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The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials--truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.

Identifying replicable signals across different studies provides stronger scientific evidence and more powerful inference. Existing literature on high dimensional applicability analysis either imposes strong modeling assumptions or has low power. We develop a powerful and robust empirical Bayes approach for high dimensional replicability analysis. Our method effectively borrows information from different features and studies while accounting for heterogeneity. We show that the proposed method has better power than competing methods while controlling the false discovery rate, both empirically and theoretically. Analyzing datasets from the genome-wide association studies reveals new biological insights that otherwise cannot be obtained by using existing methods.

This article presents factor copula approaches to model temporal dependency of non-Gaussian (continuous/discrete) longitudinal data. Factor copula models are canonical vine copulas which explain the underlying dependence structure of a multivariate data through latent variables, and therefore can be easily interpreted and implemented to unbalanced longitudinal data. We develop regression models for continuous, binary and ordinal longitudinal data including covariates, by using factor copula constructions with subject-specific latent variables. Considering homogeneous within-subject dependence, our proposed models allow for feasible parametric inference in moderate to high dimensional situations, using two-stage (IFM) estimation method. We assess the finite sample performance of the proposed models with extensive simulation studies. In the empirical analysis, the proposed models are applied for analysing different longitudinal responses of two real world data sets. Moreover, we compare the performances of these models with some widely used random effect models using standard model selection techniques and find substantial improvements. Our studies suggest that factor copula models can be good alternatives to random effect models and can provide better insights to temporal dependency of longitudinal data of arbitrary nature.

Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal.

Phase-field models of fatigue are capable of reproducing the main phenomenology of fatigue behavior. However, phase-field computations in the high-cycle fatigue regime are prohibitively expensive, due to the need to resolve spatially the small length scale inherent to phase-field models and temporally the loading history for several millions of cycles. As a remedy, we propose a fully adaptive acceleration scheme based on the cycle jump technique, where the cycle-by-cycle resolution of an appropriately determined number of cycles is skipped while predicting the local system evolution during the jump. The novelty of our approach is a cycle-jump criterion to determine the appropriate cycle-jump size based on a target increment of a global variable which monitors the advancement of fatigue. We propose the definition and meaning of this variable for three general stages of the fatigue life. In comparison to existing acceleration techniques, our approach needs no parameters and bounds for the cycle-jump size, and it works independently of the material, specimen or loading conditions. Since one of the monitoring variables is the fatigue crack length, we introduce an accurate, flexible and efficient method for its computation, which overcomes the issues of conventional crack tip tracking algorithms and enables the consideration of several cracks evolving at the same time. The performance of the proposed acceleration scheme is demonstrated with representative numerical examples, which show a speedup reaching four orders of magnitude in the high-cycle fatigue regime with consistently high accuracy.

Longitudinal studies are frequently used in medical research and involve collecting repeated measures on individuals over time. Observations from the same individual are invariably correlated and thus an analytic approach that accounts for this clustering by individual is required. While almost all research suffers from missing data, this can be particularly problematic in longitudinal studies as participation often becomes harder to maintain over time. Multiple imputation (MI) is widely used to handle missing data in such studies. When using MI, it is important that the imputation model is compatible with the proposed analysis model. In a longitudinal analysis, this implies that the clustering considered in the analysis model should be reflected in the imputation process. Several MI approaches have been proposed to impute incomplete longitudinal data, such as treating repeated measurements of the same variable as distinct variables or using generalized linear mixed imputation models. However, the uptake of these methods has been limited, as they require additional data manipulation and use of advanced imputation procedures. In this tutorial, we review the available MI approaches that can be used for handling incomplete longitudinal data, including where individuals are clustered within higher-level clusters. We illustrate implementation with replicable R and Stata code using a case study from the Childhood to Adolescence Transition Study.

In toxicology research, experiments are often conducted to determine the effect of toxicant exposure on the behavior of mice, where mice are randomized to receive the toxicant or not. In particular, in fixed interval experiments, one provides a mouse reinforcers (e.g., a food pellet), contingent upon some action taken by the mouse (e.g., a press of a lever), but the reinforcers are only provided after fixed time intervals. Often, to analyze fixed interval experiments, one specifies and then estimates the conditional state-action distribution (e.g., using an ANOVA). This existing approach, which in the reinforcement learning framework would be called modeling the mouse's "behavioral policy," is sensitive to misspecification. It is likely that any model for the behavioral policy is misspecified; a mapping from a mouse's exposure to their actions can be highly complex. In this work, we avoid specifying the behavioral policy by instead learning the mouse's reward function. Specifying a reward function is as challenging as specifying a behavioral policy, but we propose a novel approach that incorporates knowledge of the optimal behavior, which is often known to the experimenter, to avoid specifying the reward function itself. In particular, we define the reward as a divergence of the mouse's actions from optimality, where the representations of the action and optimality can be arbitrarily complex. The parameters of the reward function then serve as a measure of the mouse's tolerance for divergence from optimality, which is a novel summary of the impact of the exposure. The parameter itself is scalar, and the proposed objective function is differentiable, allowing us to benefit from typical results on consistency of parametric estimators while making very few assumptions.

We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.

Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since Hall, M\"uller and Wang (2006)'s result for a fixed number of eigenfunction estimates. In this work, we aim to establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the $\mathcal{L}^2$ and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the $\mathcal{L}^{2}$ bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. We also develop a double truncation technique to handle the uniform convergence of estimated covariance and eigenfunctions. The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed functional data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.