E.g. 1, Deriving PCA by minimizing MSE
, 
is the 
-th sample with m dimension. Assume for simplicity that 
has zero mean.
, 
is the 
-th basis vector with m dimension.
, 
is the low dimension representation of .
The optimization problem of PCA is
We can simplify the above problem by using 
and 
, as
Introducing the Lagrange multipliers 
and 
, the optimization problem is equivalent to
where we use 
, and let 
.
Next, we will derive 
.
Therefore 
. Let it be 
, we get
Left multiply the equation by 
and use eq. (2) - and eq. (3) - 
, we get
from which we can see 
is the eigenvalue of 
and 
is the corresponding eigenvector.
Substitute eq. (4) into eq. (1),
therefore 
, 
, ..., 
should be the largest k eigenvalues. 
E.g. 1, 
, where 
is a 
matrix
or
In the above, we haven't used any differential technique, because we haven't defined the derivative of vector-by-matrix 
which could be a 3D tensor. However, in some cases such as 
(w.r.t. 
), the differential technique still works (see this example).
E.g. 2, 
, where 
is a 
matrix and is a matrix
In the above, we haven't used any differential technique, because we haven't defined the derivative of matrix-by-matrix 
which could be a 4D tensor. However, in some cases such as 
, the differential technique still works (see this example). Besides, there is another excellent example of : derivative of SVD - https://arxiv.org/pdf/1509.07838.pdf.