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The scattering transform is a multilayered wavelet-based architecture that acts as a model of convolutional neural networks. Recently, several works have generalized the scattering transform to graph-structured data. Our work builds on these constructions by introducing windowed and nonwindowed geometric scattering transforms for graphs based on two very general classes wavelets, which are in most cases based on asymmetric matrices. We show that these transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. Therefore, it helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties. 

We propose a new graph neural network (GNN) module, based on relaxations of recently proposed geometric scattering transforms, which consist of a cascade of graph wavelet filters. Our learnable geometric scattering (LEGS) module enables adaptive tuning of the wavelets to encourage band-pass features to emerge in learned representations. The incorporation of our LEGS-module in GNNs enables the learning of longer-range graph relations compared to many popular GNNs, which often rely on encoding graph structure via smoothness or similarity between neighbors. Further, its wavelet priors result in simplified architectures with significantly fewer learned parameters compared to competing GNNs. We demonstrate the predictive performance of LEGS-based networks on graph classification benchmarks, as well as the descriptive quality of their learned features in biochemical graph data exploration tasks. Our results show that LEGS-based networks match or outperforms popular GNNs, as well as the original geometric scattering construction, on many datasets, in particular in biochemical domains, while retaining certain mathematical properties of handcrafted (non-learned) geometric scattering. 

Patient-derived organoids (PDOs) can model personalized therapy responses; however, current screening technologies cannot reveal drug response mechanisms or how tumor microenvironment cells alter therapeutic performance. To address this, we developed a highly multiplexed mass cytometry platform to measure post- translational modification (PTM) signaling, DNA damage, cell-cycle activity, and apoptosis in >2,500 colorectal cancer (CRC) PDOs and cancer-associated fibroblasts (CAFs) in response to clinical therapies at single-cell resolution. To compare patient- and microenvironment-specific drug responses in thousands of single-cell da- tasets, we developed ‘‘Trellis’’—a highly scalable, tree-based treatment effect analysis method. Trellis single- cell screening revealed that on-target cell-cycle blockage and DNA-damage drug effects are common, even in chemorefractory PDOs. However, drug-induced apoptosis is rarer, patient-specific, and aligns with cancer cell PTM signaling. We find that CAFs can regulate PDO plasticity—shifting proliferative colonic stem cells (proCSCs) to slow-cycling revival colonic stem cells (revCSCs) to protect cancer cells from chemotherapy. 

Abstract Due to commonalities in pathophysiology, age-related macular degeneration (AMD) represents a uniquely accessible model to investigate therapies for neurodegenerative diseases, leading us to examine whether pathways of disease progression are shared across neurodegenerative conditions. Here we use single-nucleus RNA sequencing to profile lesions from 11 postmortem human retinas with age-related macular degeneration and 6 control retinas with no history of retinal disease. We create a machine-learning pipeline based on recent advances in data geometry and topology and identify activated glial populations enriched in the early phase of disease. Examining single-cell data from Alzheimer’s disease and progressive multiple sclerosis with our pipeline, we find a similar glial activation profile enriched in the early phase of these neurodegenerative diseases. In late-stage age-related macular degeneration, we identify a microglia-to-astrocyte signaling axis mediated by interleukin-1 β which drives angiogenesis characteristic of disease pathogenesis. We validated this mechanism using in vitro and in vivo assays in mouse, identifying a possible new therapeutic target for AMD and possibly other neurodegenerative conditions. Thus, due to shared glial states, the retina provides a potential system for investigating therapeutic approaches in neurodegenerative diseases. 

In modern relational machine learning it is common to encounter large graphs that arise via interactions or similarities between observations in many domains. Further, in many cases the target entities for analysis are actually signals on such graphs. We propose to compare and organize such datasets of graph signals by using an earth mover’s distance (EMD) with a geodesic cost over the underlying graph. Typically, EMD is computed by optimizing over the cost of transporting one probability distribution to another over an underlying metric space. However, this is inefficient when computing the EMD between many signals. Here, we propose an unbalanced graph EMD that efficiently embeds the unbalanced EMD on an underlying graph into an L1 space, whose metric we call unbalanced diffusion earth mover’s distance (UDEMD). Next, we show how this gives distances between graph signals that are robust to noise. Finally, we apply this to organizing patients based on clinical notes, embedding cells modeled as signals on a gene graph, and organizing genes modeled as signals over a large cell graph. In each case, we show that UDEMD-based embeddings find accurate distances that are highly efficient compared to other methods. 

We propose a method called integrated diffusion for combining multimodal data, gathered via different sensors on the same system, to create a integrated data diffusion operator. As real world data suffers from both local and global noise, we introduce mechanisms to optimally calculate a diffusion operator that reflects the combined information in data by maintaining low frequency eigenvectors of each modality both globally and locally. We show the utility of this integrated operator in denoising and visualizing multimodal toy data as well as multi-omic data generated from blood cells, measuring both gene expression and chromatin accessibility. Our approach better visualizes the geometry of the integrated data and captures known cross-modality associations. More generally, integrated diffusion is broadly applicable to multimodal datasets generated by noisy sensors collected in a variety of fields. 

We propose a new graph neural network (GNN) module, based on relaxations of recently proposed geometric scattering transforms, which consist of a cascade of graph wavelet filters. Our learnable geometric scattering (LEGS) module enables adaptive tuning of the wavelets to encourage band-pass features to emerge in learned representations. The incorporation of our LEGS-module in GNNs enables the learning of longer-range graph relations compared to many popular GNNs, which often rely on encoding graph structure via smoothness or similarity between neighbors. Further, its wavelet priors result in simplified architectures with significantly fewer learned parameters compared to competing GNNs. We demonstrate the predictive performance of LEGS-based networks on graph classification benchmarks, as well as the descriptive quality of their learned features in biochemical graph data exploration tasks.