Low-rank Tucker and CP tensor decompositions are powerful tools in data analytics. The widely used alternating least squares (ALS) method, which solves a sequence of over-determined least squares subproblems, is costly for large and sparse tensors. We propose a fast and accurate sketched ALS algorithm for Tucker decomposition, which solves a sequence of sketched rank-constrained linear least squares subproblems. Theoretical sketch size upper bounds are provided to achieve $O(\epsilon)$ relative error for each subproblem with two sketching techniques, TensorSketch and leverage score sampling. Experimental results show that this new ALS algorithm, combined with a new initialization scheme based on randomized range finder, yields up to $22.0\%$ relative decomposition residual improvement compared to the state-of-the-art sketched randomized algorithm for Tucker decomposition of various synthetic and real datasets. This Tucker-ALS algorithm is further used to accelerate CP decomposition, by using randomized Tucker compression followed by CP decomposition of the Tucker core tensor. Experimental results show that this algorithm not only converges faster, but also yields more accurate CP decompositions.
翻译:低端塔克和CP 温度分解是数据分析的有力工具。 广泛使用的交替最小方块(ALS)方法(ALS)解决了一组定额最小方块的分问题,对于大和稀有的分解器来说成本很高。 我们为塔克分解提出了快速和准确的草图 ALS 算法,它解决了一组素描的按级排列的线性最小方块子子子子问题。 提供了理论草图大小的上界线线以达到美元( epsilon) 相对差错。 提供了这种塔克- ALS 算法, 以两种素描技术( TensorSketch 和 杠杆分数抽样) 来加速每种子问题中的CP解析。 实验结果表明, 新的ALS 算法, 加上一种基于随机测距的测距的测距法的新的初始化方法, 将产生22. 0 $ 相对的分解析剩余法, 与最先进的塔克分解算法相比, 各种合成和真实的数据集的分解。 这种塔克- 算法, 也进一步用于加速解剖,, 通过使用随机的测算方法, 而不是更精确的实验性测算法, 也显示了卡质变。