数据降维--SVD&CUR

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 V^T 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 V^T 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