# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
###### Abstract  
Most of the dimensionality reduction algorithms assume that data are independent and identically distributed (i.i.d.). In real-world applications, however, sometimes there exist relationships between data. Some relational learning methods have been proposed, but those with discriminative relationship analysis are lacking yet, as important supervisory information is usually ignored. In this paper, we propose a novel and general framework, called relational Fisher analysis (RFA), which successfully integrates relational information into the dimensionality reduction model. For nonlinear data representation learning, we adopt the kernel trick to RFA and propose the kernelized RFA (KRFA). In addition, the convergence of the RFA optimization algorithm is proved theoretically. By leveraging suitable strategies to construct the relational matrix, we conduct extensive experiments to demonstrate the superiority of our RFA and KRFA methods over related approaches. relational learning; dimensionality reduction; graph embedding; trace ratio; document understanding and recognition; face recognition +
Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/)]([https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/))). Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/0](https://creativecommons.org/licenses/py/4.0/0)]([https://creativecommons.org/licenses/py/4.0/0](https://creativecommons.org/licenses/py/4.0/0))). structure of data. Nevertheless, most of these manifold learning methods disregard label information when recovering the low-dimensional manifold structure, in that they are inherently unsupervised. Although the above-mentioned methods are defined from different perspectives of dimensionality reduction, graph embedding provide a unified framework for understanding and comparing them [14]. Furthermore, by integrating label information in the computation of intrinsic and penalty graphs within the graph embedding framework, a supervised dimensionality reduction method called marginal Fisher analysis (MFA) is proposed [14]. In traditional dimensionality reduction algorithms as described above, a data distribution assumption is generally applied that data are independent and identically distributed (i.i.d.). However, in real-world applications, there are often certain relativity or links between certain data, for instance, geometrical or semantic similarity, links among web pages, citation relations between scientific papers. Relationships usually indicate that these related samples are likely to have similarities or belonging to the same class. Nevertheless, although some dimensionality reduction methods consider to preserve the locality of data [15; 16; 17; 18], the useful relationships are often simply ignored during the learning process of most existing dimensionality reduction methods. Recently, relational learning has often been used in practical applications, for instance, web mining [19] and social network analysis [20]. In addition, relational information is also considered in social network discovery, document classification, sequential data analysis and semi-supervised graph embedding [21; 22; 23]. In the domain of dimensionality reduction, some algorithms have already been proposed in which the relational information is integrated into the representation learning process. In [24], [PERSON] et al. propose the relational discriminant analysis (RDA). In RDA, relationships among data are measured by the Euclidean distance between objects and prototypes or support objects of each class. However, RDA uses mean squared error as the objective function, and cannot perform well on multi-class learning problems. In [25], [PERSON] et al.