Jayakrishnan Unnikrishnan

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Unlabeled Sensing with Random Linear Measurements

Martin Vetterli, Jayakrishnan Unnikrishnan, Saeid Haghighatshoar

We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations y = Ax when the order of the observations in the vector y is unknown. Focusing on the setting in which A is a random matrix with i.i.d. entries, we show that if the sensing matrix A admits an oversampling ratio of 2 or higher, then, with probability 1, it is possible to recover x exactly without the knowledge of the order of the observations in y. Furthermore, if x is of dimension K, then any 2K entries of y are sufficient to recover x. This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. The result is universal in that conditioned on the realization of matrix A, recovery is guaranteed for all possible choices of x. While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. We also analyze a noisy version of the problem and show that local stability is guaranteed by the solution. In particular, for every x, the recovery error tends to zero as the signal-to-noise ratio tends to infinity. The question of universal stability is unclear. In addition, we obtain a converse of the result in the noiseless case: If the number of observations in y is less than 2K, then with probability 1, universal recovery fails, i.e., with probability 1, there exist distinct choices of x which lead to the same unordered list of observations in y. We also present extensions of the result of the noiseless case to special cases with non-i.i.d. entries in A, and to a different setting in which the labels of a portion of the observations y are known. In terms of applications, the unlabeled sensing problem is related to data association problems encountered in different domains including robotics where it is appears in a method called “simultaneous localization and mapping”, multi-target tracking applications, and in sampling signals in the presence of jitter.

Institute of Electrical and Electronics Engineers2018

Unlabeled Sensing With Random Linear Measurements

Martin Vetterli, Jayakrishnan Unnikrishnan

2018

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Most users of online services have unique behavioral or usage patterns. These behavioral patterns can be used to identify and track users by using only the observed patterns in the behavior. We study the task of identifying users from statistics of their behavioral patterns. Specifically, we focus on the setting in which we are given histograms of users’ data collected in two different experiments. In the first dataset, we assume that the users’ identities are anonymized or hidden and in the second dataset we assume that their identities are known. We study the task of identifying the users in the first dataset by matching the histograms of their data with the histograms from the second dataset. In a recent work [1], [2] the optimal algorithm for this user identification task was introduced. In this paper, we evaluate the effectiveness of this method on a wide range of datasets with up to 50, 000 users, and in a wide range of scenarios. Using datasets such as call data records, web browsing histories, and GPS trajectories, we demonstrate that a large fraction of users can be easily identified given only histograms of their data, and hence these histograms can act as users’ fingerprints. We also show that simultaneous identification of users achieves better performance compared to one-by-one user identification. Furthermore, we show that using the optimal method for identification does indeed give higher identification accuracies than heuristics-based approaches in such practical scenarios. The accuracies obtained under this optimal method can thus be used to quantify the maximum level of user identification that is possible in such settings. We show that the key factors affecting the accuracy of the optimal identification algorithm are the duration of the data collection, the number of users in the anonymized dataset, and the resolution of the dataset. We also analyze the effectiveness of k-anonymization in resisting user identification attacks on these datasets.

Institute of Electrical and Electronics Engineers2016