# Almost Optimal Tensor Sketch

3 Sep 2019Thomas D. AhleJakob B. T. Knudsen

We construct a matrix $M\in R^{m\otimes d^c}$ with just $m=O(c\,\lambda\,\varepsilon^{-2}\text{poly}\log1/\varepsilon\delta)$ rows, which preserves the norm $\|Mx\|_2=(1\pm\varepsilon)\|x\|_2$ of all $x$ in any given $\lambda$ dimensional subspace of $R^d$ with probability at least $1-\delta$. This matrix can be applied to tensors $x^{(1)}\otimes\dots\otimes x^{(c)}\in R^{d^c}$ in $O(c\, m \min\{d,m\})$ time -- hence the name "Tensor Sketch"... (read more)

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