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Register a new userNormalized Information Distance is Not Semicomputable
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Added by
Paul Vitanyi
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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Normalized information distance (NID) uses the theoretical notion of Kolmogorov complexity, which for practical purposes is approximated by the length of the compressed version of the file involved, using a real-world compression program. This practical application is called normalized compression distance and it is trivially computable. It is a parameter-free similarity measure based on compression, and is used in pattern recognition, data mining, phylogeny, clustering, and classification. The complexity properties of its theoretical precursor, the NID, have been open. We show that the NID is neither upper semicomputable nor lower semicomputable.
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The normalized information distance is a universal distance measure for objects of all kinds. It is based on Kolmogorov complexity and thus uncomputable, but there are ways to utilize it. First, compression algorithms can be used to approximate the Kolmogorov complexity if the objects have a string representation. Second, for names and abstract concepts, page count statistics from the World Wide Web can be used. These practical realizations of the normalized information distance can then be applied to machine learning tasks, expecially clustering, to perform feature-free and parameter-free data mining. This chapter discusses the theoretical foundations of the normalized information distance and both practical realizations. It presents numerous examples of successful real-world applications based on these distance measures, ranging from bioinformatics to music clustering to machine translation.
In this paper we apply different techniques of information distortion on a set of classical books written in English. We study the impact that these distortions have upon the Kolmogorov complexity and the clustering by compression technique (the latter based on Normalized Compression Distance, NCD). We show how to decrease the complexity of the considered books introducing several modifications in them. We measure how the information contained in each book is maintained using a clustering error measure. We find experimentally that the best way to keep the clustering error is by means of modifications in the most frequent words. We explain the details of these information distortions and we compare with other kinds of modifications like random word distortions and unfrequent word distortions. Finally, some phenomenological explanations from the different empirical results that have been carried out are presented.
Despite great popularity of applying softmax to map the non-normalised outputs of a neural network to a probability distribution over predicting classes, this normalised exponential transformation still seems to be artificial. A theoretic framework that incorporates softmax as an intrinsic component is still lacking. In this paper, we view neural networks embedding softmax from an information-theoretic perspective. Under this view, we can naturally and mathematically derive log-softmax as an inherent component in a neural network for evaluating the conditional mutual information between network output vectors and labels given an input datum. We show that training deterministic neural networks through maximising log-softmax is equivalent to enlarging the conditional mutual information, i.e., feeding label information into network outputs. We also generalise our informative-theoretic perspective to neural networks with stochasticity and derive information upper and lower bounds of log-softmax. In theory, such an information-theoretic view offers rationality support for embedding softmax in neural networks; in practice, we eventually demonstrate a computer vision application example of how to employ our information-theoretic view to filter out targeted objects on images.
We introduce a uniform representation of general objects that captures the regularities with respect to their structure. It allows a representation of a general class of objects including geometric patterns and images in a sparse, modular, hierarchical, and recursive manner. The representation can exploit any computable regularity in objects to compactly describe them, while also being capable of representing random objects as raw data. A set of rules uniformly dictates the interpretation of the representation into raw signal, which makes it possible to ask what pattern a given raw signal contains. Also, it allows simple separation of the information that we wish to ignore from that which we measure, by using a set of maps to delineate the a priori parts of the objects, leaving only the information in the structure. Using the representation, we introduce a measure of information in general objects relative to structures defined by the set of maps. We point out that the common prescription of encoding objects by strings to use Kolmogorov complexity is meaningless when, as often is the case, the encoding is not specified in any way other than that it exists. Noting this, we define the measure directly in terms of the structures of the spaces in which the objects reside. As a result, the measure is defined relative to a set of maps that characterize the structures. It turns out that the measure is equivalent to Kolmogorov complexity when it is defined relative to the maps characterizing the structure of natural numbers. Thus, the formulation gives the larger class of objects a meaningful measure of information that generalizes Kolmogorov complexity.
A recent article by Mathur attempts a precise formulation for the paradox of black hole information loss [S. D. Mathur, arXiv:1108.0302v2 (hep-th)]. We point out that a key component of the above work, which refers to entangled pairs inside and outside of the horizon and their associated entropy gain or information loss during black hole evaporation, is a presumptuous false outcome not backed by the very foundation of physics. The very foundation of Mathurs above work is thus incorrect. We further show that within the framework of Hawking radiation as tunneling the so-called small corrections are sufficient to resolve the information loss problem.
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