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Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonlyused sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only one of these measures satisfies all six: the Gini Index.

For recommender systems that base their product rankings primarily on a measure of similarity between items and the user query, it can often happen that products on the recommendation list are highly similar to each other and lack diversity. In this article we argue that the motivation of diversity research is to increase the probability of retrieving unusual or novel items which are relevant to the user and introduce a methodology to evaluate their performance in terms of novel item retrieval. Moreover, noting that the retrieval of a set of items matching a user query is a common problem across many applications of information retrieval, we formulate the trade-off between diversity and matching quality as a binary optimization problem, with an input control parameter allowing explicit tuning of this trade-off. We study solution strategies to the optimization problem and demonstrate the importance of the control parameter in obtaining desired system performance. The methods are evaluated for collaborative recommendation using two datasets and case-based recommendation using a synthetic dataset constructed from the public-domain Travel dataset.

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Collaborative recommendation has emerged as an effective technique for personalized information access. However, there has been relatively little theoretical analysis of the conditions under which the technique is effective. To explore this issue, we analyse the <i>robustness</i> of collaborative recommendation: the ability to make recommendations despite (possibly intentional) noisy product ratings. There are two aspects to robustness: recommendation <i>accuracy</i> and <i>stability</i>. We formalize recommendation accuracy in machine learning terms and develop theoretically justified models of accuracy. In addition, we present a framework to examine recommendation stability in the context of a widely-used collaborative filtering algorithm. For each case, we evaluate our analysis using several real-world data-sets. Our investigation is both practically relevant for enterprises wondering whether collaborative recommendation leaves their marketing operations open to attack, and theoretically interesting for the light it sheds on a comprehensive theory of collaborative recommendation.

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