Quantum mean centering for block-encoding-based quantum algorithm
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Abstract
Mean Centering (MC) is an important data preprocessing technique, which has a wide range of applications in data mining, machine learning, and multivariate statistical analysis. When the data set is large, this process will be time-consuming. In this paper, we propose an efficient quantum MC algorithm based on the block-encoding technique, which enables the existing quantum algorithms can get rid of the assumption that the original data set has been classically mean-centered. Specifically, we first adopt the strategy that MC can be achieved by multiplying by the centering matrix C , i.e., removing the row means, column means and row-column means of the original data matrix X can be expressed as XC , CX and CXC , respectively. This allows many classical problems involving MC, such as Principal Component Analysis (PCA), to directly solve the matrix algebra problems related to XC , CX or CXC . Next, we can employ the block-encoding technique to realize MC. To achieve it, we first show how to construct the block-encoding of the centering matrix C , and then further obtain the block-encodings of XC , CX and CXC . Finally, we describe one by one how to apply our MC algorithm to PCA and other algorithms.