
Sparse HighDimensional Linear Regression. Algorithmic Barriers and a Local Search Algorithm
We consider a sparse high dimensional regression model where the goal is...
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Optimal Multistage Group Testing Algorithm for 3 Defectives
Group testing is a wellknown search problem that consists in detecting ...
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Individual testing is optimal for nonadaptive group testing in the linear regime
We consider nonadaptive probabilistic group testing in the linear regime...
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NonAdaptive Group Testing in the Linear Regime: Strong Converse and Approximate Recovery
In this paper, we consider the nonadaptive group testing problem in the...
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The Overlap Gap Property in Principal Submatrix Recovery
We study support recovery for a k × k principal submatrix with elevated ...
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On the AllOrNothing Behavior of Bernoulli Group Testing
In this paper, we study the problem of group testing, in which one seeks...
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Secure Adaptive Group Testing
Group Testing (GT) addresses the problem of identifying a small subset o...
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Group testing and local search: is there a computationalstatistical gap?
In this work we study the fundamental limits of approximate recovery in the context of group testing. One of the most wellknown, theoretically optimal, and easy to implement testing procedures is the nonadaptive Bernoulli group testing problem, where all tests are conducted in parallel, and each item is chosen to be part of any certain test independently with some fixed probability. In this setting, there is an observed gap between the number of tests above which recovery is information theoretically (IT) possible, and the number of tests required by the currently best known efficient algorithms to succeed. Often times such gaps are explained by a phase transition in the landscape of the solution space of the problem (an Overlap Gap Property phase transition). In this paper we seek to understand whether such a phenomenon takes place for Bernoulli group testing as well. Our main contributions are the following: (1) We provide first moment evidence that, perhaps surprisingly, such a phase transition does not take place throughout the regime for which recovery is IT possible. This fact suggests that the model is in fact amenable to local search algorithms ; (2) we prove the complete absence of "bad" local minima for a part of the "hard" regime, a fact which implies an improvement over known theoretical results on the performance of efficient algorithms for approximate recovery without falsenegatives, and finally (3) we present extensive simulations that strongly suggest that a very simple local algorithm known as Glauber Dynamics does indeed succeed, and can be used to efficiently implement the wellknown (theoretically optimal) Smallest Satisfying Set (SSS) estimator.
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