高阶奇异值分解(HOSVD)理解

在基于情境上下文的推荐系统中，HOSVD是常用方法，这里通过一篇文章简单理解下HOSVD。

1、适用场景：

This decomposition plays an important role in various domains, such as:
? Spectral analysis,
? Non-linear modeling,
? Communication and Radar processing,
? blind source separation,
? image processing,
? biomedical applications (magnetic resonance imaging and electrocardiography),
? web search,
? computer facial recognition,
? handwriting analysis,

2、HOSVD定义：

3、张量分解

1）standard unfoldings

巨大的计算成本。

2）Higer PCA

3）Kernel decomposition in Volterra series

4）1-D harmonic retrieval problem

5）Oblique unfoldings to decrease the complexity

6）Complexities of the HOSVD algorithms

HOSVD主要在于张量分解方法。

4、展望

•Structured tensors imply strongly structured modes if oblique unfoldings are used. Not true for standard unfoldings !
• Increasing the structure of the modes allows to exploit fast techniques from numerical linear algebra based on - the column-redundancy property
- fast products vector/matrix for Toeplitz or Hankel matrices.
• Fastest implementation of the rank-truncated HOSVD (dedicated to Hankel tensors) has a quasilinear complexity w.r.t. the tensor dimensions.

• Generalize to tensors of order > 3.

• Extend to other HOSVD (constrained HOSVD, cross-HOSVD,...)?

最基础的可以参考维基https://en.wikipedia.org/wiki/Higher-order_singular_value_decomposition