M ANIFOLD M ICRO -S URGERY WITH L INEARLY N EARLY E UCLIDEAN M ETRICS
Anonymous authors
Paper under double-blind review
A BSTRACT
The Ricci ﬂow is a method of manifold surgery, which can trim manifolds to more regular.
However, in most cases, the Rich ﬂow tends to develop singularities and lead to divergence of the solution.
In this paper, we propose linearly nearly
Euclidean metrics to assist manifold micro-surgery, which means that we prove the dynamical stability and convergence of such metrics under the Ricci-DeTurck ﬂow.
From the information geometry and mirror descent points of view, we give the approximation of the steepest descent gradient ﬂow on the linearly nearly Euclidean manifold with dynamical stability.
In practice, the regular shrinking or expanding of Ricci solitons with linearly nearly Euclidean metrics will provide a geometric optimization method for the solution on a manifold.
I NTRODUCTION
In general relativity (Wald, 2010), a complete Riemannian manifold ( M , g ) endowed with a linearly nearly ﬂat spacetime metric g ij is considered for linearized gravity to solve the Newtonian limit.
The form of this metric is g ij =
η ij + γ ij , where η ij represents a ﬂat Minkowski metric whose background is special relativity and γ ij is small from η ij .
An adequate deﬁnition of “smallness” in this context is that the components of γ ij are much smaller than 1 in some global inertial coordinate system of η ij .
Now, let us step out of the physical world and give a similar metric g ij =
δ ij + γ ij in Riemannian n -manifold ( M n , g ) , i.e. the linearly nearly Euclidean metric, where δ ij represents a ﬂat Euclidean metric and γ ij is small from δ ij .
A natural problem for such a linearly nearly Euclidean metric is: how does the metric evolve over time with respect to the Ricci ﬂow while ensuring the constant topological structure?
Let us review some stability analyses of different manifolds along with the Ricci ﬂow.
For the Riemannian n -dimensional manifold ( M n , g ) that is isometric to the Euclidean ndimensional space ( R n , δ ) , Schn ¨ urer et al.
(Schn ¨ urer et al., 2007) have showed the stability of
Euclidean space under the Ricci ﬂow for a small C 0 perturbation.
Koch et al.
(Koch & Lamm, 2012) have given the stability of the Euclidean space along with the Ricci ﬂow in the L ∞ -Norm.
Moreover, for the decay of the L ∞ -Norm on Euclidean space, Appleton (Appleton, 2018) has given the proof of a different method.
Considering the stability of integrable and closed Ricci-ﬂat metrics, Sesum (Sesum, 2006) has proved that the convergence rate is exponential because the spectrum of the Lichnerowicz operator is discrete.
Furthermore, Deruelle et al.
(Deruelle & Kr¨oncke, 2021) have proved that an asymptotically locally Euclidean Ricci-ﬂat metric is dynamically stable under the Ricci ﬂow, for an L 2 ∩ L ∞ perturbation on non-ﬂat and non-compact Ricci-ﬂat manifolds.
In this paper, we consider a complete Riemannian n -dimensional manifold ( M n , g ) endowed with linearly nearly Euclidean metrics g ( t )
=
δ + γ ( t ) .
First of all , we prove the stability of linearly nearly Euclidean manifolds under the Ricci-DeTurck ﬂow in the L 2 -Norm if initial metrics are integrable and linearly stable, i.e. has a manifold structure of ﬁnite dimension.
We mean that any Ricci-DeTurck ﬂow which starts from near g exists for all time and converges to a linearly nearly Euclidean metric near g .
Note that we use the Einstein summation convention and denote generic constants by C or C 1 .
Furthermore , we deﬁne and construct linearly nearly Euclidean manifolds based on information geometry and mirror descent algorithm.
Based on a symmetrized convex function, we obtain the linearly nearly Euclidean divergence which is used to calculate the steepest descent gradient in lin- early nearly Euclidean manifolds.
Experimentally , when we use the approximated steepest descent gradient ﬂow to learn several neural networks on classiﬁcation tasks, we observe the evolution of its metric is consistent with the micro-surgery process under the Ricci-DeTurck ﬂow.
R ICCI F LOW
Let us introduce a partial differential equation, the Ricci ﬂow, without explanation.
