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Multivariate functional data present theoretical and practical complications which are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. We introduce a novel model for multivariate functional data that incorporates such mutual time warping via nonlinear transport functions. This model allows for meaningful interpretation and is well suited to represent functional vector data. The proposed approach combines a random amplitude factor for each component with population based registration across the components of a multivariate functional data vector and also includes a latent population function, which corresponds to a common underlying trajectory as well as subject-specific warping component. We also propose estimators for all components of the model. The proposed approach not only leads to a novel representation for multivariate functional data, but is also useful for downstream analyses such as Fr\'echet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations and practical aspects are illustrated in simulations and with application to multivariate human growth curves as well as multivariate environmental pollution data.

Time-series generated by end-users, edge devices, and different wearables are mostly unlabelled. We propose a method to auto-generate labels of un-labelled time-series, exploiting very few representative labelled time-series. Our method is based on representation learning using Auto Encoded Compact Sequence (AECS) with a choice of best distance measure. It performs self-correction in iterations, by learning latent structure, as well as synthetically boosting representative time-series using Variational-Auto-Encoder (VAE) to improve the quality of labels. We have experimented with UCR and UCI archives, public real-world univariate, multivariate time-series taken from different application domains. Experimental results demonstrate that the proposed method is very close to the performance achieved by fully supervised classification. The proposed method not only produces close to benchmark results but outperforms the benchmark performance in some cases.

In this paper, we extend the epsilon admissible subsets (EAS) model selection approach, from its original construction in the high-dimensional linear regression setting, to an EAS framework for performing group variable selection in the high-dimensional multivariate regression setting. Assuming a matrix-Normal linear model we show that the EAS strategy is asymptotically consistent if there exists a sparse, true data generating set of predictors. Nonetheless, our EAS strategy is designed to estimate a posterior-like, generalized fiducial distribution over a parsimonious class of models in the setting of correlated predictors and/or in the absence of a sparsity assumption. The effectiveness of our approach, to this end, is demonstrated empirically in simulation studies, and is compared to other state-of-the-art model/variable selection procedures.

This work is concerned with approximating matrix functions for banded matrices, hierarchically semiseparable matrices, and related structures. We develop a new divide-and-conquer method based on (rational) Krylov subspace methods for performing low-rank updates of matrix functions. Our convergence analysis of the newly proposed method proceeds by establishing relations to best polynomial and rational approximation. When only the trace or the diagonal of the matrix function is of interest, we demonstrate -- in practice and in theory -- that convergence can be faster. For the special case of a banded matrix, we show that the divide-and-conquer method reduces to a much simpler algorithm, which proceeds by computing matrix functions of small submatrices. Numerical experiments confirm the effectiveness of the newly developed algorithms for computing large-scale matrix functions from a wide variety of applications.

Structured Latent Attribute Models (SLAMs) are a family of discrete latent variable models widely used in education, psychology, and epidemiology to model multivariate categorical data. A SLAM assumes that multiple discrete latent attributes explain the dependence of observed variables in a highly structured fashion. Usually, the maximum marginal likelihood estimation approach is adopted for SLAMs, treating the latent attributes as random effects. The increasing scope of modern assessment data involves large numbers of observed variables and high-dimensional latent attributes. This poses challenges to classical estimation methods and requires new methodology and understanding of latent variable modeling. Motivated by this, we consider the joint maximum likelihood estimation (MLE) approach to SLAMs, treating latent attributes as fixed unknown parameters. We investigate estimability, consistency, and computation in the regime where sample size, number of variables, and number of latent attributes all can diverge. We establish the statistical consistency of the joint MLE and propose efficient algorithms that scale well to large-scale data for several popular SLAMs. Simulation studies demonstrate the superior empirical performance of the proposed methods. An application to real data from an international educational assessment gives interpretable findings of cognitive diagnosis.

Knowledge graph (KG) embeddings learn low-dimensional representations of entities and relations to predict missing facts. KGs often exhibit hierarchical and logical patterns which must be preserved in the embedding space. For hierarchical data, hyperbolic embedding methods have shown promise for high-fidelity and parsimonious representations. However, existing hyperbolic embedding methods do not account for the rich logical patterns in KGs. In this work, we introduce a class of hyperbolic KG embedding models that simultaneously capture hierarchical and logical patterns. Our approach combines hyperbolic reflections and rotations with attention to model complex relational patterns. Experimental results on standard KG benchmarks show that our method improves over previous Euclidean- and hyperbolic-based efforts by up to 6.1% in mean reciprocal rank (MRR) in low dimensions. Furthermore, we observe that different geometric transformations capture different types of relations while attention-based transformations generalize to multiple relations. In high dimensions, our approach yields new state-of-the-art MRRs of 49.6% on WN18RR and 57.7% on YAGO3-10.

Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

Meta learning is a promising solution to few-shot learning problems. However, existing meta learning methods are restricted to the scenarios where training and application tasks share the same out-put structure. To obtain a meta model applicable to the tasks with new structures, it is required to collect new training data and repeat the time-consuming meta training procedure. This makes them inefficient or even inapplicable in learning to solve heterogeneous few-shot learning tasks. We thus develop a novel and principled HierarchicalMeta Learning (HML) method. Different from existing methods that only focus on optimizing the adaptability of a meta model to similar tasks, HML also explicitly optimizes its generalizability across heterogeneous tasks. To this end, HML first factorizes a set of similar training tasks into heterogeneous ones and trains the meta model over them at two levels to maximize adaptation and generalization performance respectively. The resultant model can then directly generalize to new tasks. Extensive experiments on few-shot classification and regression problems clearly demonstrate the superiority of HML over fine-tuning and state-of-the-art meta learning approaches in terms of generalization across heterogeneous tasks.

Purpose: MR image reconstruction exploits regularization to compensate for missing k-space data. In this work, we propose to learn the probability distribution of MR image patches with neural networks and use this distribution as prior information constraining images during reconstruction, effectively employing it as regularization. Methods: We use variational autoencoders (VAE) to learn the distribution of MR image patches, which models the high-dimensional distribution by a latent parameter model of lower dimensions in a non-linear fashion. The proposed algorithm uses the learned prior in a Maximum-A-Posteriori estimation formulation. We evaluate the proposed reconstruction method with T1 weighted images and also apply our method on images with white matter lesions. Results: Visual evaluation of the samples showed that the VAE algorithm can approximate the distribution of MR patches well. The proposed reconstruction algorithm using the VAE prior produced high quality reconstructions. The algorithm achieved normalized RMSE, CNR and CN values of 2.77\%, 0.43, 0.11; 4.29\%, 0.43, 0.11, 6.36\%, 0.47, 0.11 and 10.00\%, 0.42, 0.10 for undersampling ratios of 2, 3, 4 and 5, respectively, where it outperformed most of the alternative methods. In the experiments on images with white matter lesions, the method faithfully reconstructed the lesions. Conclusion: We introduced a novel method for MR reconstruction, which takes a new perspective on regularization by using priors learned by neural networks. Results suggest the method compares favorably against the other evaluated methods and can reconstruct lesions as well. Keywords: Reconstruction, MRI, prior probability, MAP estimation, machine learning, variational inference, deep learning