Browsing by Subject "Matrix completion"

Now showing 1 - 3 of 3
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    Efficient matrix completion with Gaussian models
    (University of Minnesota. Institute for Mathematics and Its Applications, 2010-10) Léger, Flavien; Yu, Guoshen; Sapiro, Guillermo
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    Low dimensional approximations: problems and algorithms
    (2014-05) Ngo, Thanh Trung
    High dimensional data usually have intrinsic low rank representations. These low rank representations not only reveal the hidden structure of the data but also reduce the computational cost of data analysis. Therefore, finding low dimensional approximations of the data is an essential task in many data mining applications.Classical low dimensional approximations rely on two universal tools: the eigenvalue decomposition and the singular value decomposition. These two different but related decompositions are of high importance in a large number of areas in science and engineering. As a result, research in numerical linear algebra has been conducted to derive efficient algorithms for solving eigenvalue and singular value problems. Because available solvers for these problems are so well developed, they are often used as black boxes in data analysis.This thesis explores numerical linear algebra techniques and extends well-known methods for low rank approximations to solve new problems in data analysis. Specifically, we carefully analyze the trace ratio optimization and argue that solving this problem can be done efficiently. We also propose efficient algorithms for low rank matrix approximations with missing entries. We also reformulate and analyze classical problems from a different perspective. This reveals the connection between the proposed methods and traditional methods in numerical linear algebra. The performance of the proposed algorithms is established both theoretically and through extensive experiments in dimension reduction and collaborative filtering.
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    Preference modeling and Accuracy in Recommender Systems
    (2017-09) Sharma, Mohit
    Recommender systems are widely used to recommend the most appealing items to users. In this thesis, we focus on analyzing the accuracy of the state-of-the-art matrix completion-based recommendation methods and develop methods to model users' preferences to address different problems that arise in recommender systems. Collaborative filtering-based methods are widely used to generate item recommendations to the user. The low-rank matrix completion method is the state-of-the-art collaborative filtering method. We will show that the accuracy and the ranking performance of matrix completion-based methods are affected by the skewed distribution of ratings in the user-item rating matrix. Additionally, we will illustrate that the number of ratings an item has positively correlates with the prediction accuracy and the ranking performance of the matrix completion approach for the item. Furthermore, we show that the users or the items that are present in the tail, i.e., those having few ratings in real datasets, may not have sufficient ratings to estimate the low-rank models accurately by matrix completion approach. We use these insights to develop TruncatedMF, a matrix completion-based approach that outperforms the state-of-the-art matrix completion method for the users and the items in the tail. Since for new items we do not have any prior preferences from existing users, it is hard to recommend these items to the users. We can use non-collaborative methods that rely on similarities between the new item and the items preferred by a user in the past to model the user preference for the new item. However, these methods consider the item features independently and ignore the interactions among the features of the items while computing the similarities. Modeling the interactions among features can provide more information towards the relevance of an item in comparison to the scenario when the features are considered independently. We develop a new method called User-specific Feature-based factorized Bilinear Similarity Model (UFBSM), that uses all available information across users to capture these interactions among features and learns a low-rank user personalized bilinear similarity model for the Top-n recommendation of new items. In addition to providing ratings over individual items, the users can also provide ratings on sets of items. A rating provided by a user on a set of items conveys some preference information about the items in the set and enables us to acquire a user’s preferences for more items that the number of ratings that the user provided. Moreover, users may have privacy concerns and hence may not be willing to indicate their preferences on individual items explicitly but may be willing to provide a rating to a set of items, as it provides some level of information hiding. We will investigate how do users’ item-level preferences relate to their set-level preferences. Also, we will introduce collaborative filtering-based methods that explicitly model the user behavior of providing ratings on sets of items and can be used to recommend items to users.