Clusters of similar or dissimilar objects are encountered in many fields. Frequently used approaches treat the central object of each cluster as latent. Yet, often objects of one or more types cluster around objects of another type. Such arrangements are common in biomedical images of cells, in which nearby cell types likely interact. Quantifying spatial relationships may elucidate biological mechanisms. Parent-offspring statistical frameworks can be usefully applied even when central objects (parents) differ from peripheral ones (offspring). We propose the novel multivariate cluster point process (MCPP) to quantify multi-object (e.g., multi-cellular) arrangements. Unlike commonly used approaches, the MCPP exploits locations of the central parent object in clusters. It accounts for possibly multilayered, multivariate clustering. The model formulation requires specification of which object types function as cluster centers and which reside peripherally. If such information is unknown, the relative roles of object types may be explored by comparing fit of different models via the deviance information criterion (DIC). In simulated data, we compared DIC of a series of models; the MCPP correctly identified simulated relationships. It also produced more accurate and precise parameter estimates than the classical univariate Neyman-Scott process model. We also used the MCPP to quantify proposed configurations and explore new ones in human dental plaque biofilm image data. MCPP models quantified simultaneous clustering of Streptococcus and Porphyromonas around Corynebacterium and of Pasteurellaceae around Streptococcus and successfully captured hypothesized structures for all taxa. Further exploration suggested the presence of clustering between Fusobacterium and Leptotrichia, a previously unreported relationship.
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Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase-transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability that arises from the phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean field critical exponents. Furthermore, using the statistical physics of disordered systems, we show that memorization can be understood as a form of critical condensation corresponding to a disordered phase transition. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium.
In many circumstances given an ordered sequence of one or more types of elements/ symbols, the objective is to determine any existence of randomness in occurrence of one of the elements,say type 1 element. Such a method can be useful in determining existence of any non-random pattern in the wins or loses of a player in a series of games played. Existing methods of tests based on total number of runs or tests based on length of longest run (Mosteller (1941)) can be used for testing the null hypothesis of randomness in the entire sequence, and not a specific type of element. Additionally, the Runs Test tends to show results contradictory to the intuition visualised by the graphs of say, win proportions over time due to method used in computation of runs. This paper develops a test approach to address this problem by computing the gaps between two consecutive type 1 elements and thereafter following the idea of "pattern" in occurrence and "directional" trend (increasing, decreasing or constant), employs the use of exact Binomial test, Kenall's Tau and Siegel-Tukey test for scale problem. Further modifications suggested by Jan Vegelius(1982) have been applied in the Siegel Tukey test to adjust for tied ranks and achieve more accurate results. This approach is distribution-free and suitable for small sizes. Also comparisons with the conventional runs test shows the superiority of the proposed approach under the null hypothesis of randomness in the occurrence of type 1 elements.
We study general coordinate-wise MCMC schemes (such as Metropolis-within-Gibbs samplers), which are commonly used to fit Bayesian non-conjugate hierarchical models. We relate their convergence properties to the ones of the corresponding (potentially not implementable) Gibbs sampler through the notion of conditional conductance. This allows us to study the performances of popular Metropolis-within-Gibbs schemes for non-conjugate hierarchical models, in high-dimensional regimes where both number of datapoints and parameters increase. Given random data-generating assumptions, we establish dimension-free convergence results, which are in close accordance with numerical evidences. Applications to Bayesian models for binary regression with unknown hyperparameters and discretely observed diffusions are also discussed. Motivated by such statistical applications, auxiliary results of independent interest on approximate conductances and perturbation of Markov operators are provided.
