Manifold Micro-Surgery with Linearly Nearly Euclidean Metrics
Authors: Jun Chen
Tianxin Huang
Wenzhou Chen
Yong Liu
Venue: ICLR 2022 Submitted (2022)
Keywords: Geometry
Ricci flow
Neural network
Metric learning
Information geometry

Abstract: 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.

Under review as a conference paper at ICLR 2022
MANIFOLD MICRO-SURGERY WITH LINEARLY NEARLY EUCLIDEAN METRICS
Anonymous authors Paper under double-blind review
ABSTRACT
The Ricci flow is a method of manifold surgery, which can trim manifolds to more regular. However, in most cases, the Rich flow tends develop singularities and lead to divergence of the solution. In this paper, we propose linearly nearly Euclidean metries to assist manifold micro-surgery, which means that we prove the dynamical stability and convergence of such metrics under the Ricci-De'Turck flow. From the information geometry and mirror descent points of view, we give the approximation of the steepest descent gradient flow the linearly nearly Eu- clidean manifold with dynamical stability. In practice, the regular shrinking or expanding of Ricci solitons with linearly nearly Euclidean metrics will provide geometric optimization method for the solution on a manifold.
INTRODUCTION
In general relativity (Wald 2010 complete Riemannian manifold (M, g) endowed with linearly nearly flat spacetime metric 9sy is considered for linearized gravity to solve the Newtonian limit. The form of this metric is 9ij Tky + Vaj. where Tk represents a flat Minkowski metric whose background is special relativity and Yis is small from 1/j-- An adequate definition of "smallness" in this context is that the components of Rr are much smaller than some global inertial coordinate system of Mij. Now, let us step out of the physical world and give similar metric Pūj = 5.ş + Tii in Riemannian n-manifold (M",9) .g), i.e. the linearly nearly Euclidean metric, where d., represents a flat Euclidean metric and Yi is small from dij.
natural problem for such 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 review some stability analyses of different manifolds along with the Ricci flow.
For the Riemannian n-dimensional manifold (M",.) that is isometric to the Euclidean n- dimensional space (R",5) 5)) Schnürer et al. (Schnürer et al., 2007) have showed the stability of Euclidean space under the Ricci flow for small C 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 T,2o-Norm on Euclidean space, Appleton (Appleton] 2018) has given the proof of a different method. Considering the stability of integrable and closed Ricci-flat met- rics, Sesum (Sesum, 2006) has proved that the convergence rate exponential because the spectrum of the Lichnerowicz operator is discrete. Furthermore, Deruelle et al. (Deruelle & Kröncke (02T) have proved that an asymptotically locally Euclidean Ricci-flat metric dynamically stable under the Ricci flow, for an L n L perturbation on non-flat and non-compact Ricci-flat manifolds.
If we embed Riemannian n-dimensional manifold in the neural network, then we are training the neural network on the dynamic manifold. 'The most famous method for training neural networks on manifolds the natural gradient (Amari 1998). However, for the Riemannian manifold corre- sponding the KL divergence (representing the natural gradient), its stability and convergence still unknown (Martens] 2020). As a way of manifold evolution, Ricci flow seems to be an excellent choice to ensure that neural networks trained on dynamic and stable manifolds (Glass et al. 2020 Jejjala et al. 2020). But the research on the relationship between the two has not yet sprouted.
In this paper, consider complete Riemannian n--dimensional manifold (M" g) endowed with linearly nearly Euclidean metrics g() = 8 + 7t).. First of all, we prove the stability of linearly nearly Fuclidean manifolds under the Ricci-DeTurre flow in the 1, 1.2NNrmm if initial metrics

