Basis function learning is the stepping stone towards effective three-dimensional (3D) sound speed field (SSF) inversion for various acoustic signal processing tasks, including ocean acoustic tomography, underwater target localization/tracking, and underwater communications. Classical basis functions include the empirical orthogonal functions (EOFs), Fourier basis functions, and their combinations. The unsupervised machine learning method, e.g., the K-singular value decomposition (K-SVD) algorithm, has recently tapped into the basis function design, showing better representation performance than the EOFs. However, existing methods do not consider basis function learning approaches that treat 3D SSF data as a third-order tensor, and, thus, cannot fully utilize the 3D interactions/correlations therein. To circumvent such a drawback, basis function learning is linked to tensor decomposition in this paper, which is the primary drive for recent multi-dimensional data mining. In particular, a tensor-based basis function learning framework is proposed, which can include the classical basis functions (using EOFs and/or Fourier basis functions) as its special cases. This provides a unified tensor perspective for understanding and representing 3D SSFs. Numerical results using the South China Sea 3D SSF data have demonstrated the excellent performance of the tensor-based basis functions.