This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

The Analysis of Time Series: An Introduction, 4th edn. By C. Chatfield. ISBN 0 412 31820 2. Chapman and Hall, London, 1989. 242 pp. £13.50.

The Analysis of Time Series: An Introduction

Digital processing of speech signals

eMedicine创建于1996年，由近万名临床医师作为作者或编辑参与此临床医学知识库的建设，其中编辑均是来自美国哈佛、耶鲁、斯坦福、芝加哥、德克萨斯、加州大学等各分校医学院的教授或副教授。

eMedicine

An unsupervised learning algorithm for the separation of sound sources in one-channel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a time-varying gain. Each sound source, in turn, is modeled as a sum of one or more components. The parameters of the components are estimated by minimizing the reconstruction error between the input spectrogram and the model, while restricting the component spectrograms to be nonnegative and favoring components whose gains are slowly varying and sparse. Temporal continuity is favored by using a cost term which is the sum of squared differences between the gains in adjacent frames, and sparseness is favored by penalizing nonzero gains. The proposed iterative estimation algorithm is initialized with random values, and the gains and the spectra are then alternatively updated using multiplicative update rules until the values converge. Simulation experiments were carried out using generated mixtures of pitched musical instrument samples and drum sounds. The performance of the proposed method was compared with independent subspace analysis and basic nonnegative matrix factorization, which are based on the same linear model. According to these simulations, the proposed method enables a better separation quality than the previous algorithms. Especially, the temporal continuity criterion improved the detection of pitched musical sounds. The sparseness criterion did not produce significant improvements

Monaural Sound Source Separation by Nonnegative Matrix Factorization With Temporal Continuity and Sparseness Criteria

An algorithm for Non-negative Tensor Factorisation is introduced which extends current matrix factorisation techniques to deal with tensors. The effectiveness of the algorithm is then demonstrated through tests on synthetic data. The algorithm is then employed as a means of performing sound source separation on two channel mixtures, and the separation capabilities of the algorithm demonstrated on a two channel mixture containing saxophone, strings and bass guitar. Keywords - Non-negative tensor factorisation, sound source separation.

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Non-negative Tensor Factorisation for Sound Source Separation

Recently, shift-invariant tensor factorisation algorithms have been proposed for the purposes of sound source separation of pitched musical instruments. However, in practice, existing algorithms require the use of log-frequency spectrograms to allow shift invariance in frequency which causes problems when attempting to resynthesise the separated sources. Further, it is difficult to impose harmonicity constraints on the recovered basis functions. This paper proposes a new additive synthesis-based approach which allows the use of linear-frequency spectrograms as well as imposing strict harmonic constraints, resulting in an improved model. Further, these additional constraints allow the addition of a source filter model to the factorisation framework, and an extended model which is capable of separating mixtures of pitched and percussive instruments simultaneously.

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Extended nonnegative tensor factorisation models for musical sound source separation.

In this paper we present a novel sound source separation algorithm which requires no prior knowledge, no learning, assisted or otherwise, and performs the task of separation based purely on azimuth discrimination within the stereo field. The algorithm exploits the use of the pan pot as a means to achieve image localisation within stereophonic recordings. As such, only an interaural intensity difference exists between left and right channels for a single source. We use gain scaling and phase cancellation techniques to expose frequency dependent nulls across the azimuth domain, from which source separation and resynthesis is carried out. We present results obtained from real recordings, and show that for musical recordings, the algorithm improves upon the output quality of current source separation schemes.

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Sound Source Separation: Azimuth Discrimination and Resynthesis

While Independent Subspace Analysis provides a means of separating sound sources from a single channel signal, making it an effective tool for drum transcription, it does have a number of problems. Not least of these is that the amount of information required to allow separation of sound sources varies from signal to signal. To overcome this indeterminacy and improve the robustness of transcription an extension of Independent Subspace Analysis to include sub-band processing is proposed. The use of this approach is demonstrated by its application in a simple drum transcription algorithm.

/pdf/sub-band-independent-subspace-analysis-for-drum-2bqfnfknnk.pdf

Sub-band independent subspace analysis for drum transcription

Non-negative Tensor Factorisation based methods have found use in the context of musical sound source separation. These techniques require the use of a suitable cost function to determine the optimal factorisation, and most work has focused on the use of the generalised Kullback-Liebler divergence, and more recently the Itakura-Saito divergence. These divergences can be regarded as limiting cases of the parameterised Beta divergence. This paper looks at the use of the Beta Divergence in the context of musical source separation with a view to determining an optimal value of Beta for this problem. This is considered for both magnitude and power spectrograms. In an effort to avoid potential local minima in the Beta divergence, the use of a "tempered" Beta Divergence is also explored. Also presented are the update equations for the generalised non-negative tensor factorisation model described in this paper which were previously unpublished.

https://arrow.tudublin.ie/cgi/viewcontent.cgi?article=1076&context=argcon

Eugene Coyle

Papers

Non-negative Tensor Factorisation for Sound Source Separation

Extended nonnegative tensor factorisation models for musical sound source separation.

Sound Source Separation: Azimuth Discrimination and Resynthesis

Sub-band independent subspace analysis for drum transcription

On the use of the beta divergence for musical source separation