by Poornima Balakrishna Rajesh Ganesan George Mason University A Comparison of Classical Wavelet with Diffusion Wavelets

2 Classical Vs Diffusion Wavelet Special case of diffusion wavelet Represents 1D and 2D signal/data effectively The basis function must be pre-specified (such as Haar, Daubechies, Symlets etc Is a generic wavelet transform Represents n-D data effectively The best basis functions are obtained by exploring the structure of the data Classical Diffusion

3 Potential Diffusion Wavelet Applications Diffusion Wavelets can be used for Performing analysis of data in multi-dimension Multi-dimension: time x space x attributes Aiding Functional Data Analysis Analysis of contours, 2 D and 3 D images Function Approximation (Our Research) Mitigation of curse of dimensionality in reinforcement learning systems (Artificial Intelligence) Multi-dimensional data compression Denoising Statistical Process Monitoring of multi-dimension data

4 Classical Wavelet Theory j = dilation index k = translation index  Scaling function, and w= wavelet function f(t) V0V0 W1W1 V1V1

5 Diffusion Wavelet Multiscale representation on a graph manifold Inputs to perform a Diffusion Wavelet decomposition A directed graph (G,E,W) and a precision parameter  G-Graph vertices: A point in n-D Euclidean space E: Edges connecting vertices W: Weights on edges. Outputs are Best basis functions for representing large data sets Compact set of scaling and wavelet coefficients

6 Conversion of Data X into a Graph (G,E,W) Many procedures exist Simplest being a Gaussian Kernel W x~y = e -(||x-y||/  )^2 x, y are any 2 data points (n-D vector) from the data sample

7 From Data X obtain (G,E,W) From (G,E,W) obtain P= D -1 W Obtain Laplacian L of (G,E,W) I-L = D 1/2 P D 1/2 = D -1/2 WD -1/2 = T Where L is the combinatorial Laplacian Lf(x)=(D-W)f The relation shows that the eigenvectors values of T and I-L are the same Hence T can be derived from the Laplacian of (G,E,W) Spectral Graph Theory How to get Diffusion Operator T?

8 Concept behind DW: Scaling Functions For level j=0, derive the Diffusion Operator T 1 on the finite multi- dimensional data X Assume Perform QR factorization on T 1 =QR The columns of Q give the orthogonal basis functions on space V 1 R is an upper triangle matrix Using self-adjoint property of T (T=T* = R*Q*, complex conjugate of T) T 2 j = RxR*, j=1 Perform QR factorization on T 2 =Q 1 R 1 The columns of Q 1 give the orthogonal basis functions on space V 2 V0V0 W1W1 V1V1 W2W2 V2V2

9 Concept behind DW: Wavelet Functions Diffusion wavelet basis functions w 1 are obtained via Sparse factorization of = Q 0 ’R 0 ’ w 1 = Q 0 ’ In summary, the powers of T support the dilation and downsampling and translation is achieved via the QR factorization.

10 An Illustration to obtain Best Basis Function Ex: Signal Haar is the best basis for approximating this signal The graph using Gaussian kernal is W= The normalized Laplacian I-L =

11 An Illustration to obtain Best Basis Function The diffusion operator T = Sparse factorization of T yields Q = Scaling function Translation and downsampling

12 An Illustration to obtain Best Basis Function Sparse factorization of yields Q 1 ’ = wavelet function = sqrt(2) (1/2) = sqrt(2) (-1/2) Haar wavelet Translation and downsampling

13 Application in Data Compression There are significant gains in data compression with perfect reconstruction Decompose-store the basis function and coefficients- reconstruct the original signal For example, Data size Basis functions coefficients d (wavelet) c (scaling) f=64x15 V0 64x64 (Identity) 0 15x64 V1 64x30 W1 64x34 15x34 15x30 V2 64x6 W2 64x24 15x2415x6 V3 64x 2 W3 64x4 15x415x2 V4 64x 1 W4 64x1 15x115x1 f=Vc’+Wd’