High-Performance Tensor-Train Primitives Using GPU Tensor Cores

Learning tensor-train (TT) structure (a.k.a matrix product state (MPS) representation) from large-scale high-dimensional data has been a common task in big data analysis, deep learning, and quantum machine learning. However, tensor-train algorithms are compute-intensive, which hinders their real-world applications. In this paper, we present high-performance tensor-train primitives using GPU tensor cores and demonstrate three applications. First, we use GPU tensor cores to optimize tensor-train primitives, including tensor contraction, singular value decomposition, and data transfer and computing. Second, we utilize the optimized primitives to accelerate tensor-train decomposition algorithms for big data analysis. Further, we propose a shard mode for high-order tensor computations on multiple GPUs. Third, we apply the optimized primitives to accelerate the tensor-train layer for compressing deep neural networks. Last, we utilize the optimized primitives to accelerate a quantum machine learning algorithm called <i>Density Matrix Renormalization Group (DMRG)</i>. In performance evaluations, our third-order TT tensor decomposition achieves up to <inline-formula><tex-math notation="LaTeX">$3.34\times$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$6.91\times$</tex-math></inline-formula> speedups over two popular libraries (namely T3F and tntorch) on an A100 GPU, respectively. The proposed sixth-order tensor-train decomposition achieves up to a speedup of <inline-formula><tex-math notation="LaTeX">$5.01\times$</tex-math></inline-formula> over T3F on multiple A100 GPUs. Our tensor-train layer for a fully connected neural network achieves a compression ratio of <inline-formula><tex-math notation="LaTeX">$65.3\times$</tex-math></inline-formula> at the cost of <inline-formula><tex-math notation="LaTeX">$0.3\%$</tex-math></inline-formula> drop in accuracy and a speedup of <inline-formula><tex-math notation="LaTeX">$1.53\times$</tex-math></inline-formula> over a PyTorch implementation on CUDA cores. The optimized <i>DMRG</i> algorithm achieves up to a speedup of <inline-formula><tex-math notation="LaTeX">$14.0\times$</tex-math></inline-formula> over TensorNetwork, indicating the potential of the optimized tensor primitives for the classical simulation of quantum machine learning algorithms.