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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?

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  • $\begingroup$ 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 $j$. $\endgroup$ Aug 14, 2017 at 2:50
  • $\begingroup$ @MatthewLeingang I added some more information from paper. And yes, $j$ is like index of vector. Does it help? $\endgroup$
    – Jan
    Aug 14, 2017 at 3:38
  • $\begingroup$ It seems like you're not quoting directly from the paper, but instead substituting your versions of equations (8)–(12). $\endgroup$ Aug 15, 2017 at 13:43

1 Answer 1

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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?

It seems to me that these ambiguities should have been caught by an editor. You might get full clarity by emailing one of the authors.

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