• Title, Summary, Keyword: Orthogonal Matrix

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A SIMPLE CONSTRUCTION FOR THE SPARSE MATRICES WITH ORTHOGONAL ROWS

  • Cheon, Gi-Sang;Lee, Gwang-Yeon
    • Communications of the Korean Mathematical Society
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    • v.15 no.4
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    • pp.587-595
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    • 2000
  • We contain a simple construction for the sparse n x n connected orthogonal matrices which have a row of p nonzero entries with 2$\leq$p$\leq$n. Moreover, we study the analogous sparsity problem for an m x n connected row-orthogonal matrices.

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A Mathematical Implementation of OFDM System with Orthogonal Basis Matrix (직교 기저행렬을 이용하는 직교 주파수분할다중화의 수학적 구현)

  • Kang, Seog-Geun
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.13 no.12
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    • pp.2731-2736
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    • 2009
  • In this paper, a new implementation method of OFDM (orthogonal frequency division multiplexing) system with an orthogonal basis matrix is developed mathematically. The basis matrix is based on the Haar basis but has an appropriate form for modulation of multiple subchannel signals of OFDM. It is verified that the new basis matrix can be expanded with a simple recursive algorithm. The order of synthesis matrix in the transmitter is increased by the factor of two with every expansion. Demodulation in the receiver is carried out by its inverse matrix, which can be generated recursively with the orthogonal basis matrix. It is shown that perfect reconstruction of original signals is possibly achieved in the proposed OFDMsystem.

Nonnegative Matrix Factorization with Orthogonality Constraints

  • Yoo, Ji-Ho;Choi, Seung-Jin
    • Journal of Computing Science and Engineering
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    • v.4 no.2
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    • pp.97-109
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    • 2010
  • Nonnegative matrix factorization (NMF) is a popular method for multivariate analysis of nonnegative data, which is to decompose a data matrix into a product of two factor matrices with all entries restricted to be nonnegative. NMF was shown to be useful in a task of clustering (especially document clustering), but in some cases NMF produces the results inappropriate to the clustering problems. In this paper, we present an algorithm for orthogonal nonnegative matrix factorization, where an orthogonality constraint is imposed on the nonnegative decomposition of a term-document matrix. The result of orthogonal NMF can be clearly interpreted for the clustering problems, and also the performance of clustering is usually better than that of the NMF. We develop multiplicative updates directly from true gradient on Stiefel manifold, whereas existing algorithms consider additive orthogonality constraints. Experiments on several different document data sets show our orthogonal NMF algorithms perform better in a task of clustering, compared to the standard NMF and an existing orthogonal NMF.

CONSTRUCTIONS FOR SPARSE ROW-ORTHOGONAL MATRICES WITH A FULL ROW

  • Cheon, Gi-Sang;Park, Se-Won;Seol, Han-Guk
    • Journal of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.333-344
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    • 1999
  • In [4], it was shown that an n by n orthogonal matrix which has a row of nonzeros has at least ( log2n + 3)n - log2n +1 nonzero entries. In this paper, the matrices achieving these bounds are constructed. The analogous sparsity problem for m by n row-orthogonal matrices which have a row of nonzeros in conjectured.

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An Efficient Computing Method of the Orthogonal Projection Matrix for the Balanced Factorial Design

  • Kim, Byung-Chun;Park, Jong-Tae
    • Journal of the Korean Statistical Society
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    • v.22 no.2
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    • pp.249-258
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    • 1993
  • It is well known that design matrix X for any factorial design can be represented by a product $X = TX_o$ where T is replication matrix and $X_o$ is the corresponding balanced design matrix. Since $X_o$ consists of regular arrangement of 0's and 1's, we can easily find the spectral decomposition of $X_o',X_o$. Also using this we propose an efficient algorithm for computing the orthogonal projection matrix for a balanced factorial design.

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CONSTRUCTIONS FOR THE SPARSEST ORTHOGONAL MATRICES

  • Cheon, Gi-Sang;Shader, Bryan L.
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.119-129
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    • 1999
  • In [1], it was shown that for $n\geq 2$ the least number of nonzero entries in an $n\times n$ orthogonal matrix is not direct summable is 4n-4, and zero patterns of the $n\times n$ orthogonal matrices with exactly 4n-4 nonzero entries were determined. In this paper, we construct $n\times n$ orthogonal matrices with exactly 4n-r nonzero entries. furthermore, we determine m${\times}$n sparse row-orthogonal matrices.

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An Orthogonal Representation of Estimable Functions

  • Yi, Seong-Baek
    • Communications for Statistical Applications and Methods
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    • v.15 no.6
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    • pp.837-842
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    • 2008
  • Students taking linear model courses have difficulty in determining which parametric functions are estimable when the design matrix of a linear model is rank deficient. In this note a special form of estimable functions is presented with a linear combination of some orthogonal estimable functions. Here, the orthogonality means the least squares estimators of the estimable functions are uncorrelated and have the same variance. The number of the orthogonal estimable functions composing the special form is equal to the rank of the design matrix. The orthogonal estimable functions can be easily obtained through the singular value decomposition of the design matrix.

THE EXISTENCE THEOREM OF ORTHOGONAL MATRICES WITH p NONZERO ENTRIES

  • CHEON, GI-SANG;LEE, SANG-GU;SONG, SEOK-ZUN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.1
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    • pp.109-119
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    • 2000
  • It was shown that if Q is a fully indecomposable $n{\times}n$ orthogonal matrix then Q has at least 4n-4 nonzero entries in 1993. In this paper, we show that for each integer p with $4n-4{\leq}p{\leq}n^2$, there exist a fully indecomposable $n{\times}n$ orthogonal matrix with exactly p nonzero entries. Furthermore, we obtain a method of construction of a fully indecomposable $n{\times}n$ orthogonal matrix which has exactly 4n-4 nonzero entries. This is a part of the study in sparse matrices.

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AN INVERSE HOMOGENEOUS INTERPOLATION PROBLEM FOR V-ORTHOGONAL RATIONAL MATRIX FUNCTIONS

  • Kim, Jeon-Gook
    • Journal of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.717-734
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    • 1996
  • For a scalar rational function, the spectral data consisting of zeros and poles with their respective multiplicities uniquely determines the function up to a nonzero multiplicative factor. But due to the richness of the spectral structure of a rational matrix function, reconstruction of a rational matrix function from a given spectral data is not that simple.

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On the Design of the Linear Phase Lapped Orthogonal Transform Bases Using the Prefilter Approach (전처리 필터를 이용한 선형 위성 LOT 기저의 설계에 관한 연구)

  • 이창우;이상욱
    • Journal of the Korean Institute of Telematics and Electronics B
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    • v.31B no.7
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    • pp.91-100
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    • 1994
  • The lapped orthogonal transform(LOT) has been recently proposed to alleviate the blocking effects in transform coding. The LOT is known to provide an improved coding gain than the conventional transform. In this paper, we propose a prefilter approach to design the LOT bases with the view of maximizing the transform coding gain. Since the nonlinear phase basis is inappropriate to the image coding only the linear phase basis is considered in this paper. Our approach is mainly based on decomposing the transform matrix into the orthogonal matrix and the prefilter matrix. And by assuming that the input is the 1st order Markov source we design the prefilter matrix and the orthogonal matirx maximizing the transform coding gain. The computer simulation results show that the proposed LOT provides about 0.6~0.8 dB PSNR gain over the DCT and about 0.2~0.3 dB PSNR gain over the conventional LOT [7]. Also, the subjective test reveals that the proposed LOT shows less blocking effect than the DCT.

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