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# Lecture1 - Adaptive Algorithms for PCA Review of Principal...

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Review of Principal Component Analysis Solves the matrix equation Diagonalizes the covariance matrix of the input signal, The eigenvectors become eigenfilters and they span the frequency spectrum of the input signal. This is true only if the dimensionality of the data is very high. (From the spectral decomposition property of eigenvectors) The eigenvectors being linearly independent form the most optimal basis for any vector space. This is the motivation for using eigenvectors in data compression (KLT) Λ = V RV Λ = RV V T
How to solve PCA using adaptive structures? Although there are many numerical techniques (SVD) to solve PCA, for real-time applications, we need iterative algorithms that solve PCA using one sample of data at a time. MATLAB programmers use eig function. I METHOD - Minimization of mean square reconstruction error N N M Input=x Y=W T x Wy x = ˆ Desired response is the input itself. { } 2 ˆ x x E J - =

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Minimize the cost function using gradient method with constraint The weight matrix will converge to a rotation of the eigenvector
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