A quantum Jensen-Shannon graph kernel using discrete-time quantum walks

In this paper, we develop a new graph kernel by using the quantum Jensen-Shannon divergence and the discrete-time quantum walk. To this end, we commence by performing a discrete-time quantum walk to compute a density matrix over each graph being compared. For a pair of graphs, we compare the mixed quantum states represented by their density matrices using the quantum Jensen-Shannon divergence. With the density matrices for a pair of graphs to hand, the quantum graph kernel between the pair of graphs is defined by exponentiating the negative quantum Jensen-Shannon divergence between the graph density matrices. We evaluate the performance of our kernel on several standard graph datasets, and demonstrate the effectiveness of the new kernel

kernInt : A Kernel Framework for Integrating Supervised and Unsupervised Analyses in Spatio-Temporal Metagenomic Datasets

The advent of next-generation sequencing technologies allowed relative quantification of microbiome communities and their spatial and temporal variation. In recent years, supervised learning (i.e., prediction of a phenotype of interest) from taxonomic abundances has become increasingly common in the microbiome field. However, a gap exists between supervised and classical unsupervised analyses, based on computing ecological dissimilarities for visualization or clustering. Despite this, both approaches face common challenges, like the compositional nature of next-generation sequencing data or the integration of the spatial and temporal dimensions. Here we propose a kernel framework to place on a common ground the unsupervised and supervised microbiome analyses, including the retrieval of microbial signatures (taxa importances). We define two compositional kernels (Aitchison-RBF and compositional linear) and discuss how to transform non-compositional beta-dissimilarity measures into kernels. Spatial data is integrated with multiple kernel learning, while longitudinal data is evaluated by specific kernels. We illustrate our framework through a single point soil dataset, a human dataset with a spatial component, and a previously unpublished longitudinal dataset concerning pig production. The proposed framework and the case studies are freely available in the kernInt package at https://github.com/elies-ramon/kernInt

Fast depth-based subgraph kernels for unattributed graphs

In this paper, we investigate two fast subgraph kernels based on a depth-based representation of graph-structure. Both methods gauge depth information through a family of K-layer expansion subgraphs rooted at a vertex [1]. The first method commences by computing a centroid-based complexity trace for each graph, using a depth-based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices [2]. This subgraph kernel is computed by measuring the Jensen-Shannon divergence between centroid-based complexity entropy traces. The second method, on the other hand, computes a depth-based representation around each vertex in turn. The corresponding subgraph kernel is computed using isomorphisms tests to compare the depth-based representation rooted at each vertex in turn. For graphs with n vertices, the time complexities for the two new kernels are O(n 2) and O(n 3), in contrast to O(n 6) for the classic Gärtner graph kernel [3]. Key to achieving this efficiency is that we compute the required Shannon entropy of the random walk for our kernels with O(n 2) operations. This computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state-of-the-art graph kernels. Experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels