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I am reading an article entitled "Diffuse Interface Models on Graphs for Classification of High Dimensional Data." Seems like the idea is to use the Ginzburg-Landau functional, in association with graph partitioning methods to apply classification on high-dimensional data.

The Ginzburg-Landau functional looks like: $$ GL(u) = \frac{\epsilon}{2}\int |\nabla u|^2dx + \frac{1}{\epsilon}\int W(u)dx $$

Where the function $W(u)$ is a double well potential like $\frac{1}{4}(u^2 - 1)^2$. So the first term in the functional is just the familiar Dirichlet energy which applies smoothness on the solution. But the second term is the double well potential which looks like a $w$ with wells corresponding to -1 and +1.

The paper makes a couple of claims.

  1. That the GL functional can be used instead of the Total Variation norm, which is $\int |\nabla u|dx$.

  2. That minimizing the functional aids in some image processing applications like segmentation.

So I was trying to understand both of these claims.

First, why can the GL function be a replacement for the Total Variation norm, when the total variation norm has no double well? The paper says something about Gamma convergence, but I was not clear on what that meant.

Second. can someone explain the intuition or motivation behind using the GL functional for this type of classification. I mean I understand the benefit of the Dirichlet energy in applications like image denoising and such. The double well potential $W(u)$ is not clear. This is not a difference operator like the gradient. So the function $W(u)$ will just try and adjust the interface between the two wells so that pixels fall at lower energy points either in the -1 or +1 wells. But I don't understand what this kind of separation or partitioning does? Is the resulting GL functional used to calculate the edge weights in a graph partitioning scheme, like spectral clustering?

Any insights would be appreciated.

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I would rely on reasoning from PDE and imaging, since it is easier to visualize. However, I believe with some adaptations it holds true for graphs.

GL potential leads to 2 things: coarsening and sharp interface. TVN usually does coarsening and smoothing.

First, why can the GL function be a replacement for the Total Variation norm, when the total variation norm has no double well?

I wouldn't say it's a replacement, because they do things differently. Double well helps in keeping the interface sharp. In all 3 cases below, $\int|\nabla u|$ is the same, so TVN doesn't mind shallow slopes.

enter image description here

On the contrary, double well term in GL will try to thin out shallow slopes, which leads to nicely formed clusters

enter image description here

Second. can someone explain the intuition or motivation behind using the GL functional for this type of classification.

I am not an expert, so take this with a grain of salt. I believe one would use GL in types of problems where clusters are very intricate, so TVN or similar low order approaches tend to smooth everything out.

You can think of double well potential as a “rounding” operator. Is it slighlty larger than zero? — It's 1 then. Is it a little bit negative? — It's -1 now. With this operator alone, you would get just a black and white grained image (infinitely sharp interfaces). But the addition of laplacian term smoothes this out and coarses, while keeping the interfaces sharp enough (see the pic above)

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  • $\begingroup$ Vasily, this is very very helpful. I understand the intuition now. So this explains why the GL functional--or similar functionals--are called edge preserving. So the data or image is smoothed but in a way that keeps distinct edges between grouping. Haha, in the papers there is a lot of confusing terms like "non-local" smoothing, etc., but your explanation finally makes sense. Thank you again. $\endgroup$ – krishnab Sep 12 at 2:37

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