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My goal here is to make some notes on a specific approach to PCA that we could extend to sparse PCA.
In brief let \(X\) be an \(n \times p\) data matrix, and consider seeking matrices \(L\) (\(n \times k\)) and \(F\) (\(p \times k\)) to \[\text{min}_{L,F} ||X - LF'||_2^2 \text{ subject to } L'L=I_k\] Here \(||A||_2^2\) denotes the squared Frobenius norm of \(A\) (the sum of squared entries of the matrix).
Obviously one can write the above as \[\text{min}_{F} \text{min}_{L:L'L=I_k} ||X - LF'||_2^2 \] We will show that the inner part of this minimization:
\[h(F; X):= \text{min}_{L:L'L=I_k} ||X - LF'||_2^2\] depends on \(F\) only through \(FF'\) and depends on \(X\) only through \(X'X\). Furthermore \(h(F;X)=0\) if \(FF'=X'X\). This demonstrates that the above formulation of PCA is finding an \(F\) such that \(FF'\) approximates \(X'X\).
For a matrix \(A=(a_{ij})\) let \(A'\) denote its transpose, and \(||A||_2^2\) denote the squared Frobenius norm, \[||A||_2^2 = \sum a_{ij}^2 = \tr(A'A)\] If \(A\) has svd \(A=UDV'\) then let \(\sigma(A)=diag(D)\) denote the vector of singular values, and \(\text{Polar}(A):=UV'\) (which is the \(U\) part of the polar decomposition of \(A=UP\)). If \(A\) is psd then let \(\sqrt{A}\) denote the matrix \(A=UD^{0.5}V'\) (\(=UD^{0.5}U'\) since \(U=V\) for psd \(A\)). Thus if \(A=UDV'\) then \(\sqrt(AA')= UDU'\).
Let \(||A||_*\) denote the trace norm (nuclear norm) of \(A\). \[||A||_* = \sum_i \sigma_i(A) = \sum_i \sigma_i(\sqrt{A'A}) = tr(\sqrt{A'A}) = ||\sqrt{A'A}||_*\] Note that the trace norm is unitarily invariant (eg see https://nhigham.com/2021/02/02/what-is-a-unitarily-invariant-norm/). That is, if \(U\) and \(V\) are unitary (meaning \(UU'=I\), and \(U'U=I\)) then \[||U'AV||_* = ||A||_*.\]
Recall that we defined \[h(F; X) := min_{L:L'L=I_k} ||X - LF'||_2^2\]
We state two key results. First, the minimum over \(L\) is attained by \(\hat{L}=Polar(XF)\). Second \(h(F;X)\) depends on \(F\) only through \(FF'\) and on \(X\) only through \(X'X\). Also if \(FF'=X'X\) then \(h(F;X)=0\). So \(h\) is a measure of difference between \(FF'\) and \(X'X\).
The first result follows directly from Theorem 4 in Zou et al ("Sparse Principal Components analysis), so we focus on the second.
First note that if \(XF=UDV'\) then \(\hat{L}=Polar(XF)=UV'\) so \[\hat{L}F'X' = UV'VDU'=UDU'= \sqrt{XFF'X'}\]
Also for any \(L'L=I\) we have \[||X - LF'||_2^2 = \text{Tr}{(X-LF')'(X-LF')} = \text{Tr}(X'X - 2X'LF' + FL'LF') = \text{Tr}(X'X - 2LF'X' + FF')\]
Putting this together: \[||X - \hat{L}F'||_2^2 = \text{Tr}(X'X - 2\sqrt{XFF'X'} + FF')\]
Note that \[Tr(\sqrt{XFF'X'}) = ||XF||_* = ||\sqrt{X'X}\sqrt{FF'}||_*\] This can be proved by the unitary property of \(||.||_*\).
Note: initially I mistakenly thought that this could be further simplified to \(||\sqrt{X'X}\sqrt{FF'}||_* = \text{Tr}(\sqrt{X'X}\sqrt{FF'})\). However, this is not true because \(\sqrt{X'X}\sqrt{FF'}\) is generally not SPD (and indeed, not symmetric).
Here was some code I used to make some numeric checks of some of these results.
n = 10
p = 5
k= 6
X = matrix(rnorm(n*p),nrow=n,ncol=p)
F = matrix(rnorm(k*p), nrow= p, ncol=k)
XF = X %*% F
norm = function(A){sum(svd(A)$d)}
sqrt_AtA = function(A){A.e = eigen(t(A)%*%A); d = A.e$values; v = A.e$vectors; return(v %*% diag(ifelse(d>0,sqrt(d),0)) %*% t(v))}
tr = function(A){return(sum(diag(A)))}
norm(X %*% F) - norm(sqrt_AtA(X) %*% sqrt_AtA(t(F)))[1] -2.4869e-14X.svd = svd(X)
X.u = X.svd$u
X.d = X.svd$d
X.v = X.svd$v
norm(X)[1] 11.8675norm(X.u %*% diag(X.d) %*% t(X.v))[1] 11.8675norm(diag(X.d) %*% t(X.v))[1] 11.8675norm(X.v %*% diag(X.d) %*% t(X.v))[1] 11.8675tr(X.v %*% diag(X.d) %*% t(X.v))[1] 11.8675F.svd = svd(F)
F.u = F.svd$u
F.d = F.svd$d
F.v = F.svd$v
norm(X %*% F)[1] 24.88342norm(X.u %*% diag(X.d) %*% t(X.v) %*% F.u %*% diag(F.d) %*% t(F.v))[1] 24.88342norm(diag(X.d) %*% t(X.v) %*% F.u %*% diag(F.d))[1] 24.88342norm(X.v %*% diag(X.d) %*% t(X.v) %*% F.u %*% diag(F.d) %*% t(F.u))[1] 24.88342
sessionInfo()R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Big Sur 11.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
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