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MaximumMargin Matrix Factorization
 Advances in Neural Information Processing Systems 17
, 2005
"... We present a novel approach to collaborative prediction, using lownorm instead of lowrank factorizations. The approach is inspired by, and has strong connections to, largemargin linear discrimination. We show how to learn lownorm factorizations by solving a semidefinite program, and discuss ..."
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Cited by 260 (20 self)
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We present a novel approach to collaborative prediction, using lownorm instead of lowrank factorizations. The approach is inspired by, and has strong connections to, largemargin linear discrimination. We show how to learn lownorm factorizations by solving a semidefinite program, and discuss generalization error bounds for them.
Fast maximum margin matrix factorization for collaborative prediction
 In Proceedings of the 22nd International Conference on Machine Learning (ICML
, 2005
"... Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to lowrank approximations and standard factor models. MMMF can be formulated as a semidefinite programming (SDP) and learned using standard SDP solvers. However, cu ..."
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Cited by 241 (8 self)
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Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to lowrank approximations and standard factor models. MMMF can be formulated as a semidefinite programming (SDP) and learned using standard SDP solvers. However, current SDP solvers can only handle MMMF problems on matrices of dimensionality up to a few hundred. Here, we investigate a direct gradientbased optimization method for MMMF and demonstrate it on large collaborative prediction problems. We compare against results obtained by Marlin (2004) and find that MMMF substantially outperforms all nine methods he tested. 1.
Nonnegative matrix factorization for polyphonic music transcription
 in Proc. IEEE Workshop Applications of Signal Processing to Audio and Acoustics
, 2003
"... In this paper we present a methodology for analyzing polyphonic musical passages comprised by notes that exhibit a harmonically fixed spectral profile (such as piano notes). Taking advantage of this unique note structure we can model the audio content of the musical passage by a linear basis transfo ..."
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Cited by 240 (14 self)
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In this paper we present a methodology for analyzing polyphonic musical passages comprised by notes that exhibit a harmonically fixed spectral profile (such as piano notes). Taking advantage of this unique note structure we can model the audio content of the musical passage by a linear basis transform and use nonnegative matrix decomposition methods to estimate the spectral profile and the temporal information of every note. This approach results in a very simple and compact system that is not knowledge based, but rather learns notes by observation.
The Missing Link  A Probabilistic Model of Document Content and Hypertext Connectivity
, 2001
"... We describe a joint probabilistic model for modeling the contents and interconnectivity of document collections such as sets of web pages or research paper archives. The model is based on a probabilistic factor decomposition and allows identifying principal topics of the collection as well as autho ..."
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Cited by 217 (3 self)
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We describe a joint probabilistic model for modeling the contents and interconnectivity of document collections such as sets of web pages or research paper archives. The model is based on a probabilistic factor decomposition and allows identifying principal topics of the collection as well as authoritative documents within those topics. Furthermore, the relationships between topics is mapped out in order to build a predictive model of link content. Among the many applications of this approach are information retrieval and search, topic identification, query disambiguation, focused web crawling, web authoring, and bibliometric analysis.
Efficient learning of sparse representations with an energybased model
 ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS (NIPS 2006
, 2006
"... We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying nonlinearity that turns a code vector into a quasibinary sparse code vector. Given an input, the optimal code minimizes the distance b ..."
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Cited by 213 (16 self)
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We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying nonlinearity that turns a code vector into a quasibinary sparse code vector. Given an input, the optimal code minimizes the distance between the output of the decoder and the input patch while being as similar as possible to the encoder output. Learning proceeds in a twophase EMlike fashion: (1) compute the minimumenergy code vector, (2) adjust the parameters of the encoder and decoder so as to decrease the energy. The model produces “stroke detectors ” when trained on handwritten numerals, and Gaborlike filters when trained on natural image patches. Inference and learning are very fast, requiring no preprocessing, and no expensive sampling. Using the proposed unsupervised method to initialize the first layer of a convolutional network, we achieved an error rate slightly lower than the best reported result on the MNIST dataset. Finally, an extension of the method is described to learn topographical filter maps. 1
A DataDriven Reflectance Model
 ACM TRANSACTIONS ON GRAPHICS
, 2003
"... We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space o ..."
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Cited by 212 (7 self)
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We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space of acquired BRDFs to create new BRDFs. We treat each acquired BRDF as a single highdimensional vector taken from a space of all possible BRDFs. We apply both linear (subspace) and nonlinear (manifold) dimensionality reduction tools in an effort to discover a lowerdimensional representation that characterizes our measurements. We let users define perceptually meaningful parametrization directions to navigate in the reduceddimension BRDF space. On the lowdimensional manifold, movement along these directions produces novel but valid BRDFs.
Algorithms and applications for approximate nonnegative matrix factorization
 Computational Statistics and Data Analysis
, 2006
"... In this paper we discuss the development and use of lowrank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both spars ..."
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Cited by 199 (7 self)
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In this paper we discuss the development and use of lowrank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for largescale and timevarying datasets. Key words: nonnegative matrix factorization, text mining, spectral data analysis, email surveillance, conjugate gradient, constrained least squares.
Learning Spatially Localized, PartsBased Representation
, 2001
"... In this paper, we propose a novel method, called local nonnegative matrix factorization (LNMF), for learning spatially localized, partsbased subspace representation of visual patterns. An objective function is defined to impose localization constraint, in addition to the nonnegativity constraint i ..."
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Cited by 196 (7 self)
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In this paper, we propose a novel method, called local nonnegative matrix factorization (LNMF), for learning spatially localized, partsbased subspace representation of visual patterns. An objective function is defined to impose localization constraint, in addition to the nonnegativity constraint in the standard NMF [1]. This gives a set of bases which not only allows a nonsubtractive (partbased) representation of images but also manifests localized features. An algorithm is presented for the learning of such basis components. Experimental results are presented to compare LNMF with the NMF and PCA methods for face representation and recognition, which demonstrates advantages of LNMF.
When Does NonNegative Matrix Factorization Give Correct Decomposition into Parts?
, 2003
"... We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the f ..."
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Cited by 191 (1 self)
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We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of `parts' and NMF correctly identifies the `parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.
Image SuperResolution via Sparse Representation
"... This paper presents a new approach to singleimage superresolution, based on sparse signal representation. Research on image statistics suggests that image patches can be wellrepresented as a sparse linear combination of elements from an appropriately chosen overcomplete dictionary. Inspired by th ..."
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Cited by 189 (9 self)
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This paper presents a new approach to singleimage superresolution, based on sparse signal representation. Research on image statistics suggests that image patches can be wellrepresented as a sparse linear combination of elements from an appropriately chosen overcomplete dictionary. Inspired by this observation, we seek a sparse representation for each patch of the lowresolution input, and then use the coefficients of this representation to generate the highresolution output. Theoretical results from compressed sensing suggest that under mild conditions, the sparse representation can be correctly recovered from the downsampled signals. By jointly training two dictionaries for the low resolution and high resolution image patches, we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries. Therefore, the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch. The learned dictionary pair is a more compact representation of the patch pairs, compared to previous approaches which simply sample a large amount of image patch pairs, reducing the computation cost substantially. The effectiveness of such a sparsity prior is demonstrated for general image superresolution and also for the special case of face hallucination. In both cases, our algorithm can generate highresolution images that are competitive or even superior in quality to images produced by other similar SR methods, but with faster processing speed.