TASK DEFINITION: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
PROBLEM: The paper addresses the problem of learning a regression model parameterized by a fixedrank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks.

SOLUTION: Regression on fixed-rank positive semidefinite matrices: a Riemannian approach

PROBLEM: Our experience of the world is multimodal we see objects, hear sounds, feel texture, smell odors, and taste flavors. Modality refers to the way in which something happens or is experienced and a research problem is characterized as multimodal when it includes multiple such modalities. In order for Artificial Intelligence to make progress in understanding the world around us, it needs to be able to interpret such multimodal signals together. Multimodal machine learning aims to build models that can process and relate information from multiple modalities. It is a vibrant multi-disciplinary field of increasing importance and with extraordinary potential. Instead of focusing on specific multimodal applications, this paper surveys the recent advances in multimodal machine learning itself and presents them in a common taxonomy. We go beyond the typical early and late fusion categorization and identify broader challenges that are faced by multimodal machine learning, namely: representation, translation, alignment, fusion, and co-learning. This new taxonomy will enable researchers to better understand the state of the field and identify directions for future research.

SOLUTION: Multimodal Machine Learning: A Survey and Taxonomy

PROBLEM: Composition of low-dimensional distribu­ tions, whose foundations were laid in the pa­ per published in the Proceedings of UAI'97 (Jirousek 1997), appeared to be an alterna­ tive apparatus to describe multidimensional probabilistic models. In contrast to Graphi­ cal Markov Models, which define multidimen­ sional distributions in a declarative way, this approach is rather procedural. Ordering of low-dimensional distributions into a proper sequence fully defines the respective compu­ tational procedure; therefore, a study of dif­ ferent types of generating sequences is one of the central problems in this field. Thus, it ap­ pears that an important role is played by spe­ cial sequences that are called perfect. Their main characterization theorems are presented in this paper. However, the main result of this paper is a solution to the problem of marginalization for general sequences. The main theorem describes a way to obtain a generating sequence that defines the model corresponding to the marginal of the distri­ bution defined by an arbitrary generating se­ quence. From this theorem the reader can see to what extent these computations are lo­ cal; i.e., the sequence consists of marginal dis­ tributions whose computation must be made by summing up over the values of the vari­ able eliminated (the paper deals with a finite model) .

SOLUTION:
Marginalization in Composed Probabilistic Models