# The phrase “coordinate-wise” and its meaning

I was reading through a paper, Learning Fast Approximations of Sparse Coding (LeCunn, ICML 2010) and ran across a phrase I'm not terribly familiar with. Apologetically, I quote with very little context (though I'll link to a copy of the paper):

[a function described in the paper] is a coordinate-wise shrinking function (...)

[Postscript version (ps.gz)][1], and a [google-created HTML version][2].

The question is, what does coordinate-wise mean preferably in a general sense, not just in the context of this paper.

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I've removed the previous edit, as it doesn't appear that the second part of my question applies to the original question.

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Doesn't it generally mean that the elements are manipulated one-by-one instead of being dealt with as a whole? For example, if you have a point in $\mathbb{R}^2$ such as $(x,y)$, then instead of mapping both coordinates to another one with one operation (e.g. a matrix-vector multiply) you first change the first coordinate then the second coordinate (or the opposite order). – JasonMond Mar 22 '11 at 17:17
pointwise, see Wikipedia – beroal Mar 22 '11 at 17:23
I didn't mean that $(x,y)$ are dependent on each other. I admit, I made a hash of my comment so that it could be misinterpreted. As far as I remember, in the context of optimization, "coordinate wise" means that you are manipulating each coordinate independently with the only criterion being that it improves your objective function. So you're zigzagging instead of taking a direct path. I can't remember where I read this, but in some cases, you can go through your coordinates in any order you want and still hit the optimum. – JasonMond Mar 22 '11 at 20:04
Ah. I see your point. So, if for example I have a scalar function $\sigma$ I want to apply to the elements of a vector/matrix (eg, $\mathbf{Y} = \mathbf{\sigma}(\mathbf{X})$ ), I could say $\mathbf{\sigma}$ is evaluated "coordinate-wise" on elements of $\mathbf{x}$. You should write up your response as an answer so I can accept it. – Brian Vandenberg Mar 22 '11 at 20:17

So, another potential interpretation of that, it sounds like, is you can apply a function $f: \Re^1 \mapsto \Re^1$ coordinate-wise to a matrix or array by applying it element-wise. – Brian Vandenberg Mar 22 '11 at 21:37