It is a challenge to manage infinite- or high-dimensional data in situations where storage, transmission, or computation resources are constrained. In the simplest scenario when the data consists of a noisy infinite-dimensional signal, we introduce the notion of local \emph{effective dimension} (i.e., pertinent to the underlying signal), formulate and study the problem of its recovery on the basis of noisy data. This problem can be associated to the problems of adaptive quantization, (lossy) data compression, oracle signal estimation. We apply a Bayesian approach and study frequentists properties of the resulting posterior, a purely frequentist version of the results is also proposed. We derive certain upper and lower bounds results about identifying the local effective dimension which show that only the so called \emph{one-sided inference} on the local effective dimension can be ensured whereas the \emph{two-sided inference}, on the other hand, is in general impossible. We establish the \emph{minimal} conditions under which two-sided inference can be made. Finally, connection to the problem of smoothness estimation for some traditional smoothness scales (Sobolev scales) is considered.

翻译：在存储、传输或计算资源受限的情况下，处理无穷维或高维数据是一项挑战。针对含噪无穷维信号这一最简单场景，我们引入局部有效维数（即与潜在信号相关的维数）的概念，提出基于含噪数据恢复该维数的问题并加以研究。该问题可与自适应量化、（有损）数据压缩及先知信号估计等问题相关联。我们采用贝叶斯方法，分析后验分布的频率学派性质，同时提出纯频率学派版本的结论。通过推导局部有效维数识别的上下界结果，我们发现仅能保证所谓的局部有效维数"单侧推断"，而"双侧推断"在一般情况下不可实现。我们进一步建立了实现双侧推断所需的极小条件，最后探讨了与经典光滑性尺度（Sobolev尺度）光滑性估计问题的关联。