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Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a set of merge trees using techniques from matrix sketching. That is, given a large set T of merge trees, we would like to find a much smaller set of basis trees S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be approximated via matrix sketching techniques such as principal component analysis and column subset selection. However, until now, there has not been any work on sketching a set of merge trees. We develop a framework for sketching a set of merge trees that combines matrix sketching with tools from optimal transport. In particular, we vectorize a set of merge trees into high-dimensional vectors while preserving their structures and structural relations. We demonstrate the applications of our framework in sketching merge trees that arise from time-varying scientific simulations. Specifically, our framework obtains a set of basis trees as representatives that capture the “modes” of physical phenomena for downstream analysis and visualization. 

Abstract PremiseThe selection ofArabidopsisas a model organism played a pivotal role in advancing genomic science. The competing frameworks to select an agricultural‐ or ecological‐based model species were rejected, in favor of building knowledge in a species that would facilitate genome‐enabled research. MethodsHere, we examine the ability of models based onArabidopsisgene expression data to predict tissue identity in other flowering plants. Comparing different machine learning algorithms, models trained and tested onArabidopsisdata achieved near perfect precision and recall values, whereas when tissue identity is predicted across the flowering plants using models trained onArabidopsisdata, precision values range from 0.69 to 0.74 and recall from 0.54 to 0.64. ResultsThe identity of belowground tissue can be predicted more accurately than other tissue types, and the ability to predict tissue identity is not correlated with phylogenetic distance fromArabidopsis.k‐nearest neighbors is the most successful algorithm, suggesting that gene expression signatures, rather than marker genes, are more valuable to create models for tissue and cell type prediction in plants. DiscussionOur data‐driven results highlight that the assertion that knowledge fromArabidopsisis translatable to other plants is not always true. Considering the current landscape of abundant sequencing data, we should reevaluate the scientific emphasis onArabidopsisand prioritize plant diversity. 

Since they emerged approximately 125 million years ago, flowering plants have evolved to dominate the terrestrial landscape and survive in the most inhospitable environments on earth. At their core, these adaptations have been shaped by changes in numerous, interconnected pathways and genes that collectively give rise to emergent biological phenomena. Linking gene expression to morphological outcomes remains a grand challenge in biology, and new approaches are needed to begin to address this gap. Here, we implemented topological data analysis (TDA) to summarize the high dimensionality and noisiness of gene expression data using lens functions that delineate plant tissue and stress responses. Using this framework, we created a topological representation of the shape of gene expression across plant evolution, development, and environment for the phylogenetically diverse flowering plants. The TDA-based Mapper graphs form a well-defined gradient of tissues from leaves to seeds, or from healthy to stressed samples, depending on the lens function. This suggests that there are distinct and conserved expression patterns across angiosperms that delineate different tissue types or responses to biotic and abiotic stresses. Genes that correlate with the tissue lens function are enriched in central processes such as photosynthetic, growth and development, housekeeping, or stress responses. Together, our results highlight the power of TDA for analyzing complex biological data and reveal a core expression backbone that defines plant form and function. 

Abstract Deep neural networks such as GoogLeNet, ResNet, and BERT have achieved impressive performance in tasks such as image and text classification. To understand how such performance is achieved, we probe a trained deep neural network by studying neuron activations, i.e.combinations of neuron firings, at various layers of the network in response to a particular input. With a large number of inputs, we aim to obtain a global view of what neurons detect by studying their activations. In particular, we develop visualizations that show the shape of the activation space, the organizational principle behind neuron activations, and the relationships of these activations within a layer. Applying tools from topological data analysis, we presentTopoAct, a visual exploration system to study topological summaries of activation vectors. We present exploration scenarios usingTopoActthat provide valuable insights into learned representations of neural networks. We expectTopoActto give a topological perspective that enriches the current toolbox of neural network analysis, and to provide a basis for network architecture diagnosis and data anomaly detection.