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I have some data that exist on a graph $G=(V,E)$. The vertices belong to one of two classes $y_i\in\{-1,1\}$, and I'm interested in training an SVM to distinguish between the two classes. One appropriate kernel for this is the diffusion kernel, $K=\exp(-\beta L),$ where $L$ is the Laplacian of $G$ and $\beta$ is a tuning parameter.

Tuning the SVM requires selection of hyperparameters, so I have to tune $\theta=(\beta, C).$ Conventionally, we use cross-validation for this problem, but that doesn't seem appropriate here, since omitting a vertex $i$ from $G$ changes the whole graph, possibly even increasing the number of connected components! If the number of connected components changes, some vertices become unreachable from others, and we are faced with a very different set of data than we began with. That is, not only are we missing the removed vertex $i$, but we're also missing information about all other vertices $j$ in the graph which were adjacent to that vertex.

The basic notion of cross-validation is that we would like to approximate how the model will perform when it is presented with new data. In standard problems, the omission of some of your data for testing does not change the values of the remaining training data. However, in the case of graph data, it's not clear what it means for the model to see "new" data in the CV setting. Omitting vertices or edges has the potential to entirely change the data. For example, imagine a graph $S=(V_S,E_S)$ which is a $k$-star graph, in which one vertex has $k$ edges to $k$ vertices, and all other vertices have 1 edge. Omitting the central vertex to construct the training data $S^*$ will entirely disconnect the graph, and the kernel matrix will be diagonal! But of course, it will be possible to train a model on this training data provided in $S^*$. What is less clear is what it means to then test the out-of-sample performance of the resulting model. Does one recompute the kernel matrix for $S$, and provide that to make predictions?

Or, alternatively, does one begin by computing the kernel matrix of $S$ in its entirety and omit rows and columns as necessary to produce the kernel matrix used for estimating the SVM? This presents its own conceptual problems, since the inclusion of the central node in $S$ means that every vertex is reachable from every other vertex, and the kernel matrix is dense. Will this inclusion mean that there is information leakage across folds, and bias the cross-validation output? On the one hand, data about the omitted central nodes is still present, as the omitted central node makes the graph connected. On the other hand, we know nothing about the labels $y$ of the omitted nodes, so we may be comfortable that we are getting reasonably unbiased out-of-sample estimates from performing CV in this manner.

How does one select hyperparameters for problems of this type? Is CV imperfect-but-acceptable, or do we need specialized methods? Is hyperparameter tuning even possible at all in my context?

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  • $\begingroup$ Look for out-of-sample extension for spectral methods. I applied some of the techniques for image classification in my thesis (looking back now, i've would done that differently though). The results were interesting but the models themselves were pretty brittle and not easy to tune. $\endgroup$ – Vladislavs Dovgalecs Sep 11 '15 at 21:56
  • $\begingroup$ @xeon Any recommendations of where to start looking at this literature? $\endgroup$ – Sycorax Sep 13 '15 at 4:00
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Disclaimer: I'm not very familiar with graph kernels, so this answer might be based on wrong assumptions. I agree that omitting vertices while computing the kernel matrix is suboptimal. That said, I'm not sure that cross-validation is necessarily problematic. Is your learning context transduction or induction?

Overall, I am not convinced that computing the kernel matrix for a given $\beta$ based on all data (i.e., both train and test) necessarily creates an information leak. If computing the kernel based on all data turns out to be okay, you can then train models in a typical cv-setup, using the relevant blocks of the (precomputed) full kernel matrix for training/testing.

This approach would enable you to jointly optimize $\beta$ and $C$, for example via libraries like Optunity, where $\beta$ is used to compute the kernel based on all data and $C$ is used to train models on the training folds exclusively.

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  • $\begingroup$ Marc, thanks for your attention. My problem is inductive. My instinct is that your are correct, and that we compute the full kernel matrix for all of the data and then omit rows and columns as needed to produce the CV'd kernel. Would you happen to be aware of any references to this effect? $\endgroup$ – Sycorax Sep 14 '15 at 15:18
  • $\begingroup$ Not immediately, but I have some colleagues working on comparable problems (kernel spectral clustering) that I can ask. Maybe they have references and/or better ideas. Will update when I know more. $\endgroup$ – Marc Claesen Sep 14 '15 at 15:24

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