A Riemannian Optimization Approach to Clustering Problems
This paper considers the optimization problem min X ∈ F v f ( X ) + λ ‖ X ‖ 1 , where f is smooth, F v = { X ∈ R n × q : X T X = I q , v ∈ span ( X ) } , and v is a given positive vector. The clustering models including but not limited to the models used by k -means, community detection, and normali...
Gespeichert in:
| Veröffentlicht in: | Journal of scientific computing Jg. 103; H. 1; S. 8 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.04.2025
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0885-7474, 1573-7691 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | This paper considers the optimization problem
min
X
∈
F
v
f
(
X
)
+
λ
‖
X
‖
1
,
where
f
is smooth,
F
v
=
{
X
∈
R
n
×
q
:
X
T
X
=
I
q
,
v
∈
span
(
X
)
}
, and
v
is a given positive vector. The clustering models including but not limited to the models used by
k
-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain
F
v
forms a compact embedded submanifold of
R
n
×
q
and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of
F
v
are derived. A Riemannian proximal gradient method that allows an adaptive step size is proposed. The proposed Riemannian proximal gradient method solves its subproblem inexactly and still guarantees its global convergence. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0885-7474 1573-7691 |
| DOI: | 10.1007/s10915-025-02806-3 |