We study the problem of compression for the purpose of similarity identification, where similarity is measured by the mean square Euclidean distance between vectors. While the asymptotical fundamental limits of the problem-the minimal compression rate and the error exponent-were found in a previous work, in this paper, we focus on the nonasymptotic domain and on practical, implementable schemes. We first present a finite blocklength achievability bound based on shape-gain quantization: the gain (amplitude) of the vector is compressed via scalar quantization, and the shape (the projection on the unit sphere) is quantized using a spherical code. The results are numerically evaluated, and they converge to the asymptotic values, as predicted by the error exponent. We then give a nonasymptotic lower bound on the performance of any compression scheme, and compare to the upper (achievability) bound. For a practical implementation of such a scheme, we use wrapped spherical codes, studied by Hamkins and Zeger, and use the Leech lattice as an example for an underlying lattice. As a side result, we obtain a bound on the covering angle of any wrapped spherical code, as a function of the covering radius of the underlying lattice.

中文翻译：

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二次相似性查询的压缩：有限块长度和实用方案

我们研究压缩问题以进行相似性识别，其中相似性是通过向量之间的均方欧几里德距离来衡量的。虽然在之前的工作中发现了问题的渐近基本限制——最小压缩率和误差指数，但在本文中，我们专注于非渐近域和实用的、可实现的方案。我们首先提出了一个基于形状增益量化的有限块长度可达性界限：向量的增益（幅度）通过标量量化进行压缩，形状（单位球面上的投影）使用球形码进行量化。结果被数值评估，并且它们收敛到渐近值，正如误差指数所预测的那样。然后我们给出任何压缩方案的性能的非渐近下界，并与上限（可实现性）进行比较。对于这种方案的实际实现，我们使用由 Hamkins 和 Zeger 研究的包裹球码，并使用 Leech 格作为底层格的示例。作为附带的结果，我们获得了任何包裹球面代码的覆盖角的界限，作为底层晶格覆盖半径的函数。