The concept of the Ricci ﬂow ﬁrst published by Hamilton (Hamilton et al., 1982) on the manifold M of a timedependent Riemannian metric g ( t ) with the initial metric g 0 :
[Equation]
where Ric denotes the Ricci curvature tensor whose deﬁnition can be found in Appendix A.
The purpose of the Ricci ﬂow is to prove Thurston’s Geometrization Conjecture and Poincar´e Conjecture because the Ricci ﬂow is like a surgical scalpel, trimming irregular manifolds into regular manifolds to facilitate observation and discussion (Sheridan & Rubinstein, 2006).
In general, in order to possess good geometric and topological properties, we expect the metric to become converge and round with the help of the Ricci ﬂow.
“become round” means that the solution will not shrink to a point but converge to a constant circle.
However, in most cases, we do not even know the convergence of the solution and whether the solution will develop a singularity.
Next, we will discuss these issues for brevity.
2. S HORT T IME E XISTENCE
To show that there exists a unique solution for a short time, we must check if the system of the Ricci ﬂow is strongly parabolic.
Theorem 1 When u : M× [0 , T ) → E is a time-dependent section of the vector bundle E where M is some Riemannian manifold, if the system of the Ricci ﬂow is strongly parabolic at u 0 then there exists a solution on some time interval [0 , T ) , and the solution is unique for as long as it exists.
Proof .
The proofs can be found in (Ladyzhenskaia et al., 1988).
[Equation]
Deﬁnition
The Ricci ﬂow is strongly parabolic if there exists δ > 0 such that for all covectors ϕ (cid:54)
=
and all symmetric h ij = ∂g ij ( t ) ∂t (cid:54) = 0 , the principal symbol of − 2 Ric satisﬁes [
[Equation]
Since the inequality cannot always be satisﬁed, the Ricci ﬂow is not strongly parabolic, which makes us unable to prove the existence of the solution based on Theorem 1.
It is possible to understand which parts have an impact on its non-parabolic by the linearization of the Ricci curvature tensor.
Lemma
The linearization of − 2 Ric can be rewritten as
[Equation]
where
[Equation]
Proof .
The proofs can be found in Appendix B.1.
(cid:3)
The term O ( h ij ) will have no contribution to the principal symbol of − 2 Ric , so ignoring it will not affect our discussion of this problem.
By carefully observing the above equation, one ﬁnds that the impact on the non-parabolic of the Ricci ﬂow comes from the terms in V , not the term g pq ∇ p ∇ q h ij .
The solution is followed by the DeTurck Trick (DeTurck, 1983) that has a timedependent reparameterization of the manifold:
[Equation]
where d is a symmetric (0,2)-tensor on M .
See Appendix B.2 for details.
By choosing ∂ϕ ( t ) ∂t to cancel the effort of the terms in V , the reparameterized Ricci ﬂow is strongly parabolic.
Thus, one can say that the Ricci-DeTurck ﬂow has a unique solution, the pullback metric, for a short time.
2. C URVATURE E XPLOSION AT S INGULARITY
In this subsection, we will present the behavior of the Ricci ﬂow in ﬁnite time and show that the evolution of the curvature is close to divergence.
The core demonstration is followed with Theorem 4, which requires some other proof as a foreshadowing.
Theorem 2 Given a smooth Riemannian metric g 0 on a closed manifold M , there exists a maximal time interval [0 , T ) such that a solution g ( t ) of the Ricci ﬂow, with g (0)
=
g 0 , exists and is smooth on [0 , T ) , and this solution is unique.
Proof .
The proofs can be found in (Sheridan & Rubinstein, 2006).
[Equation]
Theorem 3 Let M be a closed manifold and g ( t ) a smooth time-dependent metric on M , deﬁned for t ∈ [0 , T ) .
If there exists a constant C < ∞ for all x ∈ M such that
[Equation]
then the metrics g ( t ) converge uniformly as t approaches T to a continuous metric g ( T ) that is uniformly equivalent to g (0) and satisﬁes
[Equation]
Proof .
The proofs can be found in Appendix B.3.
[Equation]
Corollary 1 Let ( M , g ( t )) be a solution of the Ricci ﬂow on a closed manifold.
If | Rm | g ( t ) is bounded on a ﬁnite time [0 , T ) , then g ( t ) converges uniformly as t approaches T to a continuous metric g ( T ) which is uniformly equivalent to g (0) .
