Locality sensitive hashing (LSH) is a fundamental algorithmic toolkit used by data scientists for approximate nearest neighbour search problems that have been used extensively in many large scale data processing applications such as near duplicate detection, nearest neighbour search, clustering, etc. In this work, we aim to propose faster and space efficient locality sensitive hash functions for Euclidean distance and cosine similarity for tensor data. Typically, the naive approach for obtaining LSH for tensor data involves first reshaping the tensor into vectors, followed by applying existing LSH methods for vector data $E2LSH$ and $SRP$. However, this approach becomes impractical for higher order tensors because the size of the reshaped vector becomes exponential in the order of the tensor. Consequently, the size of LSH parameters increases exponentially. To address this problem, we suggest two methods for LSH for Euclidean distance and cosine similarity, namely $CP-E2LSH$, $TT-E2LSH$, and $CP-SRP$, $TT-SRP$, respectively, building on $CP$ and tensor train $(TT)$ decompositions techniques. Our approaches are space efficient and can be efficiently applied to low rank $CP$ or $TT$ tensors. We provide a rigorous theoretical analysis of our proposal on their correctness and efficacy.

翻译：局部敏感哈希（LSH）是数据科学家用于近似最近邻搜索问题的基础算法工具集，已在众多大规模数据处理应用（如近似重复检测、最近邻搜索、聚类等）中得到广泛使用。本文旨在针对张量数据提出更快速且空间高效的欧氏距离与余弦相似度局部敏感哈希函数。通常，获得张量数据LSH的朴素方法先将张量重塑为向量，再对向量数据应用现有LSH方法（如$E2LSH$和$SRP$）。但该方法对高阶张量不实用，因为重塑后向量的维度随张量阶数呈指数增长，导致LSH参数规模也呈指数增长。为解决此问题，我们分别提出两种基于欧氏距离和余弦相似度的LSH方法：$CP-E2LSH$、$TT-E2LSH$以及$CP-SRP$、$TT-SRP$，这些方法构建在CP分解和张量列（TT）分解技术之上。我们的方法具有空间高效性，可有效应用于低秩CP或TT张量。我们通过严谨的理论分析验证了所提方法的正确性与有效性。