> For the complete documentation index, see [llms.txt](https://kerasnoone.gitbook.io/garnet/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kerasnoone.gitbook.io/garnet/zi-ran-yu-yan-chu-li/zhu-ti-mo-xing/lsa/0x01-lsa.md).

# LSA

LSA(Latent Semantic Analysis)潜在语义分析, 又称为LSI(Latent Semantic Index), 是一种常用的主题模型.

## 观点

文本和词汇之间存在着某种相关关系, 若干篇文本和若干个词汇根据这种相关关系构成了一定的语义结构. 去除结构中冗余次要的影响因素, 达到优化该结构的目的.

## 做法

将高维度的**词汇-文本共现矩阵**, 通过**奇异值分解**(SVD)法, 将原来的文章向量映射到低维度的潜在语义空间中, 即主题空间中, 空间的维度等于主题的数量.

## 意义

* 维度降低, 缩小了问题的规模
* 使得表面毫不相关的词, 体现出深层次的联系

## 数学表示

$$d$$篇文本, $$t$$个词汇, 构成一个大小为$$(t, d)$$的**词汇-文本共现矩阵**$$X$$. 其中的每个元素$$X\_{ij}$$值可以是:

* 第$$j$$个词在第$$i$$篇文本中出现的次数
* **tf-idf**值

**LSA步骤**如下:

* **SVD**将矩阵$$X$$分解分$$X=T\_0S\_0D\_0^T$$, $$T\_0$$大小为$$(t, r)$$, $$S\_0$$大小为$$(r,r)$$对角矩阵, 对角元素为**奇异值**, $$D\_0^T$$大小为$$(r,d)$$
* 考虑$$S\_0$$中最大的$$k$$个元素, $$k\lt{r}$$, $$k$$即是降维后的维度, 也是主题的数量. 取$$S\_0$$中相应的$$k$$个值组成$$k$$阶对角矩阵, 同时取出$$T\_0$$中对应的$$k$$列, $$D\_0^T$$中对应的$$k$$行, 得到$$X\approx{\hat{X}}=TSD^T$$, 其中$$T$$大小为$$(t,k)$$, $$S$$大小为$$(k,k)$$, $$D^T$$大小为$$(k,d)$$, $$\hat{X}$$即是优化后的语义结构
* 对于新文本, 先将其转化为词汇频率或tf-idf向量$$X\_q$$, 即一个列向量$$(t,1)$$. 对$$X\_q$$进行转换得到$$D\_q=X\_q^TTS^{-1}$$, 大小为$$(1,k)$$
* $$D$$, $$(d,k)$$保存了训练中所有$$d$$篇文本的降维后由主题组成的向量, 将$$D\_q$$与$$D$$比较产生相似度的度量

## LSA算法优点

* 反映的不再是简单的词汇出现的频率和分布关系, 而是利用主题表现的强化的语义关系
* 低位, 有效处理大规模文本库

## LSA算法缺点

* SVD对数据变化较敏感, 缺乏先验, 显得太机械
* **bag-of-word**模型, 忽略了语法, 词语顺序等信息
* 超参数: 主题数量对结果有较大影响, 而且模型的表现随着参数变化无规律波动大, 难以调参


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