Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity

Abstract

We study the tensor structure of two operations: the transformation of a given multidimensional vector into a multilevel Toeplitz matrix and the convolution of two given multidimensional vectors. We show that the low-rank tensor structure of the input is preserved in the output and propose efficient algorithms for these operations in the newly introduced quantized tensor train (QTT) format. Consider a $d$-dimensional $2n \times\cdots\times 2n$-vector $\boldsymbol{x}$. If it is represented elementwise, the number of parameters is $(2n)^{d}$. However, if we assume that $\boldsymbol{x}$ is given in a QTT representation with ranks bounded by $p$, the number of parameters is reduced to $\mathcal{O}\left(dp^{2} \log n\right)$. Under this assumption we show how the multilevel Toeplitz matrix generated by $\boldsymbol{x}$ can be obtained in the QTT format with ranks bounded by $2p$ in $\mathcal{O}\left(dp^{2} \log n\right)$ operations. We also describe how the convolution $\boldsymbol{x}\star\boldsymbol{y}$ of $\...