Cs 229 Final Project Reduced Rank Regression
نویسنده
چکیده
where A is an unknown p× n matrix of coefficients and E is an unobserved m× n random noise matrix with independent mean zero and variance σ. We want to find an estimate  such that ||Y −XÂ|| is small. If we use standard least square estimation directly to estimate A in (1.1) without adding any constraints, then it is just the same as regressing each response on the predictors separately. In this way, we actually ignore the possibility that the responses may be correlated among themselves. Besides, when there are many attributes (p is large) and many different kinds of responses (n is large), the number of unknowns can be larger than the sample size m. We may then need much more effort to collect more samples to increase m or the least square method simply cannot be applied. To address this problem, one popular way to handle it is reduced rank regression. Let r(M) be the rank of a matrix M . If we expect r(A) = r < min (p, n) or A can be well approximated by a low rank matrix, we can write A as a product of two matrices with rank r, see [1]. That is A = BrCr, Br ∈ Rp×r and Cr ∈ Rr×n which have total r× (n+ p) unknowns needed to be estimated. It can be much less than m if r(A) is very small. The model (1.1) then become
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