Quantum algorithm for neighborhood preserving embedding

Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at the local manifold structure. NPE contains three steps, i.e. , finding nearest neighbors of each data point, constructing weight matrix, and obtaining transformation matrix. Liang et al . proposed a variational quantum algorithm (VQA) for [ Phys. Rev. A 101 032323 (2020)]. The consists sub-algorithms, corresponding to steps NPE, was expected have exponential speedup on n However, has two disadvantages: (i) It not known how efficiently obtain input third sub-algorithm from output second one. (ii) Its complexity cannot be rigorously analyzed because in it VQA. In this paper, we propose complete which redesign sub-algorithms give rigorous analysis. shown our can achieve polynomial number points m under certain conditions over classical algorithm, significant compared .’s even without considering