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# Singular value decomposition

SVD seeks to decompose a matrix X of dimension m x n into these three component matrices:

• U of dimension m x m
• S, a diagonal matrix of size m x n; the entries of S are referred to as the singular values
• VT of dimension n x n
X = U * S * V T

Looking at the preceding formula, it appears that we have not reduced the dimensionality of the problem at all, as by multiplying U, S, and V, we reconstruct the original matrix. In practice, the truncated SVD is usually computed. That is, only the top k singular values, which represent the most variation in the data, are kept, while the rest are discarded. The formula to reconstruct X based on the component matrices is then approximate, and is given as follows:

X ~ Uk * ...

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