Naming the Most Anomalous Cluster in Hilbert Space for Structures with Attribute Information

Abstract

We consider datasets consisting of arbitrarily structured entities (e.g., molecules, sequences, graphs, etc) whose similarity can be assessed with a reproducing ker- nel (or a family thereof). These entities are assumed to additionally have a set of named attributes (e.g.: number_of_atoms, stock_price, etc). These attributes can be used to classify the structured entities in discrete sets (e.g., ‘number_of_atoms < 3’, ‘stock_price ≤ 100’, etc) and can effectively serve as Boolean predicates. Our goal is to use this side-information to provide explain- able kernel-based clustering. To this end, we propose a method which is able to find among all possible entity subsets that can be described as a conjunction of the available predicates either a) the optimal cluster within the Reproducing Kernel Hilbert Space, or b) the most anomalous subset within the same space. Our method works employs combinatorial optimisation via an adaptation of the Maximum-Mean-Discrepancy measure that captures the above intuition. Finally, we propose a criterion to select the optimal one out of a family of kernels in a way that preserves the available side-information. We provide several real world datasets that demonstrate the usefulness of our proposed method.

Brain fingerprinting using EEG graph inference

“…A major paradigm in graph structure learning exploits this notion of smoothness to infer a sparse graph from a given set of observations [11], [21]. Specifically, a graph combinatorial Laplacian matrix can be inferred via the following optimization [21]: where F is an N × M matrix of M graph signals, α is regularization parameter,∥·∥ F denotes the Frobenius norm and 1 = [1, …, 1] T .…”

“…Noting that . replacing the ℓ 2 -norm with a logarithmic barrier, the optimization in (2) can be solved more efficiently via a more general-purpose formulation with respect to the graph’s adjacency matrix [11]: where Z is an N × N matrix with elements Z i,j = ∥ F i ,: − F j ,: ∥ 2 , i.e., Euclidean distance between signal values on electrodes i and j . The first term in (3) enforces the smoothness constraint in similar way as in the first term in (2), which is based on the equivalence trace( F T LF ) = 0.5 ∥ A ◦ Z ∥ 1 , where ◦ is the Hadamard product [19].…”

“…Therefore, FC denoising remedies have been proposed based on principal componentbased reconstruction [9] or eigenspace embedding [10], each of which require the learning of latent space-based FCs from a population of subjects. Here we propose an alternative strategy, that is, to directly infer a sparse graph structure [11] from an individual’s EEG data—thus, an alternative to a functional connectome—using principles from graph signal processing (GSP) [12], [13]. The fundamental objective of the learning strategy is to infer a graph on which brain maps are seen as smooth signals—that is, the majority of signal energy lies at the lower end of the graph spectrum, which is an intrinsic property of functional brain organization supported by prior results in brain GSP [14]–[18].…”

Joint subject-identification and task-decoding from inferred functional brain graphs via a multi-task neural network

“…Let F denote an N x M matrix where each column is an observation, a graph signal, and Z an N x N matrix with elements Z i,j = ‖ F i ,: - F j ,: ‖ 2 , i.e., Euclidean distance between signal values on vertices i and j . A graph structure can be inferred from F via the optimization [4]: Where o is the Hadamard product, α and β are regularization parameters, ‖ · ‖ F the Frobenius norm, and 1 = [1,…,1] T . The first term in (2) enforces smoothness by invoking that [4]: Smooth graph signals reside on strongly-connected vertices, thus, the vertices are expected to have smaller distances.…”

“…A graph structure can be inferred from F via the optimization [4]: Where o is the Hadamard product, α and β are regularization parameters, ‖ · ‖ F the Frobenius norm, and 1 = [1,…,1] T . The first term in (2) enforces smoothness by invoking that [4]: Smooth graph signals reside on strongly-connected vertices, thus, the vertices are expected to have smaller distances. The second term enforces degrees to be positive and improves overall connectivity.…”