Unlabeled Sensing in Linear Measurement Systems

Unlabeled sensing addresses the challenge of reconstructing a signal from linear measurements when the correspondence between observed values and sensing operators is unknown or scrambled. In a canonical formulation, one seeks to recover both an unknown vector and an unknown permutation of the measurement ordering from y = ΠA x + η, where A is a known sensing matrix, Π is an unknown permutation matrix and η denotes noise. This problem arises wherever data association is imperfect, for instance in multi-sensor networks, point-cloud registration, array processing and high-throughput biological assays. The absence of labelling introduces combinatorial complexity, as exhaustive search over all permutations grows factorially with the number of measurements. Over the past decade, researchers have developed conditions guaranteeing unique recovery under model assumptions such as sparsity, low noise and restricted permutation classes. Algorithmic strategies include combinatorial optimisation, convex and spectral relaxations, robust regression formulations and graph-matching lifts. Recent efforts have focused on balancing statistical guarantees, computational tractability and resilience to real-world artefacts such as local misalignments and partial mismatches. Applications in robotics, target tracking and medical imaging underscore the global importance of reliably solving unlabeled sensing problems in diverse practical contexts.

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Unlabeled Sensing in Linear Measurement Systems publication trend

The graph below shows the total number of articles in unlabeled sensing in linear measurement systems across all publications each year (not limited to Nature Index journals).

Technical terms

Unlabeled sensing: Recovery of a signal and an unknown reordering of its measurements from linear observations.

Permutation matrix: A binary matrix that reorders elements of a vector or rows of a matrix when multiplied.

Convex relaxation: A technique that replaces a non-convex optimisation problem with a convex surrogate to enable efficient algorithms.

Graph matching: The process of aligning vertices between two graphs by maximising a correspondence score, often used to infer permutations.

References

1. Linear regression with sparsely permuted data. Electronic Journal of Statistics (2019).
2. Linear regression with partially mismatched data: local search with theoretical guarantees. Mathematical Programming (2022).
3. R-Local Unlabeled Sensing: A Novel Graph Matching Approach for Multiview Unlabeled Sensing Under Local Permutations. IEEE Open Journal of Signal Processing (2021).