  Power iteration is a fundamental algorithm in data analysis. It extracts the
eigenvector corresponding to the largest eigenvalue of a given matrix.
Applications include ranking algorithms, recommendation systems, principal
component analysis (PCA), among many others. In this paper, we introduce
multiplication-avoiding power iteration (MAPI), which replaces the standard
$\ell_2$-inner products that appear at the regular power iteration (RPI) with
multiplication-free vector products which are Mercer-type kernel operations
related with the $\ell_1$ norm. Precisely, for an $n\times n$ matrix, MAPI
requires $n$ multiplications, while RPI needs $n^2$ multiplications per
iteration. Therefore, MAPI provides a significant reduction of the number of
multiplication operations, which are known to be costly in terms of energy
consumption. We provide applications of MAPI to PCA-based image reconstruction
as well as to graph-based ranking algorithms. When compared to RPI, MAPI not
only typically converges much faster, but also provides superior performance.
