## Joint Harmonic Functions and Their Supervised Connections

** Mark Vere Culp, Kenneth Joseph Ryan**; 14(117):3721−3752, 2013.

### Abstract

The cluster assumption had a significant impact on the reasoning behind semi-supervised classification methods in graph-based learning. The literature includes numerous applications where harmonic functions provided estimates that conformed to data satisfying this well-known assumption, but the relationship between this assumption and harmonic functions is not as well- understood theoretically. We investigate these matters from the perspective of supervised kernel classification and provide concrete answers to two fundamental questions. (i) Under what conditions do semi-supervised harmonic approaches satisfy this assumption? (ii) If such an assumption is satisfied then why precisely would an observation sacrifice its own supervised estimate in favor of the cluster? First, a harmonic function is guaranteed to assign labels to data in harmony with the cluster assumption if a specific condition on the boundary of the harmonic function is satisfied. Second, it is shown that any harmonic function estimate within the interior is a probability weighted average of supervised estimates, where the weight is focused on supervised kernel estimates near labeled cases. We demonstrate that the uniqueness criterion for harmonic estimators is sensitive when the graph is sparse or the size of the boundary is relatively small. This sets the stage for a third contribution, a new regularized joint harmonic function for semi-supervised learning based on a joint optimization criterion. Mathematical properties of this estimator, such as its uniqueness even when the graph is sparse or the size of the boundary is relatively small, are proven. A main selling point is its ability to operate in circumstances where the cluster assumption may not be fully satisfied on real data by compromising between the purely harmonic and purely supervised estimators. The competitive stature of the new regularized joint harmonic approach is established.

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