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.
翻译:Kronecker 产品是支持快速线性变换的重要矩阵操作, 包括信号处理、 图形理论、 量数计算和深层学习。 在这项工作中, 我们引入了快速 Johnson- Lindenstraus 投影为 Kronecker 产品结构, Kronecker 快速 Johnson- Lindenstraus 变换 (KFJLT) 。 KFJLT 将嵌入成本降低到标准快速 Johnson- Lindenstraus 变换( FJLT) 的成本指数系数, 用于 Kronecker 结构的矢量, 避免 明确形成完整的 Kronecker 产品。 我们证明, 嵌入电源的速率只有小价: $=\ prod ⁇ k=1 nk. kFLTLT; 将硬质产值的量定值数设置为 美元构成 Ecloidea 空间 $\\\ d= d\\\\\ d\\\\\ mathbx 的 Oiral mal max $ R=r= dal= kl= dal maxal maxl= dalx max maxl= dal maxl= dal= sal= dal= dalxxxxl= dalxxxxal= dir= dir= dir= ml=l=l=l=l=l=l=l=l=l=xxxxxxxxxxl=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxal=xal=xal=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxl=