Let ˆf: X 7 Y where ˆf = ˆh g with g: X 7 Z and ˆh: Z 7 Y . Corollary 2. Consider a domainD = (P,f) with data distributionP and ground-truth labeling functionf. A hypothesis is ˆf: X 7 Y, where ˆf = ˆh g withg: X 7 Z and ˆh: Z 7 Y . Here, thiskind ofbound isdeveloped using data distributionPoninput space andlabeling functionf from input tolabel space, which arenot convenient in understanding representation learning, sincePT,PS are data nature and therefore fixed. Theorem 3. (Theorem 1 in the main paper) Consider a mixture of source domainsDπ = Next, we relate the loss on targetDTg to hybrid domain Dhyg, which differs only at the feature marginals. In other words, the equality happens when all distributions are the sameQ1=...=QC.

Domain adaptation (DA) is a specific case of MSDA when we need to transfer from a single source domain to an another target domain available during training.

In 1869, the first draft of the periodic table was published by Russian chemist Dmitri Mendeleev. In terms of data science, his achievement can be viewed as a successful example of feature embedding based on human cognition: chemical properties of all known elements at that time were compressed onto the two-dimensional grid system for tabular display. In this study, we seek to answer the question of whether machine learning can reproduce or recreate the periodic table by using observed physicochemical properties of the elements. To achieve this goal, we developed a periodic table generator (PTG). The PTG is an unsupervised machine learning algorithm based on the generative topographic mapping (GTM), which can automate the translation of high-dimensional data into a tabular form with varying layouts on-demand. The PTG autonomously produced various arrangements of chemical symbols, which organized a two-dimensional array such as Mendeleev's periodic table or three-dimensional spiral table according to the underlying periodicity in the given data. We further showed what the PTG learned from the element data and how the element features, such as melting point and electronegativity, are compressed to the lower-dimensional latent spaces.

Although many algorithms have been designed to construct Bayesian network structures using different approaches and principles, they all employ only two methods: those based on independence criteria, and those based on a scoring function and a search procedure (although some methods combine the two). Within the score+search paradigm, the dominant approach uses local search methods in the space of directed acyclic graphs (DAGs), where the usual choices for defining the elementary modifications (local changes) that can be applied are arc addition, arc deletion, and arc reversal. In this paper, we propose a new local search method that uses a different search space, and which takes account of the concept of equivalence between network structures: restricted acyclic partially directed graphs (RPDAGs). In this way, the number of different configurations of the search space is reduced, thus improving efficiency. Moreover, although the final result must necessarily be a local optimum given the nature of the search method, the topology of the new search space, which avoids making early decisions about the directions of the arcs, may help to find better local optima than those obtained by searching in the DAG space. Detailed results of the evaluation of the proposed search method on several test problems, including the well-known Alarm Monitoring System, are also presented.