Quantum nonlinear dimensionality reduction based on maximum variance unfolding
Maximum variance unfolding is a nonlinear dimensionality reduction algorithm based on the local equidistance of man-ifolds.However,it is quite computationally expensive,especially when dealing with large-scale datasets.Therefore,an efficient quantum maximum variance unfolding algorithm is proposed herein.First,we propose a quantum matrix square root algorithm with exponential speedup,which can be used to obtain the square root of the involved matrix.As a result,we can efficiently obtain the density operator of the Laplacian matrix.Then,a complete quantum dimensionality reduction algorithm is proposed to map the original high-dimensional data onto a low-dimensional data space using Hamiltonian simulation and phase estimation.Finally,our research reveals that compared with its classical counterpart,the proposed quantum dimensionality reduction algorithm achieves an exponential speedup in terms of dimension and a polynomial speedup in terms of number of samples.
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