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In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning. First, we discuss the structure of the search space over causal graphs. Second, we discuss the structure of equivalence classes over causal graphs, i.e., sets of graphs which represent what can be learned from observational data alone, and how these equivalence classes can be refined by adding interventional data. [ABSTRACT FROM AUTHOR]

We consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, n(.), on L. We derive from the supermodularity of n(.) some generalized Fréchet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from n(.), and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fréchet inequalities for multidimensional contingency tables. [ABSTRACT FROM AUTHOR]

A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed, and bidirected edges, we give a construction of this polytope, up to equivalence of normal fans, as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight to construct a new ordering-based search algorithm for causal inference via directed acyclic graphical models. [ABSTRACT FROM AUTHOR]

The 3D structure of the genome plays a key role in regulatory control of the cell. Experimental methods such as high-throughput chromosome conformation capture (Hi-C) have been developed to probe the 3D structure of the genome. However, it remains a challenge to deduce from these data chromosome regions that are colocalized and coregulated. Here, we present an integrative approach that leverages 1D functional genomic features (e.g., epigenetic marks) with 3D interactions from Hi-C data to identify functional interchromosomal interactions. We construct a weighted network with 250-kb genomic regions as nodes and Hi-C interactions as edges, where the edge weights are given by the correlation between 1D genomic features. Individual interacting clusters are determined using weighted correlation clustering on the network. We show that intermingling regions generally fall into either active or inactive clusters based on the enrichment for RNA polymerase II (RNAPII) and H3K9me3, respectively. We show that active clusters are hotspots for transcription factor binding sites. We also validate our predictions experimentally by 3D fluorescence in situ hybridization (FISH) experiments and show that active RNAPII is enriched in predicted active clusters. Our method provides a general quantitative framework that couples 1D genomic features with 3D interactions from Hi-C to probe the guiding principles that link the spatial organization of the genome with regulatory control. [ABSTRACT FROM AUTHOR]

Cells sense physical and chemical signals from their local microenvironment and transduce them to the nucleus to regulate genomic programs. In this review, we first discuss different modes of mechanotransduction to the nucleus. We then highlight the role of the spatial organization of chromosomes for integrating these signals. In particular, we emphasize the importance of chromosome intermingling for gene regulation. We also discuss various geometric models and recent advances in microscopy and genomics that have allowed access to these nanoscale chromosome intermingling regions. Taken together, the recent work summarized in this review culminates in the hypothesis that chromosome intermingling regions are mechanical hotspots for genome regulation. Maintenance of such mechanical hotspots is crucial for cellular homeostasis, and alterations in them could be precursors for various cellular reprogramming events, including diseases. [ABSTRACT FROM AUTHOR]

An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC algorithm and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow ties, tripartite graphs, and complete graphs. [ABSTRACT FROM AUTHOR]

The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application--chromosome organization in the human cell nucleus--is discussed briefly, and some illustrative results are presented. [ABSTRACT FROM AUTHOR]

Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on determining associations between single loci and a specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. In this paper, we describe a two-stage method for detecting epistasis by combining the traditionally used single-locus search with a search for multiway interactions. Our method is based on an extended version of Fisher's exact test. To perform this test, a Markov chain is constructed on the space of multidimensional contingency tables using the elements of a Markov basis as moves. We test our method on simulated data and compare it to a two-stage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a genome-wide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs). [ABSTRACT FROM AUTHOR]

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of statistics and optimization are here examined from the perspective of convex algebraic geometry. [ABSTRACT FROM AUTHOR]

The article discusses a study that examined the diagnostic efficiency of various mastitis diagnostic tests, including a bacterial test, somatic cell count (SCC) and measurement of electrical conductivity (MEC). The study was intended to enhance the efficiency of milk production. About 25 cows were tested at udder quarters with the mastitis diagnostic tests. It then estimated the disease prevalence, sensitivity and specificity of the tests.

