We present a geometric formulation of the Multiple Kernel Learning (MKL) problem. To do so, we reinterpret the problem of learning kernel weights as searching for a kernel that maximizes the minimum (kernel) distance between two convex polytopes. This interpretation combined with additional structural insights from our geometric formulation allows us to reduce the MKL problem to a simple optimization routine that yields provable convergence as well as quality guarantees. As a result our method scales efficiently to much larger data sets than most prior methods can handle. Empirical evaluation on eleven datasets shows that we are significantly faster and even compare favorably with an uniform unweighted combination of kernels.

In this paper, we present a method to attach affinity scores to the implicit labels of individual points in a clustering. The affinity scores capture the confidence level of the cluster that claims to "own" the point. We demonstrate that these scores accurately capture the quality of the label assigned to the point. We also show further applications of these scores to estimate global measures of clustering quality, as well as accelerate clustering algorithms by orders of magnitude using active selection based on affinity. This method is very general and applies to clusterings derived from any geometric source. It lends itself to easy visualization and can prove useful as part of an interactive visual analytics framework. It is also efficient: assigning an affinity score to a point depends only polynomially on the number of clusters and is independent both of the size and dimensionality of the data. It is based on techniques from the theory of interpolation, coupled with sampling and estimation algorithms from high dimensional computational geometry.

This paper proposes a new distance metric between clusterings that incorporates information about the spatial distribution of points and clusters. Our approach builds on the idea of a Hilbert space-based representation of clusters as a combination of the representations of their constituent points. We use this representation and the underlying metric to design a spatially-aware consensus clustering procedure. This consensus procedure is implemented via a novel reduction to Euclidean clustering, and is both simple and efficient. All of our results apply to both soft and hard clusterings. We accompany these algorithms with a detailed experimental evaluation that demonstrates the efficiency and quality of our techniques.