用Kronecker积实现更快的Johnson-Lindenstrauss变换
Faster Johnson-Lindenstrauss Transforms via Kronecker Products
摘要：
克罗内克积是一种重要的矩阵运算，在支持快速线性变换方面具有广泛的应用，包括信号处理、图论、量子计算和深度学习。在这项工作中，我们介绍了一个推广的快速约翰逊-林登施特劳斯投影嵌入向量与克罗内克积结构，克罗内克快速约翰逊-林登施特劳斯变换（KFJLT）。KFJLT通过避免显式形成完整的Kronecker乘积，将嵌入成本降低到标准快速Johnson-Lindenstrauss变换（FJLT）应用于具有Kronecker结构的向量时的成本的指数因子。我们证明，这种计算增益来只有一个小的代价嵌入功率：给定$N = \prod_{k=1}^d n_k$，考虑一组有限的$p$点的$d$构成欧氏空间$\bigotimes_{k=d}^{1}\mathbb{R}^{n_k} \subset \mathbb{R}^{N}$的张量积。以高概率，维数$N \times m$的随机KFJLT矩阵嵌入点集直到由$m \gtrsim \varepsilon^{-2} \cdot \log^{2d - 1} (p) \cdot \log N$提供的乘性失真$(1\pm \varepsilon)$。最后，我们描述了一个直接应用KFJLT的大规模克罗内克结构的最小二乘问题的有效解决方案，适合CP张量分解。
英文摘要：
The Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson-Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson-Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost to an exponential factor of the standard fast Johnson-Lindenstrauss transform (FJLT)'s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: given $N = \prod_{k=1}^d n_k$, consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes_{k=d}^{1}\mathbb{R}^{n_k} \subset \mathbb{R}^{N}$. With high probability, a random KFJLT matrix of dimension $N \times m$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon)$ provided by $m \gtrsim \varepsilon^{-2} \cdot \log^{2d - 1} (p) \cdot \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.
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