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We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlanet al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.

This paper studies the classical problem of finding all $k$ nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. The well-known work by Beygelzimer, Kakade, and Langford in 2006 introduced cover trees and claimed to guarantee a near linear time complexity in the size $|R|$ of the reference set for $k=1$. Our previous work defined compressed cover trees and corrected the key arguments for $k\geq 1$ and previously unknown challenging data cases. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a similar complexity for $k = 1$. Our work fills all previous gaps and substantially improves the neighbor search based on pairs of new compressed cover trees. The novel imbalance parameter of paired trees allowed us to prove a better time complexity for any number of neighbors $k\geq 1$.

We propose a novel method for computing $p$-values based on nested sampling (NS) applied to the sampling space rather than the parameter space of the problem, in contrast to its usage in Bayesian computation. The computational cost of NS scales as $\log^2{1/p}$, which compares favorably to the $1/p$ scaling for Monte Carlo (MC) simulations. For significances greater than about $4\sigma$ in both a toy problem and a simplified resonance search, we show that NS requires orders of magnitude fewer simulations than ordinary MC estimates. This is particularly relevant for high-energy physics, which adopts a $5\sigma$ gold standard for discovery. We conclude with remarks on new connections between Bayesian and frequentist computation and possibilities for tuning NS implementations for still better performance in this setting.

Multi-fidelity modeling and calibration are data fusion tasks that ubiquitously arise in engineering design. In this paper, we introduce a novel approach based on latent-map Gaussian processes (LMGPs) that enables efficient and accurate data fusion. In our approach, we convert data fusion into a latent space learning problem where the relations among different data sources are automatically learned. This conversion endows our approach with attractive advantages such as increased accuracy, reduced costs, flexibility to jointly fuse any number of data sources, and ability to visualize correlations between data sources. This visualization allows the user to detect model form errors or determine the optimum strategy for high-fidelity emulation by fitting LMGP only to the subset of the data sources that are well-correlated. We also develop a new kernel function that enables LMGPs to not only build a probabilistic multi-fidelity surrogate but also estimate calibration parameters with high accuracy and consistency. The implementation and use of our approach are considerably simpler and less prone to numerical issues compared to existing technologies. We demonstrate the benefits of LMGP-based data fusion by comparing its performance against competing methods on a wide range of examples.

The COVID-19 pandemic has emphasized the need for a robust understanding of epidemic models. Current models of epidemics are classified as either mechanistic or non-mechanistic: mechanistic models make explicit assumptions on the dynamics of disease, whereas non-mechanistic models make assumptions on the form of observed time series. Here, we introduce a simple mixture-based model which bridges the two approaches while retaining benefits of both. The model represents time series of cases and fatalities as a mixture of Gaussian curves, providing a flexible function class to learn from data compared to traditional mechanistic models. Although the model is non-mechanistic, we show that it arises as the natural outcome of a stochastic process based on a networked SIR framework. This allows learned parameters to take on a more meaningful interpretation compared to similar non-mechanistic models, and we validate the interpretations using auxiliary mobility data collected during the COVID-19 pandemic. We provide a simple learning algorithm to identify model parameters and establish theoretical results which show the model can be efficiently learned from data. Empirically, we find the model to have low prediction error. The model is available live at covidpredictions.mit.edu. Ultimately, this allows us to systematically understand the impacts of interventions on COVID-19, which is critical in developing data-driven solutions to controlling epidemics.

Influence maximization is the task of selecting a small number of seed nodes in a social network to maximize the spread of the influence from these seeds, and it has been widely investigated in the past two decades. In the canonical setting, the whole social network as well as its diffusion parameters is given as input. In this paper, we consider the more realistic sampling setting where the network is unknown and we only have a set of passively observed cascades that record the set of activated nodes at each diffusion step. We study the task of influence maximization from these cascade samples (IMS), and present constant approximation algorithms for this task under mild conditions on the seed set distribution. To achieve the optimization goal, we also provide a novel solution to the network inference problem, that is, learning diffusion parameters and the network structure from the cascade data. Comparing with prior solutions, our network inference algorithm requires weaker assumptions and does not rely on maximum-likelihood estimation and convex programming. Our IMS algorithms enhance the learning-and-then-optimization approach by allowing a constant approximation ratio even when the diffusion parameters are hard to learn, and we do not need any assumption related to the network structure or diffusion parameters.

The information bottleneck (IB) method is a technique for extracting information that is relevant for predicting the target random variable from the source random variable, which is typically implemented by optimizing the IB Lagrangian that balances the compression and prediction terms. However, the IB Lagrangian is hard to optimize, and multiple trials for tuning values of Lagrangian multiplier are required. Moreover, we show that the prediction performance strictly decreases as the compression gets stronger during optimizing the IB Lagrangian. In this paper, we implement the IB method from the perspective of supervised disentangling. Specifically, we introduce Disentangled Information Bottleneck (DisenIB) that is consistent on compressing source maximally without target prediction performance loss (maximum compression). Theoretical and experimental results demonstrate that our method is consistent on maximum compression, and performs well in terms of generalization, robustness to adversarial attack, out-of-distribution detection, and supervised disentangling.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.

In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.