张力对张力回归的计算和统计保证与张力列车分解
概括
本研究分析了基于张量列车 (TT) 的张量对张量 (ToT) 回归模型,解决计算挑战. 它提出了代硬值 (IHT) 和里曼梯度下降 (RGD) 算法,提供理论错误边界和收率.
科学领域:
- 机器学习 机器学习
- 数字分析 数字分析
- 张量计计算 张量计计算
背景情况:
- 张量对张量 (ToT) 回归概括了张量恢复,但面临着存储和计算方面的挑战.
- 张量分解,特别是张量列车 (TT) 格式,为ToT回归提供了效率改进.
- 在基于 TT 的 ToT 模型的理论分析和实际表现之间存在差距.
研究的目的:
- 从理论上分析基于TT的ToT回归模型,专注于受限制异位数属性 (RIP) 下的误差边界.
- 开发和评估高效的优化算法,以解决基于TT的ToT回归中的受约束最小平方问题.
- 为了比较拟议的算法的性能和复杂性,即代硬值 (IHT) 和里曼梯度下降 (RGD).
主要方法:
- 对受约束最小平方优化问题的错误分析,假设回归运算符满足RIP.
- 开发了两个优化算法:使用TT-单数值分解 (TT-SVD) 和里曼梯度下降 (RGD) 的代硬值 (IHT).
- 在RIP条件下分析IHT和RGD算法的光谱初始化和线性收率.
主要成果:
- 导出了理论上的上限误差和最小值的下限误差,显示了多项式依赖于 $N+M$ 的顺序.
- 无论是IHT还是RGD算法,在满足RIP时,都能实现带有光谱初始化的线性收率.
- 与IHT相比,RGD通过优化Stiefel分流器上的因素来降低存储复杂性,但恢复性能略差.
结论:
- 阐明了基于TT的ToT回归模型的理论特性和算法解决方案.
- IHT和RGD提供了有效的方法来解决基于TT的ToT回归,在存储和回收方面有明显的权衡.
- 实验结果验证了理论发现和算法的有效性.
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