-varying graph. We propose an iterative method to learn time-varying graph topologies within a semi-online framework where only a mini-batch of data is used to update the graph. Simulations with both synthetic and real datasets demonstrate the efficacy of our model in analyzing heavy-tailed data, particularly those found in financial markets.
Paper Structure (14 sections, 4 theorems, 70 equations, 7 figures, 7 tables, 1 algorithm)

This paper contains 14 sections, 4 theorems, 70 equations, 7 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of Existing Works
  3. Notation
  4. Problem Statement
  5. Related Works
  6. Contributions
  7. Proposed Approach
  8. Numerical Results
  9. Synthetic data
  10. Real Data
  11. Conclusion
  12. Proof of Proposition \ref{['prop:1']}
  13. Proof of Proposition \ref{['prop:2']}
  14. Proof of Theorem \ref{['thm:convergence']}

Key Result

Proposition 1

Let ${\bf{w}}_{n-1}$ in eq:MAP_Online be replaced by an estimate of the graph weights from the previous time frame, denoted as $\hat{{\bf{w}}}_{n-1}$. By expanding the posterior probability for MAP estimation and simplifying, we obtain the following formulation for learning the time-varying graph: where $\alpha = \frac{2}{T_n\sigma_\epsilon}$, $\beta = \frac{2\log\sigma_\epsilon}{T_n}$, and $\gam

Figures (7)

  • Figure 1: Illustration of the concept of time-varying graphs.
  • Figure 2: Illustration of the time frames.
  • Figure 3: Visualization of the learned (weighted) adjacency matrices at different time intervals, with frames of length $T_n=200$ (top) and $T_n=100$ (bottom).
  • Figure 4: The $k$-component graphs learned from financial data corresponding to the log-returns of 100 stocks in the S&P 500 index (comprising $k = 8$ sectors). The graphs are shown for the last data frame (with length $T_n = 200$).
  • Figure 5: The time evolution of the graphs learned from S&P500 data via the proposed method for the frame length of $T_n = 200$. Colors represent the inferred clusters ($k=8$).
  • ...and 2 more figures

Theorems & Definitions (11)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • proof
  • proof
  • proof
  • Lemma 1
  • ...and 1 more