Proof .
The bound on | Rm | g ( t ) implies one on | Ric | g ( t ) .
Based on Equation (1), we can extend the bound on | ∂ ∂t g ( t ) | g ( t ) .
Therefore, we obtain an integral of a bounded quantity over a ﬁnite interval is also bounded, by Theorem 3.
[Equation]
Theorem 4 If g 0 is a smooth metric on a compact manifold M , the Ricci ﬂow with g (0) =
g 0 has a unique solution g ( t ) on a maximal time interval t ∈ [0 , T ) .
If T < ∞ , then
[Equation]
Proof .
For a contradiction, we assume that | Rm x ( t ) | is bounded by a constant.
It follows from
Corollary 1 that the metrics g ( t ) converge uniformly in the norm induced by g ( t ) to a smooth metric g ( T ) .
Based on Theorem 2, it is possible to ﬁnd a solution to the Ricci ﬂow on t ∈ [0 , T ) because the smooth metric g ( T ) is uniformly equivalent to initial metric g (0) .
Hence, one can extend the solution of the Ricci ﬂow after the time point t
= T , which is the result for continuous derivatives at t = T .
This tell us that the time T of existence of the Ricci ﬂow has not been maximal, which contradicts our assumption.
In other words, | Rm x ( t ) | is unbounded.
(cid:3)
As approaching the singular time T , the Riemann curvature | Rm | g ( t ) becomes no longer convergent and tends to explode.
E VOLUTION OF L INEARLY N EARLY E UCLIDEAN M ETRICS
Next, this paper will focus on linearly nearly Euclidean metrics, proving that them can have a good performance in terms of stability, i.e., the convergence of a Ricci-DeTurck ﬂow ¯ g ( t ) to a linearly nearly Euclidean metric ¯ g ( ∞ ) .
Before that, we have to construct a family ¯ g 0 of linearly nearly
Euclidean reference metrics such that ∂∂t ¯ g 0 ( t ) =
O ( [Equation]
g
[Equation]
t ) −
[Equation]
g 0 ( t )) 2 ) .
Let[-us give the deﬁnition of linearly nearly Euclidean metrics without further explanation: Deﬁnition 
 A complete Riemannian n -manifold ( M n , g 0 ) is said to be linearly nearly Euclidean with one end of order τ > 0 if there exists a compact set K ⊂ M , a radius r , a point x in M and a diffeomorphism satisfying φ : M\ K-]
[-Deﬁnition 
-]
[Equation]
be the set of stationary points under the Ricci-DeTurck ﬂow.
We are able to establish a manifold
[Equation]
where U is an L 2 -neighbourhood of integral ¯ g 0 .
Before starting the discussion about long time stability of linearly nearly Euclidean metrics, we need some prior knowledge about short time existence (Appendix D and Appendix E).
In particular, we yield
Lemma 2 Let ¯ g ( t ) be a Ricci–DeTurck ﬂow on a maximal time interval t ∈ (0 , T ) in an L 2 neighbourhood of ¯ g 0 .
We have the following estimate such that:
[Equation]
Proof .
Let { e 1 ( t ) , e 2 ( t ) , . . .
, e n ( t ) } be a family of L 2 -orthonormal bases of the kernel ker Lsuch that ∂∂t e i ( t ) depends linearly on ∂∂t d 0 ( t ) .
For an isomorphism orthogonal projection Π :
T ¯ g 0 ˜ F → ker L 2 , by the Hardy inequality (Minerbe, 2009), one has similar proofs by referring the details (Deruelle & Kr ¨ oncke, 2021).
[Equation]
Theorem 5 Let ( M n , ¯ g 0 ) be a linearly nearly Euclidean n -manifold which is linearly stable and integrable.
Furthermore, there exists a constant α ¯ g 0 such that
[Equation]
for all ¯ g ( t ) ∈ ˜ F which is as in (6).
Proof .
The similar proofs can be found in (Devyver, 2014) with some minor modiﬁcations.
Due to the linear stability requirement of linearly nearly Euclidean manifolds in Deﬁnition 8, − L ¯ g 0 is non-negative.
