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This paper extends various results related to the Gaussian product inequality (GPI) conjecture to the setting of disjoint principal minors of Wishart random matrices. This includes product-type inequalities for matrix-variate analogs of completely monotone functions and Bernstein functions of Wishart disjoint principal minors, respectively. In particular, the product-type inequalities apply to inverse determinant powers. Quantitative versions of the inequalities are also obtained when there is a mix of positive and negative exponents. Furthermore, an extended form of the GPI is shown to hold for the eigenvalues of Wishart random matrices by virtue of their law being multivariate totally positive of order~2 ($\mathrm{MTP}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.

The Shallow Ice Approximation (SIA) model on strong form is commonly used for inferring the flow dynamics of grounded ice sheets. The solution to the SIA model is a closed-form expression for the velocity field. When that velocity field is used to advance the ice surface in time, the time steps have to take small values due to quadratic scaling in terms of the horizontal mesh size. In this paper we write the SIA model on weak form, and add in the Free Surface Stabilization Algorithm (FSSA) terms. We find numerically that the time step restriction scaling is improved from quadratic to linear, but only for large horizontal mesh sizes. We then extend the weak form by adding the initially neglected normal stress terms. This allows for a linear time step restriction across the whole range of the horizontal mesh sizes, leading to an improved efficiency. Theoretical analysis demonstrates that the inclusion of FSSA stabilization terms transitions the explicit time stepping treatment of second derivative surface terms to an implicit approach. Moreover, a computational cost analysis, combined with numerical results on stability and accuracy, advocates for preferring the SIA models written on weak form over the standard SIA model.

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.

We present ReCAT, a recursive composition augmented Transformer that is able to explicitly model hierarchical syntactic structures of raw texts without relying on gold trees during both learning and inference. Existing research along this line restricts data to follow a hierarchical tree structure and thus lacks inter-span communications. To overcome the problem, we propose a novel contextual inside-outside (CIO) layer that learns contextualized representations of spans through bottom-up and top-down passes, where a bottom-up pass forms representations of high-level spans by composing low-level spans, while a top-down pass combines information inside and outside a span. By stacking several CIO layers between the embedding layer and the attention layers in Transformer, the ReCAT model can perform both deep intra-span and deep inter-span interactions, and thus generate multi-grained representations fully contextualized with other spans. Moreover, the CIO layers can be jointly pre-trained with Transformers, making ReCAT enjoy scaling ability, strong performance, and interpretability at the same time. We conduct experiments on various sentence-level and span-level tasks. Evaluation results indicate that ReCAT can significantly outperform vanilla Transformer models on all span-level tasks and baselines that combine recursive networks with Transformers on natural language inference tasks. More interestingly, the hierarchical structures induced by ReCAT exhibit strong consistency with human-annotated syntactic trees, indicating good interpretability brought by the CIO layers.

The Bell regression model (BRM) is a statistical model that is often used in the analysis of count data that exhibits overdispersion. In this study, we propose a Bayesian analysis of the BRM and offer a new perspective on its application. Specifically, we introduce a G-prior distribution for Bayesian inference in BRM, in addition to a flat-normal prior distribution. To compare the performance of the proposed prior distributions, we conduct a simulation study and demonstrate that the G-prior distribution provides superior estimation results for the BRM. Furthermore, we apply the methodology to real data and compare the BRM to the Poisson regression model using various model selection criteria. Our results provide valuable insights into the use of Bayesian methods for estimation and inference of the BRM and highlight the importance of considering the choice of prior distribution in the analysis of count data.

The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using reproducing kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function.

We consider the design of a new class of passive iFIR controllers given by the parallel action of an integrator and a finite impulse response filter. iFIRs are more expressive than PID controllers but retain their features and simplicity. The paper provides a model-free data-driven design for passive iFIR controllers based on virtual reference feedback tuning. Passivity is enforced through constrained optimization (three different formulations are discussed). The proposed design does not rely on large datasets or accurate plant models.

The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where $P$ is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of $P$. This provides some theoretical justification for the use of such estimators for cluster analysis in case that $P$ has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.

We present a new way to summarize and select mixture models via the hierarchical clustering tree (dendrogram) constructed from an overfitted latent mixing measure. Our proposed method bridges agglomerative hierarchical clustering and mixture modeling. The dendrogram's construction is derived from the theory of convergence of the mixing measures, and as a result, we can both consistently select the true number of mixing components and obtain the pointwise optimal convergence rate for parameter estimation from the tree, even when the model parameters are only weakly identifiable. In theory, it explicates the choice of the optimal number of clusters in hierarchical clustering. In practice, the dendrogram reveals more information on the hierarchy of subpopulations compared to traditional ways of summarizing mixture models. Several simulation studies are carried out to support our theory. We also illustrate the methodology with an application to single-cell RNA sequence analysis.

Multi-contrast (MC) Magnetic Resonance Imaging (MRI) reconstruction aims to incorporate a reference image of auxiliary modality to guide the reconstruction process of the target modality. Known MC reconstruction methods perform well with a fully sampled reference image, but usually exhibit inferior performance, compared to single-contrast (SC) methods, when the reference image is missing or of low quality. To address this issue, we propose DuDoUniNeXt, a unified dual-domain MRI reconstruction network that can accommodate to scenarios involving absent, low-quality, and high-quality reference images. DuDoUniNeXt adopts a hybrid backbone that combines CNN and ViT, enabling specific adjustment of image domain and k-space reconstruction. Specifically, an adaptive coarse-to-fine feature fusion module (AdaC2F) is devised to dynamically process the information from reference images of varying qualities. Besides, a partially shared shallow feature extractor (PaSS) is proposed, which uses shared and distinct parameters to handle consistent and discrepancy information among contrasts. Experimental results demonstrate that the proposed model surpasses state-of-the-art SC and MC models significantly. Ablation studies show the effectiveness of the proposed hybrid backbone, AdaC2F, PaSS, and the dual-domain unified learning scheme.