Prabhakar.G.Vaidya and Swarnali Majumder

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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

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Global 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.

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Covering the Trajectory by Local Patches
Farmer,J.D. and Sidorowich,J.J., Predicting
Chaotic Time Series, Physical review Letters 59,
1987.
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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.

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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.
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We take data from the 3 dim differential equation
of mobius strip
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H is the data matrix of mobius strip. It is 100
by 3.
u
v
y
x
y
z
v
z
z

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.
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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 HsU sVt
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We get bijection between 3 dim data and 2 dim
local co-ordinates in each local patch.
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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.

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We create a global dynamics
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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.
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In a specific patch we get the following dynamics
a .999998, b .0007, c -.00478, d .999998, e
.012, f -.0000028
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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.
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Reference

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

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• Thank You