Chapter 3.Neural Dynamics II:Activation Models X3.6.2 Autoassociative OLAM Filtering Then(3-80)also implies that the additive dual matrix I-behaves as a projection operator: (I-X*X)2=(I-X*X)(I-X*X) =1-X'X-X'X+X'XX'X =I-2XX+(XXXX =I-2X'X+X'Y =I-XX We can represent a projection matrix M as the mapping M:R"→L 2002.10.8
2002.10.8 Chapter 3. Neural Dynamics II:Activation Models ※3.6.2 Autoassociative OLAM Filtering Then (3-80) also implies that the additive dual matrix I X X behaves as a projection operator: * − I - X X I - 2X X X X I - 2X X X XX X I - X X - X X X XX X I X X I X X I X X * * * * 2 * * * * = = + = + = + − = − − ( ) ( ) ( )( ) * * * 2 * * We can represent a projection matrix M as the mapping M R L : n →
Chapter 3.Neural Dynamics II:Activation Models X3.6.2 Autoassociative OLAM Filtering The Pythagorean theorem underlies projection operators. The known signal vectors 1,,span some unique linear subspace L(1,..,X)of R" L equals{∑"c,X,:for all c,∈R;,the set of all linear combinations of the m known signal vectors. L denotes the orthogonal complement space {x∈R":xy=0 for all y∈L the set of all real n-vectors x orthogonal to every n-vector y in L. 2002.10.8
2002.10.8 Chapter 3. Neural Dynamics II:Activation Models ※3.6.2 Autoassociative OLAM Filtering The Pythagorean theorem underlies projection operators. The known signal vectors span some unique linear subspace of X1 , , X m ( , , ) L X1 X m n R L equals , the set of all linear combinations of the m known signal vectors. { : } m i ci Xi for all ci R denotes the orthogonal complement space ⊥ L {x R : xy 0 for all y L} n T = ,the set of all real n-vectors x orthogonal to every n-vector y in L
Chapter 3.Neural Dynamics II:Activation Models X3.6.2 Autoassociative OLAM Filtering 1.Operator projects B"onto L. 2.The dual operator I-X*Yprojects R"onto L. Projection Operator'and I-uniquely decompose every R"vector x into a summed signa/ vector x and a noise or nove/tyvectorx: x=xX'X+x(I-X'X) X =x+x x X 2002.10.8
2002.10.8 Chapter 3. Neural Dynamics II:Activation Models ※3.6.2 Autoassociative OLAM Filtering 1. Operator projects onto L. 2. The dual operator projects onto . X X * n R n I X X R * − ⊥ L Projection Operator and uniquely decompose every vector x into a summed signal vector and a noise or novelty vector : X X * I X X * − n R x ˆ x ~ x x x xX X x I X X ~ ˆ ( ) * * = + = + − x x ˆ x ~