In this Current Opinion, we highlight the importance of the material properties of tissues and how alterations therein, which influence epithelial-to-mesenchymal transitions, represent an important layer of regulation in a number of diseases and potentially also play a critical role in host-pathogen interactions. In light of the current SARS-CoV-2 pandemic, we here highlight the possible role of lung tissue stiffening with ageing and how this might facilitate increased SARS-CoV-2 replication through matrix-stiffness dependent epithelial-to-mesenchymal transitions of the lung epithelium. This emphasizes the need for integrating material properties of tissues in drug discovery programs. [ABSTRACT FROM AUTHOR]

Human life expectancy is constantly increasing and aging has become a major risk factor for many diseases, although the underlying gene regulatory mechanisms are still unclear. Using transcriptomic and chromosomal conformation capture (Hi‐C) data from human skin fibroblasts from individuals across different age groups, we identified a tight coupling between the changes in co‐regulation and co‐localization of genes. We obtained transcription factors, cofactors, and chromatin regulators that could drive the cellular aging process by developing a time‐course prize‐collecting Steiner tree algorithm. In particular, by combining RNA‐Seq data from different age groups and protein–protein interaction data we determined the key transcription regulators and gene regulatory changes at different life stage transitions. We then mapped these transcription regulators to the 3D reorganization of chromatin in young and old skin fibroblasts. Collectively, we identified key transcription regulators whose target genes are spatially rearranged and correlate with changes in their expression, thereby providing potential targets for reverting cellular aging. [ABSTRACT FROM AUTHOR]

We consider the problem of learning a Bayesian network or directed acyclic graph model from observational data. A number of constraint‐based, score‐based and hybrid algorithms have been developed for this purpose. Statistical consistency guarantees of these algorithms rely on the faithfulness assumption, which has been shown to be restrictive especially for graphs with cycles in the skeleton. We here propose the sparsest permutation (SP) algorithm, showing that learning Bayesian networks is possible under strictly weaker assumptions than faithfulness. This comes at a computational price, thereby indicating a statistical‐computational trade‐off for causal inference algorithms. In the Gaussian noiseless setting, we prove that the SP algorithm boils down to finding the permutation of the variables with the sparsest Cholesky decomposition of the inverse covariance matrix, which is equivalent to ℓ0‐penalized maximum likelihood estimation. We end with a simulation study showing that in line with the proven stronger consistency guarantees, and the SP algorithm compares favourably to standard causal inference algorithms in terms of accuracy for a given sample size. Copyright © 2018 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Identifying computational mechanisms for memorization and retrieval of data is a long-standing problem at the intersection of machine learning and neuroscience. Our main finding is that standard overparameterized deep neural networks trained using standard optimization methods implement such a mechanism for real-valued data. We provide empirical evidence that 1) overparameterized autoencoders store training samples as attractors and thus iterating the learned map leads to sample recovery, and that 2) the same mechanism allows for encoding sequences of examples and serves as an even more efficient mechanism for memory than autoencoding. Theoretically, we prove that when trained on a single example, autoencoders store the example as an attractor. Lastly, by treating a sequence encoder as a composition of maps, we prove that sequence encoding provides a more efficient mechanism for memory than autoencoding. [ABSTRACT FROM AUTHOR]

Summary Designing interventions to control gene regulation necessitates modeling a gene regulatory network by a causal graph. Currently, large-scale gene expression datasets from different conditions, cell types, disease states, and developmental time points are being collected. However, application of classical causal inference algorithms to infer gene regulatory networks based on such data is still challenging, requiring high sample sizes and computational resources. Here, we describe an algorithm that efficiently learns the differences in gene regulatory mechanisms between different conditions. Our difference causal inference (DCI) algorithm infers changes (i.e. edges that appeared, disappeared, or changed weight) between two causal graphs given gene expression data from the two conditions. This algorithm is efficient in its use of samples and computation since it infers the differences between causal graphs directly without estimating each possibly large causal graph separately. We provide a user-friendly Python implementation of DCI and also enable the user to learn the most robust difference causal graph across different tuning parameters via stability selection. Finally, we show how to apply DCI to single-cell RNA-seq data from different conditions and cell states, and we also validate our algorithm by predicting the effects of interventions. Availability and implementation Python package freely available at http://uhlerlab.github.io/causaldag/dci. Supplementary information Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

The space of discrete copulas admits a representation as a convex polytope, and this has been exploited in entropy-copula methods used in hydrology and climatology. In this paper, we focus on the class of component-wise convex copulas, i.e., ultramodular copulas, which describe the joint behavior of stochastically decreasing random vectors. We show that the family of ultramodular discrete copulas and its generalization to component-wise convex discrete quasi-copulas also admit representations as polytopes. In doing so, we draw connections to the Birkhoff polytope, the alternating sign matrix polytope, and their generalizations, thereby unifying and extending results on these polytopes from both the statistics and the discrete geometry literature. [ABSTRACT FROM AUTHOR]

Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on Rd that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems. [ABSTRACT FROM AUTHOR]

DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G . Such models are used to model complex cause–effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence . Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e., skeleton) and the same set of the induced subDAGs i → j ← k , known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, we introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and we studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some well-studied families of graphs including paths, stars, cycles, spider graphs, caterpillars, and balanced binary trees. In doing so, we recover connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictate the number and size of MECs. [ABSTRACT FROM AUTHOR]

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane. [ABSTRACT FROM AUTHOR]

Long-term sustained mechano-chemical signals in tissue microenvironment regulate cell-state transitions. In recent work, we showed that laterally confined growth of fibroblasts induce dedifferentiation programs. However, the molecular mechanisms underlying such mechanically induced cell-state transitions are poorly understood. In this paper, we identify Lef1 as a critical somatic transcription factor for the mechanical regulation of de-differentiation pathways. Network optimization methods applied to time-lapse RNA-seq data identify Lef1 dependent signaling as potential regulators of such cell-state transitions. We show that Lef1 knockdown results in the down-regulation of fibroblast de-differentiation and that Lef1 directly interacts with the promoter regions of downstream reprogramming factors. We also evaluate the potential upstream activation pathways of Lef1, including the Smad4, Atf2, NFkB and Beta-catenin pathways, thereby identifying that Smad4 and Atf2 may be critical for Lef1 activation. Collectively, we describe an important mechanotransduction pathway, including Lef1, which upon activation, through progressive lateral cell confinement, results in fibroblast de-differentiation. [ABSTRACT FROM AUTHOR]

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Blog post, https://11011110.github.io/blog/2006/10/13/another-gratuitously-nonplanar-drawing.html 2 Eppstein D. Happy endings for flip graphs. [Extracted from the article]

Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semisupervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience. [ABSTRACT FROM AUTHOR]

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. Graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime. [ABSTRACT FROM AUTHOR]

We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log‐supermodular (MTP2) distributions and log‐L♮‐concave (LLC) distributions. In both cases we also assume log‐concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}d or in ℝ2 under MTP2, and for samples in ℚd under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate. [ABSTRACT FROM AUTHOR]

Lineage tracing involves the identification of all ancestors and descendants of a given cell, and is an important tool for studying biological processes such as development and disease progression. However, in many settings, controlled time-course experiments are not feasible, for example when working with tissue samples from patients. Here we present ImageAEOT, a computational pipeline based on autoencoders and optimal transport for predicting the lineages of cells using time-labeled datasets from different stages of a cellular process. Given a single-cell image from one of the stages, ImageAEOT generates an artificial lineage of this cell based on the population characteristics of the other stages. These lineages can be used to connect subpopulations of cells through the different stages and identify image-based features and biomarkers underlying the biological process. To validate our method, we apply ImageAEOT to a benchmark task based on nuclear and chromatin images during the activation of fibroblasts by tumor cells in engineered 3D tissues. We further validate ImageAEOT on chromatin images of various breast cancer cell lines and human tissue samples, thereby linking alterations in chromatin condensation patterns to different stages of tumor progression. Our results demonstrate the promise of computational methods based on autoencoding and optimal transport principles for lineage tracing in settings where existing experimental strategies cannot be used. Author summary: Many key biological processes, such as development and disease progression, require analyzing lineages of cells backwards as well as forwards in time. However, current single-cell experiments tend to be destructive to cells, so that a single lineage can only be measured at one point in time. In this work, we introduce a computational framework for predicting the lineage of cells from a single snapshot in time based on measurements of other cells at other time points. The method generates these lineages by computing the most plausible path for a population of cells to transition from one time point to the next, assuming that a cell is more likely to transition to similar cells compared to dissimilar cells. We validate our computational method on imaging data of fibroblasts and cancer cells, though our method could also be applied to other modalities of single-cell data such as genomics and transcriptomics as well as multi-modal single-cell datasets. [ABSTRACT FROM AUTHOR]

Summary: A complete mitochondrial (mt) genome sequence was reconstructed from a 38,000 year-old Neandertal individual with 8341 mtDNA sequences identified among 4.8 Gb of DNA generated from ∼0.3 g of bone. Analysis of the assembled sequence unequivocally establishes that the Neandertal mtDNA falls outside the variation of extant human mtDNAs, and allows an estimate of the divergence date between the two mtDNA lineages of 660,000 ± 140,000 years. Of the 13 proteins encoded in the mtDNA, subunit 2 of cytochrome c oxidase of the mitochondrial electron transport chain has experienced the largest number of amino acid substitutions in human ancestors since the separation from Neandertals. There is evidence that purifying selection in the Neandertal mtDNA was reduced compared with other primate lineages, suggesting that the effective population size of Neandertals was small. [Copyright &y& Elsevier]