Learning vector quantization for proximity data

Hofmann D. Learning vector quantization for proximity data. Bielefeld: Universität Bielefeld; 2016.Prototype-based classifiers such as learning vector quantization (LVQ) often display intuitive and flexible classification and learning rules. However, classical techniques are restricted to vectorial data only, and hence not suited for more complex data structures. Therefore, a few extensions of diverse LVQ variants to more general data which are characterized based on pairwise similarities or dissimilarities only have been proposed recently in the literature. In this contribution, we propose a novel extension of LVQ to similarity data which is based on the kernelization of an underlying probabilistic model: kernel robust soft LVQ (KRSLVQ). Relying on the notion of a pseudo-Euclidean embedding of proximity data, we put this specific approach as well as existing alternatives into a general framework which characterizes different fundamental possibilities how to extend LVQ towards proximity data: the main characteristics are given by the choice of the cost function, the interface to the data in terms of similarities or dissimilarities, and the way in which optimization takes place. In particular the latter strategy highlights the difference of popular kernel approaches versus so-called relational approaches. While KRSLVQ and alternatives lead to state of the art results, these extensions have two drawbacks as compared to their vectorial counterparts: (i) a quadratic training complexity is encountered due to the dependency of the methods on the full proximity matrix; (ii) prototypes are no longer given by vectors but they are represented in terms of an implicit linear combination of data, i.e. interpretability of the prototypes is lost. We investigate different techniques to deal with these challenges: We consider a speed-up of training by means of low rank approximations of the Gram matrix by its Nyström approximation. In benchmarks, this strategy is successful if the considered data are intrinsically low-dimensional. We propose a quick check to efficiently test this property prior to training. We extend KRSLVQ by sparse approximations of the prototypes: instead of the full coefficient vectors, few exemplars which represent the prototypes can be directly inspected by practitioners in the same way as data. We compare different paradigms based on which to infer a sparse approximation: sparsity priors while training, geometric approaches including orthogonal matching pursuit and core techniques, and heuristic approximations based on the coefficients or proximities. We demonstrate the performance of these LVQ techniques for benchmark data, reaching state of the art results. We discuss the behavior of the methods to enhance performance and interpretability as concerns quality, sparsity, and representativity, and we propose different measures how to quantitatively evaluate the performance of the approaches. We would like to point out that we had the possibility to present our findings in international publication organs including three journal articles [6, 9, 2], four conference papers [8, 5, 7, 1] and two workshop contributions [4, 3]. References [1] A. Gisbrecht, D. Hofmann, and B. Hammer. Discriminative dimensionality reduction mappings. Advances in Intelligent Data Analysis, 7619: 126–138, 2012. [2] B. Hammer, D. Hofmann, F.-M. Schleif, and X. Zhu. Learning vector quantization for (dis-)similarities. Neurocomputing, 131: 43–51, 2014. [3] D. Hofmann. Sparse approximations for kernel robust soft lvq. Mittweida Workshop on Computational Intelligence, 2013. [4] D. Hofmann, A. Gisbrecht, and B. Hammer. Discriminative probabilistic prototype based models in kernel space. New Challenges in Neural Computation, TR Machine Learning Reports, 2012. [5] D. Hofmann, A. Gisbrecht, and B. Hammer. Efficient approximations of kernel robust soft lvq. Workshop on Self-Organizing Maps, 198: 183–192, 2012. [6] D. Hofmann, A. Gisbrecht, and B. Hammer. Efficient approximations of robust soft learning vector quantization for non-vectorial data. Neurocomputing, 147: 96–106, 2015. [7] D. Hofmann and B. Hammer. Kernel robust soft learning vector quantization. Artificial Neural Networks in Pattern Recognition, 7477: 14–23, 2012. [8] D. Hofmann and B. Hammer. Sparse approximations for kernel learning vector quantization. European Symposium on Artificial Neural Networks, 549–554, 2013. [9] D. Hofmann, F.-M. Schleif, B. Paaßen, and B. Hammer. Learning interpretable kernelized prototype-based models. Neurocomputing, 141: 84–96, 2014

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Crafting chaos: computational design of contraptions with complex behaviour

The 2010s saw the democratisation of digital fabrication technologies. Although this phenomenon made fabrication more accessible, physical assemblies displaying a complex behaviour are still difficult to design. While many methods support the creation of complex shapes and assemblies, managing a complex behaviour is often assumed to be a tedious aspect of the design process. As a result, the complex parts of the behaviour are either deemed negligible (when possible) or managed directly by the software, without offering much fine-grained user control. This thesis argues that efficient methods can support designers seeking complex behaviours by increasing their level of control over these behaviours. To demonstrate this, I study two types of artistic devices that are particularly challenging to design: drawing machines, and chain reaction contraptions. These artefacts’ complex behaviour can change dramatically even as their components are moved by a small amount. The first case study aims to facilitate the exploration and progressive refinement of complex patterns generated by drawing machines under drawing-level user-defined constraints. The approach was evaluated with a user study, and several machines drawing the expected pattern were fabricated. In the second case study, I propose an algorithm to optimise the layout of complex chain reaction contraptions described by a causal graph of events in order to make them robust to uncertainty. Several machines optimised with this method were successfully assembled and run. This thesis makes the following contributions: (1) support complex behaviour specifications; (2) enable users to easily explore design variations that respect these specifications; and (3) optimise the layout of a physical assembly to maximise the probability of real-life success

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version