TY - GEN

T1 - Discrete regression methods on the cone of positive-definite matrices

AU - Boumal, Nicolas

AU - Absil, P. A.

PY - 2011

Y1 - 2011

N2 - We consider the problem of fitting a discrete curve to time-labeled data points on the set ℙn of all n-by-n symmetric positive-definite matrices. The quality of a curve is measured by a weighted sum of a term that penalizes its lack of fit to the data and a regularization term that penalizes speed and acceleration. The corresponding objective function depends on the choice of a Riemannian metric on ℙn. We consider the Euclidean metric, the Log-Euclidean metric and the affine-invariant metric. For each, we derive a numerical algorithm to minimize the objective function. We compare these in terms of reliability and speed, and we assess the visual appearance of the solutions on examples for n = 2. Notably, we find that the Log-Euclidean and the affine-invariant metrics tend to yield similar - and sometimes identical - results, while the former allows for much faster and more reliable algorithms than the latter.

AB - We consider the problem of fitting a discrete curve to time-labeled data points on the set ℙn of all n-by-n symmetric positive-definite matrices. The quality of a curve is measured by a weighted sum of a term that penalizes its lack of fit to the data and a regularization term that penalizes speed and acceleration. The corresponding objective function depends on the choice of a Riemannian metric on ℙn. We consider the Euclidean metric, the Log-Euclidean metric and the affine-invariant metric. For each, we derive a numerical algorithm to minimize the objective function. We compare these in terms of reliability and speed, and we assess the visual appearance of the solutions on examples for n = 2. Notably, we find that the Log-Euclidean and the affine-invariant metrics tend to yield similar - and sometimes identical - results, while the former allows for much faster and more reliable algorithms than the latter.

KW - Positive-definite matrices

KW - Riemannian metrics

KW - finite differences

KW - non-parametric regression

UR - http://www.scopus.com/inward/record.url?scp=80051604087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80051604087&partnerID=8YFLogxK

U2 - 10.1109/ICASSP.2011.5947287

DO - 10.1109/ICASSP.2011.5947287

M3 - Conference contribution

AN - SCOPUS:80051604087

SN - 9781457705397

T3 - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings

SP - 4232

EP - 4235

BT - 2011 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011 - Proceedings

T2 - 36th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011

Y2 - 22 May 2011 through 27 May 2011

ER -