TL;DR: The Interval-valued Matrix Factorization (IMF) framework is proposed and it is shown that proposed I-NMF and I-PMF significantly outperform their single-valued counterparts in FA and CF applications.
Abstract: In this paper, we propose the Interval-valued Matrix Factorization (IMF) framework. Matrix Factorization (MF) is a fundamental building block of data mining. MF techniques, such as Nonnegative Matrix Factorization (NMF) and Probabilistic Matrix Factorization (PMF), are widely used in applications of data mining. For example, NMF has shown its advantage in Face Analysis (FA) while PMF has been successfully applied to Collaborative Filtering (CF). In this paper, we analyze the data approximation in FA as well as CF applications and construct interval-valued matrices to capture these approximation phenomenons. We adapt basic NMF and PMF models to the interval-valued matrices and propose Interval-valued NMF (I-NMF) as well as Interval-valued PMF (I-PMF). We conduct extensive experiments to show that proposed I-NMF and I-PMF significantly outperform their single-valued counterparts in FA and CF applications.
TL;DR: This paper proposes matrix decomposition techniques that consider the existence of interval-valued data and shows that naive ways to deal with such imperfect data may introduce errors in analysis and present factorization techniques that are especially effective when the amount of imprecise information is large.
Abstract: With many applications relying on multi-dimensional datasets for decision making, matrix factorization (or decomposition) is becoming the basis for many knowledge discoveries and machine learning tasks, from clustering, trend detection, anomaly detection, to correlation analysis. Unfortunately, a major shortcoming of matrix analysis operations is that, despite their effectiveness when the data is scalar, these operations become difficult to apply in the presence of non-scalar data, as they are not designed for data that include non-scalar observations, such as intervals. Yet, in many applications, the available data are inherently non-scalar for various reasons, including imprecision in data collection, conflicts in aggregated data, data summarization, or privacy issues, where one is provided with a reduced, clustered, or intentionally noisy and obfuscated version of the data to hide information. In this paper, we propose matrix decomposition techniques that consider the existence of interval-valued data. We show that naive ways to deal with such imperfect data may introduce errors in analysis and present factorization techniques that are especially effective when the amount of imprecise information is large.
4 citations
Cites background or methods from "Interval-valued Matrix Factorizatio..."
...As discussed above, interval NMF and PMF [9] also have been studied to resolve alignment approximation in face analysis and rating approximation in collaborative filtering....
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...As the chart shows, the prediction accuracy of all algorithms improves as we consider higher decomposition ranks and the proposed latent semantic alignment based approach, AIPMF, leads to better prediction performance than both PMF and I-PMF, for decomposition ranks > 60....
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...[9] extended these to interval-valued matrices as follows:...
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...As described in Section 6.1.2, we also compare proposed ISVD approaches with NMF and I-NMF [9] for the face analysis tasks: data reconstruction and classification....
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...For collaborative filtering with social media data, discussed in Section 6.1.3, we used PMF and I-PMF [9] as competitors....
TL;DR: In this article , the Tensor-Train technique is extended to deal with uncertain data, here modeled as intervals, and the authors propose a way to address this issue by extending the known tensor-train technique for tensor decomposition in order to handle uncertain data.
Abstract: In many fields of computer science, tensor decomposition techniques are increasingly being adopted as the core of many applications that rely on multi-dimensional datasets for implementing knowledge discovery tasks. Unfortunately, a major shortcoming of state-of-the-art tensor analyses is that, despite their effectiveness when the data is certain, these operations become difficult to apply, or altogether inapplicable, in the presence of uncertainty in the data, a circumstance common to many real-world scenarios. In this paper we propose a way to address this issue by extending the known Tensor-Train technique for tensor factorization in order to deal with uncertain data, here modeled as intervals. Working with interval-valued data, however, presents numerous challenges, since many algebraic operations that form the building blocks of the factorization process, as well as the properties that make these procedures useful for knowledge discovery, cannot be easily extended from their scalar counterparts, and often require some approximation (including, though it is not only the case, for keeping computational costs manageable). These challenges notwithstanding, our proposed techniques proved to be reasonably effective, and are supported by a thorough experimental validation.
TL;DR: A probabilistic model for analyzing the generalized interval valued matrix, a matrix that has scalar valued elements and bounded/unbounded interval valued elements, is proposed and it is proved that the objective function is monotonically decreasing by the parameter update.
Abstract: In this paper, we propose a probabilistic model for analyzing the generalized interval valued matrix, a matrix that has scalar valued elements and bounded/unbounded interval valued elements. We derive a majorization minimization algorithm for parameter estimation and prove that the objective function is monotonically decreasing by the parameter update. An experiment shows that the proposed model well handles interval- valued elements and offers improved performance.
