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Table 2.1 Gradient flows and corresponding Riemannian metric
ODE
Riemannian Metric
Tangent Space
T
T
w
S
n
−
1
OJAn
2
ξ
η
T
2
n
LUO
2
ξ
η/
w
T
w
(
− {
0
}
)
2
2
ξ
T
n
EXIN
2
w
η
T
w
(
− {
0
}
)
The same analysis about LUO can be applied to MCA EXIN. In this case the
derivative is taken parallel to the Frechet derivative of
r
(w
,
R
)
:
n
− {
0
} →
4
[i.e., eq. (2.25) multiplied by
w
2
] and the Riemannian metric on each tangent
n
n
space
T
w
(
− {
0
}
] is,
∀
ξ
,
η
∈
T
w
(
− {
0
}
)
,
d
w (
t
)
dt
(
)
⇒
2
.
42
2
2
T
=−
w
−
2
2
ξ
,
η
≡
2
w
ξ
η
∇
(w)
(2.46)
ODE MCA EXIN
The analysis above is summarized in Table 2.1. Convergence and stability
properties of gradient flows may depend on the choice of the Riemannian metric.
In case of a nondegenerate critical point of
, the local stability properties of
the gradient flow around that point do not change with the Riemannian metric.
However, in case of a degenerate critical point, the qualitative picture of the
local phase portrait of the gradient around that point may well change with the
Riemannian metric [84,169].
Proposition 51 (Equivalence)
The ODEs of LUO, OJAn, and MCA EXIN are
equivalent
because they only differ from the Riemannian metric. This fact implies
a similar stability analysis
(
except for the critical points
)
.
As seen in Proposition 44, the Rayleigh quotient critical points are degenerate.
As a consequence, the phase portrait of the gradient flow has only degenerate
straight lines [also recall the RQ homogeneity property (2.2) for
β
=
1] in the
direction of the RQ eigenvectors (i.e., the critical points are not isolated). This
fundamental observation, together with the analysis above, implies the following
proposition and justifies the creation of MCA EXIN [24,30].
Proposition 52 (Local Stability)
The fact that the Rayleigh quotient is not a
Morse function implies that the local stability properties
(
local phase portrait
)
of the gradient flow around the RQ critical points change with the Riemannian
metric.
2.5 MCA EXIN ODE STABILITY ANALYSIS
This analysis deals with the MCA EXIN ODE and is therefore constrained by
the ODE approximation assumptions: It will be considered as a theory about the
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