The Experts below are selected from a list of 408 Experts worldwide ranked by ideXlab platform
Ali Sekmen - One of the best experts on this subject based on the ideXlab platform.
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Reduced Row Echelon Form and non-linear approximation for subspace segmentation and high-dimensional data clustering
Applied and Computational Harmonic Analysis, 2014Co-Authors: Akram Aldroubi, Ali SekmenAbstract:Abstract Given a set of data W = { w 1 , … , w N } ∈ R D drawn from a union of subspaces, we focus on determining a nonlinear model of the Form U = ⋃ i ∈ I S i , where { S i ⊂ R D } i ∈ I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced Row Echelon Form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace S i . We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise.
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Reduction and Null Space Algorithms for the Subspace Clustering Problem
2010Co-Authors: Akram Aldroubi, Ali SekmenAbstract:This paper presents two algorithms for clustering high-dimensional data points that are drawn from a union of lower dimensional subspaces. The first algorithm is based on binary reduced Row Echelon Form of a data matrix. It can solve the subspace segmentation problem perfectly for noise free data, however, it is not reliable for noisy cases. The second algorithm is based on Null Space representation of data. It is devised for the cases when the subspace dimensions are equal. Such cases occur in applications such as motion segmentation and face recognition. This algorithm is reliable in the presence of noise, and applied to the Hopkins 155 Dataset it generates the best results to date for motion segmentation. The recognition rates for two and three motion video sequences are 99.15% and 98.85%, respectively.
Akram Aldroubi - One of the best experts on this subject based on the ideXlab platform.
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Reduced Row Echelon Form and non-linear approximation for subspace segmentation and high-dimensional data clustering
Applied and Computational Harmonic Analysis, 2014Co-Authors: Akram Aldroubi, Ali SekmenAbstract:Abstract Given a set of data W = { w 1 , … , w N } ∈ R D drawn from a union of subspaces, we focus on determining a nonlinear model of the Form U = ⋃ i ∈ I S i , where { S i ⊂ R D } i ∈ I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced Row Echelon Form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace S i . We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise.
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Reduction and Null Space Algorithms for the Subspace Clustering Problem
2010Co-Authors: Akram Aldroubi, Ali SekmenAbstract:This paper presents two algorithms for clustering high-dimensional data points that are drawn from a union of lower dimensional subspaces. The first algorithm is based on binary reduced Row Echelon Form of a data matrix. It can solve the subspace segmentation problem perfectly for noise free data, however, it is not reliable for noisy cases. The second algorithm is based on Null Space representation of data. It is devised for the cases when the subspace dimensions are equal. Such cases occur in applications such as motion segmentation and face recognition. This algorithm is reliable in the presence of noise, and applied to the Hopkins 155 Dataset it generates the best results to date for motion segmentation. The recognition rates for two and three motion video sequences are 99.15% and 98.85%, respectively.
Natalia Silberstein - One of the best experts on this subject based on the ideXlab platform.
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Representation of Subspaces and Enumerative Encoding of the Grassmannian Space
2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:Codes in the Grassmannian space have found recently application in network coding. Representation of k- dimensional subspaces of F n has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different represen- tations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on a Ferrers diagram representation of subspaces; and another is based on an identifying vector and a reduced Row Echelon Form representation of subspaces. A third method which combines the previous two is more efficient than the other two enumerative encodings. Each enumerative encoding is induced by some ordering of the Grassmannian. These orderings also induce lexicographic codes in the Grassmannian. Some of these codes suggest a new method to generate error-correcting codes in the Grassmannian with larger size than the current known codes.
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Enumerative Coding for Grassmannian Space
arXiv: Information Theory, 2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of $\F_q^n$. One enumerative coding method is based on a Ferrers diagram representation and on an order for $\Gr$ based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced Row Echelon Form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.
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Enumerative Encoding in the Grassmannian Space
arXiv: Information Theory, 2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:Codes in the Grassmannian space have found recently application in network coding. Representation of $k$-dimensional subspaces of $\F_q^n$ has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different representations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on Ferrers diagrams representation of subspaces; and another is based on identifying vector and reduced Row Echelon Form representation of subspaces. A third method which combine the previous two is more efficient than the other two enumerative encodings.
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Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams
arXiv: Information Theory, 2008Co-Authors: Tuvi Etzion, Natalia SilbersteinAbstract:Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced Row Echelon Form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced Row Echelon Form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction Form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank-metric codes. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension.
