by Poornima Balakrishna Rajesh Ganesan George Mason University

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

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) V0 W1 V1

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 e • 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 Wx~y = e-(||x-y||/d)^2 x, y are any 2 data points (n-D vector) from the data sample

7. Spectral Graph Theory How to get Diffusion Operator T? • From Data X obtain (G,E,W) • From (G,E,W) obtain P= D-1W • Obtain Laplacian L of (G,E,W) • I-L = D1/2P D1/2= D-1/2WD-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)

8. Concept behind DW: Scaling Functions • For level j=0, derive the Diffusion Operator T1 on the finite multi-dimensional data X • Assume • Perform QR factorization on T1=QR • The columns of Q give the orthogonal basis functions on space V1 • R is an upper triangle matrix • Using self-adjoint property of T (T=T* = R*Q*, complex conjugate of T) • T2j= RxR*, j=1 • Perform QR factorization on T2=Q1R1 • The columns of Q1 give the orthogonal basis functions on space V2 V1 V0 V2 W2 W1

9. Concept behind DW: Wavelet Functions • Diffusion wavelet basis functions w1 are obtained via • Sparse factorization of = Q0’R0’ • w1= Q0’ • 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 4 4 -4 -4 • 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 = Translation and downsampling Scaling function

12. An Illustration to obtain Best Basis Function • Sparse factorization of yields Q1’ = Translation and downsampling Haar wavelet wavelet function 0.7071 = sqrt(2) (1/2) -0.7071 = sqrt(2) (-1/2)

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 15x24 15x6 V3 64x 2 W3 64x4 15x4 15x2 V4 64x 1 W4 64x1 15x1 15x1 f=V(c/2)’+W(d/2)’