FFT backwards error

SVT backward error for Low-rank structure
Above is other problem which I meet in the same work I do.
What’s more, for tensor-SVD, we usually transform the original data to the Fourier domain (using FFT or DCT), then do matrix-SVT for each frontal slice, and at last transform back (using inverse-FFT or some inverse function), to obtain a tensor-SVD-based Low-rank approximation of the original data.

But when I use the autograd and expect to learn the thresholding operator tau, the error occurs: svd_backward:The singular vectors in the complex case are specified up to multiplication by e^{i phi}. The specified loss function depends on this phase term, making it ill-defined.

Is the reason of this error that the complex case leads to incorrect gradients?