prabhakar.g.vaidya and swarnali majumder

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Prabhakar.G.Vaidya and Swarnali Majumder A preliminary investigation of the feasibility of using SVD and algebraic topology to study dynamics on a manifold

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A preliminary investigation of the feasibility of using SVD and algebraic topology to study dynamics on a manifold. Prabhakar.G.Vaidya and Swarnali Majumder. Global Method and Atlas Method. - PowerPoint PPT Presentation

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Page 1: Prabhakar.G.Vaidya and Swarnali Majumder

Prabhakar.G.Vaidya and Swarnali Majumder

A preliminary investigation of the feasibility of using SVD and algebraic topology to study dynamics on a manifold

Page 2: Prabhakar.G.Vaidya and Swarnali Majumder

Global Method and Atlas MethodGlobal Method and Atlas Method

In global methods we get maps or equations for the whole state space, but in case the of the atlas methods we cover the trajectory by overlapping patches and get maps or equations for each one of them separately.

Page 3: Prabhakar.G.Vaidya and Swarnali Majumder

Covering the Trajectory by Local Patches

Farmer,J.D. and Sidorowich,J.J., “Predicting Chaotic Time Series”,Physical review Letters 59, 1987.

Page 4: Prabhakar.G.Vaidya and Swarnali Majumder

Role of singular value decomposition in studying algebraic topology

Finding local dimension of the manifold where data resides. Local dimension is equal to the number of nonzero singular values.

Locally we model high dim data by a low dim manifold. SVD gives us local co ordinates of a manifold when it is embedded in higher dim.

Page 5: Prabhakar.G.Vaidya and Swarnali Majumder

Let us consider a local patch on mobius strip

Mobius strip is 2 dim manifold, but it is embedded in 3 dim, so we get data in 3 dim. By SVD we find local dimension of this patch. Also it is a natural way of getting local co ordinates.

Page 6: Prabhakar.G.Vaidya and Swarnali Majumder

tx0

d

dx1

v

2sin

t

2

cos t( )

tx1

d

dx0

v

2sin

t

2

sin t( )

tx2

d

d

v

2cos

t

2

We take data from the 3 dim differential equation of mobius strip

Page 7: Prabhakar.G.Vaidya and Swarnali Majumder

H is the data matrix of mobius strip. It is 100 by 3.

=

H = UWVt

Number of nonzero diagonal element in W gives the local dimension. In case of mobius strip it is 2. The above relationship gives a 1-1 transformation from 3D to 2D.

yx y z u vzzv

Page 8: Prabhakar.G.Vaidya and Swarnali Majumder

svds H 0( )( )

3.2241129399

0.0489748943

0.0043051604

Since the 3rd singular value in W is very small, we consider only first two columns of UW. Let us call it sU. Let us consider first two column of V and let us call it as sV. So we have a local bijective relation

H=sU sVt

Page 9: Prabhakar.G.Vaidya and Swarnali Majumder

We get bijection between 3 dim data and 2

dim local co-ordinates in each local patch.

Page 10: Prabhakar.G.Vaidya and Swarnali Majumder

Non-linear singular value decomposition When we want to do local approximation in a

bigger area we do generalization of singular value decomposition.

We consider non linear combinations of x,y,z and do svd on the matrix.

Page 11: Prabhakar.G.Vaidya and Swarnali Majumder

We create a global dynamics

Page 12: Prabhakar.G.Vaidya and Swarnali Majumder

Dynamics is created in the lower dim of each chart and going to the higher dim when overlapping region comes.

We have transformation from higher to lower dimension and also from lower to higher dimension in each chart.

Page 13: Prabhakar.G.Vaidya and Swarnali Majumder

In a specific patch we get the following dynamics

we want to create a map like

U0

n 1

U1n 1

a

c

b

d

U0n

U1n

e

f

a= .999998, b= .0007, c= -.00478, d= .999998, e= .012, f= -.0000028

Page 14: Prabhakar.G.Vaidya and Swarnali Majumder

We consider first two columns of U, which are the local coordinates. Using this U we do rectification.

Aligning two charts together

We continue this alignment for every chart and get a low dimensional manifolds. It is the covering space of the original manifold, once we make identification.

Page 15: Prabhakar.G.Vaidya and Swarnali Majumder

0.1 0 0.10.2

0

0.2Qmm nn

Q1mm1 nn1

Q2mm2 nn2

Pmm nn P1mm1 nn1 P2mm2 nn2

Page 16: Prabhakar.G.Vaidya and Swarnali Majumder

2 1 0 120

0

20

40

QQm n

PPm n

Page 17: Prabhakar.G.Vaidya and Swarnali Majumder

Reference:

1. Farmer,J.D. and Sidorowich,J.J., “Predicting Chaotic Time Series”,

Physical review Letters 59, 1987.

Page 18: Prabhakar.G.Vaidya and Swarnali Majumder

Thank You