Ridge regression (RR), also called regularized linear regression, is an important machine learning technique which introduces a regularization hyperparameter to ordinary multiple linear regression for analyzing data suffering from multicollinearity. Here we provide an efficient quantum algorithm for RR. Specifically, by giving the technique of parallel Hamiltonian simulation that can simulate a number of Hermitian matrices in parallel, we develop a quantum version of
K-fold cross-validation approach that can efficiently estimate the predictive performance of RR. Our algorithm involves two phases: (1) we first use the quantum
K-fold cross-validation to efficiently determine an approximately optimal regularization hyperparameter for RR with which RR can achieve very good predictive performance, and (2) then generate a quantum state encoding the optimal fitting parameters of RR with such hyperparameter, which can be further utilized to predict new data. Our algorithm can handle nonsparse data matrices, and is exponentially faster than the classical algorithm for (low-rank) design matrices with relatively small elements and low condition numbers.
We propose a quantum algorithm based on ridge regression model, which get the optimal fitting parameters w and a regularization hyperparameter α by analysing the training dataset. The algorithm consists of two subalgorithms. One is generating predictive value for a new input, the way is to apply the phase estimation algorithm to the initial state |Xi and apply the controlled rotation to the eigenvalue register. The other is finding an optimal regularization hyperparameter α , the way is to apply the phase estimation algorithm to the initial state |yi and apply the controlled rotation to the eigenvalue register. The second subalgorithm can compute the whole training dataset in parallel that improve the efficiency. Compared with the classical ridge regression algorithm, our algorithm overcome multicollinearity and overfitting. Moreover, it have exponentially faster. What's more, our algorithm can deal with the non-sparse matrices in comparison to some existing quantum algorithms and have slightly speedup than the existing quantum counterpart. At present, the quantum algorithm has a wide range of application and the proposed algorithm can be used as a subroutine of other quantum algorithms.
Ridge regression (RR) is an important machine learning technique which introduces a regularization hyperparameter
αto ordinary multiple linear regression for analyzing data suffering from multicollinearity. In this paper, we present a quantum algorithm for RR, where the technique of parallel Hamiltonian simulation to simulate a number of Hermitian matrices in parallel is proposed and used to develop a quantum version of
K-fold cross-validation approach, which can efficiently estimate the predictive performance of RR. Our algorithm consists of two phases: (1) using quantum
K-fold cross-validation to efficiently determine a good
αwith which RR can achieve good predictive performance, and then (2) generating a quantum state encoding the optimal fitting parameters of RR with such
α, which can be further utilized to predict new data. Since indefinite dense Hamiltonian simulation has been adopted as a key subroutine, our algorithm can efficiently handle non-sparse data matrices. It is shown that our algorithm can achieve exponential speedup over the classical counterpart for (low-rank) data matrices with low condition numbers. But when the condition numbers of data matrices are large to be amenable to full or approximately full ranks of data matrices, only polynomial speedup can be achieved.