prabhakar.g.vaidya and swarnali majumder a preliminary investigation of the feasibility of using svd...

Prabhakar.G.Vaidya and Swarnali Majumder

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

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.

Covering the Trajectory by Local Patches

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

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.

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.

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

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

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

We get bijection between 3 dim data and 2

dim local co-ordinates in each local patch.

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.

We create a global dynamics

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.

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

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.

0.1 0 0.10.2

0

0.2Qmm nn

Q1mm1 nn1

Q2mm2 nn2

Pmm nn P1mm1 nn1 P2mm2 nn2

2 1 0 120

0

20

40

QQm n

PPm n

Reference:

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

Physical review Letters 59, 1987.

Thank You