Unlabeled Sensing: Reconstruction Algorithm and Theoretical Guarantees

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 the samples 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 out of 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

Sampling at unknown locations: Uniqueness and reconstruction under constraints

Traditional sampling results assume that the sample locations are known. Motivated by simultaneous localization and mapping (SLAM) and structure from motion (SfM), we investigate sampling at unknown locations. Without further constraints, the problem is often hopeless. For example, we recently showed that, for polynomial and bandlimited signals, it is possible to find two signals, arbitrarily far from each other, that fit the measurements. However, we also showed that this can be overcome by adding constraints to the sample positions. In this paper, we show that these constraints lead to a uniform sampling of a composite of functions. Furthermore, the formulation retains the key aspects of the SLAM and SfM problems, whilst providing uniqueness, in many cases. We demonstrate this by studying two simple examples of constrained sampling at unknown locations. In the first, we consider sampling a periodic bandlimited signal composite with an unknown linear function. We derive the sampling requirements for uniqueness and present an algorithm that recovers both the bandlimited signal and the linear warping. Furthermore, we prove that, when the requirements for uniqueness are not met, the cases of multiple solutions have measure zero. For our second example, we consider polynomials sampled such that the sampling positions are constrained by a rational function. We previously proved that, if a specific sampling requirement is met, uniqueness is achieved. In addition, we present an alternate minimization scheme for solving the resulting non-convex optimization problem. Finally, fully reproducible simulation results are provided to support our theoretical analysis

Combining Range and Direction for Improved Localization

Self-localization of nodes in a sensor network is typically achieved using either range or direction measurements; in this paper, we show that a constructive combination of both improves the estimation. We propose two localization algorithms that make use of the differences between the sensors’ coordinates, or edge vectors; these can be calculated from measured distances and angles. Our first method improves the existing edge-multidimensional scaling algorithm (E-MDS) by introducing additional constraints that enforce geometric consistency between the edge vectors. On the other hand, our second method decomposes the edge vectors onto 1-dimensional spaces and introduces the concept of coordinate difference matrices (CDMs) to independently regularize each projection. This solution is optimal when Gaussian noise is added to the edge vectors. We demonstrate in numerical simulations that both algorithms outperform state-of-the-art solutions

Unlabeled Sensing

Like other data sensing problems, in unlabeled sensing, the target is to solve the equation y = Φx by finding vector x given a set of sample values in vector y as well as the matrix Φ. However, the main challenge in unlabeled sensing is that the correct order of the sample values in vector y is not available. We only have access to the unordered sample values and not to their sorted indices and labellings. Thus, in unlabeled sensing, we have to recover both the vector x as well as the correct ordering of the sample values in y. In this project, we want to answer two main questions regarding data recovery in unlabeled sensing. First, given a set of unordered sample values and specific matrix Φ, is there a unique solution? By considering different configurations in 2-dimensional plane, we prove that there exists no unique solution for few special configurations of Φ where we exactly quantify. However, by simulations we show that in some other configurations uniqueness is guaranteed. In the noisy case, uniqueness depends on configuration of Φ as well as noise level over sample values. Therefore, we consider probability of existence of unique solution in noisy case. Whether unique solution exists or not, the second investigated question is how to define an efficient algorithm to find all possible solutions with low complexity. We propose Distance-Based Algorithm (DBA) as an efficient algorithm for finding all possible solutions with noiseless sample values. Generalized DBA is also proposed as an extension of DBA which can be utilized when noisy sample values are accessible. The main advantages of DBA are its low complexity compared to combinatorial algorithms and also applicability in noisy cases. Simulation results and theoretical proofs are provided in each section to validate our claims