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Tensor decompositions for signal processing applications. From Twoway to Multiway Component Analysis
 ESATSTADIUS INTERNAL REPORT
, 2014
"... The widespread use of multisensor technology and the emergence of big datasets has highlighted the limitations of standard flatview matrix models and the necessity to move towards more versatile data analysis tools. We show that higherorder tensors (i.e., multiway arrays) enable such a fundame ..."
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The widespread use of multisensor technology and the emergence of big datasets has highlighted the limitations of standard flatview matrix models and the necessity to move towards more versatile data analysis tools. We show that higherorder tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrixbased methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced causeeffect and multiview data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization.
Cemgil, “Score guided musical source separation using generalized coupled tensor factorization
 in EUSIPCO
, 2012
"... Providing prior knowledge about sources to guide source separation is known to be useful in many audio applications. In this paper we present two tensor factorization models for musical source separation where musical information is incorporated by using the Generalized Coupled Tensor Factorizati ..."
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Providing prior knowledge about sources to guide source separation is known to be useful in many audio applications. In this paper we present two tensor factorization models for musical source separation where musical information is incorporated by using the Generalized Coupled Tensor Factorization (GCTF) framework. The approach is an extension of Nonnegative Matrix Factorization where more than one matrix or tensor object is simultaneously factorized. The first model uses a temporally aligned transcription of the mixture and incorporates spectral knowledge via coupling. In contrast of using a temporally aligned transcription, the second model incorporates harmonic information by taking an approximate, incomplete, and not necessarily aligned transcription of the musical piece as input. We evaluate our models on piano and cello duets where the experiments show that instead of using a temporally aligned transcription, we can achieve competitive results by using only a partial and incomplete transcription.
SCORE GUIDED AUDIO RESTORATION VIA GENERALISED COUPLED TENSOR FACTORISATION
"... Generalised coupled tensor factorisation is a recently proposed algorithmic framework for simultaneously estimating tensor factorisation models where several observed tensors can share a set of latent factors. This paper proposes a model in this framework for coupled factorisation of piano spectrogr ..."
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Generalised coupled tensor factorisation is a recently proposed algorithmic framework for simultaneously estimating tensor factorisation models where several observed tensors can share a set of latent factors. This paper proposes a model in this framework for coupled factorisation of piano spectrograms and piano roll representations to solve audio interpolation and restoration problem. The model incorporates temporal and harmonic information from an approximate musical score (not necessarily belonging to the played piece), and spectral information from isolated piano sounds. The performance of the proposed approach is evaluated on the restoration of classical music pieces where we get about 5dB SNR improvement when50 % of data frames are missing. Index Terms — Audio Restoration, Coupled Tensor Factorisation 1.
PROBABILISTIC LATENT TENSOR FACTORIZATION FRAMEWORK FOR AUDIO MODELING
"... This paper introduces probabilistic latent tensor factorization (PLTF) as a general framework for hierarchical modeling of audio. This framework combines practical aspects of graphical modeling of machine learning with tensor factorization models. Once a model is constructed in the PLTF framework, t ..."
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This paper introduces probabilistic latent tensor factorization (PLTF) as a general framework for hierarchical modeling of audio. This framework combines practical aspects of graphical modeling of machine learning with tensor factorization models. Once a model is constructed in the PLTF framework, the estimation algorithm is immediately available. We illustrate our approach using several popular models such as NMF or NMF2D and provide extensions with simulation results on real data for key audio processing tasks such as restoration and source separation.
prediction via generalized coupled tensor factorisation
 in ECML/PKDDWorkshop on Collective Learning and Inference on Structured Data
, 2012
"... Abstract. This study deals with the missing link prediction problem: the problem of predicting the existence of missing connections between entities of interest. We address link prediction using coupled analysis of relational datasets represented as heterogeneous data, i.e., datasets in the form of ..."
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Abstract. This study deals with the missing link prediction problem: the problem of predicting the existence of missing connections between entities of interest. We address link prediction using coupled analysis of relational datasets represented as heterogeneous data, i.e., datasets in the form of matrices and higherorder tensors. We propose to use an approach based on probabilistic interpretation of tensor factorisation models, i.e., Generalised Coupled Tensor Factorisation, which can simultaneously fit a large class of tensor models to higherorder tensors/matrices with common latent factors using different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorisation improves the link prediction performance and the selection of right loss function and tensor model is crucial for accurately predicting missing links.
Role Discovery in Networks
, 2014
"... Roles represent nodelevel connectivity patterns such as starcenter, staredge nodes, nearcliques or nodes that act as bridges to different regions of the graph. Intuitively, two nodes belong to the same role if they are struturally similar. Roles have been mainly of interest to sociologists, b ..."
