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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.

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 Shallow Ice Approximation (SIA) model written on strong form is commonly used for inferring the dynamics of ice sheets and glaciers. The model describes non-Newtonian, viscous, and gravity driven flow of ice in grounded ice sheets. The solution to the SIA model is a closed-form expression for the velocity field. A disadvantage is that when using the SIA velocities to advance the ice surface in time, the time step restriction has a 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 formulation by adding in the normal stress terms which are originally neglected. This allows for a linear time step restriction across the whole range of the horizontal mesh sizes and as such leads to a computationally more efficient SIA model. To support the numerical results we theoretically show that the addition of the FSSA stabilization terms switches the explicit time stepping treatment of the second derivative surface terms to an implicit time stepping treatment. In addition we perform a computational cost analysis, which, when combined with the numerical results on stability properties and accuracy, speaks for favouring SIA models on weak form over the standard SIA model.

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.

Dual-path is a popular architecture for speech separation models (e.g. Sepformer) which splits long sequences into overlapping chunks for its intra- and inter-blocks that separately model intra-chunk local features and inter-chunk global relationships. However, it has been found that inter-blocks, which comprise half a dual-path model's parameters, contribute minimally to performance. Thus, we propose the Single-Path Global Modulation (SPGM) block to replace inter-blocks. SPGM is named after its structure consisting of a parameter-free global pooling module followed by a modulation module comprising only 2% of the model's total parameters. The SPGM block allows all transformer layers in the model to be dedicated to local feature modelling, making the overall model single-path. SPGM achieves 22.1 dB SI-SDRi on WSJ0-2Mix and 20.4 dB SI-SDRi on Libri2Mix, exceeding the performance of Sepformer by 0.5 dB and 0.3 dB respectively and matches the performance of recent SOTA models with up to 8 times fewer parameters. Model and weights are available at huggingface.co/yipjiaqi/spgm

In this manuscript, we combine non-intrusive reduced order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM. The prediction of the mixed formulation is given by a convex linear combination of the predictions of some previously-trained ROMs, where we assign to each model a space-dependent weight. The ROMs taken into account to build the mixed model exploit different reduction techniques, such as Proper Orthogonal Decomposition (POD) and AutoEncoders (AE), and/or different approximation techniques, namely a Radial Basis Function Interpolation (RBF), a Gaussian Process Regression (GPR) or a feed-forward Artificial Neural Network (ANN). The contribution of each model is retained with higher weights in the regions where the model performs best, and, vice versa, with smaller weights where the model has a lower accuracy with respect to the other models. Finally, a regression technique, namely a Random Forest, is exploited to evaluate the weights for unseen conditions. The performance of the aggregated model is evaluated on two different test cases: the 2D flow past a NACA 4412 airfoil, with an angle of attack of 5 degrees, having as parameter the Reynolds number varying between 1e5 and 1e6 and a transonic flow over a NACA 0012 airfoil, considering as parameter the angle of attack. In both cases, the mixed-ROM has provided improved accuracy with respect to each individual ROM technique.

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.

Innovative solution for addressing the challenges in the legal records management system through a blockchain-based eVault platform. Our objective is to create a secure, transparent, and accessible ecosystem that caters to the needs of all stakeholders, including lawyers, judges, clients, and registrars. First and foremost, our solution is built on a robust blockchain platform like Ethereum harnessing the power of smart contracts to manage access, permissions, and transactions effectively. This ensures the utmost security and transparency in every interaction within the system. To make our eVault system user-friendly, we've developed intuitive interfaces for all stakeholders. Lawyers, judges, clients, and even registrars can effortlessly upload and retrieve legal documents, track changes, and share information within the platform. But that's not all; we've gone a step further by incorporating a document creation and saving feature within our app and website. This feature allows users to generate and securely store legal documents, streamlining the entire documentation process.

This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.