Stability and Rigidity of the Leibniz and the Chain Rules

2018 ◽  
pp. 75-90
Author(s):  
Hermann König ◽  
Vitali Milman
Keyword(s):  
Chain Rules

scholarly journals Derivations, derivatives and chain rules

1999 ◽  
Vol 302-303 ◽  
pp. 231-244 ◽  
Author(s):  
Rajendra Bhatia ◽  
Kalyan B. Sinha
Keyword(s):  
Chain Rules

Chain rules for multivariate cumulant coefficients

Stat ◽  
2022 ◽  
Author(s):  
Christopher S. Withers ◽  
Saralees Nadarajah
Keyword(s):  
Chain Rules

Chain Rules for Linear Openness in General Banach Spaces

2012 ◽  
Vol 22 (3) ◽  
pp. 899-913 ◽  
Author(s):  
Marius Durea ◽  
Radu Strugariu
Keyword(s):  
Banach Spaces ◽  
Linear Openness ◽  
Chain Rules

Blockchain Transforms the Retail Level by Using a Supply chain Rules and Regulation

2019 ◽  
Author(s):  
Rana M. Amir Latif ◽  
Samar Iqbal ◽  
Osama Rizwan ◽  
Syed Umair Aslam Shah ◽  
Muhammad Farhan ◽  
...  
Keyword(s):  
Supply Chain ◽  
Chain Rules

scholarly journals Enabling quaternion derivatives: the generalized HR calculus

2015 ◽  
Vol 2 (8) ◽  
pp. 150255 ◽  
Author(s):  
Dongpo Xu ◽  
Cyrus Jahanchahi ◽  
Clive C. Took ◽  
Danilo P. Mandic
Keyword(s):  
Quaternion Algebra ◽  
Optimization Problems ◽  
Adaptive Signal Processing ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Regular Functions ◽  
Complex Optimization ◽  
Left And Right ◽  
Chain Rules ◽  
Adaptive Signal

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

Do the chain rules for matrix functions hold without commutativity?

2010 ◽  
Vol 58 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Boris Shekhtman
Keyword(s):  
Matrix Functions ◽  
Chain Rules
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