pdSplineReg {pdSpecEst}

Cubic smoothing spline regression for HPD matrices

Description

pdSplineReg() performs cubic smoothing spline regression in the space of HPD matrices equipped with the affine-invariant Riemannian metric through minimization of a penalized regression objective function using a geometric conjugate gradient descent method as outlined in (Boumal and Absil 2011a) and (Boumal and Absil 2011b). This is a specific implementation of the more general algorithm in (Boumal and Absil 2011a) and (Boumal and Absil 2011b), setting the part in the objective function based on the first-order finite geometric differences to zero, such that the solutions of the regression problem are approximating cubic splines.

Usage

pdSplineReg(P, f0, lam = 1, Nd, ini_step = 1, max_iter = 100,
  eps = 0.001, ...)

Arguments

P

a (d,d,n)-dimensional array corresponding to a length n sequence of (d, d)-dimensional noisy HPD matrices.

f0

a (d,d,n)-dimensional array corresponding to an initial estimate of the smooth target curve of (d, d)-dimensional HPD matrices.

lam

a smoothness penalty, defaults to lam = 1. If lam = 0, the penalized curve estimate coincides with geodesic interpolation of the data points with respect to the Riemannian metric. If lam increases to \infty, the penalized regression estimator is approximately a fitted geodesic curve.

Nd

a numeric value (Nd <= n) determining a lower resolution of the cubic spline regression estimator to speed up computation time, defaults to n.

ini_step

initial candidate step size in a backtracking line search based on the Armijo-Goldstein condition, defaults to ini_step = 1.

max_iter

maximum number of gradient descent iterations, defaults to max_iter = 100.

eps

optional tolerance parameter in gradient descent algorithm. The gradient descent procedure exits if the absolute difference between consecutive evaluations of the objective function is smaller than eps, defaults to eps = 1E-3.

...

additional arguments for internal use.

Value

A list with three components:

f

a (d, d, N_d)-dimensional array corresponding to a length Nd estimated cubic smoothing spline curve of (d, d)-dimensional HPD matrices.

cost

a numeric vector containing the costs of the objective function at each gradient descent iteration.

total_iter

total number of gradient descent iterations.

Note

This function does not check for positive definiteness of the matrices given as input, and may fail if matrices are close to being singular.

References

Boumal N, Absil P (2011a). “Discrete regression methods on the cone of positive-definite matrices.” In IEEE ICASSP, 2011, 4232–4235.

Boumal N, Absil P (2011b). “A discrete regression method on manifolds and its application to data on SO(n).” IFAC Proceedings Volumes, 44(1), 2284–2289.

Examples

## Not run: 
set.seed(2)
P <- rExamples1D(50, example = 'gaussian', noise.level = 0.1)
P.spline <- pdSplineReg(P$P, P$P, lam = 0.5, Nd = 25)

## Examine matrix-component (1,1)
plot((1:50)/50, Re(P$P[1, 1, ]), type = "l", lty = 2) ## noisy observations
lines((1:25)/25, Re(P.spline$f[1, 1, ])) ## estimate
lines((1:50)/50, Re(P$f[1, 1, ]), col = 2, lty = 2) ## smooth target

## End(Not run)