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

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

Correlation Clustering

Knowledge Discovery in Databases (KDD) is the non-trivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data. The core step of the KDD process is the application of a Data Mining algorithm in order to produce a particular enumeration of patterns and relationships in large databases. Clustering is one of the major data mining techniques and aims at grouping the data objects into meaningful classes (clusters) such that the similarity of objects within clusters is maximized, and the similarity of objects from different clusters is minimized. This can serve to group customers with similar interests, or to group genes with related functionalities. Currently, a challenge for clustering-techniques are especially high dimensional feature-spaces. Due to modern facilities of data collection, real data sets usually contain many features. These features are often noisy or exhibit correlations among each other. However, since these effects in different parts of the data set are differently relevant, irrelevant features cannot be discarded in advance. The selection of relevant features must therefore be integrated into the data mining technique. Since about 10 years, specialized clustering approaches have been developed to cope with problems in high dimensional data better than classic clustering approaches. Often, however, the different problems of very different nature are not distinguished from one another. A main objective of this thesis is therefore a systematic classification of the diverse approaches developed in recent years according to their task definition, their basic strategy, and their algorithmic approach. We discern as main categories the search for clusters (i) w.r.t. closeness of objects in axis-parallel subspaces, (ii) w.r.t. common behavior (patterns) of objects in axis-parallel subspaces, and (iii) w.r.t. closeness of objects in arbitrarily oriented subspaces (so called correlation cluster). For the third category, the remaining parts of the thesis describe novel approaches. A first approach is the adaptation of density-based clustering to the problem of correlation clustering. The starting point here is the first density-based approach in this field, the algorithm 4C. Subsequently, enhancements and variations of this approach are discussed allowing for a more robust, more efficient, or more effective behavior or even find hierarchies of correlation clusters and the corresponding subspaces. The density-based approach to correlation clustering, however, is fundamentally unable to solve some issues since an analysis of local neighborhoods is required. This is a problem in high dimensional data. Therefore, a novel method is proposed tackling the correlation clustering problem in a global approach. Finally, a method is proposed to derive models for correlation clusters to allow for an interpretation of the clusters and facilitate more thorough analysis in the corresponding domain science. Finally, possible applications of these models are proposed and discussed.Knowledge Discovery in Databases (KDD) ist der Prozess der automatischen Extraktion von Wissen aus großen Datenmengen, das gültig, bisher unbekannt und potentiell nützlich für eine gegebene Anwendung ist. Der zentrale Schritt des KDD-Prozesses ist das Anwenden von Data Mining-Techniken, um nützliche Beziehungen und Zusammenhänge in einer aufbereiteten Datenmenge aufzudecken. Eine der wichtigsten Techniken des Data Mining ist die Cluster-Analyse (Clustering). Dabei sollen die Objekte einer Datenbank in Gruppen (Cluster) partitioniert werden, so dass Objekte eines Clusters möglichst ähnlich und Objekte verschiedener Cluster möglichst unähnlich zu einander sind. Hier können beispielsweise Gruppen von Kunden identifiziert werden, die ähnliche Interessen haben, oder Gruppen von Genen, die ähnliche Funktionalitäten besitzen. Eine aktuelle Herausforderung für Clustering-Verfahren stellen hochdimensionale Feature-Räume dar. Reale Datensätze beinhalten dank moderner Verfahren zur Datenerhebung häufig sehr viele Merkmale (Features). Teile dieser Merkmale unterliegen oft Rauschen oder Abhängigkeiten und können meist nicht im Vorfeld ausgesiebt werden, da diese Effekte in Teilen der Datenbank jeweils unterschiedlich ausgeprägt sind. Daher muss die Wahl der Features mit dem Data-Mining-Verfahren verknüpft werden. Seit etwa 10 Jahren werden vermehrt spezialisierte Clustering-Verfahren entwickelt, die mit den in hochdimensionalen Feature-Räumen auftretenden Problemen besser umgehen können als klassische Clustering-Verfahren. Hierbei wird aber oftmals nicht zwischen den ihrer Natur nach im Einzelnen sehr unterschiedlichen Problemen unterschieden. Ein Hauptanliegen der Dissertation ist daher eine systematische Einordnung der in den letzten Jahren entwickelten sehr diversen Ansätze nach den Gesichtspunkten ihrer jeweiligen Problemauffassung, ihrer grundlegenden Lösungsstrategie und ihrer algorithmischen Vorgehensweise. Als Hauptkategorien unterscheiden wir hierbei die Suche nach Clustern (1.) hinsichtlich der Nähe von Cluster-Objekten in achsenparallelen Unterräumen, (2.) hinsichtlich gemeinsamer Verhaltensweisen (Mustern) von Cluster-Objekten in achsenparallelen Unterräumen und (3.) hinsichtlich der Nähe von Cluster-Objekten in beliebig orientierten Unterräumen (sogenannte Korrelations-Cluster). Für die dritte Kategorie sollen in den weiteren Teilen der Dissertation innovative Lösungsansätze entwickelt werden. Ein erster Lösungsansatz basiert auf einer Erweiterung des dichte-basierten Clustering auf die Problemstellung des Korrelations-Clustering. Den Ausgangspunkt bildet der erste dichtebasierte Ansatz in diesem Bereich, der Algorithmus 4C. Anschließend werden Erweiterungen und Variationen dieses Ansatzes diskutiert, die robusteres, effizienteres oder effektiveres Verhalten aufweisen oder sogar Hierarchien von Korrelations-Clustern und den entsprechenden Unterräumen finden. Die dichtebasierten Korrelations-Cluster-Verfahren können allerdings einige Probleme grundsätzlich nicht lösen, da sie auf der Analyse lokaler Nachbarschaften beruhen. Dies ist in hochdimensionalen Feature-Räumen problematisch. Daher wird eine weitere Neuentwicklung vorgestellt, die das Korrelations-Cluster-Problem mit einer globalen Methode angeht. Schließlich wird eine Methode vorgestellt, die Cluster-Modelle für Korrelationscluster ableitet, so dass die gefundenen Cluster interpretiert werden können und tiefergehende Untersuchungen in der jeweiligen Fachdisziplin zielgerichtet möglich sind. Mögliche Anwendungen dieser Modelle werden abschließend vorgestellt und untersucht

