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In group testing, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether at least one defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, communication protocols and many more. In this paper, we study (i) a sparsity-constrained version of the problem, in which the testing procedure is subjected to one of the following two constraints: items are finitely divisible and thus may participate in at most $\gamma $ tests; or tests are size-constrained to pool no more than $\rho $ items per test; and (ii) a noisy version of the problem, where each test outcome is independently flipped with some constant probability. Under each of these settings, considering the for-each recovery guarantee with asymptotically vanishing error probability, we introduce a fast splitting algorithm and establish its near-optimality not only in terms of the number of tests, but also in terms of the decoding time. While the most basic formulations of our algorithms require $\varOmega (n)$ storage for each algorithm, we also provide low-storage variants based on hashing, with similar recovery guarantees.

 

Location estimation is one of the most basic questions in parametric statistics. Suppose we have a known distribution density f, and we get n i.i.d. samples from f(x − µ) for some unknown shift µ. The task is to estimate µ to high accuracy with high probability. The maximum likelihood estimator (MLE) is known to be asymptotically optimal as n → ∞, but what is possible for finite n? In this paper, we give two location estimators that are optimal under different criteria: 1) an estimator that has minimax-optimal estimation error subject to succeeding with probability 1 − δ and 2) a confidence interval estimator which, subject to its output interval containing µ with probability at least 1 − δ, has the minimum expected squared interval width among all shift-invariant estimators. The latter construction can be generalized to minimizing the expectation of any loss function on the interval width.