Then there exists a positive constant α ¯ g 0 such that
[Equation]
By Taylor expansion, one repeatedly uses elliptic regularity and Sobolev embedding (Pacini, 2010) to obtain the estimate.
[Equation]
Corollary 2 Let ( M n , ¯ g 0 ) be a linearly nearly Euclidean n -manifold which is integrable.
For a
Ricci–DeTurck ﬂow ¯ g ( t ) on a maximal time interval t ∈ [0 , T ] , if it satisﬁes (cid:107)
¯ g ( t ) − ¯ g 0 (cid:107) L ∞ < (cid:15) where (cid:15) > 0 , then there exists a constant C < ∞ for t ∈ [0 , T ] such that the evolution inequality satisﬁes
[Equation]
Proof .
Based on Equation (16)
[Equation]
we know
[Equation]
Followed by Lemma 2 and Theorem 5, we further obtain
[Equation]
[-L(cid:107) d ( t ) − d 0 (cid:107) L≤ ( − 
-]
Let (cid:15) be small enough that −
α ¯ g 0 + C · (cid:15) < 0 holds, we can ﬁnd
[Equation]
holds.
[Equation]
Theorem 6 Let ( M n , ¯ g 0 ) be a linearly nearly Euclidean n -manifold which is linearly stable and integrable.
For every (cid:15) 1 > 0 , there exists a (cid:15) 2 > 0 satisfying:
For any metric ¯ g ( t ) ∈ B L 2 (¯ g 0 , (cid:15) 2 ) , there is a complete Ricci–DeTurck ﬂow ( M n , ¯ g ( t )) starting from ¯ g ( t ) converging to a linearly nearly
Euclidean metric ¯ g ( ∞ ) ∈
B L 2 (¯ g 0 , (cid:15) 1 ) .
Note that B L 2 (¯ g 0 , (cid:15) ) is the (cid:15) -ball with respect to the L 2 -Norm induced by ¯ g 0 and centred at ¯ g 0 .
Proof .
By Lemma 4, one can ﬁnd so small (cid:15) 2 >
such that d ( t ) ∈
B
L 2 (0 , (cid:15) 2 ) holds.
By Lemmaand Corollary 2, we have
[Equation]
Furthermore, we obtain
[Equation]
By the triangle inequality, we get
[Equation]
Followed by Corollary 4 and Lemma
[Equation]
such T should be pushed further outward
[Equation]
because
[Equation]
Thus, as t approaches to + ∞
[Equation]
¯ g ( t ) converges to ¯ g ( ∞ ) =
¯ g 0 + d 0 ( ∞ ) .
By the Euclidean Sobolev inequality (Minerbe, 2009)
[Equation]
d ( t ) − d 0 ( t ) converges to 0 as t goes to + ∞ ,
[Equation]
We now conclude a result for linearly nearly Euclidean manifolds under the Ricci-DeTurck ﬂow, which ensures a inﬁnite time existence.
[Equation]
G RADIENT F LOW WITH L INEARLY N EARLY E UCLIDEAN M ETRICS
Consequently, we have clariﬁed the convergence of linearly nearly Euclidean manifolds under the Ricci-DeTurck ﬂow.
Furthermore, we will consider the solution of gradient ﬂow with linearly nearly
Euclidean metrics, which will allow us to observe the neural network behavior for back-propagation in the linearly nearly Euclidean manifold.
Empirically, we introduce information geometry (Amari & Nagaoka, 2000; Amari, 2016) and mirror descent algorithm (Bubeck et al., 2015) to construct the gradient ﬂow with the help of divergences.
4. L INEARLY N EARLY E UCLIDEAN D IVERGENCE
From the perspective of information geometry and mirror descent algorithm, the metric ¯ g can be deduced by the divergence that needs to satisfy certain criteria (Basseville, 2013).
We now consider two nearby points P and Q in a manifold M , where these two points are expressed in coordinates as ξ P and ξ Q , where ξ is a column vector.
Moreover, the divergence is deﬁned as half the square of an inﬁnitesimal distance between two sufﬁciently close points in Deﬁnition 2.
Deﬁnition 2 D [ P : Q ] is called a divergence when it satisﬁes the following criteria: [Equation]
P [Equation]
Q ] [-= 
-][Equation]
0 when and only when P = Q , (3) When P and Q are sufﬁciently close, by denoting their coordinates by ξ P and ξ Q = ξ P + d ξ , the Taylor expansion of D is written as
[Equation]
and metric ¯ g ij is positive-deﬁnite, depending on ξ P .
