Unlabeled sensing: Reconstruction algorithm and theoretical guarantees

Abstract

It often happens that we are interested in reconstructing an unknown signal from partial measurements. Also, it is typically assumed that the location (temporal or spatial) of each sample is known and that the only distortion present in the observations is due to additive measurement noise. However, there are some applications where such location information is lost. In this paper, we consider the situation in which the order of noisy samples, taken from a linear measurement system, is missing. Previous work on this topic has only considered the noiseless case and exhaustive search combinatorial algorithms. We propose a much more efficient algorithm based on a geometrical viewpoint of the problem. We also study the uniqueness of the solution under different choices of the sampling matrix and its robustness to noise for the case of two-dimensional signals. Finally we provide simulation results to confirm the theoretical findings of the paper.

An Algebraic-Geometric Approach for Linear Regression Without Correspondences

“…A more efficient algorithm is that of [8], which is able to reduce the complexity as a function of m to a factor of at least m 7 . However, as the authors of [8] write, their algorithm strongly exploits the assumption of noiseless measurements and is also very brittle and very likely fails in the presence of noise; the same is true for the O(m n ) complexity algorithm of [38]. Finally, the authors in [8] write we are not aware of previous algorithms for the average-case problem in general dimension n. In the same paper a (1 + ǫ) approximation algorithm with theoretical guarantees and of complexity O((m/ǫ) n ) is proposed, which however, is not meant for practical deployment, but instead is 2.…”

Homomorphic Sensing

“…Instead, we are aware of only two relevant algorithms, which nevertheless are suitable under strong structural assumptions on the data. The O(nm n+1 ) method of [25] applies a brute-force solution, which explicitly relies on the data being noiseless and whose theoretical guarantees require a particular exponentially spaced structure on A. On the other hand, [24] attempt to solve min…”

“…Algorithms. Inspired by [24], [25] and [26] we make three algorithmic contributions. First, we introduce a branch-and-bound algorithm for the unlabeled sensing problem by globally minimizing (3).…”

“…Instead of branching over the space of selection matrices, which is known to be intractable [27], our algorithm only branches over the space of x ∈ R n , relying on a locally optimal computation of the selection matrix via dynamic programming. Second, it is this dynamic programming feature that also allows us to modify the purely theoretical algorithm of [25] into a robust and efficient method for small dimensions n. These two algorithms constitute to the best of our knowledge the first working solutions for the unlabeled sensing problem. Third, when customized for image registration under an affine transformation, our branch-andbound algorithm is on par with or outperforms state-of-the-art methods [10], [28], [29] on benchmark datasets.…”

“…It turns out that the dynamic programming trick of §3.1 is also the key to a robust version of the theoretical algorithm of [25]: we randomly select a sub-vector ȳ of y of length n, and for each A i out of the m!/(m − n)! many n × n matrices that can be made by concatenating different rows of A in any order we let…”