Quantum algorithm for Help-Training semi-supervised support vector machine

Abstract

Semi-supervised support vector machine ( $$S^3VM$$ ) is a popular strategy for many machine learning tasks due to the expensiveness of getting enough labeled data. In this paper, we propose a quantum Help-Training $$S^3VM$$ and design a quantum Parzen window model to select $$n_1+n_2$$ unlabeled data from l labeled and n unlabeled data set in each iteration, the time complexity is $$O(\tau \sqrt{nn_1}+\tau \sqrt{nn_2}+\tau \sqrt{n})$$ for $$\tau $$ iterations, which exhibits a quadratic speed-up over classical algorithm, we adopt quantum linear system to build Lagrangian multipliers with accuracy $$\varepsilon $$ , the time complexity is $$O(\tau \kappa ^3$$ $$\varepsilon ^{-3} \hbox {polylog}(N(n+l)))$$ , where condition number is $$\kappa $$ and feature dimension is N, it is exponentially faster than classical $$S^3VM$$ algorithm. Our scheme has two significant merits, (i) we provide the first quantum method for semi-supervised learning, which uses multiple unlabeled data with quantum superposition to predict Lagrangian multipliers at the same time, (ii) quantum matrix decomposition method avoids building matrices of different dimensions in one iteration; specially, this work provides inspiration to explore the potential quantum machine learning applications.