In order to construct a linearly nearly Euclidean metric, according to Deﬁnition 2, one can construct the divergence to obtain the expression of the metric indirectly.
And the advantage is that the constructed divergence can be used to calculate the gradient ﬂow under linearly nearly Euclidean metrics.
Therefore, with the assist of Deﬁnition 3, we introduce a symmetrized convex function to construct the needed divergence:
[Equation]
where τ is a constant parameter.
Deﬁnition
The Bregman divergence (Bregman, 1967) D B [ ξ : ξ (cid:48) ] is deﬁned as the difference between a convex function φ ( ξ ) and its tangent hyperplane z = φ ( ξ (cid:48) )
+ ( ξ − ξ (cid:48) ) ∇ φ ( ξ (cid:48) ) , depending on the Taylor expansion at the point ξ (cid:48) :
[Equation]
[-Deﬁnition 
-]
Theorem 7 For a convex function φ deﬁned by Equation (7), the linearly nearly Euclidean divergence between two points ξ and ξ (cid:48) is
[Equation]
where the Riemannian metric is
[Equation]
[Equation]
Proof .
The proofs can be found in Appendix C.1.
(cid:3)
Based on Theorem 7, the form of the metrics ¯ g ( t ) constructed by the linearly nearly Euclidean divergence is consistent with the deﬁnition of linearly nearly Euclidean metrics, as long as we adjust parameter τ to satisfy Deﬁnition 7.
Moreover, we have also proven that the linearly nearly Euclidean divergence satisﬁes the criteria of divergence followed by Deﬁnition 2.
4. W EAK A PPROXIMATION OF THE G RADIENT F LOW
By the linearly nearly Euclidean divergence, we can perform micro-surgery on the neural manifold under the gradient descent.
Speciﬁcally, we dynamically consider the gradient ﬂow toward the optimal descent direction on a manifold endowed with linearly nearly Euclidean metrics.
Lemma
The steepest descent gradient ﬂow measured by the linearly nearly Euclidean divergence is deﬁned as
[Equation]
Proof .
The proofs can be found in Appendix C.2.
[Equation]
However, Lemma 3 involves inversion, which greatly consumes computing resources.
In particular, we propose two methods for approximating the gradient ﬂow: weak approximation and strong approximation respectively.
For the weak approximation of the gradient ﬂow, we put forward higher requirements for this metric on the basis of Deﬁnition 7, which requires that the linearly nearly Euclidean metric is also a strictly diagonally-dominant matrix based on Corollary 3.
Corollary
The weak approximation of the gradient ﬂow measured by the linearly nearly Euclidean divergence is deﬁned as
[Equation]
if the metric satisﬁes strictly diagonally-dominant.
Proof .
The proofs can be found in Appendix C.3.
[Equation]
4. S TRONG A PPROXIMATION WITH N EURAL N ETWORKS
Bypassing the requirement of weak approximation in Corollary 3, our goal is to approximate the gradient ﬂow, ¯ g − 1 ( t ) ∂ ξ in Lemma 3, from the assist of multi-layer perceptron (MLP) neural network. Using the neural network, we can easily yield the strong approximation of the gradient ﬂow because a neural network with a single hidden layer and a ﬁnite number of neurons can be used to approximate a continuous function on compact subsets (Jejjala et al., 2020), which is stated by the universal approximation theorem (Cybenko, 1989; Hornik, 1991).
As an n × n symmetric matrix, ¯ g ( t ) can be decomposed in terms of the combination of entries P and A , where P is the entries made up of the elements of the lower triangular matrix that contains n ( n − 1) / 2 real parameters and A is the entries made up of the elements of the diagonal matrix that contains n real parameters.
During the training in Figure 1, ˜ g ( t ) can be used as strong approximation of ¯ g − 1 ( t ) in the gradient ﬂow. [Equation]  E XPERIMENT [Figure]
CIFAR datasets. The two CIFAR datasets Krizhevsky et al. (2009) consist of natural color images with 32 × 32 pixels, respectively 50,000 training and 10,000 test images, and we hold out 5,  training images as a validation set from the training set. CIFAR10 consists of images organized into 10 classes and CIFAR100 into 100 classes.
