Node embedding is a powerful approach for representing the structural role of each node in a graph. $textit{Node2vec}$ is a widely used method for node embedding that works by exploring the local neighborhoods via biased random walks on the graph. Ho
wever, $textit{node2vec}$ does not consider edge weights when computing walk biases. This intrinsic limitation prevents $textit{node2vec}$ from leveraging all the information in weighted graphs and, in turn, limits its application to many real-world networks that are weighted and dense. Here, we naturally extend $textit{node2vec}$ to $textit{node2vec+}$ in a way that accounts for edge weights when calculating walk biases, but which reduces to $textit{node2vec}$ in the cases of unweighted graphs or unbiased walks. We empirically show that $textit{node2vec+}$ is more robust to additive noise than $textit{node2vec}$ in weighted graphs using two synthetic datasets. We also demonstrate that $textit{node2vec+}$ significantly outperforms $textit{node2vec}$ on a commonly benchmarked multi-label dataset (Wikipedia). Furthermore, we test $textit{node2vec+}$ against GCN and GraphSAGE using various challenging gene classification tasks on two protein-protein interaction networks. Despite some clear advantages of GCN and GraphSAGE, they show comparable performance with $textit{node2vec+}$. Finally, $textit{node2vec+}$ can be used as a general approach for generating biased random walks, benefiting all existing methods built on top of $textit{node2vec}$. $textit{Node2vec+}$ is implemented as part of $texttt{PecanPy}$, which is available at https://github.com/krishnanlab/PecanPy .
This article discusses a generalization of the 1-dimensional multi-reference alignment problem. The goal is to recover a hidden signal from many noisy observations, where each noisy observation includes a random translation and random dilation of the
hidden signal, as well as high additive noise. We propose a method that recovers the power spectrum of the hidden signal by applying a data-driven, nonlinear unbiasing procedure, and thus the hidden signal is obtained up to an unknown phase. An unbiased estimator of the power spectrum is defined, whose error depends on the sample size and noise levels, and we precisely quantify the convergence rate of the proposed estimator. The unbiasing procedure relies on knowledge of the dilation distribution, and we implement an optimization procedure to learn the dilation variance when this parameter is unknown. Our theoretical work is supported by extensive numerical experiments on a wide range of signals.
We present a machine learning model for the analysis of randomly generated discrete signals, which we model as the points of a homogeneous or inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by S. Mallat
, our construction is a mathematical model of convolutional neural networks and is naturally invariant to translations and reflections. Our model replaces wavelets with Gabor-type measurements and therefore decouples the roles of scale and frequency. We show that, with suitably chosen nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes based on the first and second moments of the arrival intensity $lambda(t)$, as well as the absolute moments of the charges associated to each point.
