<p style='text-indent:20px;'>This work proposes a novel technique for clustering multimodal data according to their information content. Statistical correlations present in data that contain similar information are exploited to perform the clustering task. Specifically, multiset canonical correlation analysis is equipped with norm-one regularization mechanisms to identify clusters within different types of data that share the same information content. A pertinent minimization formulation is put forth, while block coordinate descent is employed to derive a batch clustering algorithm which achieves better clustering performance than existing alternatives. Relying on subgradient descent, an online clustering approach is derived which substantially lowers computational complexity compared to the batch approach, while not compromising significantly the clustering performance. It is established that for an increasing number of data the novel regularized multiset framework is able to correctly cluster the multimodal data entries. Further, it is proved that the online clustering scheme converges with probability one to a stationary point of the ensemble regularized multiset correlations cost having the potential to recover the correct clusters. Extensive numerical tests demonstrate that the novel clustering scheme outperforms existing alternatives, while the online scheme achieves substantial computational savings.</p>

<p style='text-indent:20px;'>Nonlocal models have recently had a major impact in nonlinear continuum mechanics and are used to describe physical systems/processes which cannot be accurately described by classical, calculus based "local" approaches. In part, this is due to their multiscale nature that enables aggregation of micro-level behavior to obtain a macro-level description of singular/irregular phenomena such as peridynamics, crack propagation, anomalous diffusion and transport phenomena. At the core of these models are <i>nonlocal</i> differential operators, including nonlocal analogs of the gradient/Hessian. This paper initiates the use of such nonlocal operators in the context of optimization and learning. We define and analyze the convergence properties of nonlocal analogs of (stochastic) gradient descent and Newton's method on Euclidean spaces. Our results indicate that as the nonlocal interactions become less noticeable, the optima corresponding to nonlocal optimization converge to the "usual" optima. At the same time, we argue that nonlocal learning is possible in situations where standard calculus fails. As a stylized numerical example of this, we consider the problem of non-differentiable parameter estimation on a non-smooth translation manifold and show that our <i>nonlocal</i> gradient descent recovers the unknown translation parameter from a non-differentiable objective function.</p>

<p style='text-indent:20px;'>In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.</p>

<p style='text-indent:20px;'>Given an undirected measurement graph <inline-formula><tex-math id="M1">\begin{document}$ G = ([n], E) $\end{document}</tex-math></inline-formula>, the classical angular synchronization problem consists of recovering unknown angles <inline-formula><tex-math id="M2">\begin{document}$ \theta_1, \dots, \theta_n $\end{document}</tex-math></inline-formula> from a collection of noisy pairwise measurements of the form <inline-formula><tex-math id="M3">\begin{document}$ (\theta_i - \theta_j) \mod 2\pi $\end{document}</tex-math></inline-formula>, for each <inline-formula><tex-math id="M4">\begin{document}$ \{i, j\} \in E $\end{document}</tex-math></inline-formula>. This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from preference relationships. In this paper, we consider a generalization to the setting where there exist <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> unknown groups of angles <inline-formula><tex-math id="M6">\begin{document}$ \theta_{l, 1}, \dots, \theta_{l, n} $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M7">\begin{document}$ l = 1, \dots, k $\end{document}</tex-math></inline-formula>. For each <inline-formula><tex-math id="M8">\begin{document}$ {\left\{{{i, j}}\right\}} \in E $\end{document}</tex-math></inline-formula>, we are given noisy pairwise measurements of the form <inline-formula><tex-math id="M9">\begin{document}$ \theta_{\ell, i} - \theta_{\ell, j} $\end{document}</tex-math></inline-formula> for an <i>unknown</i> <inline-formula><tex-math id="M10">\begin{document}$ \ell \in \{1, 2, \ldots, k\} $\end{document}</tex-math></inline-formula>. This can be thought of as a natural extension of the angular synchronization problem to the heterogeneous setting of multiple groups of angles, where the measurement graph has an unknown edge-disjoint decomposition <inline-formula><tex-math id="M11">\begin{document}$ G = G_1 \cup G_2 \ldots \cup G_k $\end{document}</tex-math></inline-formula>, where the <inline-formula><tex-math id="M12">\begin{document}$ G_i $\end{document}</tex-math></inline-formula>'s denote the subgraphs of edges corresponding to each group. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise. The theoretical findings are complemented by a comprehensive set of numerical experiments, showcasing the efficacy of our algorithm under various parameter regimes. Finally, we consider an application of bi-synchronization to the graph realization problem, and provide along the way an iterative graph disentangling procedure that uncovers the subgraphs <inline-formula><tex-math id="M13">\begin{document}$ G_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$ i = 1, \ldots, k $\end{document}</tex-math></inline-formula> which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.</p>