We adopt a standard data augmentation scheme (random corner cropping and random ﬂipping) that is widely used for these two datasets.
We normalize the images using the channel means and standard deviations in preprocessing.
Settings.
We set total training epochs as 200 where the learning strategy is lowered by 10 times at epoch 80, 150, and 190, with the initial 0.1.
The learning strategy is a weight decay of 0.0001, a batch size of 128, SGD optimization.
On CIFAR10 and CIFAR100 datasets, we apply ResNet20,
ResNet32, ResNet44, ResNet56 and ResNet110 models (He et al., 2016) to observe the evolution of neural manifold, i.e., the convergence of metrics depended on time.
As far as we deﬁne the metric ¯ g ( t ) , we can use the length | ds 2 | = (cid:113)(cid:80) i,j ¯ g ij ( t ) dξ
i dξ j to intuitively reﬂect the change of metrics.
Speciﬁcally, we deﬁne a ball whose radius is equal to | ds 2 | :
[Equation]
Details.
We embed the linearly nearly Euclidean manifold into a neural network, which means that a neural networks uses Corollary 3 for back-propagation.
No other parts of the neural network need to be modiﬁed.
Neural Network Behavior.
By observing the change of the ball in Figure 2, we can know the change of the metric.
Through simple observation, metrics ¯ g ( t ) on CIFAR10 converge in aboutepochs and metrics ¯ g ( t ) on CIFAR100 converge in about 150 epochs.
For CIFAR10, metrics ¯ g ( t ) in ResNet32 and ResNet44 seem to converge the fastest.
For CIFAR100, metrics ¯ g ( t ) in ResNet seem to converge the fastest.
In general, experiments show that all metrics in neural manifolds converges to a linearly nearly Euclidean metric.
It is consistent with the evolution of Ricci-DeTurck ﬂow in Section 3.
Remark.
For a neural network that is speciﬁed by connection weights, the set of all such weighs forms a manifold.
When we use the gradient ﬂow to learn a neural network ( ξ ( t ) is composed of weights), we observe the evolution of its metric is consistent with the micro-surgery process under the Ricci-DeTurck ﬂow.
Consequently, the training of a neural manifold is also a surgery, i.e., the manifold is gradually regular, whose process is stable and eventually converges.
C ONCLUSION
In this paper, we have analysed the evolution of linearly nearly Euclidean metrics under the RicciDeTurck ﬂow, including proof of convergence in short and inﬁnite time.
Furthermore, we construct a linearly nearly Euclidean metric with the assist of the information geometry and use it as a springboard to reach the approximation of gradient ﬂow.
In view of the convergence and stability of metrics, the neural network trained by the approximated gradient ﬂow allows us to observe the relevance between Ricci ﬂow and neural network behavior under the manifold microsurgery .
We hope that this paper will open an exciting future direction which will use Ricci ﬂow to assist neural network training in a manifold.
A D IFFERENTIAL G EOMETRY
Riemann curvature tensor (Rm) is a (
3)-tensor deﬁned for a 1-form ω :
[Equation]
where the covariant derivative of F satisﬁes
[Equation]
In particular, coordinate form of the Riemann curvature tensor is:
[Equation]
Christoffel symbol in terms of an ordinary derivative operator is:
[Equation]
Ricci curvature tensor (Ric) is a (0,2)-tensor:
[Equation]
Scalar curvature is the trace of the Ricci curvature tensor:
[Equation]
Lie derivative of F in the direction dϕ ( t ) dt :
[Equation]
where ϕ ( t ) :
M → M for t ∈ ( − (cid:15), (cid:15) ) is a time-dependent diffeomorphism of M to M .
B P ROOF FOR THE R ICCI F LOW B. P ROOF FOR L EMMA
Deﬁnition
The linearization of the Ricci curvature tensor is given by
[Equation]
Proof .
Based on Appendix A, we have
[Equation]
Combining with Deﬁnition 4, we can obtain the deformation equation because of ∇ k g ij = 0 ,
D [ − 2Ric]( h
[Equation]
ij
[Equation]
B. D ESCRIPTION OF THE D E T URCK T RICK
Using a time-dependent diffeomorphism ϕ
( t ) , we express the pullback metrics ¯ g ( t ) :
[Equation]
is a solution of the Ricci ﬂow.
