The field of information retrieval (IR) has received much attention in recent years mainly due to the enormous growth and financial success of Internet search engine companies such as Google and Yahoo.  This slim book (150 pages) provides a refreshing new approach to IR research by explaining how the theorems of quantum mechanics (QM) can be applied to IR.  The author, C.J. “Keith” van Rijsbergen, has spent more than thirty years in IR research and is currently a professor of and leader of the IR group at the University of Glasgow, Scotland. He is the author of the widely cited 1979 classic: Information Retrieval.


But what is the advantage of using the language of QM for IR?  One main advantage is that the QM paradigm can be used to combine the three main current IR models (viz. the probabilistic, logical and vector space models).  In addition, the approach described in this book is not text specific and is applicable to other data types, such as images, video and music.  One of the central ideas presented is that an IR object (such as a document) can be represented in Hilbert space (the mathematical foundation for QM), and Hermitian operators (in QM, these are quantities that be measured experimentally) can represent the relevance of queries to that object.                    


The Geometry of Information Retrieval consists of a prolog, followed by an introduction and five short and concise main chapters.  Three appendices are also included.  The bibliography is one highlight of this interesting book, with the author providing a brief description of each referenced work.    


The prolog introduces the main ideas in the book with a fictional discussion between a senior academic K (a thinly disguised van Rijsbergen) and two other critical academics.  This is an interesting approach (reminding me of Douglas Hofstadter’s narratives in Gödel, Escher, Bach), which give the reader motivation to continue reading through the rest of the book.


Chapter two is an overview of IR using set theory.  It provides definitions of the most commonly used IR measurements (precision, recall, etc.).  The final part of this chapter explains why Boolean logic is unable to handle object classes in IR, and discusses limitations in the traditional inverted file Boolean approach where keywords are not available (for example,  when image features need to be indexed).     


The third and forth chapters provide background mathematics on vector spaces and operators.  Chapter three introduces vector and Hilbert spaces and the Dirac notation.  The cosine coefficient (often used in IR as a measure of similarity between two document vectors) is also discussed.  Chapter four is concerned with linear transformations, operators (such as projectors), eigenvalues and eigenvectors, and the spectral theorem (which shows that if an observable is seen as a question, it can be reduced to a set of yes/no questions).    


Chapter five presents a formal connection between conditional logic in IR (that is: if a query is implied by a document then the document is assumed to be about the query) and quantum logic.  According to the author this allows conditional logic to be interpreted in Hilbert (or vector) space for the first time. 


The background work is then tied together in chapter six, which is the longest and most interesting chapter.   A language, based on the Dirac notation, is introduced which uses a small set of operators (such as the density operator) and functions.  This language is then discussed with application to IR topics such as co-ordination level matching, pseudo-relevance and relevance feedback, dynamic clustering and ostensive retrieval.   


As mentioned earlier, the book contains three useful appendices.  The first of these provides an introduction to linear algebra using Hilbert spaces, and also introduces the Dirac notation.  The second appendix presents a brief introduction to QM and includes sections on physical states, observables, measurements and the famous Heisenberg Uncertainty Principle.  The last appendix describes classical and quantum probability.  These appendices contain a great deal of condensed information, and each concludes with a well-written section on further reading.     


This is a very interesting book, which appears to provide a solid foundation for future research.  The math required some effort, particularly as I was not familiar with QM.  However van Rijsbergen’s explanations are well written and logical. In addition, there are many pointers to other sources of relevant information. 


This book would be of particular interest to those conducting IR, Artificial Intelligence or Cognitive Science research.  The novel approach may also be of interest to web search engine developers.  As the author points out, this book could also provide a useful introduction to quantum computation, as most of the required mathematics is included.  It would be great to see a discussion of implementation issues for an IR system based on van Rijsbergen’s mathematical foundation, covering such topics as performance, scalability and reliability.


In conclusion, this is a clearly written and thought-provoking book that has been a pleasure to read.  It is highly recommended.



The Geometry of Information Retrieval


C.J. van Rijsbergen

Cambridge University Press, 2004


Reviewed by:  Jason Dowling

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