Abstract
For a natural number \(m\), a distribution is called \(m\)-grained, if each element appears with probability that is an integer multiple of \(1/m\). We prove that, for any constant \(c<1\), testing whether a distribution over \([\Theta(m)]\) is \(m\)-grained requires \(\Omega(m^c)\) samples, where testing a property of distributions means distinguishing between distributions that have the property and distributions that are far (in total variation distance) from any distribution that has the property.
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Acknowledgements
The research presented in this paper was partially supported by the Israel Science Foundation (grant No. 1041/18). O.G. received additional funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 819702). We are grateful to Clément Canonne for communicating to us the lower bound pre sented in Section 3.1 and allowing us to include it in this paper.
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Goldreich , O., Ron, D. A Lower Bound on the Complexity of Testing Grained Distributions. comput. complex. 32, 11 (2023). https://doi.org/10.1007/s00037-023-00245-w
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DOI: https://doi.org/10.1007/s00037-023-00245-w