AI 中文总结
研究针对规范多adic分解的随机交替最小二乘优化，提出基于杠杆得分的采样策略，通过随机强秩揭示QR分解近似杠杆得分分布，该方法能以特定采样数实现精度，且优于先前方案，减少计算和存储开销。
AI 中文摘要
我们提出了一种用于规范多adic分解（CPD-ALS）的随机交替最小二乘优化的基于杠杆得分的新型采样策略。与以前的策略不同，我们根据正在分解的目标张量的杠杆得分来确定CPD-ALS问题的逐行采样。我们证明，当根据矩阵化目标张量的杠杆得分分布进行行采样时，CPD-ALS问题的每个最小二乘子问题在残差范数上以至少1-δ的概率实现(1+ε)-相对精度，采样数为s = \(\frac{Rγ}{β}\) max(\(\frac{4}{δε}\), \(\frac{144\ln(2R/δ)}{ε_{0}^{2}}\))，其中\(ε_{0}\)是常数，β是杠杆得分的近似常数，R是目标秩，γ捕获CPD因子矩阵的Khatri Rao乘积（KRP）与精确KRP之间的相干性；随着ALS迭代收敛，γ减小。为了在不明确计算杠杆得分的情况下有效近似目标张量每个矩阵化的杠杆得分分布，我们使用随机强秩揭示QR（sRRQR）分解，即SE-QRCS。通过构造，这种基于QR的杠杆得分采样方法优于先前发布的方案，因为原则上它不需要对目标张量进行重新采样或重新计算KRP的杠杆得分，从而最小化了CPD-ALS过程的计算和存储开销。
英文摘要
We present a novel leverage score-based sampling strategy for the randomized alternating least squares optimization (ALS) of the canonical polyadic decomposition (CPD-ALS). Unlike previous strategies, we determine row-wise samples for the CPD-ALS problem from the leverage scores of the target tensor which is being decomposed. We demonstrate that, when rows are sampled according to the leverage score distribution of the matricized target tensor, each least squares subproblem of the CPD-ALS problem achieves $(1+ε)-$relative accuracy in the residual norm with probability at least $1-δ$ using a sampling $s=\frac{Rγ}β \max\left(\frac{4}{δε}, \frac{144\ln(2R/δ)}{ε_{0}^{2}}\right)$, where $ε_{0}$ is a constant, $β$ is leverage score's approximation constant, $R$ is the target rank and $γ$ captures the coherence between the Khatri Rao product (KRP) of the CPD factor matrices and the exact KRP; $γ$ decreases as the ALS iterates converge. To efficiently approximate the leverage score distribution for each matricization of the target tensor without explicitly computing leverage scores we use a randomized strong rank-revealing QR (sRRQR) factorizations, SE-QRCS. By construction, this QR-based leverage score sampling method outperforms previously published schemes as it does not, in principle, require the resampling of the target tensor or recomputing the leverage scores of the KRP, minimizing the computational and storage overhead of the CPD-ALS procedure.
