# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
## 2 Related Work
### Relational Learning  
In many real-world applications, data generally share some kinds of relations, such as geometrical or semantic similarity, links or citations. This relation information encodedinside data provides valuable evidence for some issues, such as classification and retrieval. To the end, relational learning is generally integrated into the representation learning models. In [36], [PERSON] et al. prove that it is possible to use only proximity measure (distances or similarities) to represent the samples rather than mapping the feature vectors to the low-dimensional space. In addition, they propose a proximity description-based dimensionality reduction method called relational discriminant analysis (RDA) in [24]. Instead of data, RDA uses similarities to a subset of objects in the training data as features. In this case, dimensionality reduction can be conducted either by selection methods (such as random selection [37], systematic selection [38]), or by feature extraction methods (such as multi-dimensional scaling [39], Sammon mapping [40] and Niemann mapping [41]). In [25], [PERSON] et al. model the covariance of data with the relationships between instances and propose a Gaussian latent variable model which successfully integrates relational information into the dimensionality reduction process, called probabilistic relational PCA (PRPCA). In PRPCA, relational information is defined by the relevance between data samples. We take the scientific paper citation as an example. If there is a quoting between the papers, it means that these papers most likely have similar topics. To take the inter-influence between cited papers into account, [PERSON] et al. further construct a matrix \(\mathbf{\Phi}=\mathbf{\Delta}^{-1}\), which satisfy the condition that similar instances often have a lower probability density at the latent space. To the end, PRPCA, based on the relational covariance \(\mathbf{\Phi}\), successfully applies the relational information to the dimensionality reduction algorithms. Relational learning is also commonly used for data mining, information retrieval and other machine learning-related applications. [PERSON] et al. [42] propose a method, called linear relational embedding, for the distributed representations of data, where data consist of the relationship of concepts. [PERSON] et al. [43] utilize the characteristic that existing relations between items are often useful in recommendation systems and propose a model called relational collaborative topic regression (RCTR), which expand the traditional CTR model by integrating feedback information, item content information and relational information. [PERSON] et al. [44] propose a nonparametric relational topic model using stochastic processes instead of fixed-dimensional probability distributions. Based on the classical works mentioned above, we propose a general and effective dimensionality reduction framework named relational Fisher analysis (RFA). This framework uses graph embedding [14] as theoretical foundation and integrates the relational information [24; 25] encoded inside data into the dimensionality reduction process. Besides the intrinsic graph and the penalty graph as defined using graph embedding, we further construct a relational graph based on the existing relationships between data, which enables the desired low-dimensional space to preserve the intrinsic information, reduce the penalty information and further learn and preserve the relational information among the data samples. In addition, through the derivation and equivalent transformation operations, the objective function of our proposed method can be transformed into the trace ratio form for optimization. Based on a systematic analysis of two optimization method for trace ratio problems [33], we propose a novel iterative algorithm which uses the value of the trace ratio as criterion for the algorithmic convergence.