Van Rijsbergen C.J. The Geometry of Information Retrieval

Издательство Cambridge University Press, 2004, -163 pp.This book begins and ends in information retrieval, but travels through a route constructed in an abstract way. In particular it goes through some of the most interesting and important models for information retrieval, a vector space model, a probabilistic model and a logical model, and shows how these three and possibly others can be described and represented in Hilbert space. The reasoning that occurs within each one of these models is formulated algebraically and can be shown to depend essentially on the geometry of the information space. The geometry can be seen as a ‘language’ for expressing the different models of information retrieval.
The approach taken is to structure these developments firmly in terms of the mathematics of Hilbert spaces and linear operators. This is of course the approach used in quantum mechanics. It is remarkable that the application of Hilbert space mathematics to information retrieval is very similar to its application to quantum mechanics. A document in IR can be represented as a vector in Hilbert space, and an observable such as ‘relevance’ or ‘aboutness’ can be represented by a Hermitian operator. However, this is emphatically not a book about quantum mechanics but about using the same language, the mathematical language of quantum mechanics, for the description of information retrieval. It turns out to be very convenient that quantum mechanics provides a ready-made interpretation of this language. It is as if in physics we have an example semantics for the language, and as such it will be used extensively to motivate a similar but different interpretation for IR.We introduce an appropriate logic and probability theory for information spaces guided by their introduction into quantum mechanics. Gleason’s Theorem, which specifies an algorithm for computing probabilities associated with subspaces in Hilbert space, is of critical importance in quantum mechanics and will turn out to be central for the same reasons in information retrieval. Whereas quantum theory is about a theory of measurement for natural systems, The Geometry of Information Retrieval is about such a theory for artificial systems, and in particular for information retrieval.
The important notions in quantum mechanics, state vector, observable, uncertainty, complementarity, superposition and compatibility readily translate into analogous notions in information retrieval, and hence the theorems of quantum theory become available as theorems in IR.
One of the main aims of this book is to present the requisite mathematics to explore in detail the foundation of information retrieval as a parallel to that of quantum mechanics. The material is principally addressed to students and researchers in information retrieval but will also be of interest to those working in such disciplines as AI and quantum computation. An attempt is made to lay a sound mathematical foundation for reasoning about existing models in IR sufficient for their modification and extension. The hope is that the treatment will inspire and enable the invention of new models. All the mathematics is introduced in an elementary fashion, step-by-step, making copious references to matching developments in quantum mechanics. Any reader with a good grasp of high school mathematics, or A-level equivalent, should be able to follow the mathematics from first principles. One exception to this is the material in the Prologue, where some more advanced notions are rapidly introduced, as is often the case in dialogue, but even there a quick consultation of the appropriate appendices would clarify the mathematics.Onsets and kinds for IR
Vector and Hilbert spaces
Linear transformations, operators and matrices
Conditional logic in IR
The geometry of IR