Fractional Extreme Value Adaptive Training Method: Fractional Steepest Descent Approach

The application of fractional calculus to signal processing and adaptive learning is an emerging area of research. A novel fractional adaptive learning approach that utilizes fractional calculus is presented in this paper. In particular, a fractional steepest descent approach is proposed. A fractional quadratic energy norm is studied, and the stability and convergence of our proposed method are analyzed in detail. The fractional steepest descent approach is implemented numerically and its stability is analyzed experimentally.

[1]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[2]  Daniel Graupe,et al.  Identification of Systems , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  Yangquan Chen,et al.  A new IIR-type digital fractional order differentiator , 2003, Signal Process..

[4]  西本 勝之,et al.  Fractional calculus : integrations and differentiations of arbitrary order , 1984 .

[5]  Pu Yi-Fei,et al.  Implement Any Fractional Order Neural-type Pulse Oscillator with Net-grid Type Analog Fractance Circuit , 2006 .

[6]  Yi-Fei Pu,et al.  Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement , 2010, IEEE Transactions on Image Processing.

[7]  N. Engheta On the role of fractional calculus in electromagnetic theory , 2016 .

[8]  Weixing Wang,et al.  Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation , 2008, Science in China Series F: Information Sciences.

[9]  Kenneth S. Miller,et al.  Derivatives of Noninteger Order , 1995 .

[10]  Pu Yifei,et al.  A Recursive Net-Grid-Type Analog Fractance Circuit for Any Order Fractional Calculu , 2005, IEEE International Conference Mechatronics and Automation, 2005.

[11]  E. R. Love,et al.  Fractional Derivatives of Imaginary Order , 1971 .

[12]  R. Bitmead Convergence properties of LMS adaptive estimators with unbounded dependent inputs , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[13]  S. Douglas A family of normalized LMS algorithms , 1994, IEEE Signal Processing Letters.

[14]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[15]  R. Hilfer Applications Of Fractional Calculus In Physics , 2000 .

[16]  S. Holm,et al.  Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. , 2004, The Journal of the Acoustical Society of America.

[17]  R. Koeller Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[18]  Nasir Ahmed,et al.  Sequential regression considerations of adaptive filtering , 1977 .

[19]  Yiguang Liu,et al.  Fractional partial differential equation denoising models for texture image , 2014, Science China Information Sciences.

[20]  Yi-Fei Pu,et al.  A fractional partial differential equation based multiscale denoising model for texture image , 2014 .

[21]  B. Anderson,et al.  Performance of adaptive estimation algorithms in dependent random environments , 1980 .

[22]  Chien-Cheng Tseng Design of fractional order digital FIR differentiators , 2001, IEEE Signal Processing Letters.

[23]  Rachid Harba,et al.  nth-order fractional Brownian motion and fractional Gaussian noises , 2001, IEEE Trans. Signal Process..

[24]  Zhou Ji-liu,et al.  Research on Application of Fractional Calculus , 2011 .

[25]  D Rowe Adaptive Filtering and Signal Analysis , 2005 .

[26]  Luiz W. P. Biscainho,et al.  Optimal variable step size for the LMS/Newton algorithm with application to subband adaptive filtering , 1992, IEEE Trans. Signal Process..

[27]  Jian Bai,et al.  Fractional-Order Anisotropic Diffusion for Image Denoising , 2007, IEEE Transactions on Image Processing.

[28]  Yifei Pu,et al.  Fractional Calculus Approach to Texture of Digital Image , 2006, 2006 8th international Conference on Signal Processing.

[29]  B. West Fractional Calculus in Bioengineering , 2007 .

[30]  N. Engheta On fractional calculus and fractional multipoles in electromagnetism , 1996 .

[31]  Paulo S. R. Diniz,et al.  Adaptive Filtering: Algorithms and Practical Implementation , 1997 .

[32]  Ingo Schäfer,et al.  Fractional Calculus via Functional Calculus: Theory and Applications , 2002 .

[33]  Jan A Snyman,et al.  Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms , 2005 .

[34]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[35]  P. Butzer,et al.  AN INTRODUCTION TO FRACTIONAL CALCULUS , 2000 .

[36]  S. D. Stearns,et al.  Digital Signal Analysis , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[37]  S. Manabe A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control , 2002 .

[38]  Yuan Xiao,et al.  Structuring analog fractance circuit for 1/2 order fractional calculus , 2005, 2005 6th International Conference on ASIC.

[39]  S. Thomas Alexander,et al.  Adaptive Signal Processing , 1986, Texts and Monographs in Computer Science.

[40]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[41]  Shyang Chang,et al.  Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification , 1997, IEEE Trans. Image Process..

[42]  Pu Yifei Implement any fractional order multilayer dynamics associative neural network , 2005, 2005 6th International Conference on ASIC.

[43]  Jiliu Zhou,et al.  A novel approach for multi-scale texture segmentation based on fractional differential , 2011, Int. J. Comput. Math..

[44]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[45]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[46]  Zhou Ji-liu,et al.  Five Numerical Algorithms of Fractional Calculus Applied in Modern Signal Analyzing and Processing , 2005 .

[47]  Yiguang Liu,et al.  Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image , 2013 .

[48]  S. T. Alexander,et al.  Adaptive Signal Processing: Theory and Applications , 1986 .

[49]  迈克尔·维尔格特 An identification system , 2004 .

[50]  N. Engheia On the role of fractional calculus in electromagnetic theory , 1997 .

[51]  Hari M. Srivastava,et al.  Fractional calculus operators and their applications involving power functions and summation of series , 1997 .