RiemBase (version 0.1.0)

mean: Fr<U+00E9>chet Mean of Manifold-valued Data

Description

For manifold-valued data, Fr<U+00E9>chet mean is the solution of following cost function, $$\textrm{min}_x \sum_{i=1}^n \rho^2 (x, x_i),\quad x\in\mathcal{M}$$ for a given data \(\{x_i\}_{i=1}^n\) and \(\rho(x,y)\) is the geodesic distance between two points on manifold \(\mathcal{M}\). It uses a gradient descent method with a backtracking search rule for updating.

Usage

mean(input, maxiter = 496, eps = 1e-06, parallel = FALSE)

Arguments

input

a S3 object of riemdata class. See riemfactory for more details.

maxiter

maximum number of iterations for gradient descent algorithm.

eps

stopping criterion for the norm of gradient.

parallel

a flag for enabling parallel computation.

Value

a named list containing

x

an estimate Fr<U+00E9>chet mean.

iteration

number of iterations until convergence.

References

karcher_riemannian_1977RiemBase

kendall_probability_1990RiemBase

afsari_convergence_2013RiemBase

Examples

# NOT RUN {
### Generate 10 data points on Sphere S^2 near (0,0,1).
ndata = 10
theta = seq(from=-0.99,to=0.99,length.out=ndata)*pi
tmpx  = cos(theta) + rnorm(ndata,sd=0.1)
tmpy  = sin(theta) + rnorm(ndata,sd=0.1)

### Wrap it as 'riemdata' class
data  = list()
for (i in 1:ndata){
  tgt = c(tmpx[i],tmpy[i],1)
  data[[i]] = tgt/sqrt(sum(tgt^2)) # project onto Sphere
}
data = riemfactory(data, name="sphere")

### Compute Fr<U+00E9>chet Mean
out1 = RiemBase::mean(data)
out2 = RiemBase::mean(data,parallel=TRUE) # test parallel implementation

# }