An Evaluation of Popular Copy-Move Forgery Detection Approaches

A copy-move forgery is created by copying and pasting content within the same image, and potentially post-processing it. In recent years, the detection of copy-move forgeries has become one of the most actively researched topics in blind image forensics. A considerable number of different algorithms have been proposed focusing on different types of postprocessed copies. In this paper, we aim to answer which copy-move forgery detection algorithms and processing steps (e.g., matching, filtering, outlier detection, affine transformation estimation) perform best in various postprocessing scenarios. The focus of our analysis is to evaluate the performance of previously proposed feature sets. We achieve this by casting existing algorithms in a common pipeline. In this paper, we examined the 15 most prominent feature sets. We analyzed the detection performance on a per-image basis and on a per-pixel basis. We created a challenging real-world copy-move dataset, and a software framework for systematic image manipulation. Experiments show, that the keypoint-based features SIFT and SURF, as well as the block-based DCT, DWT, KPCA, PCA and Zernike features perform very well. These feature sets exhibit the best robustness against various noise sources and downsampling, while reliably identifying the copied regions.Comment: Main paper: 14 pages, supplemental material: 12 pages, main paper appeared in IEEE Transaction on Information Forensics and Securit

Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods

Feature extraction and dimensionality reduction are important tasks in many fields of science dealing with signal processing and analysis. The relevance of these techniques is increasing as current sensory devices are developed with ever higher resolution, and problems involving multimodal data sources become more common. A plethora of feature extraction methods are available in the literature collectively grouped under the field of Multivariate Analysis (MVA). This paper provides a uniform treatment of several methods: Principal Component Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis (CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions derived by means of the theory of reproducing kernel Hilbert spaces. We also review their connections to other methods for classification and statistical dependence estimation, and introduce some recent developments to deal with the extreme cases of large-scale and low-sized problems. To illustrate the wide applicability of these methods in both classification and regression problems, we analyze their performance in a benchmark of publicly available data sets, and pay special attention to specific real applications involving audio processing for music genre prediction and hyperspectral satellite images for Earth and climate monitoring