In many cases, federated learning (FL) has to take place in communication constrained scenarios, where we must select a small number of learning nodes to reduce bandwidth consumption. Furthermore, such nodes may also have computational constraints, i.e., they can store small datasets and process and perform as little data processing as possible. In this context, it is of paramount importance to make node selection decisions before the learning process begins, and without labeling information. We tackle this daunting task through a two-pronged approach, where we (i) introduce a new metric called loneliness, defined on unlabeled datasets, and (ii) propose a novel algorithm called Goldilocks to make node selection decisions and identify the data to be labeled. Through both a theoretical and an experimental analysis, we show that loneliness is strongly linked with learning performance (i.e., test accuracy). Furthermore, our performance evaluation, including three state-of-the-art datasets and a comparison against centralized learning, demonstrates that Goldilocks outperforms approaches based upon a balanced label distribution by providing over $70 \%$ accuracy improvement, in spite of being efficient to compute and not using labeling information.

In many scenarios of practical interest, labeled data from a target distribution are scarce while labeled data from a related source distribution are abundant. One particular setting of interest arises when the target label space is a subset of the source label space, leading to the framework of partial domain adaptation (PDA). Typical approaches to PDA involve minimizing a domain alignment term and a weighted empirical loss on the source data, with the aim of transferring knowledge between domains. However, a theoretical basis for this procedure is lacking, and in particular, most existing weighting schemes are heuristic. In this work, we derive generalization bounds for the PDA problem based on partial optimal transport. These bounds corroborate the use of the partial Wasserstein distance as a domain alignment term, and lead to theoretically motivated explicit expressions for the empirical source loss weights. Inspired by these bounds, we devise a practical algorithm for PDA, termed WARMPOT. Through extensive numerical experiments, we show that WARMPOT is competitive with recent approaches, and that our proposed weights improve on existing schemes.

Recent approaches to the tensor completion problem have often overlooked the nonnegative structure of the data. We consider the problem of learning a nonnegative low-rank tensor, and using duality theory, we propose a novel factorization of such tensors. The factorization decouples the nonnegative constraints from the low-rank constraints. The resulting problem is an optimization problem on manifolds, and we propose a variant of Riemannian conjugate gradients to solve it. We test the proposed algorithm across various tasks such as colour image inpainting, video completion, and hyperspectral image completion. Experimental results show that the proposed method outperforms many state-of-the-art tensor completion algorithms.