Joint estimation of smooth graph signals from partial linear measurements

2025-05-29

AI Summary

This paper addresses the joint estimation of smooth signals over graphs under sparse sampling: given an undirected connected graph (G) with (T) vertices, each vertex (t) hosts an underlying signal (x_t in mathbb{R}^n), but only a few linear measurements are observed at a tiny fraction of vertices; signals are assumed smooth over (G) (i.e., small graph Laplacian quadratic form). We propose a smoothing-regularized least-squares estimator and establish, for the first time, non-asymptotic mean-squared error bounds—proving weak consistency even under extreme sparsity (e.g., one-dimensional measurement per vertex and vanishing sampling fraction). We further extend the framework to multi-layer pairwise ranking, where multiple sparse, potentially disconnected measurement graphs (G_t) share a common underlying smooth signal structure, enabling robust strength inference. The theory accommodates both generic and structured graphs (e.g., cycles, grids). Synthetic and multi-layer ranking experiments demonstrate high-accuracy signal recovery under ultra-sparse observations.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Given an undirected and connected graph $G$ on $T$ vertices, suppose each vertex $t$ has a latent signal $x_t in mathbb{R}^n$ associated to it. Given partial linear measurements of the signals, for a potentially small subset of the vertices, our goal is to estimate $x_t$'s. Assuming that the signals are smooth w.r.t $G$, in the sense that the quadratic variation of the signals over the graph is small, we obtain non-asymptotic bounds on the mean squared error for jointly recovering $x_t$'s, for the smoothness penalized least squares estimator. In particular, this implies for certain choices of $G$ that this estimator is weakly consistent (as $T ightarrow infty$) under potentially very stringent sampling, where only one coordinate is measured per vertex for a vanishingly small fraction of the vertices. The results are extended to a ``multi-layer'' ranking problem where $x_t$ corresponds to the latent strengths of a collection of $n$ items, and noisy pairwise difference measurements are obtained at each ``layer'' $t$ via a measurement graph $G_t$. Weak consistency is established for certain choices of $G$ even when the individual $G_t$'s are very sparse and disconnected.

Problem

• Estimating smooth graph signals from partial linear measurements
• Analyzing mean squared error for joint signal recovery
• Extending results to multi-layer ranking with sparse measurements

Innovation

• Smoothness penalized least squares estimator
• Joint recovery from partial measurements
• Weak consistency under stringent sampling