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Roles represent nodelevel connectivity patterns such as starcenter, staredge nodes, nearcliques or nodes that act as bridges to different regions of the graph. Intuitively, two nodes belong to the same role if they are struturally similar. Roles have been mainly of interest to sociologists, but more recently, roles have become increasingly useful in other domains. Traditionally, the notion of roles were defined based on graph equivalences such as structural, regular, and stochastic equivalences. We briefly revisit the notions and instead propose a more general formulation of roles based on the similarity of a feature representation (in contrast to the graph representation). This leads us to propose a taxonomy of two general classes of techniques for discovering roles which includes (i) graphbased roles and (ii) featurebased roles. This survey focuses primarily on featurebased roles. In particular, we also introduce a flexible framework for discovering roles using the notion of structural similarity on a featurebased representation. The framework consists of two fundamental components: (1) role feature construction and (2) role assignment using the learned feature representation. We discuss the relevant decisions for discovering featurebased roles and highlight the advantages and disadvantages of the many techniques that can be used for this purpose. Finally, we discuss potential applications and future directions and challenges.
Cemgil, Liver ct annotation via generalized coupled tensor factorization
 in CLEF 2014 Labs and Workshops, Notebook Papers. CEUR Workshop Proceedings (CEURWS.org
, 2014
"... Abstract. This study deals with the missing answers prediction problem. We address this problem using coupled analysis of ImageCLEF2014 dataset by representing it as a heterogeneous data, i.e., dataset in the form of matrices. We propose to use an approach based on probabilistic interpretation of t ..."
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Abstract. This study deals with the missing answers prediction problem. We address this problem using coupled analysis of ImageCLEF2014 dataset by representing it as a heterogeneous data, i.e., dataset in the form of matrices. We propose to use an approach based on probabilistic interpretation of tensor factorization models, i.e., Generalized Coupled Tensor Factorization, which can simultaneously fit a large class of matrix/tensor models to higherorder matrices/tensors with common latent factors using different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorization gives high prediction performance.
A Bayesian Tensor Factorization Model via Variational Inference for Link Prediction
"... Probabilistic approaches for tensor factorization aim to extract meaningful structure from incomplete data by postulating low rank constraints. Recently, variational Bayesian (VB) inference techniques have successfully been applied to large scale models. This paper presents full Bayesian inference ..."
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Probabilistic approaches for tensor factorization aim to extract meaningful structure from incomplete data by postulating low rank constraints. Recently, variational Bayesian (VB) inference techniques have successfully been applied to large scale models. This paper presents full Bayesian inference via VB on both single and coupled tensor factorization models. Our method can be run even for very large models and is easily implemented. It exhibits better prediction performance than existing approaches based on maximum likelihood on several realworld datasets for missing link prediction problem. 1
Generalised Coupled Tensor Factorisation
"... We derive algorithms for generalised tensor factorisation (GTF) by building upon the wellestablished theory of Generalised Linear Models. Our algorithms are general in the sense that we can compute arbitrary factorisations in a message passing framework, derived for a broad class of exponential fam ..."
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We derive algorithms for generalised tensor factorisation (GTF) by building upon the wellestablished theory of Generalised Linear Models. Our algorithms are general in the sense that we can compute arbitrary factorisations in a message passing framework, derived for a broad class of exponential family distributions including special cases such as Tweedie’s distributions corresponding to βdivergences. By bounding the step size of the Fisher Scoring iteration of the GLM, we obtain general updates for real data and multiplicative updates for nonnegative data. The GTF framework is, then extended easily to address the problems when multiple observed tensors are factorised simultaneously. We illustrate our coupled factorisation approach on synthetic data as well as on a musical audio restoration problem. 1
Probabilistic Latent Tensor Factorization for 3way Microarray Data Analysis with Missing Values
"... The recent advances in microarray technology enabled the measurement of gene expression levels of samples over a series of time points. Unlike the traditional 2D microarray data, such experiments generate 3D (genesampletime) microarray data, which require specialized methods for analysis. In this ..."
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The recent advances in microarray technology enabled the measurement of gene expression levels of samples over a series of time points. Unlike the traditional 2D microarray data, such experiments generate 3D (genesampletime) microarray data, which require specialized methods for analysis. In this study, we propose a novel tensor factorization model for modeling 3D microarray data. The model assumes the existence of certain temporal patterns that are repeated over time. One main advantage of the model is that it handles the missing data implicitly, so that the estimation process is not effected by the existence of missing values, which commonly occur in microarray data. We evaluate our model on classification of the good or bad responders to Interferon beta (INFβ) treatments by using a real genesampletime microarray data set and achieve a promising prediction performance. 1