Faithful Representations and Topographic Maps: From Distortion- to Information-Based Self-Organization

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Summary: A new perspective on topographic map formation
Review: Faithful Representations and Topographic Maps is an eye-openingguide for neural network researchers, industrial scientists involvedin data mining, and anyone interested in self-organization and topographic maps. It provides an in-depth survey of the approaches available for topographic map formation, as well as their applications in, for example, density estimation, regression, clustering, and feature extraction. It can also be used as an advanced graduate-level course for students that already dispose of a background in competitive learning, and perhaps in topographic map formation. Concerning the learning rules used in this book, not only the formulas but also the complete algorithms and the simulation details are given: in this way, the simulations can be easily repeated. Furthermore, the benchmark performances of the learning rules are listed for comparison. All this is aimed at providing the reader with a profound understanding not only of the theory behind topographic map formation but also of the performance and pitfalls of a given learning rule.

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Summary: A highly innovative but quite specialized angle!
Review: For those familiar with the technology and terminology of Kohonen's self-organizing maps, this book is a highly recommendable asset. The insights on the deficiencies of various previously developed techniques and how to improve them are brilliant. Although the matter is presented in a very bottom-up fashion, it is sometimes hard to keep focusing on the big picture while all the different aspects are in-depth explored. This makes the book sometimes hard to understand. Although it was probably never intended to be easily understandable, this is the reason why I didn't give it the full 5 stars.

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Summary: A fresh approach on topographic maps
Review: The book presents a fresh approach on Topographic Maps, emphasizing ``equiprobabilistic'' topographic maps in which all representational units participate with the same probability in the representation. However, the text goes far beyond a monograph on this particular type of topographic maps and provides an excellent exposition of the topic of self-organizing map models in general, discussing their biological motivation and explaining in depth their connections with important statistical concepts such as vector quantization, non-parametric regression and density estimation. The practicioner will find detailed performance comparisons and psoudo code listings that tremendously facilitate an implementation of the described methods. The potential of the newly introduced equiprobabilistic topographic maps is amply demonstrated with detailed treatments of a broad range of application topics. I am convinced that this book marks an important contribution to the field of topographic map representations and that it has the potential to become a major reference for many years.

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Summary: A New Frontier in Computational Geometry
Review: This is the first monograph devoted to an extremely important aspect of how nature organizes the sensory surfaces in the higher vertebrates. As this work points out, all sensory surfaces in mammals exist as two dimensional maps which project (or map) onto the (folded) two dimensional surface of the cerebral cortex. Previous authors have pointed out that in the case of the visual system, the retinal surface projects onto area V1 in the occipital cortex in a manner which approximates a "quasi-conformal" complex logarithmic function. Other authors have demonstrated that the cochlear basilar membrane also maps onto the auditory cortex in a similar quasi-conformal fashion. And most obviously, the mechano-receptors in our skin map onto the their corresponding cortical sensory areas in a manner which preserves local order of the projection. The important thing to note here is that this mapping of one 2-D surface onto another 2-D surface preserves the local orthogonality of the map coordinates (defn of quasi-conformal). (In the case of the retinal coordinates, they are the simple R, theta coordinates of the visual field and, in the case of the aural (auditory) map, the coordinates appear to be sound intensity (loudness) and frequency (pitch)). Van Hulle (Kohonen, et al.) describes, in vivid detail, how several "self-organizing" algorithms can make this mapping possible. THIS IS THE IMPORTANT POINT CONVEYED IN THIS MAGNIFICENT WORK. "The underlying INFORMATION coordinates of the data being processed by these 2-D onto 2-D mappings is what 'organizes' or defines the form of the mapping." The only point which the author does not address, is the global nature of these mappings. If we consider how the edges of these 2-D maps project onto eachother, there are three possible projections. One forms an ordinary torus and, unfolded looks like the raster scan of a TV set. This mapping is what has been assumed to be what would be followed in the neural projections of the retina onto area V1, but admits an indeterminacy in that there are two "normal" directions to approach the target surface. The second type of projection, when unfolded, resembles a Mobius strip (or Klein bottle for a closed surface). The importance of this type of projection is that there is only one way for neurons to project onto this surface which removes the ambiguity of the first mapping. (This surface is said to be non-orientable and of genus one in topological terms.) If this is in fact how nature chooses to wire the retino-topic projection, then she must also admit one singular point for each "patch" of map. (Consider what happens when you "squish" a Mobius strip made out of a paper strip between two flat surfaces. There is always a "fold".) About seven years ago, when the color distribution in the mammalian visual cortex was illucidated, it was found that the projection areas were made up of a "patch work quilt" of surfaces each with a singular point about which a set of color strips was splayed (like a peacock tail). Perhaps this is the necessary singular point for this type of retinotopic map. I would hope that the author, in his next book (or edition of this book) might address the global issues of these fascinating 2-D onto 2-D mappings, both natural and computational. (Just for closure, the third possible type of 2-D onto 2-D projection forms a hyperbolic surface whose physical significance is far from clear.)

If this review makes any sense to you, Please BUY THIS BOOK! I'm sure that you will be absolutely fascinated by it's content. However, be forwarned, it would be most helpful to have read Kohonen's "Self Organizing Maps", SV, 1995 before diving into this work. It is written in a "bottom up" fashion so a careful review of the Preface and Table of Contents will help in planning a reading strategy.