Large-Scale Discrete Fourier Transform on TPUs

In this work, we present two parallel algorithms for the large-scale discrete Fourier transform (DFT) on Tensor Processing Unit (TPU) clusters. The two parallel algorithms are associated with two formulations of DFT: one is based on the Kronecker product, to be specific, dense matrix multiplications between the input data and the Vandermonde matrix, denoted as KDFT in this work; the other is based on the famous Cooley-Tukey algorithm and phase adjustment, denoted as FFT in this work. Both KDFT and FFT formulations take full advantage of TPU's strength in matrix multiplications. In the two parallel algorithms, the same strategy of data decomposition is applied to the input data. Through the data decomposition, the dense matrix multiplications in KDFT and the Cooley-Tukey-algorithm-based transform in FFT are kept local within TPU cores and can be performed completely in parallel. The communication among TPU cores is achieved through the one-shuffle scheme in both parallel algorithms, with which sending and receiving data takes place simultaneously between two neighboring cores and along the same direction on the interconnect network. The one-shuffle scheme is designed for the interconnect topology of TPU clusters, minimizing the time required by the communication among TPU cores. Both parallel algorithms, namely, KDFT and FFT, are implemented in TensorFlow owing to its rich set of functionalities for scientific computing and simplicity in realizing parallel computing algorithms. The three-dimensional complex DFT is performed on an example of dimension $8192 \times 8192 \times 8192$ with a full TPU Pod: the run time of KDFT is 12.66 seconds and that of FFT is 8.3 seconds. Strong and weak scaling analyses are provided to demonstrate the high parallel efficiency of the two DFT implementations on TPUs.