We consider the one-bit quantizer that minimizes the mean squared error for a source living in a real Hilbert space. The optimal quantizer is a projection followed by a thresholding operation, and we provide methods for identifying the optimal direction along which to project. As an application of our methods, we characterize the optimal one-bit quantizer for a continuous-time random process that exhibits low-dimensional structure. We numerically show that this optimal quantizer is found by a neural-network-based compressor trained via stochastic gradient descent.

翻译：我们考虑的是将生活在真正的希尔伯特空间的源的正方差最小化的一位数量化器。 最佳量化器是一种预测,然后是临界操作,我们提供了确定投影最佳方向的方法。 作为我们方法的应用,我们将最佳的一位数量化器定性为连续时间随机程序,该随机程序显示的是低维结构。我们用数字显示,该最佳量化器是由通过随机梯度下降训练的神经网络压缩机找到的。