Geometric Calculi and Automatic Learning An Outline

Signal representation and processing are the backbone of mathematically-aided engineering. Among the myriad of ideas and results in that realm, many sorts of algorithms and techniques capable of learning from experience have taken the stage in the last decades, with a crescendo of great successes in a variety of fronts in recent years. This paper provides a sketchy outline of those developments that seem more relevant or promising in view of their bearing on the geometric calculus (multivector) representations of signals and the concomitant automatic learning algorithms. The corresponding artificial neurons, and their organization in networks, may be seen as a way to transcend the biologically inspired neuron networks much as the wheel or aviation transcended legs or bird flight. Recent developments suggest that there are exciting research opportunities ahead.

[1]  Felipe Cucker,et al.  Learning Theory: An Approximation Theory Viewpoint: Index , 2007 .

[2]  Eduardo Bayro-Corrochano,et al.  Geometric Computing - for Wavelet Transforms, Robot Vision, Learning, Control and Action , 2010 .

[3]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[4]  Sanjeev R. Kulkarni,et al.  An Elementary Introduction to Statistical Learning Theory: Kulkarni/Statistical Learning Theory , 2011 .

[5]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..

[6]  Christopher J. C. Burges,et al.  A Tutorial on Support Vector Machines for Pattern Recognition , 1998, Data Mining and Knowledge Discovery.

[7]  Stéphane Mallat,et al.  Understanding deep convolutional networks , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  Jan Peters,et al.  Reinforcement learning in robotics: A survey , 2013, Int. J. Robotics Res..

[9]  Michael Felsberg,et al.  The monogenic signal , 2001, IEEE Trans. Signal Process..

[10]  Helmut Pottmann,et al.  Fitting B-spline curves to point clouds by curvature-based squared distance minimization , 2006, TOGS.

[11]  Thomas Bülow,et al.  Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case , 2001, IEEE Trans. Signal Process..

[12]  Igor N. Aizenberg,et al.  Complex-Valued Neural Networks with Multi-Valued Neurons , 2011, Studies in Computational Intelligence.

[13]  Tohru Nitta.  Complex-valued Neural Networks: Utilizing High-dimensional Parameters , 2009 .

[14]  Cli McMahon,et al.  Machines Who Think : A Personal Inquiry into the History and Prospects of Artificial Intelligence , 2004 .

[15]  Rui Wang,et al.  GA-SURF: A new Speeded-Up robust feature extraction algorithm for multispectral images based on geometric algebra , 2019, Pattern Recognit. Lett..

[16]  Sebastià Xambó-Descamps,et al.  Real Spinorial Groups: A Short Mathematical Introduction , 2018 .

[17]  Avrim Blum,et al.  Foundations of Data Science , 2020 .

[18]  Isiah Zaplana,et al.  A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry , 2018 .

[19]  Corrado Segre,et al.  Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici , 1892 .

[20]  Cheng Soon Ong,et al.  Mathematics for Machine Learning , 2020, Journal of Mathematical Sciences & Computational Mathematics.

[21]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[22]  Eduardo Bayro Corrochano.  Geometric Computing for Perception Action Systems , 2001, Springer New York.

[23]  Martin Anthony,et al.  Neural Network Learning by Martin Anthony , 1999 .

[24]  Jürgen Schmidhuber,et al.  Deep learning in neural networks: An overview , 2014, Neural Networks.

[25]  Mario Geiger,et al.  Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks , 2020, ArXiv.

[26]  Stephen J. Sangwine,et al.  Quaternion and Clifford Fourier Transforms and Wavelets , 2013 .

[27]  Renato Renner,et al.  Discovering physical concepts with neural networks , 2018, Physical review letters.

[28]  Jiasong Wu,et al.  Deep Octonion Networks , 2019, Neurocomputing.

[29]  Stéphane Mallat,et al.  Invariant Scattering Convolution Networks , 2012, IEEE transactions on pattern analysis and machine intelligence.

[30]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[31]  Masaki Kobayashi,et al.  Rotational invariance of quaternionic hopfield neural networks , 2016 .

[32]  Yuan Yao,et al.  Online Learning Algorithms , 2006, Found. Comput. Math..

[33]  M. Mitrea.  Clifford Wavelets, Singular Integrals, and Hardy Spaces , 1994 .

[34]  Nils J. Nilsson,et al.  The Quest for Artificial Intelligence , 2009 .