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Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams
2008Co-Authors: Tuvi Etzion, Natalia SilbersteinAbstract:Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced Row Echelon Form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced Row Echelon Form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rankmetric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. All the proposed codes can be efficiently encoded and decoded. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension
Tuvi Etzion - One of the best experts on this subject based on the ideXlab platform.
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Representation of Subspaces and Enumerative Encoding of the Grassmannian Space
2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:Codes in the Grassmannian space have found recently application in network coding. Representation of k- dimensional subspaces of F n has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different represen- tations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on a Ferrers diagram representation of subspaces; and another is based on an identifying vector and a reduced Row Echelon Form representation of subspaces. A third method which combines the previous two is more efficient than the other two enumerative encodings. Each enumerative encoding is induced by some ordering of the Grassmannian. These orderings also induce lexicographic codes in the Grassmannian. Some of these codes suggest a new method to generate error-correcting codes in the Grassmannian with larger size than the current known codes.
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Enumerative Coding for Grassmannian Space
arXiv: Information Theory, 2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of $k$-dimensional subspaces of $\F_q^n$. One enumerative coding method is based on a Ferrers diagram representation and on an order for $\Gr$ based on this representation. The complexity of this enumerative coding is $O(k^{5/2} (n-k)^{5/2})$ digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced Row Echelon Form representation of subspaces. The complexity of the enumerative coding, based on this order, is $O(nk(n-k)\log n\log\log n)$ digits operations. A combination of the two methods reduces the complexity on average by a constant factor.
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Enumerative Encoding in the Grassmannian Space
arXiv: Information Theory, 2009Co-Authors: Natalia Silberstein, Tuvi EtzionAbstract:Codes in the Grassmannian space have found recently application in network coding. Representation of $k$-dimensional subspaces of $\F_q^n$ has generally an essential role in solving coding problems in the Grassmannian, and in particular in encoding subspaces of the Grassmannian. Different representations of subspaces in the Grassmannian are presented. We use two of these representations for enumerative encoding of the Grassmannian. One enumerative encoding is based on Ferrers diagrams representation of subspaces; and another is based on identifying vector and reduced Row Echelon Form representation of subspaces. A third method which combine the previous two is more efficient than the other two enumerative encodings.
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Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams
arXiv: Information Theory, 2008Co-Authors: Tuvi Etzion, Natalia SilbersteinAbstract:Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced Row Echelon Form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced Row Echelon Form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction Form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank-metric codes. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension.
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Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams
2008Co-Authors: Tuvi Etzion, Natalia SilbersteinAbstract:Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced Row Echelon Form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced Row Echelon Form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rankmetric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. All the proposed codes can be efficiently encoded and decoded. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension
Piotr Dniestrzański - One of the best experts on this subject based on the ideXlab platform.
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SYSTEMS OF LINEAR EQUATIONS AND REDUCED MATRIX IN A LINEAR ALGEBRA COURSE FOR ECONOMICS STUDIES
2016Co-Authors: Piotr DniestrzańskiAbstract:Abstract. The article presents two elements of the concept of a linear algebra lecture for economics studies. It attempts to demonstrate the significant role of ordering of the lectured content – with a focus on starting the lecture with systems of linear equations, and shows the considerable benefits of introducing the concept of the reduced Row Echelon Form of matrix as one of the most useful concepts of linear algebra
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Systems of Linear Equations and Reduced Matrix in a Linear Algebra Course or Economics Studies
2012Co-Authors: Piotr DniestrzańskiAbstract:The article presents two elements of the concept of a linear algebra lecture for economics studies. It attempts to demonstrate the significant role of ordering of the lectured content – with a focus on starting the lecture with systems of linear equations, and shows the considerable benefits of introducing the concept of the reduced Row Echelon Form of matrix as one of the most useful concepts of linear algebra.
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Systems of linear equations and reduced matrix in a linear algebra course for economics studies. Didactics of Mathematics, 2012, Nr 9 (13), s. 43-54
2012Co-Authors: Piotr DniestrzańskiAbstract:The article presents two elements of the concept of a linear algebra lecture for economics studies. It attempts to demonstrate the significant role of ordering of the lectured content – with a focus on starting the lecture with systems of linear equations, and shows the considerable benefits of introducing the concept of the reduced Row Echelon Form of matrix as one of the most useful concepts of linear algebra.
