We consider density estimation for Besov spaces when each sample is quantized to only a limited number of bits. We provide a noninteractive adaptive estimator that exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minimax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which arise naturally for those spaces, can be used to achieve the same performance.

翻译：我们考虑贝索夫空间的密度估计,当每个样本被量化为数量有限的位数时。 我们提供一个非互动的适应性估计器,利用波浪基的宽度,同时在通信限制下利用模拟和推算法进行模拟和推算技术。 我们显示,我们的测算器通过得出即使在允许互动协议的情况下也保持的最小最大下限,几乎是最佳的。 有趣的是,我们基于波波列的测算器对于索博列尔夫空间来说也几乎是速率和最佳的,但尚不清楚这些空间自然产生的标准四重基数是否能够用来实现同样的性能。