MOTIVATION: Accurately representing biological networks in a low-dimensional space, also known as network embedding, is a critical step in network-based machine learning and is carried out widely using node2vec, an unsupervised method based on biased random walks. However, while many networks, including functional gene interaction networks, are dense, weighted graphs, node2vec is fundamentally limited in its ability to use edge weights during the biased random walk generation process, thus under-using all the information in the network. RESULTS: Here, we present node2vec+, a natural extension of node2vec that accounts for edge weights when calculating walk biases and reduces to node2vec in the cases of unweighted graphs or unbiased walks. Using two synthetic datasets, we empirically show that node2vec+ is more robust to additive noise than node2vec in weighted graphs. Then, using genome-scale functional gene networks to solve a wide range of gene function and disease prediction tasks, we demonstrate the superior performance of node2vec+ over node2vec in the case of weighted graphs. Notably, due to the limited amount of training data in the gene classification tasks, graph neural networks such as GCN and GraphSAGE are outperformed by both node2vec and node2vec+. AVAILABILITY AND IMPLEMENTATION: The data and code are available on GitHub at https://github.com/krishnanlab/node2vecplus_benchmarks. All additional data underlying this article are available on Zenodo at https://doi.org/10.5281/zenodo.7007164. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.

The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website, and traffic networks, the vast majority of this research focuses on undirected graphs. In this paper, we propose MagNet, a GNN for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian. This matrix encodes undirected geometric structure in the magnitude of its entries and directional information in their phase. A charge parameter attunes spectral information to variation among directed cycles. We apply our network to a variety of directed graph node classification and link prediction tasks showing that MagNet performs well on all tasks and that its performance exceeds all other methods on a majority of such tasks. The underlying principles of MagNet are such that it can be adapted to other GNN architectures.

The wavelet scattering transform is an invariant and stable signal representation suitable for many signal processing and machine learning applications. We present the Kymatio software package, an easy-to-use, high-performance Python implementation of the scattering transform in 1D, 2D, and 3D that is compatible with modern deep learning frameworks, including PyTorch and TensorFlow/Keras. The transforms are implemented on both CPUs and GPUs, the latter offering a significant speedup over the former. The package also has a small memory footprint. Source code, documentation, and examples are available under a BSD license at https://www.kymat.io.

Big data often has emergent structure that exists at multiple levels of abstraction, which are useful for characterizing complex interactions and dynamics of the observations. Here, we consider multiple levels of abstraction via a multiresolution geometry of data points at different granularities. To construct this geometry we define a time-inhomogemeous diffusion process that effectively condenses data points together to uncover nested groupings at larger and larger granularities. This inhomogeneous process creates a deep cascade of intrinsic low pass filters on the data affinity graph that are applied in sequence to gradually eliminate local variability while adjusting the learned data geometry to increasingly coarser resolutions. We provide visualizations to exhibit our method as a "continuously-hierarchical" clustering with directions of eliminated variation highlighted at each step. The utility of our algorithm is demonstrated via neuronal data condensation, where the constructed multiresolution data geometry uncovers the organization, grouping, and connectivity between neurons.

We explore the generalization of scattering transforms from traditional (e.g., image or audio) signals to graph data, analogous to the generalization of ConvNets in geometric deep learning, and the utility of extracted graph features in graph data analysis. In particular, we focus on the capacity of these features to retain informative variability and relations in the data (e.g., between individual graphs, or in aggregate), while relating our construction to previous theoretical results that establish the stability of similar transforms to families of graph deformations. We demonstrate the application of our geometric scattering features in graph classification of social network data, and in data exploration of biochemistry data.

We introduce a solid harmonic wavelet scattering representation, invariant to rigid motion and stable to deformations, for regression and classification of 2D and 3D signals. Solid harmonic wavelets are computed by multiplying solid harmonic functions with Gaussian windows dilated at different scales. Invariant scattering coefficients are obtained by cascading such wavelet transforms with the complex modulus nonlinearity. We study an application of solid harmonic scattering invariants to the estimation of quantum molecular energies, which are also invariant to rigid motion and stable with respect to deformations. A multilinear regression over scattering invariants provides close to state of the art results over small and large databases of organic molecules.

A novel algorithm for endmember extraction is presented. The approach follows the linear mixture model for hyperspectral data. Endmembers are identified based on sparsity consideration. Theoretical and experimental results suggest the potential of the method.

We propose a new kernel and frame based dimension reducing algorithm by exploiting the synergy between endmembers and kernel based classification schemes. This synergy becomes available by means of utilizing the notions of frames and frame representations.