Matching correlated VAR time series

2025-11-23

AI Summary

This paper addresses the matching problem for correlated vector autoregressive (VAR) time series: given an original multivariate time series and its observation after unknown perturbation and sample-wise permutation, the goal is to recover the latent sample-level correspondence. We propose a probabilistic VAR representation and establish a theoretical connection between maximum likelihood estimation (MLE) and the linear assignment formulation, deriving sharp thresholds for exact and partial recovery under noise. Innovatively, we formulate matching as a convex relaxation over the Birkhoff polytope and design an alternating minimization algorithm to jointly estimate VAR parameters and the permutation matrix. Experiments demonstrate that our linear assignment approach matches or outperforms MLE-based baselines across most settings, offering both theoretical guarantees—via recovery bounds—and computational efficiency.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the problem of matching correlated VAR time series databases, where a multivariate time series is observed along with a perturbed and permuted version, and the goal is to recover the unknown matching between them. To model this, we introduce a probabilistic framework in which two time series $(x_t)_{tin[T]},(x^#_t)_{tin[T]}$ are jointly generated, such that $x^#_t=x_{π^*(t)}+σ ilde{x}_{π^*(t)}$, where $(x_t)_{tin[T]},( ilde{x}_t)_{tin[T]}$ are independent and identically distributed vector autoregressive (VAR) time series of order $1$ with Gaussian increments, for a hidden $π^*$. The objective is to recover $π^*$, from the observation of $(x_t)_{tin[T]},(x^#_t)_{tin[T]}$. This generalizes the classical problem of matching independent point clouds to the time series setting. We derive the maximum likelihood estimator (MLE), leading to a quadratic optimization over permutations, and theoretically analyze an estimator based on linear assignment. For the latter approach, we establish recovery guarantees, identifying thresholds for $σ$ that allow for perfect or partial recovery. Additionally, we propose solving the MLE by considering convex relaxations of the set of permutation matrices (e.g., over the Birkhoff polytope). This allows for efficient estimation of $π^*$ and the VAR parameters via alternating minimization. Empirically, we find that linear assignment often matches or outperforms MLE relaxation based approaches.

Problem

• Recovering hidden permutations between correlated VAR time series databases
• Matching perturbed and permuted multivariate time series observations
• Establishing recovery guarantees for linear assignment estimators

Innovation

• Probabilistic framework for matching perturbed VAR time series
• Maximum likelihood estimator via quadratic optimization over permutations
• Convex relaxations over Birkhoff polytope for efficient estimation