Lattice Computing in Hybrid Intelligent Systems Manuel Graa - PowerPoint PPT Presentation

Introduction Lattice Associative Memories Applications Concluding remarks Lattice Computing in Hybrid Intelligent Systems Manuel Graña Computational Intelligence Group, UPV/EHU December 4, 2012 Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Applications Concluding remarks Summary of the talk Introduce Lattice Computing paradigm Focus on Lattice Autoassociative Memories Applications involving hybridization Hyperspectral image unmixing Face recognition MRI classification fMRI processing Multivariate Mathematical Morphology Hyperspectral image brain networks on resting state fMRI Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Applications Concluding remarks Contents Introduction 1 Lattice Computing Lattice Computing Approaches Lattice Associative Memories 2 LAAM definitions and properties Lattice Independent Component Analysis Applications 3 Face Recognition Di ff usion MRI data classification Multivariate Mathematical Morphology Application resting state fMRI Spatial-spectral classification Concluding remarks 4 Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Contents Introduction 1 Lattice Computing Lattice Computing Approaches Lattice Associative Memories 2 LAAM definitions and properties Lattice Independent Component Analysis Applications 3 Face Recognition Di ff usion MRI data classification Multivariate Mathematical Morphology Application resting state fMRI Spatial-spectral classification Concluding remarks 4 Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Lattice Computing Definition Lattice Computing is the class of algorithms built on the basis of Lattice Theory. define computations in the ring of the real valued spaces endowed with some (inf, sup) lattice operators ( R n , _ , ^ , +) , or use lattice theory to produce generalizations or fusions of conventional approaches. Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Contents Introduction 1 Lattice Computing Lattice Computing Approaches Lattice Associative Memories 2 LAAM definitions and properties Lattice Independent Component Analysis Applications 3 Face Recognition Di ff usion MRI data classification Multivariate Mathematical Morphology Application resting state fMRI Spatial-spectral classification Concluding remarks 4 Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Mathematical Morphology Classical application of lattice theory to signal and image processing Filtering and detection Erosion and dilation operators corresponding to infimum and supremum non-linear convolution-like processes with structural elements Opening and closing basic filters segmentation by morphological gradient and watershed detection by top-hat, hit-and-miss Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Formal Concept Analysis Application of lattice theory to semantic analysis Ontology induction from data intensional (attributes) and extensional (instances) representation of concepts building the lattice induced by the partial order of concepts ((A,C,I,L,P,Re,Ro),( )) ((A,L,P,Re,Ro),(Cap)) ((I,L,P,Ro),(Riv)) ((C,I,Re),(Ski)) ((A,C,I,P,Ro),(Eur)) ((A,P,Ro),(Cap,Eur)) ((L,P,Ro),(Cap,Riv)) ((I,P,Ro),(Eur,Riv)) ((C,I),(Eur,Ski)) ((Re),(Cap,Ski)) ((A,Ro),(Arc,Bea,Cap,Eur)) ((P,Ro),(Cap,Eur,Riv)) ((I),(Eur,Riv,Ski)) ((Ro),(Arc,Bea,Cap,Eur,Riv)) (( ),(Arc,Bea,Cap,Eur,Riv,Ski)) Fig. 1. Concept Lattice of the European Cities context. Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Lattice Associative Memories Builiding learning algorithms with morphological operators Associative Memories Store and recall patterns Dual memories from infimum and supremum operators Nice properties: infinite storage capacity of real valued patterns robustness against erosive/dilative noise not-nice: sensitivity to general additive noise Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories Lattice Computing Applications Lattice Computing Approaches Concluding remarks Kaburlasos’ Lattice Interval Numbers A new general data type: Intervals Numbers many conventional data types can be mapped into IN the valuation function allows to define error measures define the variations of conventional learning algoritms generalization of Fuzzy-ART lattice Self Organizing Map Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks Contents Introduction 1 Lattice Computing Lattice Computing Approaches Lattice Associative Memories 2 LAAM definitions and properties Lattice Independent Component Analysis Applications 3 Face Recognition Di ff usion MRI data classification Multivariate Mathematical Morphology Application resting state fMRI Spatial-spectral classification Concluding remarks 4 Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks LAAM definitions LAAMs are auto-associative neural networks neuron functional activations built on morphological (lattice) operations. LAAMs present interesting properties such as perfect recall, unlimited storage and one-step convergence. Proposed by Ritter et al. 1 2 We found applications besides image storage and retrieval 1 G. X. Ritter, P. Sussner, and J. L. Diaz-de Leon. Morphological associa- tive memories. Neural Networks, IEEE Transactions on, 9(2):281–293, 1998. 2 G. X. Ritter, J. L. Diaz-de Leon, and P. Sussner. Morphological bidirectional associative memories. Neural Networks, 12(6):851–867, 1999. Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks LAAM definitions input/output pairs of patterns n⇣ x ξ , y ξ ⌘ o ( X , Y ) = ; ⇠ = 1 , .., k a linear heteroassociative neural network x ξ ⌘ 0 y ξ · ⇣ X W = . ξ erosive and dilative LAMs, respectively k k  � x ξ ⌘ 0 �  � x ξ ⌘ 0 � y ξ ⇥ ⇣ y ξ ⇥ ⇣ ^ _ W XY = and M XY = , ξ = 1 ξ = 1 where ⇥ is any of the _ ⇤ or ^ ⇤ operators, Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks LAAM definitions operator _ ⇤ denotes the max matrix product _ C = A _ ⇤ B = [ c ij ] , c ij = { a ik + b kj } , k = 1 .. n operator ^ ⇤ denotes the min matrix product ^ C = A ^ ⇤ B = [ c ij ] , c ij = { a ik + b kj } . k = 1 .. n Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks LAAM definitions and properties Definition When X = Y then W XX and M XX are called Lattice Auto-Associative Memories (LAAMs). perfect recall for an unlimited number of real-valued stored patterns W XX _ ⇤ X = X = M XX ^ ⇤ X convergence in one step for any input pattern if W XX _ ⇤ z = v then W XX _ ⇤ v = v if M XX _ ⇤ z = u then M XX ^ ⇤ u = u . Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks Fixed points of M XX and W XX a a G.X.Ritter,G.Urcid,“Lattice algebra approach to endmember determination in hyperspectral imagery,” in P. Hawkes (Ed.), Advances in imaging and electron physics, Vol. 160, 113–169. Elsevier, Burlington, MA (2010) x 2 x 6 5 x 5 – x 1 x 11 H v 1 ( d 12 ) x 10 x 8 x 12 x 7 x 2 x 9 x 4 u 1 x 3 v 2 u 2 5 x 1 + E u 1 ( d 12 ) 1 ( d 12 ) H u v 1 E v 1 ( d 12 ) Manuel Graña Lattice Computing in Hybrid Intelligent Systems

Introduction Lattice Associative Memories LAAM definitions and properties Applications Lattice Independent Component Analysis Concluding remarks Contents Introduction 1 Lattice Computing Lattice Computing Approaches Lattice Associative Memories 2 LAAM definitions and properties Lattice Independent Component Analysis Applications 3 Face Recognition Di ff usion MRI data classification Multivariate Mathematical Morphology Application resting state fMRI Spatial-spectral classification Concluding remarks 4 Manuel Graña Lattice Computing in Hybrid Intelligent Systems