Zeta Hull Pursuits: Learning Nonconvex Data Hulls
Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured data information into column sampling, research efforts were devoted to the cases where data points are fitted with clusters, simplices, or general convex hulls. This paper aims to study nonconvex hull learning which has rarely been investigated in the literature. In order to learn data-adaptive nonconvex hulls, a novel approach is proposed based on a graph-theoretic measure that leverages graph cycles to characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform structured column sampling. The process of pursuing a Zeta hull involves the computation of matrix inverse. To accelerate the matrix inversion computation and reduce its space complexity as well, we exploit a low-rank approximation to the graph adjacency matrix by using an efficient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can achieve state-of-the-art accuracy in text and image classification tasks.
"Zeta Hull Pursuits: Learning Non-convex Data Hulls" addresses the problem of column sampling in datasets by finding points that do not alter significantly the Ihara zeta function of a graph (presumably constructed via a kernel). They provide establish some basic facts about the zeta function and the effect that removing a vertex has on it.  These they use to provide a fast greedy algorithm for selecting 'outlying' vertices, and a faster version using anchors.  They demonstrate on some standard datasets that using these points can improve classification accuracy.    The paper is fairly well written, with good organization and only minor typos.  I have checked the proofs and did not find any mistakes.  I am not sure that the method is motivated enough to merit acceptance, while it is in fact a novel idea.The authors elude to the fact that this works for summarization of the data points without assuming that they are preserving the convex hull or enscribing ellipsoids.  I believe that this is the primary merit of this paper, that they are able to find representative points in a combinatorial fashion.  With that said, it is not exactly clear what is the motivation behind the use of the zeta function.  If they had clearly argued for the use of points that minimize their epsilon objective, perhaps via the learning of non-convex hulls (which was never clearly defined) then my review might be better.Also, I don't think that the Theorems are significant enough technical contributions to merit much attention.  Theorem 1 reiterates an established fact regarding the a determinant form of the Zeta function (generalized to weighted graphs).  Theorems 2 and 3 are straightforward results from matrix algebra and the SVD.