Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Bayesian Robust PCA <s> For $d$-dimensional tensors with possibly large $d>3$, an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given. <s> BIB001 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Bayesian Robust PCA <s> A hierarchical Bayesian model is considered for decomposing a matrix into low-rank and sparse components, assuming the observed matrix is a superposition of the two. The matrix is assumed noisy, with unknown and possibly non-stationary noise statistics. The Bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low-rank and sparse-outlier contributions; the model is robust to a broad range of noise levels, without having to change model hyperparameter settings. In addition, the Bayesian framework allows exploitation of additional structure in the matrix. For example, in video applications each row (or column) corresponds to a video frame, and we introduce a Markov dependency between consecutive rows in the matrix (corresponding to consecutive frames in the video). The properties of this Markov process are also inferred based on the observed matrix, while simultaneously denoising and recovering the low-rank and sparse components. We compare the Bayesian model to a state-of-the-art optimization-based implementation of robust PCA; considering several examples, we demonstrate competitive performance of the proposed model. <s> BIB002
In BIB002 , a stable robust PCA is modeled as: where The three terms W(DΛ)Z, B. * S and E are the low-rank, the sparse noise and the dense noise terms respectively. If we do not consider the sparse noise term B. * S, then Equation (101) is equivalent to Equation  . If all columns of B. * S are same, then the stable robust PCA becomes to be the PCA model BIB001 . The following considers the probability distributions of all matrices in the right of Equation (101). We assume that w ·i ∼ N (0, Two methods were proposed in BIB002 , that is, Gibbs sampling and VB inference. For the second method, the joint probability distribution is p (X, W, Z, D, Λ, B, S, p, τ, π, v, γ) = p (X|W, Z, D, Λ, B, S, γ)  VB inference is employed to approximating the posterior distribution p (W, Z, D, Λ, B, S, p, τ, π, v, γ|X) . Bayesian robust PCA is robust to different noise levels without changing model hyperparameters and exploits additional structure of the matrix in video applications.