# Which variable has to be with respect to for partial derivative in minimization algorithm?

The function has many variables. Which variable has to be with respect to for partial derivative in minimization algorithm? I would like to ask in general but probably, the example would be helpful.

For example from this paper, they choose $x_j$ for doing $dQ'/dx_j$. Why we know it has to be with respect to $x_j$? The paper said...

"We can derive update rules by partial derivation on this function. First, we use a auxiliary function defined as

$$Q(X,\gamma)=- \sum_{j} \frac{1}{2\gamma}|x_{j}|^2+\frac{\gamma}{2}$$

where $γ_j$ is an auxiliary variable. This cost function satisfies the three properties described in 3.1 (about $Q(θnew) ≤ Q+(θnew, φnew) ≤ Q+(θ, φnew) = Q(θ)$) and is equal to the original cost function when $$γ_j = |s_j|$$ When we solve $∂Q'/∂x_j = 0$, the update rule of separated signal $x_j$ is written as

$$x_j = 2\gamma"$$

Is it because $\gamma$ is just auxiliary variable so it is not important? If the function has more than one variables, which one will we choose for partial derivative?

• Hard to tell without the full context, but I would guess that $j$ stands for an arbitrary index, and the statements such as $\frac{\partial Q}{\partial x_j} =0$ are to hold for all $j$. Aug 14, 2017 at 2:50
• @MatthewLeingang I added some more information from paper. And yes, $j$ is like index of vector. Does it help?
– Jan
Aug 14, 2017 at 3:38
• It seems like you're not quoting directly from the paper, but instead substituting your versions of equations (8)–(12). Aug 15, 2017 at 13:43

Contextual clues indicate that the authors mean "for each $j$ between $1$ and $J$" whenever they refer to a symbol $s_j$ or $x_j$.
1. Notice how in some equations the index $j$ is summed over. This makes it a dummy variable in those equations. It would be confusing for $j$ to be a dummy variable in some equations and a fixed index in others.
2. At no point do the authors say "for some $j$", which would indicate a specific index.
3. The first-order conditions for a function of several variables $Q(x_1, \dots, x_J)$ to have a minimum are that $\frac{\partial Q}{\partial x_j} = 0$ for each $j$.
I am not sure why the authors use $\gamma_j$ in some places and $\gamma$ in others (e.g., the caption for Figure 3). Could it be that $J=1$ in those cases, so there is only a single variable?