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PublicationSimplex distributions for embedding data matrices over time( 2012)
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PublicationPre-symptomatic prediction of plant drought stress using Dirichlet-aggregation regression on hyperspectral images( 2012)
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PublicationEarly drought stress detection in cereals: Simplex volume maximization for hyperspectral image analysis( 2012)
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PublicationDescriptive matrix factorization for sustainability Adopting the principle of oppositesClimate change, the global energy footprint, and strategies for sustainable development have become topics of considerable political and public interest. The public debate is informed by an exponentially growing amount of data and there are diverse partisan interest when it comes to interpretation. We therefore believe that data analysis methods are called for that provide results which are intuitively understandable even to non-experts. Moreover, such methods should be efficient so that non-experts users can perform their own analysis at low expense in order to understand the effects of different parameters and influential factors. In this paper, we discuss a new technique for factorizing data matrices that meets both these requirements. The basic idea is to represent a set of data by means of convex combinations of extreme data points. This often accommodates human cognition. In contrast to established factorization methods, the approach presented in this paper can al so determine over-complete bases. At the same time, convex combinations allow for highly efficient matrix factorization. Based on techniques adopted from the field of distance geometry, we derive a linear time algorithm to determine suitable basis vectors for factorization. By means of the example of several environmental and developmental data sets we discuss the performance and characteristics of the proposed approach and validate that significant efficiency gains are obtainable without performance decreases compared to existing convexity constrained approaches.
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PublicationMatrix factorization as searchSimplex Volume Maximization (SiVM) exploits distance geometry for efficiently factorizing gigantic matrices. It was proven successful in game, social media, and plant mining. Here, we review the distance geometry approach and argue that it generally suggests to factorize gigantic matrices using search-based instead of optimization techniques.
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PublicationConvex non-negative matrix factorization for massive datasetsNon-negative matrix factorization (NMF) has become a standard tool in data mining, information retrieval, and signal processing. It is used to factorize a non-negative data matrix into two non-negative matrix factors that contain basis elements and linear coefficients, respectively. Often, the columns of the first resulting factor are interpreted as "cluster centroids" of the input data, and the columns of the second factor are understood to contain cluster membership indicators. When analyzing data such as collections of gene expressions, documents, or images, it is often beneficial to ensure that the resulting cluster centroids are meaningful, for instance, by restricting them to be convex combinations of data points. However, known approaches to convex-NMF suffer from high computational costs and therefore hardly apply to large-scale data analysis problems. This paper presents a new framework for convex-NMF that allows for an efficient factorization of data matrices of millions of data points. Triggered by the simple observation that each data point can be expressed as a convex combination of vertices of the data convex hull, we require the basic factors to be vertices of the data convex hull. The benefits of convex-hull NMF are twofold. First, for a growing number of data points the expected size of the convex hull, i.e. the number of its vertices, grows much slower than the dataset. Second, distance preserving low-dimensional embeddings allow us to efficiently sample the convex hull and hence to quickly determine candidate vertices. Our extensive experimental evaluation on large datasets shows that convex-hull NMF compares favorably to convex-NMF in terms of both speed and reconstruction quality. We demonstrate that our method can easily be applied to large-scale, real-world datasets, in our case consisting of 750,000 DBLP entries, 4,000,000 digital images, and 150,000,000 votes on World of Warcraft ®guilds, respectively.
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PublicationHierarchical convex NMF for clustering massive dataWe present an extension of convex-hull non-negative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization or Archetypal Analysis. CH-NMF factorizes a non-negative data matrix V into two non- negative matrix factors V WH such that the columns of W are convex combinations of certain data points so that they are readily interpretable to data analysts. There is, however, no free lunch: imposing convexity constraints on W typically prevents adaptation to intrinsic, low dimensional structures in the data. Alas, in cases where the data is distributed in a non-convex manner or consists of mixtures of lower dimensional convex distributions, the cluster representatives obtained from CH-NMF will be less meaningful. In this paper, we present a hierarchical CH-NMF that automatically adapts to internal structures of a dataset, hence it yields meaningful and interpretable clusters for non-convex datasets. This i s also confirmed by our extensive evaluation on DBLP publication records of 760,000 authors, 4,000,000 images harvested from the web, and 150,000,000 votes on World of Warcraft guilds.
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PublicationConvex NMF on non-convex massiv dataWe present an extension of convex-hull nonnegative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization (CNMF) or Archetypal Analysis (AA). CH-NMF factorizes a non-negative data matrix V into two non-negative matrix factors V WH such that the columns of W are convex combinations of certain data points so that they are readily interpretable to data analysts. There is, however, no free lunch: imposing convexity constraints on W typically prevents adaptation to intrinsic, low dimensional structures in the data. Alas, in cases where the data is distributed in a nonconvex manner or consists of mixtures of lower dimensional convex distributions, the cluster representatives obtained from CH-NMF will be less meaningful. In this paper, we present a hierarchical CH-NMF that automatically adapts to internal structures of a dataset, hence it yields meaningful and interpretable clusters for non-convex datasets . This is also conformed by our extensive evaluation on DBLP publication records of 760,000 authors, 4,000,000 images harvested from the web, and 150,000,000 votes on World of Warcraft guilds.