Tensor Gaussian graphical models (GGMs), interpreting conditional independence structures within tensor data, have important applications in numerous areas. Yet, the available tensor data in one single study is often limited due to high acquisition costs. Although relevant studies can provide additional data, it remains an open question how to pool such heterogeneous data. In this paper, we propose a transfer learning framework for tensor GGMs, which takes full advantage of informative auxiliary domains even when non-informative auxiliary domains are present, benefiting from the carefully designed data-adaptive weights. Our theoretical analysis shows substantial improvement of estimation errors and variable selection consistency on the target domain under much relaxed conditions, by leveraging information from auxiliary domains. Extensive numerical experiments are conducted on both synthetic tensor graphs and a brain functional connectivity network data, which demonstrates the satisfactory performance of the proposed method.

Normalizing Flows (NFs) are emerging as a powerful class of generative models, as they not only allow for efficient sampling, but also deliver, by construction, density estimation. They are of great potential usage in High Energy Physics (HEP), where complex high dimensional data and probability distributions are everyday's meal. However, in order to fully leverage the potential of NFs it is crucial to explore their robustness as data dimensionality increases. Thus, in this contribution, we discuss the performances of some of the most popular types of NFs on the market, on some toy data sets with increasing number of dimensions.

Locality preserving projection (LPP) is a well-known method for dimensionality reduction in which the neighborhood graph structure of data is preserved. Traditional LPP employ squared F-norm for distance measurement. This may exaggerate more distance errors, and result in a model being sensitive to outliers. In order to deal with this issue, we propose two novel F-norm-based models, termed as F-LPP and F-2DLPP, which are developed for vector-based and matrix-based data, respectively. In F-LPP and F-2DLPP, the distance of data projected to a low dimensional space is measured by F-norm. Thus it is anticipated that both methods can reduce the influence of outliers. To solve the F-norm-based models, we propose an iterative optimization algorithm, and give the convergence analysis of algorithm. The experimental results on three public databases have demonstrated the effectiveness of our proposed methods.

Two-dimensional principle component analysis (2DPCA) has been widely used for face image representation and recognition. But it is sensitive to the presence of outliers. To alleviate this problem, we propose a novel robust 2DPCA, namely 2DPCA with F-norm minimization (F-2DPCA), which is intuitive and directly derived from 2DPCA. In F-2DPCA, distance in spatial dimensions (attribute dimensions) is measured in F-norm, while the summation over different data points uses 1-norm. Thus it is robust to outliers and rotational invariant as well. To solve F-2DPCA, we propose a fast iterative algorithm, which has a closed-form solution in each iteration, and prove its convergence. Experimental results on face image databases illustrate its effectiveness and advantages.

Dimensionality reduction plays an important role in many machine learning and pattern recognition tasks. In this paper, we present a novel dimensionality reduction algorithm called multilinear maximum distance embedding (M2DE), which includes three key components. To preserve the local geometry and discriminant information in the embedded space, M2DE utilizes a new objective function, which aims to maximize the distances between some particular pairs of data points, such as the distances between nearby points and the distances between data points from different classes. To make the mapping of new data points straightforward, and more importantly, to keep the natural tensor structure of high-order data, M2DE integrates multilinear techniques to learn the transformation matrices sequentially. To provide reasonable and stable embedding results, M2DE employs the L1-norm, which is more robust to outliers, to measure the dissimilarity between data points. Experiments on various datasets demonstrate that M2DE achieves good embedding results of high-order data for classification tasks.