Advances in Self-Organizing Maps: 9th International by Barbara Hammer, Andrej Gisbrecht, Alexander Schulz (auth.),

By Barbara Hammer, Andrej Gisbrecht, Alexander Schulz (auth.), Pablo A. Estévez, José C. Príncipe, Pablo Zegers (eds.)

Self-organizing maps (SOMs) have been built via Teuvo Kohonen within the early eighties. considering the fact that then greater than 10,000 works were in accordance with SOMs. SOMs are unsupervised neural networks important for clustering and visualization reasons. Many SOM purposes were constructed in engineering and technology, and different fields.

This e-book includes refereed papers awarded on the 9th Workshop on Self-Organizing Maps (WSOM 2012) held on the Universidad de Chile, Santiago, Chile, on December 12-14, 2012. The workshop introduced jointly researchers and practitioners within the box of self-organizing structures. one of the publication chapters there are first-class examples of using SOMs in agriculture, laptop technological know-how, facts visualization, wellbeing and fitness structures, economics, engineering, social sciences, textual content and photo research, and time sequence research. different chapters current the newest theoretical paintings on SOMs in addition to studying Vector Quantization (LVQ) tools.

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6(c), it is moved up by about 5 steps compared with Fig. 6(b). Fig. 6. (a) After Torus-SOM learning, (b) the map (a) is slightly moved to the right, and (c) moved upwards. (d) The result obtained with SOM_PAK learning. Finally, the result obtained after learning, using the SOM_PAK is shown in Fig. 6(d). In the case of the Torus-SOM of Figs. 6, the map is continuous on all sides. The SOM_PAK is a usual SOM. However, this SOM does not continue at all sides. As a result, after learning with SOM_PAK, the continuity of the representation provided by the spherical SOM (blossom) and Torus-SOM cannot be observed.

In: Proc. of the World Congress on Neural Networks, Portland, pp. : Optimal unsupervised learning in a single-layer linear feedforward neural network. : Learning with Kernels. : Adaptive relevance matrices in learning vector quantization. : Nonlinear component analysis as a kernel eigenvalue problem. Neur. : On the influence of the kernel on the consistency of support vector machines. : A note on gradient based learning in vector quantization using differentiable kernels for Hilbert and Banach spaces.

Without further restrictions on the kernel κΦ , both, H and Φ are not unique. Positive kernels are of special interest because they uniquely correspond to a reproducing kernel Hilbert spaces (RKHS) H in a canonical manner [1,14]. The kernel κΦ is said to be positive definite if for all finite subsets Vm ⊆ V with cardinality #Vm = m, the Gram-Matrix Gm = [κ (vi , vj ) : i, j = 1 . . m] (16) is positive semi-definite [1]. The norm Φ(v) RKHS induces a metric dH (Φ(v), Φ(w)) = H = κΦ (Φ(v), Φ(v)) of this κΦ (v, v) − 2κΦ (v, w) + κΦ (w, w) (17) based on the kernel κΦ [18].

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