An efficient tensor regression for high-dimensional data
面向高维数据的高效张量回归AI 解读 ↓
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面向高维数据的高效张量回归
- 研究问题
- 张量阶数较高时，Tucker 分解的压缩效率可能下降；本文研究如何兼顾高阶系数压缩、数据结构解释与统计估计。
- 方法与思路
- 作者修改张量列车分解并用于张量回归，进一步扩展到张量自回归，推导最小二乘估计的非渐近性质并提出估计算法。
- 材料陈述的贡献
- 摘要称，新方法能更高效地压缩高阶系数张量，解释层级及因子式数据结构，并通过模拟和两个真实案例展示其统计与计算表现。
- 阅读关注点
- 应核对分解的可识别性与秩选择、非渐近结果的条件、算法保证，以及与 Tucker 方法比较时参数规模和计算预算是否可比。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未获核实；摘要未给出误差界、算法收敛条件或案例细节，不能认定所有数据结构下均占优。
原文摘要
Most currently used tensor regression models for high-dimensional data are based on Tucker decomposition, which has good properties but loses its efficiency in compressing tensors very quickly as the order of tensors increases, say greater than four or five. However, for the simplest tensor autoregression in handling time series data, its coefficient tensor already has the order of six. This paper revises a newly proposed tensor train (TT) decomposition and then applies it to tensor regression such that a nice statistical interpretation can be obtained. The new tensor regression can well match the data with hierarchical structures, and it even can lead to a better interpretation for the data with factorial structures, which are supposed to be better fitted by models with Tucker decomposition. More importantly, the new tensor regression can be easily applied to the case with higher order tensors since TT decomposition can compress the coefficient tensors much more efficiently. The methodology is also extended to tensor autoregression for time series data, and nonasymptotic properties are derived for the ordinary least squares estimations of both tensor regression and autoregression. A new algorithm is introduced to search for estimators, and its theoretical justification is also discussed. Theoretical and computational properties of the proposed methodology are verified by simulation studies, and the advantages over existing methods are illustrated by two real examples.