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Longitudinal data are important in numerous fields, such as healthcare, sociology and seismology, but real-world datasets present notable challenges for practitioners because they can be high-dimensional, contain structured missingness patterns, and measurement time points can be governed by an unknown stochastic process. While various solutions have been suggested, the majority of them have been designed to account for only one of these challenges. In this work, we propose a flexible and efficient latent-variable model that is capable of addressing all these limitations. Our approach utilizes Gaussian processes to capture temporal correlations between samples and their associated missingness masks as well as to model the underlying point process. We construct our model as a variational autoencoder together with deep neural network parameterised encoder and decoder models, and develop a scalable amortised variational inference approach for efficient model training. We demonstrate competitive performance using both simulated and real datasets.

The deformed energy method has shown to be a good option for dimensional synthesis of mechanisms. In this paper the introduction of some new features to such approach is proposed. First, constraints fixing dimensions of certain links are introduced in the error function of the synthesis problem. Second, requirements on distances between determinate nodes are included in the error function for the analysis of the deformed position problem. Both the overall synthesis error function and the inner analysis error function are optimized using a Sequential Quadratic Problem (SQP) approach. This also reduces the probability of branch or circuit defects. In the case of the inner function analytical derivatives are used, while in the synthesis optimization approximate derivatives have been introduced. Furthermore, constraints are analyzed under two formulations, the Euclidean distance and an alternative approach that uses the previous raised to the power of two. The latter approach is often used in kinematics, and simplifies the computation of derivatives. Some examples are provided to show the convergence order of the error function and the fulfilment of the constraints in both formulations studied under different topological situations or achieved energy levels.

The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given $\ell_2$ error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.

We consider the graphon mean-field system introduced in the work of Bayraktar, Chakraborty, and Wu. It is the large-population limit of a heterogeneously interacting diffusive particle system, where the interaction is of mean-field type with weights characterized by an underlying graphon function. Observing continuous-time trajectories of a finite-population particle system, we build plug-in estimators of the particle densities, drift coefficients, and graphon interaction weights of the mean-field system. Our estimators for the densities and drifts are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function. We prove that the estimator converges to the true graphon function as the number of particles tends to infinity, when all other parameters are properly chosen. Besides, we also justify the pointwise optimality of the density estimator via a minimax analysis over a particular class of particle systems.

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution -- as well as in practice -- it is easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the $2$-parameter case, a $1$-Wasserstein stability result for Hilbert functions with respect to the $1$-presentation distance of Bjerkevik and Lesnick.

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

Collecting large quantities of high-quality data is often prohibitively expensive or impractical, and a crucial bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources like public datasets, data collected under different circumstances, or synthesized by generative models. Blurring distinctions, we refer to such data as `surrogate data'. We define a simple scheme for integrating surrogate data into training and use both theoretical models and empirical studies to explore its behavior. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution; $(ii)$ In order to reap this benefit, it is crucial to use optimally weighted empirical risk minimization; $(iii)$ The test error of models trained on mixtures of real and surrogate data is well described by a scaling law. This can be used to predict the optimal weighting and the gain from surrogate data.

The homogeneity problem for testing if more than two different samples come from the same population is considered for the case of functional data. The methodological results are motivated by the study of homogeneity of electronic devices fabricated by different materials and active layer thicknesses. In the case of normality distribution of the stochastic processes associated with each sample, this problem is known as Functional ANOVA problem and is reduced to test the equality of the mean group functions (FANOVA). The problem is that the current/voltage curves associated with Resistive Random Access Memories (RRAM) are not generated by a Gaussian process so that a different approach is necessary for testing homogeneity. To solve this problem two different parametric and nonparametric approaches based on basis expansion of the sample curves are proposed. The first consists of testing multivariate homogeneity tests on a vector of basis coefficients of the sample curves. The second is based on dimension reduction by using functional principal component analysis of the sample curves (FPCA) and testing multivariate homogeneity on a vector of principal components scores. Different approximation numerical techniques are employed to adapt the experimental data for the statistical study. An extensive simulation study is developed for analyzing the performance of both approaches in the parametric and non-parametric cases. Finally, the proposed methodologies are applied on three samples of experimental reset curves measured in three different RRAM technologies.

Knowledge graphs (KGs) of real-world facts about entities and their relationships are useful resources for a variety of natural language processing tasks. However, because knowledge graphs are typically incomplete, it is useful to perform knowledge graph completion or link prediction, i.e. predict whether a relationship not in the knowledge graph is likely to be true. This paper serves as a comprehensive survey of embedding models of entities and relationships for knowledge graph completion, summarizing up-to-date experimental results on standard benchmark datasets and pointing out potential future research directions.