The robust tensor completion problem for third-order tensors has attracted much attention recently, which aims to recover a low-rank tensor based on incomplete and/or corrupted observations. A widely used method is to minimize the tensor nuclear norm based on tensor singular value decomposition by using the discrete Fourier transform matrix. However, the efficiency of tensor nuclear norm may be challenged due to the limitation of discrete Fourier transform matrix. In this talk, we propose to employ other unitary transform matrices for tensor singular value decomposition such that a lower tensor rank can be obtained. In particular, a data-dependent unitary transform matrix is optimally constructed via the singular value decomposition of the unfolding matrix arising from the tensor along the third-dimension. By using such data-dependent transformed tensor product and tensor singular value decomposition, we show that a low-rank tensor and a sparse tensor can be recovered with high probability under some limited observations. Extensive numerical examples including hyperspectral, video, and face data sets, are tested and the results demonstrate the superior performance of the proposed method compared with that by using discrete Fourier and wavelet transforms.