Spectral methods for reconstruction problems

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“Spectral methods” capture generally the class of algorithms which cast their input data as a matrix and then employ eigenvalue and eigenvector techniques from linear algebra. Empirically, spectral methods have been shown to perform successfully in a variety of domains, e.g., text classification [DDL90], website ranking [PB98, Kle99]. On the other hand, not many theoretical guarantees have been given for spectral algorithms, and there remains a lot of work to be done to obtain a real understanding why these approaches work as well as they do. In this thesis we use the framework of reconstruction problems to gain such an understanding of these approaches: the basic idea of reconstruction problems is given some input data which are generated according to some model, the task is to recover some of the model parameters. Clearly, algorithms for these problems do not know the hidden structure and are supposed to reconstruct the structure with the plain information of the data. One main advantage of reconstruction problems is that they allow to compare and to validate algorithms: typically, in reconstruction problems there are a few parameters that measure the difficulty of the reconstruction task. An algorithm is better, if it can provably solve the problem for a wider range of parameters (note that the generating data model is known for the purpose of the analysis of the algorithm). Moreover, many algorithms that have been proven to perform well on reconstruction problems (e.g., in computer graphics, see [Dey04]) also work nicely on real world data. We provide data models and design spectral algorithms for which we show that they often provably reconstruct the model parameters. In particular, we formulate the partitioning problem and the preference elicitation problem as reconstruction problems and provide theoretical guarantees for the correct reconstruction of the hidden structure using our algorithms.