Geodesically convex M-estimation in metric spaces

Victor Emmanuel Brunel

Geodesically convex M-estimation in metric spaces

Victor Emmanuel Brunel

ENSAE

Research output: Contribution to journal › Conference article › peer-review

Abstract

We study the asymptotic properties of geodesically convex M-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic normality, which are fundamental properties in statistical inference. Our results extend the Euclidean theory of convex M-estimation; They also generalize limit theorems on non-linear spaces which, essentially, were only known for barycenters, allowing to consider robust alternatives that are defined through non-smooth M-estimation procedures.

Original language	English
Pages (from-to)	2188-2210
Number of pages	23
Journal	Proceedings of Machine Learning Research
Volume	195
Publication status	Published - 1 Jan 2023
Externally published	Yes
Event	36th Annual Conference on Learning Theory, COLT 2023 - Bangalore, India Duration: 12 Jul 2023 → 15 Jul 2023

Keywords

CAT spaces
M-estimation
Riemannian manifolds
barycenters
geodesic convexity
metric spaces
robust location estimation

Cite this

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title = "Geodesically convex M-estimation in metric spaces",

abstract = "We study the asymptotic properties of geodesically convex M-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic normality, which are fundamental properties in statistical inference. Our results extend the Euclidean theory of convex M-estimation; They also generalize limit theorems on non-linear spaces which, essentially, were only known for barycenters, allowing to consider robust alternatives that are defined through non-smooth M-estimation procedures.",

keywords = "CAT spaces, M-estimation, Riemannian manifolds, barycenters, geodesic convexity, metric spaces, robust location estimation",

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N2 - We study the asymptotic properties of geodesically convex M-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic normality, which are fundamental properties in statistical inference. Our results extend the Euclidean theory of convex M-estimation; They also generalize limit theorems on non-linear spaces which, essentially, were only known for barycenters, allowing to consider robust alternatives that are defined through non-smooth M-estimation procedures.

AB - We study the asymptotic properties of geodesically convex M-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic normality, which are fundamental properties in statistical inference. Our results extend the Euclidean theory of convex M-estimation; They also generalize limit theorems on non-linear spaces which, essentially, were only known for barycenters, allowing to consider robust alternatives that are defined through non-smooth M-estimation procedures.

KW - CAT spaces

KW - M-estimation

KW - Riemannian manifolds

KW - barycenters

KW - geodesic convexity

KW - metric spaces

KW - robust location estimation

M3 - Conference article

AN - SCOPUS:85171559055

SN - 2640-3498

VL - 195

SP - 2188

EP - 2210

JO - Proceedings of Machine Learning Research

JF - Proceedings of Machine Learning Research

T2 - 36th Annual Conference on Learning Theory, COLT 2023

Y2 - 12 July 2023 through 15 July 2023

ER -

Geodesically convex M-estimation in metric spaces

Abstract

Keywords

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