Matrix Completion on Graphs

The problem of finding the missing values of a matrix given a few of its entries, called matrix completion, has gathered a lot of attention in the recent years. Although the problem under the standard low rank assumption is NP-hard, Cand\`es and Recht showed that it can be exactly relaxed if the number of observed entries is sufficiently large. In this work, we introduce a novel matrix completion model that makes use of proximity information about rows and columns by assuming they form communities. This assumption makes sense in several real-world problems like in recommender systems, where there are communities of people sharing preferences, while products form clusters that receive similar ratings. Our main goal is thus to find a low-rank solution that is structured by the proximities of rows and columns encoded by graphs. We borrow ideas from manifold learning to constrain our solution to be smooth on these graphs, in order to implicitly force row and column proximities. Our matrix recovery model is formulated as a convex non-smooth optimization problem, for which a well-posed iterative scheme is provided. We study and evaluate the proposed matrix completion on synthetic and real data, showing that the proposed structured low-rank recovery model outperforms the standard matrix completion model in many situations.Comment: Version of NIPS 2014 workshop "Out of the Box: Robustness in High Dimension

Robust Principal Component Analysis on Graphs

Principal Component Analysis (PCA) is the most widely used tool for linear dimensionality reduction and clustering. Still it is highly sensitive to outliers and does not scale well with respect to the number of data samples. Robust PCA solves the first issue with a sparse penalty term. The second issue can be handled with the matrix factorization model, which is however non-convex. Besides, PCA based clustering can also be enhanced by using a graph of data similarity. In this article, we introduce a new model called "Robust PCA on Graphs" which incorporates spectral graph regularization into the Robust PCA framework. Our proposed model benefits from 1) the robustness of principal components to occlusions and missing values, 2) enhanced low-rank recovery, 3) improved clustering property due to the graph smoothness assumption on the low-rank matrix, and 4) convexity of the resulting optimization problem. Extensive experiments on 8 benchmark, 3 video and 2 artificial datasets with corruptions clearly reveal that our model outperforms 10 other state-of-the-art models in its clustering and low-rank recovery tasks

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

Large Scale Graph Learning from Smooth Signals

Graphs are a prevalent tool in data science, as they model the inherent structure of the data. They have been used successfully in unsupervised and semi-supervised learning. Typically they are constructed either by connecting nearest samples, or by learning them from data, solving an optimization problem. While graph learning does achieve a better quality, it also comes with a higher computational cost. In particular, the current state-of-the-art model cost is O(n^2) for n samples. In this paper, we show how to scale it, obtaining an approximation with leading cost of O(n log(n)), with quality that approaches the exact graph learning model. Our algorithm uses known approximate nearest neighbor techniques to reduce the number of variables, and automatically selects the correct parameters of the model, requiring a single intuitive input: the desired edge density

PCA using graph total variation

Mining useful clusters from high dimensional data has received sig- nificant attention of the signal processing and machine learning com- munity in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of di- mensionality. However, often such methods are accompanied with problems such as high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix fac- torization methods) or susceptibility to gross corruptions in the data. In this paper we propose a convex, robust, scalable and efficient Prin- cipal Component Analysis (PCA) based method to approximate the low-rank representation of high dimensional datasets via a two-way graph regularization scheme. Compared to the exact recovery meth- ods, our method is approximate, in that it enforces a piecewise con- stant assumption on the samples using a graph total variation and a piecewise smoothness assumption on the features using a graph Tikhonov regularization. Futhermore, it retrieves the low-rank rep- resentation in a time that is linear in the number of data samples. Clustering experiments on 3 benchmark datasets with different types of corruptions show that our proposed model outperforms 7 state-of- the-art dimensionality reduction models

Investigating advanced transaction models for federated database systems

Transaction management in federated database systems is a very difficult problem. In the literature, there are a lot of proposals for transaction models in federated database systems. However, all these models base on different assumptions or are restricted to certain application scenarios. In this paper, we analyze the underlying basic transaction types in order to have a foundation for comparing these models and for investigating their applicability in federated database system environments. (orig.)SIGLEAvailable from TIB Hannover: RR 4487(1997,7) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Learning time varying graphs

We consider the problem of inferring the hidden structure of high-dimensional dynamic systems from the perspective of graph learning. In particular, we aim at capturing the dynamic relationships between nodes by a sequence of graphs. Our approach is motivated by the observation that imposing a meaningful graph topology to the data decreases the number of samples necessary to learn meaningful structures. To regularize learning and improve performance on dynamic graphs, we introduce a new prior that asserts that the graph edges change smoothly in time. We propose a primal-dual optimization algorithm that scales linearly with the number of allowed edges and is easy to parallelize. Our new model is shown to outperform standard graph learning and other baselines both on a synthetic and a real dataset