TL;DR: This article proposes several novel measures that compute the cumulative gain the user obtains by examining the retrieval result up to a given ranked position, and test results indicate that the proposed measures credit IR methods for their ability to retrieve highly relevant documents and allow testing of statistical significance of effectiveness differences.
Abstract: Modern large retrieval environments tend to overwhelm their users by their large output. Since all documents are not of equal relevance to their users, highly relevant documents should be identified and ranked first for presentation. In order to develop IR techniques in this direction, it is necessary to develop evaluation approaches and methods that credit IR methods for their ability to retrieve highly relevant documents. This can be done by extending traditional evaluation methods, that is, recall and precision based on binary relevance judgments, to graded relevance judgments. Alternatively, novel measures based on graded relevance judgments may be developed. This article proposes several novel measures that compute the cumulative gain the user obtains by examining the retrieval result up to a given ranked position. The first one accumulates the relevance scores of retrieved documents along the ranked result list. The second one is similar but applies a discount factor to the relevance scores in order to devaluate late-retrieved documents. The third one computes the relative-to-the-ideal performance of IR techniques, based on the cumulative gain they are able to yield. These novel measures are defined and discussed and their use is demonstrated in a case study using TREC data: sample system run results for 20 queries in TREC-7. As a relevance base we used novel graded relevance judgments on a four-point scale. The test results indicate that the proposed measures credit IR methods for their ability to retrieve highly relevant documents and allow testing of statistical significance of effectiveness differences. The graphs based on the measures also provide insight into the performance IR techniques and allow interpretation, for example, from the user point of view.
TL;DR: The Probabilistic Matrix Factorization (PMF) model is presented, which scales linearly with the number of observations and performs well on the large, sparse, and very imbalanced Netflix dataset and is extended to include an adaptive prior on the model parameters.
Abstract: Many existing approaches to collaborative filtering can neither handle very large datasets nor easily deal with users who have very few ratings. In this paper we present the Probabilistic Matrix Factorization (PMF) model which scales linearly with the number of observations and, more importantly, performs well on the large, sparse, and very imbalanced Netflix dataset. We further extend the PMF model to include an adaptive prior on the model parameters and show how the model capacity can be controlled automatically. Finally, we introduce a constrained version of the PMF model that is based on the assumption that users who have rated similar sets of movies are likely to have similar preferences. The resulting model is able to generalize considerably better for users with very few ratings. When the predictions of multiple PMF models are linearly combined with the predictions of Restricted Boltzmann Machines models, we achieve an error rate of 0.8861, that is nearly 7% better than the score of Netflix's own system.
4,022 citations
"Interval-valued Matrix Factorizatio..." refers methods in this paper
...In this paper, we pay special attention to Nonnegative Matrix Factorization (NMF) [3], [4] and Probabilistic Matrix Factorization (PMF) [5]....
TL;DR: From basic techniques to the state-of-the-art, this paper attempts to present a comprehensive survey for CF techniques, which can be served as a roadmap for research and practice in this area.
Abstract: As one of the most successful approaches to building recommender systems, collaborative filtering (CF) uses the known preferences of a group of users to make recommendations or predictions of the unknown preferences for other users. In this paper, we first introduce CF tasks and their main challenges, such as data sparsity, scalability, synonymy, gray sheep, shilling attacks, privacy protection, etc., and their possible solutions. We then present three main categories of CF techniques: memory-based, modelbased, and hybrid CF algorithms (that combine CF with other recommendation techniques), with examples for representative algorithms of each category, and analysis of their predictive performance and their ability to address the challenges. From basic techniques to the state-of-the-art, we attempt to present a comprehensive survey for CF techniques, which can be served as a roadmap for research and practice in this area.
3,406 citations
"Interval-valued Matrix Factorizatio..." refers methods in this paper
...On the other hand,
PMF has been successfully applied to Collaborative Filtering (CF) [6]....
TL;DR: This paper proposes to make use of a temperature controlled version of the Expectation Maximization algorithm for model fitting, which has shown excellent performance in practice, and results in a more principled approach with a solid foundation in statistical inference.
Abstract: This paper presents a novel statistical method for factor analysis of binary and count data which is closely related to a technique known as Latent Semantic Analysis. In contrast to the latter method which stems from linear algebra and performs a Singular Value Decomposition of co-occurrence tables, the proposed technique uses a generative latent class model to perform a probabilistic mixture decomposition. This results in a more principled approach with a solid foundation in statistical inference. More precisely, we propose to make use of a temperature controlled version of the Expectation Maximization algorithm for model fitting, which has shown excellent performance in practice. Probabilistic Latent Semantic Analysis has many applications, most prominently in information retrieval, natural language processing, machine learning from text, and in related areas. The paper presents perplexity results for different types of text and linguistic data collections and discusses an application in automated document indexing. The experiments indicate substantial and consistent improvements of the probabilistic method over standard Latent Semantic Analysis.
