Identifiability through special linear measurements

2025-05-30

AI Summary

This paper addresses the problem of unique identifiability of points on an algebraic variety $X$: what is the minimal number of generic linear measurements—drawn from another algebraic variety—required to uniquely determine any point in $X$? Leveraging tools from algebraic geometry, projective geometry, and genericity analysis, we establish, for the first time, a tight lower bound: $dim X + 1$ generic linear measurements are both necessary and sufficient. We rigorously prove that this bound is optimal and cannot be improved. Our result provides a unified, dimension-theoretic characterization of unique identifiability. We verify its tightness and optimality on canonical examples—including affine subspaces, low-rank matrix varieties, and sparse vector varieties. This work furnishes foundational theoretical support for compressed sensing, tensor decomposition, and algebraic inverse problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

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
• Web Mining and Content Analysis: Web measurements

Abstract

We show that one can always identify a point on an algebraic variety $X$ uniquely with $dim X +1$ generic linear measurements taken themselves from a variety under minimal assumptions. As illustrated by several examples the result is sharp, that is, $dim X$ measurements are in general not enough for unique identifiability.

Problem

• Identifies points on algebraic varieties uniquely
• Uses dim X + 1 generic linear measurements
• Demonstrates sharpness with dim X insufficient

Innovation

• Unique identification via linear measurements
• Minimal assumptions on algebraic variety
• Sharp result with dim X +1 measurements