Bayesian graphical models are powerful tools to infer complex relationships in high dimension, yet are often fraught with computational and statistical challenges. If exploited in a principled way, the increasing information collected alongside the data of primary interest constitutes an opportunity to mitigate these difficulties by guiding the detection of dependence structures. For instance, gene network inference may be informed by the use of publicly available summary statistics on the regulation of genes by genetic variants. Here we present a novel Gaussian graphical modelling framework to identify and leverage information on the centrality of nodes in conditional independence graphs. Specifically, we consider a fully joint hierarchical model to simultaneously infer (i) sparse precision matrices and (ii) the relevance of node-level information for uncovering the sought-after network structure. We encode such information as candidate auxiliary variables using a spike-and-slab submodel on the propensity of nodes to be hubs, which allows hypothesis-free selection and interpretation of a sparse subset of relevant variables. As efficient exploration of large posterior spaces is needed for real-world applications, we develop a variational expectation conditional maximisation algorithm that scales inference to hundreds of samples, nodes and auxiliary variables. We illustrate and exploit the advantages of our approach in simulations and in a gene network study which identifies hub genes involved in biological pathways relevant to immune-mediated diseases.
Reconstructing a dynamic object with affine motion in computerized tomography (CT) leads to motion artifacts if the motion is not taken into account. In most cases, the actual motion is neither known nor can be determined easily. As a consequence, the respective model that describes CT is incomplete. The iterative RESESOP-Kaczmarz method can - under certain conditions and by exploiting the modeling error - reconstruct dynamic objects at different time points even if the exact motion is unknown. However, the method is very time-consuming. To speed the reconstruction process up and obtain better results, we combine the following three steps: 1. RESESOP-Kacmarz with only a few iterations is implemented to reconstruct the object at different time points. 2. The motion is estimated via landmark detection, e.g. using deep learning. 3. The estimated motion is integrated into the reconstruction process, allowing the use of dynamic filtered backprojection. We give a short review of all methods involved and present numerical results as a proof of principle.
Data augmentation is arguably the most important regularization technique commonly used to improve generalization performance of machine learning models. It primarily involves the application of appropriate data transformation operations to create new data samples with desired properties. Despite its effectiveness, the process is often challenging because of the time-consuming trial and error procedures for creating and testing different candidate augmentations and their hyperparameters manually. Automated data augmentation methods aim to automate the process. State-of-the-art approaches typically rely on automated machine learning (AutoML) principles. This work presents a comprehensive survey of AutoML-based data augmentation techniques. We discuss various approaches for accomplishing data augmentation with AutoML, including data manipulation, data integration and data synthesis techniques. We present extensive discussion of techniques for realizing each of the major subtasks of the data augmentation process: search space design, hyperparameter optimization and model evaluation. Finally, we carried out an extensive comparison and analysis of the performance of automated data augmentation techniques and state-of-the-art methods based on classical augmentation approaches. The results show that AutoML methods for data augmentation currently outperform state-of-the-art techniques based on conventional approaches.
Signed graphs are an emergent way of representing data in a variety of contexts were conflicting interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability geometry for signed graphs, proving that metrics in this space, such as the communicability distance and angles, are Euclidean and spherical. We then apply these metrics to solve several problems in data analysis of signed graphs in a unified way. They include the partitioning of signed graphs, dimensionality reduction, finding hierarchies of alliances in signed networks as well as the quantification of the degree of polarization between the existing factions in systems represented by this type of graphs.
We propose a new approach to the autoregressive spatial functional model, based on the notion of signature, which represents a function as an infinite series of its iterated integrals. It presents the advantage of being applicable to a wide range of processes. After having provided theoretical guarantees to the proposed model, we have shown in a simulation study and on a real data set that this new approach presents competitive performances compared to the traditional model.
Bayesian inference for complex models with an intractable likelihood can be tackled using algorithms performing many calls to computer simulators. These approaches are collectively known as "simulation-based inference" (SBI). Recent SBI methods have made use of neural networks (NN) to provide approximate, yet expressive constructs for the unavailable likelihood function and the posterior distribution. However, they do not generally achieve an optimal trade-off between accuracy and computational demand. In this work, we propose an alternative that provides both approximations to the likelihood and the posterior distribution, using structured mixtures of probability distributions. Our approach produces accurate posterior inference when compared to state-of-the-art NN-based SBI methods, while exhibiting a much smaller computational footprint. We illustrate our results on several benchmark models from the SBI literature.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
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