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1-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study 2-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for 2-parameter persistence modules. We show that this vector representation is 1-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of 1-parameter and 2-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at https://github.com/soham0209/mpml-graph. 

ABSTRACT Along their path from source to observer, gravitational waves may be gravitationally lensed by massive objects leading to distortion in the signals. Searches for these distortions amongst the observed signals from the current detector network have already been carried out, though there have as yet been no confident detections. However, predictions of the observation rate of lensing suggest detection in the future is a realistic possibility. Therefore, preparations need to be made to thoroughly investigate the candidate lensed signals. In this work, we present some follow-up analyses that could be applied to assess the significance of such events and ascertain what information may be extracted about the lens-source system by applying these analyses to a number of O3 candidate events, even if these signals did not yield a high significance for any of the lensing hypotheses. These analyses cover the strong lensing, millilensing, and microlensing regimes. Applying these additional analyses does not lead to any additional evidence for lensing in the candidates that have been examined. However, it does provide important insight into potential avenues to deal with high-significance candidates in future observations. 

https://jmlr.org/papers/ v22/18-780 

Ecological relationships between bacteria mediate the services that gut microbiomes provide to their hosts. Knowing the overall direction and strength of these relationships within hosts, and their generalizability across hosts, is essential to learn how microbial ecology scales up to affect microbiome assembly, dynamics, and host health. Here we gain insight into these patterns by inferring thousands of correlations in bacterial abundance between pairs of gut microbiome taxa from extensive time series data, consisting of 5,534 microbiome profiles from 56 wild baboon hosts over a 13-year period. We model these time series using a statistically robust, multinomial logistic-normal modeling framework and test the degree to which bacterial abundance correlations are consistent across hosts (i.e., "univeral") or individualized to each host. We also compare these patterns to two publicly available human data sets. We find that baboon gut microbial relationships are largely universal: correlation patterns within each baboon host reflect a mixture of idiosyncratic and shared patterns, but the shared pattern dominates by almost 2-fold. Surprisingly, the strongest and most consistently correlated bacterial pairs across hosts were overwhelmingly positively correlated and typically belonged to the same family - a 3-fold enrichment compared to pairs drawn from the data set as a whole. The bias towards universal, positive bacterial correlations was also apparent in monthly samples from human infants, and bacterial families that had universal relationships in baboons also tended to be universal in human infants. Together, our results advance our understanding of the relationships that shape gut microbial ecosystems, with implications for microbiome personalization, community assembly and stability, and the feasibility of microbiome interventions to improve host health.