0
$\begingroup$

I have some function $f(\boldsymbol{X}): \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$:

$$f(\boldsymbol{X}) = \text{Tr}(S\boldsymbol{X}) - \log\det \boldsymbol{X}$$

If I now want to minimize that function with respect to $\boldsymbol{X}$, I can write its derivative with respect to the matrix $\boldsymbol{X}$:

$$\frac{df}{d\boldsymbol{X}} = S^T - X^{-T}$$

Setting it equal to $0$ we have:

$$0 = S^T - X^{-T} \Rightarrow X = S^{-1}$$

in some "appropriate" sense.

I can also think of that function as:

$$f(c_1,c_2,\ldots,c_n): \underbrace{\mathbb{R}^{m}\times\mathbb{R}^m\times\ldots\times \mathbb{R}^m}_{n \; \text{many times}} \rightarrow \mathbb{R}$$

That is, as a function of the columns.


Quick aside: If say $\boldsymbol{X}$ was triangular $(m=n)$, we could capture this as:

$$f(c_1,c_2,\ldots,c_n): \mathbb{R}^{1}\times\mathbb{R}^2\times\ldots\times \mathbb{R}^{n-1}\times \mathbb{R}^n \rightarrow \mathbb{R}$$


My Problem: I am having trouble finding the equivalent of the $\frac{df}{d\boldsymbol{X}}$ object in the new way of thinking of the problem in terms of a function of its columns (or rows).


My thoughts: In my mind, the derivative with respect to $\boldsymbol{X}$ would have to correspond to the total derivative/differential of the new function. But that would mean something like:

$$df = \frac{\partial f}{\partial c_1} dc_1 + \cdots + \frac{\partial d}{\partial c_n} d c_n$$

But here, its not clear to me what that object means, let alone what the $d c_i$ objects mean. Also, I don't know how to actually take the total derivative/differential to end up something I can easily "set equal to $0$ and solve for"


Note: I think this question is related to my previous optimization question. I'll probably delete that one, as I feel this one its a clearer way of expressing my confusion.

$\endgroup$
  • 1
    $\begingroup$ You missed a transpose in your gradient derivation. It should actually be $$\frac{\partial f}{\partial X}=S^T-X^{-T}$$ Setting it to zero and solving for $X$ yields $$X=S^{-1}$$ In your alternative view of the problem, the vectors are merely the columns of the matrix from your original formulation, i.e. $$c_k=Xe_k$$ Lastly, what you refer to as the "total derivative" is called the "differential". $\endgroup$ – greg Feb 2 '17 at 3:34
  • $\begingroup$ Thanks I've made the corrections you suggested $\endgroup$ – user79950 Feb 2 '17 at 17:53
2
$\begingroup$

The question is to be how to calculate the gradient of $f$ with respect to the $k^{th}$ column of $X$
$$\eqalign{ c_k &= X\cdot e_k \cr }$$

That's easy, it's the $k^{th}$ column of $\frac{\partial f}{\partial X}$, which is given by $$\eqalign{ \frac{\partial f}{\partial c_k} &= \frac{\partial f}{\partial X}\cdot e_k \cr }$$ where $e_k$ is the standard basis vector, whose $k^{th}$ component is equal to $1$ and all other components are equal to $0$.


The differential of the function can be expressed as the sum of the column gradients times the column differentials $$df = \sum_k \,\frac{\partial f}{\partial c_k}\cdot dc_k$$ as you observed.

You can go even further and ask about an expression in terms of the individual elements of $X$, which would be $$df = \sum_j\sum_k \,\frac{\partial f}{\partial X_{jk}}\,\,dX_{jk}$$ This can be written in a very compact form using the double-dot (aka Frobenius) product $$ df = \frac{\partial f}{\partial X}:dX $$ The differential $dX$ in this expression is completely arbitrary. It can consist of all zeros except for one column, or one row, or just the diagonal, or a single element.

$\endgroup$
  • $\begingroup$ Thanks again for the help @greg. Im still not exactly sure what the $d c_k$ actually mean, and how to calculate them. Is there some example you can give me/point me to to help make this clear? (Also, if you're ever free maybe we can join one of the math.se chat rooms to have an informal discussion along these lines?) $\endgroup$ – user79950 Feb 2 '17 at 16:51
  • $\begingroup$ What is the column differential? In this new setting, how can I "set it equal to 0" and arrive at something like $X=S^{-1}$ in terms of columns? $\endgroup$ – user79950 Feb 2 '17 at 17:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.