riem.distlp: Distance between Two Curves on Manifolds

In Riemann: Learning with Data on Riemannian Manifolds

Distance between Two Curves on Manifolds

Description

Given two curves γ_1, γ_2 : I \rightarrow \mathcal{M}, we are interested in measuring the discrepancy of two curves. Usually, data are given as discrete observations so we are offering several methods to perform the task. See the section below for detailed description.

Usage

riem.distlp(
  riemobj1,
  riemobj2,
  vect = NULL,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

riemobj1	a S3 "riemdata" class for N manifold-valued data along the curve.
riemobj2	a S3 "riemdata" class for N manifold-valued data along the curve.
vect	a vector of domain values. If given Null (default), sequence 1:N is set.
geometry	(case-insensitive) name of geometry; either geodesic ("intrinsic") or embedded ("extrinsic") geometry.
...	extra parameters including p an exponent (default: 2).

Value

the distance value.

Default Method

Trapezoidal Approximation Assume γ_1 (t_i) = X_i and γ_2 (t_i) = Y_i for i=1,2,…,N. In the Euclidean space, L_p distance between two scalar-valued functions is defined as

L_p^p (γ_1 (x), γ_2 (x) = \int_{\mathcal{X}} |γ_1 (x) - γ_2 (x)|^p dx

. We extend this approach to manifold-valued curves

L_p^p (γ_1 (t), γ_2 (t)) = \int_{t\in I} d^p (γ_1 (t), γ_2 (t)) dt

where d(\cdot,\cdot) is an intrinsic/extrinsic distance on manifolds. With the given representations, the above integral is approximated using trapezoidal rule.

Examples

#-------------------------------------------------------------------
#                          Curves on Sphere
#
#  curve1 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve2 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve3 : y = 0.5*sin(x) on the tangent space at (0,0,1)
#
# * distance between curve1 & curve2 should be close to 0.
# * distance between curve1 & curve3 should be large.
#-------------------------------------------------------------------
## GENERATION
vecx  = seq(from=-0.9, to=0.9, length.out=50)
vecy1 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy2 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy3 = 0.5*sin(vecx) + rnorm(50, sd=0.05)

## WRAP AS RIEMOBJ
mat1 = cbind(vecx, vecy1, 1); mat1 = mat1/sqrt(rowSums(mat1^2))
mat2 = cbind(vecx, vecy2, 1); mat2 = mat2/sqrt(rowSums(mat2^2))
mat3 = cbind(vecx, vecy3, 1); mat3 = mat3/sqrt(rowSums(mat3^2))

rcurve1 = wrap.sphere(mat1)
rcurve2 = wrap.sphere(mat2)
rcurve3 = wrap.sphere(mat3)

## COMPUTE DISTANCES
riem.distlp(rcurve1, rcurve2, vect=vecx)
riem.distlp(rcurve1, rcurve3, vect=vecx)

Riemann documentation built on March 18, 2022, 7:55 p.m.