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In this work we study the fundamental limits of approximate recovery in the context of group testing. One of the most well-known, theoretically optimal, and easy to implement testing procedures is the non-adaptive 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 false-negatives, 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 well-known (theoretically optimal) Smallest Satisfying Set (SSS) estimator.
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $mathbf{X}^{star}$ from $m$ (possibly noisy) observations $Y_mu = | sum_{i=1}^n Phi_{mu i} X^{star}_i/sqrt{n}|$, for a large class of correlated real a
A trade-off between accuracy and fairness is almost taken as a given in the existing literature on fairness in machine learning. Yet, it is not preordained that accuracy should decrease with increased fairness. Novel to this work, we examine fair cla
Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (suc
We consider the problem of estimating a vector of discrete variables $(theta_1,cdots,theta_n)$, based on noisy observations $Y_{uv}$ of the pairs $(theta_u,theta_v)$ on the edges of a graph $G=([n],E)$. This setting comprises a broad family of statis
In this paper, we propose a general framework for tensor singular value decomposition (tensor SVD), which focuses on the methodology and theory for extracting the hidden low-rank structure from high-dimensional tensor data. Comprehensive results are