Based on the chain rule for the Lie derivative in Appendix A, we can calculate
[Equation]
With the help of Equation (1)
[Equation]
for the reparameterized metric
[Equation]
we have
[Equation]
The diffeomorphism invariance of the Ricci curvature tensor is used in the last step.
The above equation is equivalent to
[Equation]
B. P ROOF FOR T HEOREM
Considering any x ∈ M
[Equation]
t 0 ∈
[Equation]
T )
[Equation]
V ∈ T x M
[Equation]
we have
[Equation] [Equation]
By exponentiating both sides of the above inequality, we have
[Equation]
This inequality can be rewritten as
[Equation]
because it holds for any V .
Thus, the metrics g ( t ) are uniformly equivalent to g (0) .
Now, we have the well-deﬁned integral:
[Equation]
We say that this integral is well-deﬁned because of two reasons.
Firstly, as long as the metrics are smooth, the integral exists.
Secondly, the integral is absolutely integrable.
Based on the norm inequality induced by g (0) , one has
[Equation]
For each x ∈ M , the above integral will approach to zero as t approaches T .
Because M is compact, the metrics g ( t ) converge uniformly to a continuous metric g ( T ) which is uniformly equivalent to g (0) on M .
Moreover, we can show that
[Equation]
C P ROOF FOR THE I NFORMATION G EOMETRY C. P ROOF FOR T HEOREM
The linearly nearly Euclidean divergence can be deﬁned between two nearby points ξ and ξ (cid:48) , where the ﬁrst derivative of the linearly nearly Euclidean divergence w.r
[Equation]
t. ξ (cid:48) is:
[Equation]
The second derivative of the linearly nearly Euclidean divergence w
[Equation]
r
[Equation]
t. ξ
[Equation]
is:
[Equation]
We deduce the Taylor expansion of the linearly nearly Euclidean divergence at ξ (cid:48)
=
ξ :
[Equation]
C. P ROOF FOR L EMMA
We would to know in which direction minimizes the loss function with the constraint of the linearly nearly Euclidean divergence, so that we do the minimization:
[Equation]
where c is the constant.
The loss function descends along the manifold with constant speed, regardless the curvature.
Now
[Equation]
we write the minimization in Lagrangian form.
Combined with Theorem 7, the linearly nearly
Euclidean divergence can be approximated by its second order Taylor expansion.
Approximating L ( ξ + d ξ ) with it ﬁrst order
Taylor expansion, we get:
[Equation]
To solve this minimization, we set its derivative w.r.t.
d ξ to zero:
[Equation]
where a constant factor 1 /λ can be absorbed into learning rate.
Up to now, we get the optimal descent direction, i.e., the opposite direction of gradient which takes into account the local curvature deﬁned by (cid:2) δ ij − tanh( τ ξ ( t
[Equation]
tanh [Equation] τ ξ
[Equation]
t
[Equation]
− 1 .
C. P ROOF FOR C OROLLARY
Deﬁnition {+
+} For A ∈ R n × n , A is called as the strictly diagonally-dominant matrix when it satisﬁes
[Equation]
Deﬁnition
If A ∈ R n × n is a strictly diagonally-dominant matrix, then A is a nonsingular matrix together.
Subsequently, we can consider the inverse matrix of the metric ¯ g ( t ) .
Due to the strictly diagonally-dominant feature in Deﬁnition 5 and Deﬁnition 6, we can approximate (cid:2) δ ij − tanh( τ ξ ( t )
[Equation]
( τ
[Equation]
( t )) (cid:62)
[Equation]
As for we can also ignore the fourth-order small quantity (cid:80) O ( ρ a ρ b ρ c ρ d ) because of the characteristic of the strictly diagonally-dominant, so that
[Equation]
t
[Equation]
τ
[Equation]
Note that the Euclidean metric δ ij is equal to the identity matrix I .
