Please use this identifier to cite or link to this item: http://localhost/handle/Hannan/582999
Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D;
Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe
subject: Science & Technology
Year: 2016
Publisher: IEEE
Abstract: Several corrections are necessary in our paper <xref ref-type="bibr" rid="ref1">[1]</xref>. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing above $k$ in $k(x,y,f_{1})$. On p. 7135, in <xref rid="deqn22" ref-type="disp-formula">equations (22)</xref>, (23) and two lines below equation (24), the minus sign &#x2212; in front of constant lambda $ should be changed into the plus sign &#x002B;. Also, on p. 7135, the left-hand side (LHS) of <xref rid="deqn22" ref-type="disp-formula">equation (22)</xref> should be integrated with respect to $y$. Thus, the correct form of <xref rid="deqn22" ref-type="disp-formula">(22)</xref> should be<disp-formula> \begin{align*}&amp;\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&amp;-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} </disp-formula> Because of the correction in <xref rid="deqn22" ref-type="disp-formula">(22)</xref>, several other equations in the paper have to be updated appropriately. Therefore, <xref rid="deqn67" ref-type="disp-formula">equation (67)</xref> will take the form:<disp-formula> \begin{align*}&amp;\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} </disp-formula> Similarly, <xref rid="deqn112-113" ref-type="disp-formula">equations (112) and (113)</xref> must be updated to:<disp-formula> \begin{align*}&amp;\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&amp;+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&amp;+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&amp;\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&amp;\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&amp;\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} </disp-formula> and the sentence following <xref rid="deqn112-113" ref-type="disp-formula">equation (113)</xref> should be read as: n&#x2018;The following functions satisfy the equalities in <xref rid="deqn112-113" ref-type="disp-formula">(112) and (113)</xref>:&#x2019;
URI: http://localhost/handle/Hannan/164994
http://localhost/handle/Hannan/582999
ISSN: 0018-9448
1557-9654
volume: 62
issue: 7
Appears in Collections:2016

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Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D;
Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe
subject: Science & Technology
Year: 2016
Publisher: IEEE
Abstract: Several corrections are necessary in our paper <xref ref-type="bibr" rid="ref1">[1]</xref>. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing above $k$ in $k(x,y,f_{1})$. On p. 7135, in <xref rid="deqn22" ref-type="disp-formula">equations (22)</xref>, (23) and two lines below equation (24), the minus sign &#x2212; in front of constant lambda $ should be changed into the plus sign &#x002B;. Also, on p. 7135, the left-hand side (LHS) of <xref rid="deqn22" ref-type="disp-formula">equation (22)</xref> should be integrated with respect to $y$. Thus, the correct form of <xref rid="deqn22" ref-type="disp-formula">(22)</xref> should be<disp-formula> \begin{align*}&amp;\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&amp;-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} </disp-formula> Because of the correction in <xref rid="deqn22" ref-type="disp-formula">(22)</xref>, several other equations in the paper have to be updated appropriately. Therefore, <xref rid="deqn67" ref-type="disp-formula">equation (67)</xref> will take the form:<disp-formula> \begin{align*}&amp;\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} </disp-formula> Similarly, <xref rid="deqn112-113" ref-type="disp-formula">equations (112) and (113)</xref> must be updated to:<disp-formula> \begin{align*}&amp;\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&amp;+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&amp;+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&amp;\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&amp;\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&amp;\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} </disp-formula> and the sentence following <xref rid="deqn112-113" ref-type="disp-formula">equation (113)</xref> should be read as: n&#x2018;The following functions satisfy the equalities in <xref rid="deqn112-113" ref-type="disp-formula">(112) and (113)</xref>:&#x2019;
URI: http://localhost/handle/Hannan/164994
http://localhost/handle/Hannan/582999
ISSN: 0018-9448
1557-9654
volume: 62
issue: 7
Appears in Collections:2016

