Multiplication-Avoiding Variant of Power Iteration with Applications

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, principal component analysis (PCA), among many others. Certain use cases may benefit from alternate, non-linear power methods with low complexity. In this paper, we introduce multiplication-avoiding power iteration (MAPI). MAPI replaces the standard ℓ
2
inner products that appear at the regular power iteration (RPI) with multiplication-free vector products, which are Mercer-type kernels that induce the ℓ
1
norm. For an n × 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.