First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds

Birtea, Petre; Caşu, Ioan; Comǎnescu, Dan

doi:10.1007/s11590-018-1319-x

Cited by 19 publications

(21 citation statements)

References 24 publications

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“…( 31), we need to compute the gradient of F i.e., ∇ F(R) and descent direction first. In literature [4,7,28,31], the definition of the derivative of F(R) at R n in the direction of Z is given by,…”

Section: Remark 10mentioning

confidence: 99%

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Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

Saha

Khare

2021

Int. J. Appl. Comput. Math

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We deal with the problem to compute the nearest Structured Low-Rank Approximation (SLRA) to a given matrix in this paper. This problem arises in many practical applications, such as computation of approximate GCD of polynomials, matrix completion problems, image processing and control theory etc. We reformulate this problem as an unconstrained optimization problem on an appropriately chosen Stiefel manifold. This proposed formulation is based on the condition that the computed nearest SLRA is required to have a suitable number of linearly independent vectors in its nullspace. Thus, this formulation helps in computing the nearest SLRA with exact required rank as opposed to many numerical techniques available in literature. Two different techniques to solve the optimization problem on the Stiefel manifold are discussed in this paper with several numerical case studies. The effectiveness of the proposed approach is demonstrated through numerical examples and by comparison with the benchmark methods available in the literature.

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Section: Remark 10mentioning

confidence: 99%

“…The computational cost for Algorithm 2: In Algorithm 2, the computation of G takes O(nm 3 (n − k) 4 ) operations approximately (see Remark 11). Also, computation of 3 ) number of operations.…”

Section: Algorithm 2 [26]: Projected Steepest Descent With 'Armijo-backtracking' Line Searchmentioning

confidence: 99%

Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

Saha

Khare

2021

Int. J. Appl. Comput. Math

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“…. , y N are available the a posteriori distribution of the β parameter is given by Equation (13) [10,18]:…”

Section: Model Definition and Proceduresmentioning

confidence: 99%

“…In this case, it is possible that n p. When n < p, i.e., when the number of observed signals is lower than that of source signals, we are dealing with over-complete ICA bases, but when n > p we are dealing with under-complete ICA [11,12]. From a mathematical point of view, such problem can be considered an unconstrained optimization on the Stiefel manifold [13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Single Channel Source Separation with ICA-Based Time-Frequency Decomposition

Mika

Budzik

Jóźwik

2020

Sensors

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This paper relates to the separation of single channel source signals from a single mixed signal by means of independent component analysis (ICA). The proposed idea lies in a time-frequency representation of the mixed signal and the use of ICA on spectral rows corresponding to different time intervals. In our approach, in order to reconstruct true sources, we proposed a novelty idea of grouping statistically independent time-frequency domain (TFD) components of the mixed signal obtained by ICA. The TFD components are grouped by hierarchical clustering and k-mean partitional clustering. The distance between TFD components is measured with the classical Euclidean distance and the β distance of Gaussian distribution introduced by as. In addition, the TFD components are grouped by minimizing the negentropy of reconstructed constituent signals. The proposed method was used to separate source signals from single audio mixes of two- and three-component signals. The separation was performed using algorithms written by the authors in Matlab. The quality of obtained separation results was evaluated by perceptual tests. The tests showed that the automated separation requires qualitative information about time-frequency characteristics of constituent signals. The best separation results were obtained with the use of the β distance of Gaussian distribution, a distance measure based on the knowledge of the statistical nature of spectra of original constituent signals of the mixed signal.

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“…In this case, it can be n p. When n < p, i.e., the number of observed signals is smaller than the number of source signals, the problem is known as over complete bases ICA, whereas when n > p it is called under complete bases ICA. This kind of problem can be formally considered to be unconstrained optimization on the Stiefel manifold [12,13]. It is also possible to solve ICA problems for the case p = 1.…”

Section: Model Definition (Ica Isa)mentioning

confidence: 99%

Lie Group Methods in Blind Signal Processing

Mika

Jóźwik

2020

Sensors

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This paper deals with the use of Lie group methods to solve optimization problems in blind signal processing (BSP), including Independent Component Analysis (ICA) and Independent Subspace Analysis (ISA). The paper presents the theoretical fundamentals of Lie groups and Lie algebra, the geometry of problems in BSP as well as the basic ideas of optimization techniques based on Lie groups. Optimization algorithms based on the properties of Lie groups are characterized by the fact that during optimization motion, they ensure permanent bonding with a search space. This property is extremely significant in terms of the stability and dynamics of optimization algorithms. The specific geometry of problems such as ICA and ISA along with the search space homogeneity enable the use of optimization techniques based on the properties of the Lie groups O ( n ) and S O ( n ) . An interesting idea is that of optimization motion in one-parameter commutative subalgebras and toral subalgebras that ensure low computational complexity and high-speed algorithms.

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First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds

Cited by 19 publications

References 24 publications

Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

Single Channel Source Separation with ICA-Based Time-Frequency Decomposition

Lie Group Methods in Blind Signal Processing

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