MOTIVATION: Gene expression varies across a tissue due to both the organization of the tissue into spatial domains, i.e. discrete regions of a tissue with distinct cell type composition, and continuous spatial gradients of gene expression within different spatial domains. Spatially resolved transcriptomics (SRT) technologies provide high-throughput measurements of gene expression in a tissue slice, enabling the characterization of spatial gradients and domains. However, existing computational methods for quantifying spatial variation in gene expression either model only spatial domains-and do not account for continuous gradients of expression-or require restrictive geometric assumptions on the spatial domains and spatial gradients that do not hold for many complex tissues. RESULTS: We introduce GASTON-Mix, a machine learning algorithm to identify both spatial domains and spatial gradients within each domain from SRT data. GASTON-Mix extends the mixture-of-experts (MoE) deep learning framework to a spatial MoE model, combining the clustering component of the MoE model with a neural field model that learns a separate 1D coordinate ("isodepth") within each domain. The spatial MoE is capable of representing any geometric arrangement of spatial domains in a tissue, and the isodepth coordinates define continuous gradients of gene expression within each domain. We show using simulations and real data that GASTON-Mix identifies spatial domains and spatial gradients of gene expression more accurately than existing methods. GASTON-Mix reveals spatial gradients in the striatum and lateral septum that regulate complex social behavior, and GASTON-Mix reveals localized spatial gradients of hypoxia and TNF-α signaling in the tumor microenvironment. AVAILABILITY AND IMPLEMENTATION: GASTON-Mix is available at https://github.com/raphael-group/GASTON-Mix.

MOTIVATION: Perturbations in biological tissues-e.g. due to inflammation, disease, or drug treatment-alter the composition of cell types and cell states in the tissue. These alterations are often spatially localized in different regions of a tissue, and can be measured using spatial transcriptomics technologies. However, current methods to analyze differential abundance in cell types or cell states, either do not incorporate spatial information-and thus cannot identify spatially localized alterations-or use heuristic and inaccurate approaches. RESULTS: We introduce Spatial Anomaly Region Detection in Expression Manifolds (Sardine), a method to estimate spatially localized changes in spatial transcriptomics data obtained from tissue slices from two or more conditions. Sardine estimates the probability of a cell state being at the same (relative) spatial location between different conditions using spatially localized density estimation. On simulated data, Sardine recapitulates the spatial patterning of expression changes more accurately than existing approaches. On a Visium dataset of the mouse cerebral cortex before and after injury response, as well as on a Visium dataset of a mouse spinal cord undergoing electrotherapy, Sardine identifies regions of spatially localized expression changes that are more biologically plausible than alternative approaches. AVAILABILITY AND IMPLEMENTATION: We implement Sardine in Python 3, with an open source implementation available at: https://github.com/raphael-group/spatial_anomaly_detection.

MOTIVATION: Cell-cell interactions (CCIs) consist of cells exchanging signals with themselves and neighboring cells by expressing ligand and receptor molecules and play a key role in cellular development, tissue homeostasis, and other critical biological functions. Since direct measurement of CCIs is challenging, multiple methods have been developed to infer CCIs by quantifying correlations between the gene expression of the ligands and receptors that mediate CCIs, originally from bulk RNA-sequencing data and more recently from single-cell or spatially resolved transcriptomics (SRT) data. SRT has a particular advantage over single-cell approaches, since ligand-receptor correlations can be computed between cells or spots that are physically close in the tissue. However, the transcript counts of individual ligands and receptors in SRT data are generally low, complicating the inference of CCIs from expression correlations. RESULTS: We introduce Copulacci, a count-based model for inferring CCIs from SRT data. Copulacci uses a Gaussian copula to model dependencies between the expression of ligands and receptors from nearby spatial locations even when the transcript counts are low. On simulated data, Copulacci outperforms existing CCI inference methods based on the standard Spearman and Pearson correlation coefficients. Using several real SRT datasets, we show that Copulacci discovers biologically meaningful ligand-receptor interactions that are lowly expressed and undiscoverable by existing CCI inference methods. AVAILABILITY AND IMPLEMENTATION: Copulacci is implemented in Python and available at https://github.com/raphael-group/copulacci.

Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an anomaly family. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly; however, it was recently observed that for normally distributed data, the MLE is a biased estimator for some anomaly families. In this work, we demonstrate that in the normal means setting, the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. We also provide empirical evidence that the converse is true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families. Next, we derive a new anomaly estimator using a mixture model, and we prove that our anomaly estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the advantages of our estimator versus the MLE on disease outbreak data and highway traffic data.

While social networks have increased the diversity of ideas and information available to users, they are also blamed for increasing the polarization of user opinions. Eli Pariser's "filter bubble" hypothesis [55] explains this counterintuitive phenomenon by linking user polarization to algorithmic filtering: to increase user engagement, social media companies connect users with ideas they are already likely to agree with, thus creating echo chambers of users with very similar beliefs.

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex v has a weight $\gamma_e(v)$ for each incident hyperedge e that describes the contribution of v to the hyperedge e. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.