One crucial assumption made by both principal component analysis (PCA) and probabilistic PCA (PPCA) is that the instances are independent and identically distributed (i.i.d.). In this paper, by explicitly modeling covariance between instances as derived from the relational information, we propose a novel probabilistic dimensionality reduction method, called probabilistic relational PCA (PRPCA), for relational data analysis. Experiments on real-world data sets show that PRPCA can effectively utilize the relational information to dramatically outperform PCA and achieve state-of-the-art performance.

Most high-dimensional matrix recovery problems are studied under the assumption that the target matrix has certain intrinsic structures. For image data related matrix recovery problems, approximate low-rankness and smoothness are the two most commonly imposed structures. For approximately low-rank matrix recovery, the robust principal component analysis (PCA) is well-studied and proved to be effective. For smooth matrix problem, 2d fused Lasso and other total variation based approaches have played a fundamental role. Although both low-rankness and smoothness are key assumptions for image data analysis, the two lines of research, however, have very limited interaction. Motivated by taking advantage of both features, we in this paper develop a framework named projected robust PCA (PRPCA), under which the low-rank matrices are projected onto a space of smooth matrices. Consequently, a large class of piecewise smooth matrices can be decomposed as a low-rank and smooth component plus a sparse component. A key advantage of this decomposition is that the dimension of the core low-rank component can be significantly reduced. Consequently, our framework is able to address a problematic bottleneck of many low-rank matrix problems: singular value decomposition (SVD) on large matrices. Moreover, we provide the identifiability results along with explicit statistical recovery guarantees of PRPCA. Our results include classical robust PCA as a special case.

One crucial assumption made by both principal component analysis (PCA) and probabilistic PCA (PPCA) is that the instances are independent and identically distributed (i.i.d.). In this paper, by explicitly modeling covariance between instances as derived from the relational information, we propose a novel probabilistic dimensionality reduction method, called probabilistic relational PCA (PRPCA), for relational data analysis. Experiments on real-world data sets show that PRPCA can effectively utilize the relational information to dramatically outperform PCA and achieve state-of-the-art performance. Papers published at the Neural Information Processing Systems Conference.

Probabilistic relational PCA (PRPCA) can learn a projection matrix to perform dimensionality reduction for relational data. However, the results learned by PRPCA lack interpretability because each principal component is a linear combination of all the original variables. In this paper, we propose a novel model, called sparse probabilistic relational projection (SPRP), to learn a sparse projection matrix for relational dimensionality reduction. The sparsity in SPRP is achieved by imposing on the projection matrix a sparsity-inducing prior such as the Laplace prior or Jeffreys prior. We propose an expectation-maximization (EM) algorithm to learn the parameters of SPRP. Compared with PRPCA, the sparsity in SPRP not only makes the results more interpretable but also makes the projection operation much more efficient without compromising its accuracy. All these are verified by experiments conducted on several real applications.

One crucial assumption made by both principal component analysis (PCA) and probabilistic PCA (PPCA) is that the instances are independent and identically distributed (i.i.d.). However, this common i.i.d. assumption is unreasonable for relational data. In this paper, by explicitly modeling covariance between instances as derived from the relational information, we propose a novel probabilistic dimensionality reduction method, called probabilistic relational PCA (PRPCA), for relational data analysis. Although the i.i.d. assumption is no longer adopted in PRPCA, the learning algorithms for PRPCA can still be devised easily like those for PPCA which makes explicit use of the i.i.d. assumption. Experiments on real-world data sets show that PRPCA can effectively utilize the relational information to dramatically outperform PCA and achieve state-of-the-art performance.