  Singular value decomposition is central to many problems in engineering and
scientific fields. Several quantum algorithms have been proposed to determine
the singular values and their associated singular vectors of a given matrix.
Although these algorithms are promising, the required quantum subroutines and
resources are too costly on near-term quantum devices. In this work, we propose
a variational quantum algorithm for singular value decomposition (VQSVD). By
exploiting the variational principles for singular values and the Ky Fan
Theorem, we design a novel loss function such that two quantum neural networks
(or parameterized quantum circuits) could be trained to learn the singular
vectors and output the corresponding singular values. Furthermore, we conduct
numerical simulations of VQSVD for random matrices as well as its applications
in image compression of handwritten digits. Finally, we discuss the
applications of our algorithm in recommendation systems and polar
decomposition. Our work explores new avenues for quantum information processing
beyond the conventional protocols that only works for Hermitian data, and
reveals the capability of matrix decomposition on near-term quantum devices.
