数据降维--SVD&CUR
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矩阵的秩
秩即维度
SVD
使用SVD降维
SVD示例
We have used three columns for U, Σ, and V,the columns of U and V correspond to concepts.The first is “science fiction”and the second is “romance.” It is harder to explain the third column’sconcept, but it doesn’t matter all that much, because its weight, as given by the third nonzero entry in Σ, is very low compared with the weights of the first two concepts.
Let us think of the rows of A as people and the columns of A as movies. Then matrix U connects people to concepts. For example, the person Joe, who corresponds to row 1 of A, likes only the concept science fiction. The value 0.13 in the first row and first column of U is smaller than some of the other entries in that column, because while Joe watches only science fiction, he doesn’t rate those movies highly. The second column of the first row of U is 0, because Joe doesn’t rate romance movies at all. The matrix V relates movies to concepts. The 0.56,0.59 and 0.56 in each of the first three columns of the first row of VT indicates that the first three movies –Matrix, Alien, and Star Wars – each are of the science-fiction genre, while the 0’s in the last two columns of the first row say that these movies do not partake of the concept romance at all. Likewise, the second row of VT tells us that the movies Casablanca and Titanic are exclusively romances.
Finally, the matrix Σ gives the strength of each of the concepts. In our example, the strength of the science-fiction concept is 12.4, while the strength of the romance concept is 9.5. Intuitively,the science-fiction concept is stronger because the data provides more moviesof that genre and more people who like them.
CUR分解
Computing U
CUR: Provably good approx. to SVD
SVD和CUR比较
参考文献
http://cs246.stanford.edu
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