二值化Johnson-Lindenstrauss嵌入
Binarized Johnson-Lindenstrauss embeddings
摘要：
我们考虑的问题，编码成一组向量的最小数量的位，同时保留信息的欧几里德几何。我们表明，这一任务可以通过应用约翰逊-林登施特劳斯嵌入和随后的二进制化每个向量的向量的每个条目进行比较，以一个均匀的随机阈值来完成。使用这个简单的构造，我们产生两个编码的数据集，这样就可以查询欧几里德信息的一对点使用少量的位操作，直到一个所需的添加剂错误-欧几里德距离在第一种情况下，内积和平方欧几里德距离在第二。在后一种情况下，每个点都以接近线性的时间编码。这些编码所需的比特数是量化的数据集的两个自然的复杂性参数-其覆盖数和本地化的高斯复杂性-并被证明是接近最佳的。
英文摘要：
We consider the problem of encoding a set of vectors into a minimal number of bits while preserving information on their Euclidean geometry. We show that this task can be accomplished by applying a Johnson-Lindenstrauss embedding and subsequently binarizing each vector by comparing each entry of the vector to a uniformly random threshold. Using this simple construction we produce two encodings of a dataset such that one can query Euclidean information for a pair of points using a small number of bit operations up to a desired additive error - Euclidean distances in the first case and inner products and squared Euclidean distances in the second. In the latter case, each point is encoded in near-linear time. The number of bits required for these encodings is quantified in terms of two natural complexity parameters of the dataset - its covering numbers and localized Gaussian complexity - and shown to be near-optimal.
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