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes.

<p style='text-indent:20px;'>We consider the problem of static Bayesian inference for partially observed Lévy-process models. We develop a methodology which allows one to infer static parameters and some states of the process, without a bias from the time-discretization of the afore-mentioned Lévy process. The unbiased method is exceptionally amenable to parallel implementation and can be computationally efficient relative to competing approaches. We implement the method on S & P 500 log-return daily data and compare it to some Markov chain Monte Carlo (MCMC) algorithm.</p>

<p style='text-indent:20px;'>The disparity in the impact of COVID-19 on minority populations in the United States has been well established in the available data on deaths, case counts, and adverse outcomes. However, critical metrics used by public health officials and epidemiologists, such as a time dependent viral reproductive number (<inline-formula><tex-math id="M1">\begin{document}$ R_t $\end{document}</tex-math></inline-formula>), can be hard to calculate from this data especially for individual populations. Furthermore, disparities in the availability of testing, record keeping infrastructure, or government funding in disadvantaged populations can produce incomplete data sets. In this work, we apply ensemble data assimilation techniques which optimally combine model and data to produce a more complete data set providing better estimates of the critical metrics used by public health officials and epidemiologists. We employ a multi-population SEIR (Susceptible, Exposed, Infected and Recovered) model with a time dependent reproductive number and age stratified contact rate matrix for each population. We assimilate the daily death data for populations separated by ethnic/racial groupings using a technique called Ensemble Smoothing with Multiple Data Assimilation (ESMDA) to estimate model parameters and produce an <inline-formula><tex-math id="M10000">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula> for the <inline-formula><tex-math id="M2000">\begin{document}$n^{th}$\end{document}</tex-math></inline-formula> population. We do this with three distinct approaches, (1) using the same contact matrices and prior <inline-formula><tex-math id="M30000">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula> for each population, (2) assigning contact matrices with increased contact rates for working age and older adults to populations experiencing disparity and (3) as in (2) but with a time-continuous update to <inline-formula><tex-math id="M4">\begin{document}$R_t(n)$\end{document}</tex-math></inline-formula>. We make a study of 9 U.S. states and the District of Columbia providing a complete time series of the pandemic in each and, in some cases, identifying disparities not otherwise evident in the aggregate statistics.</p>

<p style='text-indent:20px;'>We study the problem of diffeomorphometric geodesic landmark matching where the objective is to find a diffeomorphism that, via its group action, maps between two sets of landmarks. It is well-known that the motion of the landmarks, and thereby the diffeomorphism, can be encoded by an initial momentum leading to a formulation where the landmark matching problem can be solved as an optimisation problem over such momenta. The novelty of our work lies in the application of a derivative-free Bayesian inverse method for learning the optimal momentum encoding the diffeomorphic mapping between the template and the target. The method we apply is the ensemble Kalman filter, an extension of the Kalman filter to nonlinear operators. We describe an efficient implementation of the algorithm and show several numerical results for various target shapes.</p>

<p style='text-indent:20px;'>Ensemble Kalman Inversion (EnKI) [<xref ref-type="bibr" rid="b23">23</xref>] and Ensemble Square Root Filter (EnSRF) [<xref ref-type="bibr" rid="b36">36</xref>] are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter [<xref ref-type="bibr" rid="b17">17</xref>,<xref ref-type="bibr" rid="b3">3</xref>]. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b25">25</xref>]. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments.</p><p style='text-indent:20px;'>We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.</p>

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