Files in This Item:
File Description SizeFormat 
7469875.pdf106.57 kBAdobe PDFThumbnail
Preview File
Title: Correction to A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities&x201D;
Authors: Sangwoo Park;Erchin Serpedin;Khalid Qaraqe
subject: Science & Technology
Year: 2016
Publisher: IEEE
Abstract: Several corrections are necessary in our paper <xref ref-type="bibr" rid="ref1">[1]</xref>. We will next describe these corrections. On page (p.) 7134, in equation (21), a tilde is missing above $k$ in $k(x,y,f_{1})$. On p. 7135, in <xref rid="deqn22" ref-type="disp-formula">equations (22)</xref>, (23) and two lines below equation (24), the minus sign &#x2212; in front of constant lambda $ should be changed into the plus sign &#x002B;. Also, on p. 7135, the left-hand side (LHS) of <xref rid="deqn22" ref-type="disp-formula">equation (22)</xref> should be integrated with respect to $y$. Thus, the correct form of <xref rid="deqn22" ref-type="disp-formula">(22)</xref> should be<disp-formula> \begin{align*}&amp;\hspace {-2pc} \int K'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2} ) + \lambda \tilde {L}'_{f_{1}^{*}}(x,y,f^{*}_{1},f^{*}_{2}) \\[2pt]&amp;-\,\lambda (y) \tilde {k}'_{f^{*}_{1}}(x,y,f^{*}_{1}) dy =0 \qquad \qquad \qquad \text{(22)} \end{align*} </disp-formula> Because of the correction in <xref rid="deqn22" ref-type="disp-formula">(22)</xref>, several other equations in the paper have to be updated appropriately. Therefore, <xref rid="deqn67" ref-type="disp-formula">equation (67)</xref> will take the form:<disp-formula> \begin{align*}&amp;\hspace {-2.5pc}K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}}=\int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x})(- \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;\hspace {-1.5pc}+\, \log f_{\scriptscriptstyle X^{*}}(\mathbf {x})+\alpha _{0} +\boldsymbol {\zeta } \mathbf {x}^{\scriptscriptstyle T} + \mathbf {x}^{\scriptscriptstyle T}\boldsymbol {\Gamma } \mathbf {x}+1-\lambda (\mathbf {y})) d \mathbf {y} =0 \qquad \qquad \text{(67)}\end{align*} </disp-formula> Similarly, <xref rid="deqn112-113" ref-type="disp-formula">equations (112) and (113)</xref> must be updated to:<disp-formula> \begin{align*}&amp;\hspace {-1pc}\int K'_{f_{\scriptscriptstyle X}}\Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} d \mathbf {y}= \int f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) [ -\mu \log f_{\scriptscriptstyle Y^{*}}(\mathbf {y}) \\[2pt]&amp;+\,(1-\alpha _{1})\log f_{\scriptscriptstyle X^{*}}(\mathbf {x}) +\mu \left ({\mu -1}\right ) \log f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) \\[2pt]&amp;+\, \alpha _{0} + \sum _{i=1}^{n} \sum _{j=1}^{n} (\gamma _{ij} y_{i} y_{j} -\gamma _{ij} x_{i} x_{j} - \gamma _{ij} \left ({y-x}\right )_{i} \left ({y-x}\right )_{j} \\[2pt]&amp;+\,\theta x_{i} x_{j} \xi _{i} \xi _{j} +\phi _{ij} y_{i} y_{j})- \lambda (\mathbf {y})+1-\alpha _{1}] d \mathbf {y}= 0.\qquad \qquad \text{(112)}\\[2pt]&amp;\hspace {3pc}\int K'_{f_{\scriptscriptstyle Y}} d \mathbf {x}+ \tilde {K}'_{ f_{\scriptscriptstyle Y}} \Big |_{f_{\scriptscriptstyle X}=f_{\scriptscriptstyle X^{*}}, f_{\scriptscriptstyle Y}=f_{\scriptscriptstyle Y^{*}}} \\[2pt]&amp;\hspace {4pc}= -\frac {\mu \int f_{\scriptscriptstyle X}(\mathbf {x})f_{\scriptscriptstyle W}(\mathbf {y}-\mathbf {x}) d\mathbf {x}}{f_{\scriptscriptstyle Y}(\mathbf {y})} + \lambda (\mathbf {y}) \\[2pt]&amp;\hspace {4pc} = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(113)}\end{align*} </disp-formula> and the sentence following <xref rid="deqn112-113" ref-type="disp-formula">equation (113)</xref> should be read as: n&#x2018;The following functions satisfy the equalities in <xref rid="deqn112-113" ref-type="disp-formula">(112) and (113)</xref>:&#x2019;
URI: http://localhost/handle/Hannan/164994
http://localhost/handle/Hannan/582999
ISSN: 0018-9448
1557-9654
volume: 62
issue: 7
Appears in Collections:2016

Files in This Item:
File Description SizeFormat 
7469875.pdf106.57 kBAdobe PDFThumbnail
Preview File