Let us give the deﬁnition of linearly nearly Euclidean metrics without further explanation: Deﬁnition 7 A complete Riemannian n -manifold ( M n , g 0 ) is said to be linearly nearly Euclidean with one end of order τ > 0 if there exists a compact set K ⊂ M , a radius r , a point x in M and a diffeomorphism satisfying φ : M\ K → ( R n \ B ( x, r )) /SO ( n ) , where B ( x, r ) is the ball and
SO ( n ) is a ﬁnite group acting freely on R n \{ 0 } , then [Equation]
14) holds on ( R n \ B ( x, r )) /SO ( n ) . g 0 can be linearly decomposed into a form containing the Euclidean metric δ : [Equation]
In this paper, we consider the linear stability and integrability of the initial metric g 0 .
Fortunately, similar to the proof process of (Koiso, 1983; Besse, 2007), we can proceed that g 0 is integral and linearly stable.
{+Deﬁnition 
+}
A complete linearly nearly Euclidean n -manifold ( M n , g 0 ) is said to be linearly stable if the L 2 spectrum of the Lichnerowicz operator L g 0 :
=
∆ g 0 + 2 Rm( g 0 ) ∗ is in ( −∞ , 0] where ∆ gis the Laplacian, when L g 0 acting on d ij satisﬁes [Equation] Deﬁnition 9 A n -manifold ( M n , g 0 ) is said to be integrable if a neighbourhood of g 0 has a smooth structure. E S HORT T IME C ONVERGENCE IN THE L 2 -N ORM
For convenience, we rewrite the Ricci-DeTurck ﬂow (3) in terms of the difference d ( t ) : = ¯ g ( t ) − ¯ g 0 , such that [Equation] (16) where the tensors F and G depend on ¯ g − 1 and Γ(¯ g 0 ) .
Note that ¯ g 0 is a linearly nearly Euclidean metric which satisﬁes the above formula, where d 0 ( t )
=
¯ g 0 ( t ) − ¯ g 0 , so that d ( t ) − d 0 ( t ) =
¯ g ( t ) − ¯ g 0 ( t ) holds.
Note that (cid:107) · (cid:107) L 2 or (cid:107) · (cid:107) L ∞ denotes the L 2 -Norm or L ∞ -Norm with respect to the metric ¯ g 0 .
Lemma 4 Let ( M n , ¯ g 0 ) be a complete linearly nearly Euclidean n -manifold.
If ¯ g (0) is a metric satisfying (cid:107) ¯ g (0) − ¯ g 0 (cid:107) L ∞ < (cid:15) where (cid:15) > 0 , then there exists a constant C < ∞ and a unique
Ricci–DeTurck ﬂow ¯ g ( t ) that satisﬁes
[Equation] If a Ricci-DeTurck ﬂow in B L ∞ (¯ g 0 , (cid:15) ) for t ≥ 1 , there exist constants such that [Equation] Proof .
The similar statement for the case of negative Einstein metrics is given in (Bamler, 2010). The proofs can be translated easily to the case of linearly nearly Euclidean metrics by referring the details (Bamler, 2011). [Equation] Lemma 5 Let ( M n , ¯ g 0 ) be a linearly nearly Euclidean n -manifold.
For a Ricci–DeTurck ﬂow ¯ g ( t ) on a maximal time interval t ∈ [0 , T ) , if it satisﬁes (cid:107) ¯ g (0) − ¯ g 0 (cid:107) L ∞ < (cid:15) where (cid:15) > 0 , then there exists a constant C < ∞ for t ∈ (0 , T ) such that
[Equation]
[Equation]
Proof .
Based on Lemma 4, we can consider about (cid:107) ¯ g ( t ) − ¯ g 0 (cid:107) L 2 . Let κ be a function such that κ = 1 on B ( x, r ) , κ = 0 on M n \ B ( x, 2 r ) and |∇ κ | ≤ 2 /r where x ∈ M n and a radius r . Followed by Equation (16), we obtain [Equation]
Note that we can always ﬁnd a suitable C 1 to make the above formula true.
By integration in time t , we can further obtain
[Equation]
Consequently, we can ﬁnd a ﬁnite ball that satisﬁes this estimate.
[Equation] Corollary 4 Based on Lemma 5, we further have [Equation] [Equation]
Proof . We obtain [Equation] where each ball of radius 2 r on M can be covered by N balls of radius r because ( M n , ¯ g 0 ) is linearly nearly Euclidean.
By the Gronwall inequality, we have [Equation] For the L 2 -Norm, the Ricci-DeTurck ﬂow in linearly nearly Euclidean manifolds has a solution for a short time. [Equation]