Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic condensation homology. We use this intrinsic topology as well as the ambient persistent homology of the condensation process to study how the data changes over diffusion time. We demonstrate both types of topological information in well-understood toy examples. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.

We present a method called Manifold Interpolating Optimal-Transport Flow (MIOFlow) that learns stochastic, continuous population dynamics from static snapshot samples taken at sporadic timepoints. MIOFlow combines dynamic models, manifold learning, and optimal transport by training neural ordinary differential equations (Neural ODE) to interpolate between static population snapshots as penalized by optimal transport with manifold ground distance. Further, we ensure that the flow follows the geometry by operating in the latent space of an autoencoder that we call a geodesic autoencoder (GAE). In GAE the latent space distance between points is regularized to match a novel multiscale geodesic distance on the data manifold that we define. We show that this method is superior to normalizing flows, Schr\"odinger bridges and other generative models that are designed to flow from noise to data in terms of interpolating between populations. Theoretically, we link these trajectories with dynamic optimal transport. We evaluate our method on simulated data with bifurcations and merges, as well as scRNA-seq data from embryoid body differentiation, and acute myeloid leukemia treatment.

A major challenge in embedding or visualizing clinical patient data is the heterogeneity of variable types including continuous lab values, categorical diagnostic codes, as well as missing or incomplete data. In particular, in EHR data, some variables are {\em missing not at random (MNAR)} but deliberately not collected and thus are a source of information. For example, lab tests may be deemed necessary for some patients on the basis of suspected diagnosis, but not for others. Here we present the MURAL forest -- an unsupervised random forest for representing data with disparate variable types (e.g., categorical, continuous, MNAR). MURAL forests consist of a set of decision trees where node-splitting variables are chosen at random, such that the marginal entropy of all other variables is minimized by the split. This allows us to also split on MNAR variables and discrete variables in a way that is consistent with the continuous variables. The end goal is to learn the MURAL embedding of patients using average tree distances between those patients. These distances can be fed to nonlinear dimensionality reduction method like PHATE to derive visualizable embeddings. While such methods are ubiquitous in continuous-valued datasets (like single cell RNA-sequencing) they have not been used extensively in mixed variable data. We showcase the use of our method on one artificial and two clinical datasets. We show that using our approach, we can visualize and classify data more accurately than competing approaches. Finally, we show that MURAL can also be used to compare cohorts of patients via the recently proposed tree-sliced Wasserstein distances.

Graph neural networks (GNNs) in general, and graph convolutional networks (GCN) in particular, often rely on low-pass graph filters to incorporate geometric information in the form of local smoothness over neighboring nodes. While this approach performs well on a surprising number of standard benchmarks, the efficacy of such models does not translate consistently to more complex domains, such as graph data in the biochemistry domain. We argue that these more complex domains require priors that encourage learning of band-pass and high-pass features rather than oversmoothed signals of standard GCN architectures. Here, we propose an alternative GNN architecture, based on a relaxation of recently proposed geometric scattering transforms, which consists of a cascade of graph wavelet filters. Our learned geometric scattering (LEGS) architecture adaptively tunes these wavelets and their scales to encourage band-pass features to emerge in learned representations. This results in a simplified GNN with significantly fewer learned parameters compared to competing methods. We demonstrate the predictive performance of our method on several biochemistry graph classification benchmarks, as well as the descriptive quality of its learned features in biochemical graph data exploration tasks. Our results show that the proposed LEGS network matches or outperforms popular GNNs, as well as the original geometric scattering construction, while also retaining certain mathematical properties of its handcrafted (nonlearned) design.

Biomolecular graph analysis has recently gained much attention in the emerging field of geometric deep learning. While numerous approaches aim to train classifiers that accurately predict molecular properties from graphs that encode their structure, an equally important task is to organize biomolecular graphs in ways that expose meaningful relations and variations between them. We propose a geometric scattering autoencoder (GSAE) network for learning such graph embeddings. Our embedding network first extracts rich graph features using the recently proposed geometric scattering transform. Then, it leverages a semi-supervised variational autoencoder to extract a low-dimensional embedding that retains the information in these features that enable prediction of molecular properties as well as characterize graphs. Our approach is based on the intuition that geometric scattering generates multi-resolution features with in-built invariance to deformations, but as they are unsupervised, these features may not be tuned for optimally capturing relevant domain-specific properties. We demonstrate the effectiveness of our approach to data exploration of RNA foldings. Like proteins, RNA molecules can fold to create low energy functional structures such as hairpins, but the landscape of possible folds and fold sequences are not well visualized by existing methods. We show that GSAE organizes RNA graphs both by structure and energy, accurately reflecting bistable RNA structures. Furthermore, it enables interpolation of embedded molecule sequences mimicking folding trajectories. Finally, using an auxiliary inverse-scattering model, we demonstrate our ability to generate synthetic RNA graphs along the trajectory thus providing hypothetical folding sequences for further analysis.

Anomaly detection is a problem of great interest in medicine, finance, and other fields where error and fraud need to be detected and corrected. Most deep anomaly detection methods rely on autoencoder reconstruction error. However, we show that this approach has limited value. First, this approach starts to perform poorly when either noise or anomalies contaminate training data, even to a small extent. Second, this approach cannot detect anomalous but simple to reconstruct points. This can be seen even in relatively simple examples, such as feeding a black image to detectors trained on MNIST digits. Here, we introduce a new discriminator-based unsupervised Lipschitz anomaly detector (LAD). We train a Wasserstein discriminator, similar to the ones used in GANs, to detect the difference between the training data and corruptions of the training data. We show that this procedure successfully detects unseen anomalies with guarantees on those that have a certain Wasserstein distance from the data or corrupted training set. Finally, we show results of this system in an electronic medical record dataset of HIV-positive veterans from the veterans aging cohort study (VACS) to establish usability in a medical setting.

Deep neural networks can learn meaningful representations of data. However, these representations are hard to interpret. For example, visualizing a latent layer is generally only possible for at most three dimensions. Neural networks are able to learn and benefit from much higher dimensional representationsm but these are not visually interpretable because neurons have arbitrary ordering within a layer. Here, we utilize the ability of a human observer to identify patterns in structured representations to visualize higher dimensions. To do so, we propose a class of regularizations we call Graph Spectral Regularizations that impose graph structure on latent layers. This is achieved by treating activations as signals on a predefined graph and constraining those activations using graph filters, such as low pass and wavelet-like filters. This framework allows for any kind of graphs and filters to achieve a wide range of structured regularizations depending on the inference needs of the data. First, we show a synthetic example where a graph-structured layer reveals topological features of the data. Next, we show that a smoothing regularization imposes semantically consistent ordering of nodes when applied to capsule nets. Further, we show that the graph-structured layer, using wavelet-like spatially localized filters, can form local receptive fields for improved image and biomedical data interpretation. In other words, the mapping between latent layer, neurons and the output space becomes clear due to the localization of the activations. Finally, we show that when structured as a grid, the representations create coherent images that allow for image processing techniques such as convolutions.