Efficient Semidefinite Spectral Clustering via Lagrange Duality

We propose an efficient approach to semidefinite spectral clustering (SSC), which addresses the Frobenius normalization with the positive semidefinite (p.s.d.) constraint for spectral clustering. Compared with the original Frobenius norm approximation based algorithm, the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p.s.d. constraint. In this paper, SSC is formulated as a semidefinite programming (SDP) problem. In order to solve the high computational complexity of SDP, we present a dual algorithm based on the Lagrange dual formalization. Two versions of the proposed algorithm are proffered: one with less memory usage and the other with faster convergence rate. The proposed algorithm has much lower time complexity than that of the standard interior-point based SDP solvers. Experimental results on both UCI data sets and real-world image data sets demonstrate that 1) compared with the state-of-the-art spectral clustering methods, the proposed algorithm achieves better clustering performance; and 2) our algorithm is much more efficient and can solve larger-scale SSC problems than those standard interior-point SDP solvers.Comment: 13 page

Convex Relaxations for Permutation Problems

Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seriation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relaxations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe

Local, multi-resolution detection of network communities by Markovian dynamics

Complex networks are used to represent systems from many disciplines, including biology, physics, medicine, engineering and the social sciences; Many real-world networks are organised into densely connected communi- ties, whose composition gives some insight into the underlying network. Most approaches for nding such communities do so by partitioning the network into disjoint subsets, at the cost of requiring global information and that nodes belong to exactly one community. In recent years, some effort has been devoted towards the development of local methods, but these are either limited in resolution or ignore relevant network features such as directedness. Here we show that introducing a dynamic process onto the network allows us to de ne a community quality function severability which is inherently multi-resolution, takes into account edge-weight and direction, can accommodate overlapping communities and orphan nodes and crucially does not require global knowledge. Both constructive and real-world examples| drawn from elds as diverse as image segmentation, metabolic networks and word association|are used to illustrate the characteristics of this approach. We envision this approach as a starting point for the future analysis of both evolving networks and networks too large to be readily analysed as a whole (e.g. the World Wide Web).Open Acces

Data Clustering And Visualization Through Matrix Factorization

Clustering is traditionally an unsupervised task which is to find natural groupings or clusters in multidimensional data based on perceived similarities among the patterns. The purpose of clustering is to extract useful information from unlabeled data. In order to present the extracted useful knowledge obtained by clustering in a meaningful way, data visualization becomes a popular and growing area of research field. Visualization can provide a qualitative overview of large and complex data sets, which help us the desired insight in truly understanding the phenomena of interest in data. The contribution of this dissertation is two-fold: Semi-Supervised Non-negative Matrix Factorization (SS-NMF) for data clustering/co-clustering and Exemplar-based data Visualization (EV) through matrix factorization. Compared to traditional data mining models, matrix-based methods are fast, easy to understand and implement, especially suitable to solve large-scale challenging problems in text mining, image grouping, medical diagnosis, and bioinformatics. In this dissertation, we present two effective matrix-based solutions in the new directions of data clustering and visualization. First, in many practical learning domains, there is a large supply of unlabeled data but limited labeled data, and in most cases it might be expensive to generate large amounts of labeled data. Traditional clustering algorithms completely ignore these valuable labeled data and thus are inapplicable to these problems. Consequently, semi-supervised clustering, which can incorporate the domain knowledge to guide a clustering algorithm, has become a topic of significant recent interest. Thus, we develop a Non-negative Matrix Factorization (NMF) based framework to incorporate prior knowledge into data clustering. Moreover, with the fast growth of Internet and computational technologies in the past decade, many data mining applications have advanced swiftly from the simple clustering of one data type to the co-clustering of multiple data types, usually involving high heterogeneity. To this end, we extend SS-NMF to perform heterogeneous data co-clustering. From a theoretical perspective, SS-NMF for data clustering/co-clustering is mathematically rigorous. The convergence and correctness of our algorithms are proved. In addition, we discuss the relationship between SS-NMF with other well-known clustering and co-clustering models. Second, most of current clustering models only provide the centroids (e.g., mathematical means of the clusters) without inferring the representative exemplars from real data, thus they are unable to better summarize or visualize the raw data. A new method, Exemplar-based Visualization (EV), is proposed to cluster and visualize an extremely large-scale data. Capitalizing on recent advances in matrix approximation and factorization, EV provides a means to visualize large scale data with high accuracy (in retaining neighbor relations), high efficiency (in computation), and high flexibility (through the use of exemplars). Empirically, we demonstrate the superior performance of our matrix-based data clustering and visualization models through extensive experiments performed on the publicly available large scale data sets

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Novel perspectives and approaches to video summarization

The increasing volume of videos requires efficient and effective techniques to index and structure videos. Video summarization is such a technique that extracts the essential information from a video, so that tasks such as comprehension by users and video content analysis can be conducted more effectively and efficiently. The research presented in this thesis investigates three novel perspectives of the video summarization problem and provides approaches to such perspectives. Our first perspective is to employ local keypoint to perform keyframe selection. Two criteria, namely Coverage and Redundancy, are introduced to guide the keyframe selection process in order to identify those representing maximum video content and sharing minimum redundancy. To efficiently deal with long videos, a top-down strategy is proposed, which splits the summarization problem to two sub-problems: scene identification and scene summarization. Our second perspective is to formulate the task of video summarization to the problem of sparse dictionary reconstruction. Our method utilizes the true sparse constraint L0 norm, instead of the relaxed constraint L2,1 norm, such that keyframes are directly selected as a sparse dictionary that can reconstruct the video frames. In addition, a Percentage Of Reconstruction (POR) criterion is proposed to intuitively guide users in selecting an appropriate length of the summary. In addition, an L2,0 constrained sparse dictionary selection model is also proposed to further verify the effectiveness of sparse dictionary reconstruction for video summarization. Lastly, we further investigate the multi-modal perspective of multimedia content summarization and enrichment. There are abundant images and videos on the Web, so it is highly desirable to effectively organize such resources for textual content enrichment. With the support of web scale images, our proposed system, namely StoryImaging, is capable of enriching arbitrary textual stories with visual content