Fast Robust PCA on Graphs

Mining useful clusters from high dimensional data has received significant attention of the computer vision and pattern recognition community in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix factorization methods) and susceptibility to gross corruptions in the data. In this paper we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient and scalable for huge datasets with O(nlog(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data

Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Many real world datasets subsume a linear or non-linear low-rank structure in a very low-dimensional space. Unfortunately, one often has very little or no information about the geometry of the space, resulting in a highly under-determined recovery problem. Under certain circumstances, state-of-the-art algorithms provide an exact recovery for linear low-rank structures but at the expense of highly inscalable algorithms which use nuclear norm. However, the case of non-linear structures remains unresolved. We revisit the problem of low-rank recovery from a totally different perspective, involving graphs which encode pairwise similarity between the data samples and features. Surprisingly, our analysis confirms that it is possible to recover many approximate linear and non-linear low-rank structures with recovery guarantees with a set of highly scalable and efficient algorithms. We call such data matrices as \textit{Low-Rank matrices on graphs} and show that many real world datasets satisfy this assumption approximately due to underlying stationarity. Our detailed theoretical and experimental analysis unveils the power of the simple, yet very novel recovery framework \textit{Fast Robust PCA on Graphs

Scalable Low-rank Matrix and Tensor Decomposition on Graphs

In many signal processing, machine learning and computer vision applications, one often has to deal with high dimensional and big datasets such as images, videos, web content, etc. The data can come in various forms, such as univariate or multivariate time series, matrices or high dimensional tensors. The goal of the data mining community is to reveal the hidden linear or non-linear structures in the datasets. Over the past couple of decades matrix factorization, owing to its intrinsic association with dimensionality reduction has been adopted as one of the key methods in this context. One can either use a single linear subspace to approximate the data (the standard Principal Component Analysis (PCA) approach) or a union of low dimensional subspaces where each data class belongs to a different subspace. In many cases, however, the low dimensional data follows some additional structure. Knowledge of such structure is beneficial, as we can use it to enhance the representativity of our models by adding structured priors. A nowadays standard way to represent pairwise affinity between objects is by using graphs. The introduction of graph-based priors to enhance matrix factorization models has recently brought them back to the highest attention of the data mining community. Representation of a signal on a graph is well motivated by the emerging field of signal processing on graphs, based on notions of spectral graph theory. The underlying assumption is that high-dimensional data samples lie on or close to a smooth low-dimensional manifold. Interestingly, the underlying manifold can be represented by its discrete proxy, i.e. a graph. A primary limitation of the state-of-the-art low-rank approximation methods is that they do not generalize for the case of non-linear low-rank structures. Furthermore, the standard low-rank extraction methods for many applications, such as low-rank and sparse decomposition, are computationally cumbersome. We argue, that for many machine learning and signal processing applications involving big data, an approximate low-rank recovery suffices. Thus, in this thesis, we present solutions to the above two limitations by presenting a new framework for scalable but approximate low-rank extraction which exploits the hidden structure in the data using the notion of graphs. First, we present a novel signal model, called `Multilinear low-rank tensors on graphs (MLRTG)' which states that a tensor can be encoded as a multilinear combination of the low-frequency graph eigenvectors, where the graphs are constructed along the various modes of the tensor. Since the graph eigenvectors have the interpretation of \textit{non-linear} embedding of a dataset on the low-dimensional manifold, we propose a method called `Graph Multilinear SVD (GMLSVD)' to recover PCA based linear subspaces from these eigenvectors. Finally, we propose a plethora of highly scalable matrix and tensor based problems for low-rank extraction which implicitly or explicitly make use of the GMLSVD framework. The core idea is to replace the expensive iterative SVD operations by updating the linear subspaces from the fixed non-linear ones via low-cost operations. We present applications in low-rank and sparse decomposition and clustering of the low-rank features to evaluate all the proposed methods. Our theoretical analysis shows that the approximation error of the proposed framework depends on the spectral properties of the graph Laplacian