Files description :

• bumpfunction.py : Consists of a class containing the required smooth C-inf bump functions
• create_charts.py : Consists code to create individual charts of the manifold
• main.py : Consists of main function which computes the global approximated value using PoU
• regression.py : Consists of a function used to fit linear/ polynomial curves to data using OLS
• visualization.py : For visualizing spherical manifold and individual charts

Partition of unity:

A partition of unity is a useful, though technical, tool that helps us work in local coordinates. This can be a tricky matter when we’re doing things all over our manifold, since it’s almost never the case that the entire manifold fits into a single coordinate patch. A (smooth) partition of unity is a way of breaking the function with the constant value 1 up into a bunch of (smooth) pieces that are easier to work with.

More precisely, for any covering of X (a topological space; here it is a manifold, which is a topological space with additional properties) by open subsets {Uₘ},there exists a sequence of smooth nonnegative functions {ϴₙ}, on X, called a partition of unity, subordinate to the open cover {Uₘ} such that for every point, x ∈ X :

• the sum of all function values at x is 1, i.e., ∑ ϴₙ(x) = 1

• 0 ≤ ϴₙ(x) ≤ 1 and all 'n'

• supp ϴₙ ⊆ Uₘ for each m, where 'supp' is the support of the function.

• The family of supports (supp ϴₙ) is locally finite, meaning that every point has a neighbourhood that intersects supp ϴₙ for only finitely many values of n

Bump Functions:

A bump function is a function on Cartesian Space R^n, for some n ∈ R with values in the real numbers R

				b : R^n -> R  *such that*

1. b is smooth
2. b has compact support

Global approximation

Partition of unity can be used to patch together local smooth objects into global ones.

Approach:

• We divide the manifold into 6 charts correponding to 6 R^2 discs.

• Each point on the original manifold now lies in 3 different charts (by observation). Corresponding to each of these 3 charts, we have introduced 3 smooth bump functions necessary for Partitons of Unity.

• We then fit linear/ Polynomial curves, locally, on each chart.

• Final approximation for a point 'x' lying on the manifold is computed as : ∑ f(x_i)φ(x_i) where f(x_i) and φ(x_i) are the linearly fitted function value of 'x' and the bump function value of 'x' in the ith chart, respectively. The summation is taken over all the i (here i = 3) charts a point lies in.

Possible further work :

Trying to globally approximate any scalar valued function (R^n -> R) other than linearly fitted function

Spherical Dataset description:

We have taken patches/ overlapping charts such that atmost 4 patches overlap anywhere on the sphere.

Sample dataset:

1. Spherical manifold dataset Image
2. Six charts of the S^2 manifold

Evaluation:

We have reported the Average test loss, Average training loss, MSE and Standard Deviation. Average test loss comes out to be ~0.20. We have used holdout method for evaluating the results.