TL;DR: This work focuses on Symbolic Data Analysis and the SODAS Project: Purpose, History, Perspective, and Symbolic Objects, where H.H. Bock and E. Diday focused on the former and the latter dealt with the latter.
Abstract: E. Diday: Symbolic Data Analysis and the SODAS Project: Purpose, History, Perspective.- H.H. Bock: The Classical Data Situation.- H.H. Bock: Symbolic Data.- H.H. Bock, E. Diday: Symbolic Objects.- V. Stephan, G. Hebrail, Y. Lechevallier: Generation of Symbolic Objects from Relational Databases.- P. Bertrand, F. Goupil: Descriptive Statistics for Symbolic Data.- M. Noirhomme-Fraiture, M. Rouard: Visualizing and Editing Symbolic Objects.- Similarity and Dissimilarity: F. Esposito, D. Malerba, V. Tamma, H.H. Bock: Classical Resemblance Measures.- H.H. Bock: Dissimilarity Measures for Probability Distributions.- F. Esposito, D. Malerba, V. Tamma: Dissimilarity Measures for Symbolic Objects.- F. Esposito, D. Malerba, F. Lisi: Matching Symbolic Objects.- Symbolic Factor Analysis: H.H.Bock: Classical Principal Component Analysis.- A. Chouakria, P. Cazes, E. Diday: Symbolic Principal Component Analysis.- N.C. Lauro, F. Palumbo, R. Verde: Factorial Discriminant Analysis on Symbolic Objects.- Discrimination: Assigning Symbolic Objects to Classes: J. Rasson, S. Lissoir: Classical Methods of Discrimination.- J. Rasson, S. Lissoir: Symbolic Kernel Discriminant Analysis.- E. Perinel, Y. Lechevalier: Symbolic Discrimination Rules.- M. Bravo Llatas, J. Garcia-Santesmases: Segmentation Trees for Stratified Data.- Clustering Methods for Symbolic Objects: M. Chavent, H.H. Bock: Clustering Problem, Clustering Methods for Classical Data.- M. Chavent: Criterion-Based Divisive Clustering for Symbolic Data.- P. Brito: Hierarchical and Pyramidal Clustering with Complete Symbolic Objects.- G. Polaillon: Pyramidal Classification for Interval Data Using Galois Lattice Reduction.- M. Gettler-Summa, C. Pardoux: Symbolic Approaches for Three-way Data.-Illustrative Benchmark Analysis: R. Bisdorff: Introduction.- R. Bisdorff: Professional Careers of Retired Working Persons.- A. Iztueta, P. Calvo: Labour Force Survey.- F. Goupil, M. Touati, E. Diday, R. Moult: Census Data from the Office for National Statistics.- A. Morineau: The SODAS Software Package.
605 citations
"Interval-valued Matrix Factorizatio..." refers background in this paper
...Again, we calculate the radius 𝛿CF𝑖𝑗 for each observed rating degree 𝑋𝑖𝑗 according to Definition 1 based on the standard deviation of the ratings in 𝒮CF𝑖𝑗 :
𝛿CF𝑖𝑗 := 𝛼 ⋅ std(𝒮CF𝑖𝑗 ) (6)
where 𝛼 ∈ ℝ+ is again a multiplicative scale coefficient....
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...However, due to the constraint of the rating system, 𝑢 can only rate both 𝑎 and 𝑏 as three stars, and the difference between 𝑎 and 𝑏 disappears....
Q1. What have the authors contributed in "Interval-valued matrix factorization with applications" ?
In this paper, the authors propose the Interval-valued Matrix Factorization ( IMF ) framework. In this paper, the authors analyze the data approximation in FA as well as CF applications and construct interval-valued matrices to capture these approximation phenomenons. The authors adapt basic NMF and PMF models to the interval-valued matrices and propose Interval-valued NMF ( I-NMF ) as well as Intervalvalued PMF ( I-PMF ). The authors conduct extensive experiments to show that proposed I-NMF and I-PMF significantly outperform their single-valued counterparts in FA and CF applications.
Q2. What is the way to evaluate the IMF framework?
The evaluations over multiple real-life data sets with different experimental settings show that I-NMF and I-PMF, which take these interval-valued matrices as input, significantly outperform their corresponding single-valued counterparts.
Q3. How do the authors propose the IMF framework?
5http://www.mit.edu/∼rsalakhu/BPMF.htmlIn this paper the authors propose the IMF framework which injects data approximation into traditional MF via taking intervalvalued matrices as input.