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integrable and linearly stable, ie. has manifold structure of finite dimension. We mean that any Ricci-DeTurek flow which starts from near exists for all time and converges to a linearly nearly Euclidean metric near g. Note that use the Einstein summation convention and denote generic constants by C or C1.
Furthermore, define and construct linearly nearly Euclidean manifolds based information geometry and mirror descent algorithm. Based on a symmetrized convex function, we obtain the linearly nearly Euclidean divergence which is used calculate the steepest descent gradient in lin- early nearly Euclidean manifolds. Experimentally. when we use the approximated steepest descent gradient flow learn several neural networks on classification tasks, we observe the evolution of its metric is consistent with the micro-surgen process under the Ricci-DeTuur flow.
2 RICCI FLOW
Let us introduce a partial differential equation, the Ricci flow, without explanation. The concept of the Ricci flow first published by Hamilton (Hamilton al., 1982) on the manifold M of a time- dependent Riemannian metric g(t) with the initial metric go:
i Ot g (t) Ric(g(t))
g go
where Ric denotes the Ricci curvature tensor whose definition can be found in Appendix A
The purpose of the Ricci flow to prove Thurston's Geometrization Conjecture and Poincaré Con- jecture because the Ricci flow is like surgical scalpel. trimming irregular manifolds into regular manifolds 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 point but converge to constant circle. However, in most cases, we do not even know the convergence the solution and whether the solution will develop a singularity. Next, we will discuss these issues for brevity.
2.1 SHORT TIME EXISTENCE
To show that there exists unique solution for a short time, we must check the system of the Ricci flow is strongly parabolic.
Theorem When u M x [0, T) + E is time- dependen section of the vector bundle & where M is some Riemannian manifold, the system of the Ricci flow is strongly parabolic at uo then there exists a solution some time interval [0, (), and the solution is unique for long as it exists.
Proof. The proofs be found in (Ladyzhenskaia al. 1988
Definition 1 The Ricci flow is strongly parabolic if there exists 0 such that for all covectors >p * and all symmetric hig 0g,(())
+ the principal symbol of -2 Ric satisfies
St
[-2 (p)
-9 (e + 1pisylipg Ppppilipp hsp) h > h ht.
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 [
It is possible to understand which parts have impact its non-parabolic by the linearization of the Ricci curvature tensor.
Lemma The linearization of -2 Ric can be rewritten
Dic2 = V + V.V, + V +0
(2)
where
V, = n9
V.hpa
Volpi
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Proof. The proofs be found in Appendix B.1
The term (hhy will have no contribution to the principal symbol of -2 Ric, so ignoring will not affect discussion of this problem. By carefully observing the above equation, finds that the impact on the non-parabolic of the Ricci flow comes from the terms in V, not the term V The solution is followed by the DeTTrrkk Trick (DeTurck 1983) that has time- dependent reparameterizztioo of the manifold:
a
σt (1) Ric(g(()) Logta g(1)
(3)
((0) go + d,
where is a symmetric (0,2)-tensor on M. See Appendix B.2 for details. By choosing (e(()) to at
cancel the effort of the terms in V, the reparameterized Ricei flow is strongly parabolic. Thus, one can say that the
flow has unique solution, the pullback metric, for a short time.
2.2 CURVATURE EXPLOSION AT SINGULARITY
In this subsection, we will present the behavior of the Ricci flow in finite time and show that the evo- lution of the curvature is close to divergence. The core demonstration is followed with Theorem 4 which requires some other proof as a foreshadowing.
Given a smooth Riemannian metric 9o on a closed manifold M, there exists a maximal time interval [0, T) such that solution g(t) of the Ricci flow, with g00) go. exists and is smooth on [0, T), and this solution is unique.
Theorem
Proof. The proofs can be found in (Sheridan & Rubinstein, 2006).
Theorem Let M be closed manifold and g(t) smooth time time- denendennddnt metric on M, defined fort E T). If there exists a constant C' < 0o for all x E M such that
J." (00
dt sC
(9(t)
then the metrics g(t) converge uniformly as t approaches T continuous metric g(T) that is uniformly equivalent to g(0) and satisfies
(0) S gz) e (0).
Proof. The proofs can be found in Appendix B.3
Corollary Let (t)) be solution of the Ricci flow a closed manifold. If |Rm late) is bounded on finite time [0, T), then g(t) converges uniformly as approaches T to continuous metric (TT which uniformly equivalent to g()).
Proof. The bound on Rm la(e) implies one on Ric la(e)- Based on Equation I we can extend the bound on 2 (t)((()9)))) Therefore, we obtain an integral of bounded quantity over finite interval also bounded, by Theorem
Theorem If go is smooth metric on a compact manifold M, the Ricci flow with g(0) = 9u has unique solution g(t) maximal time interval [O,T). If < sc, then
lim E-
sup Rm, oc. AE
(5)
Proof. For a contradiction, we assume that Rm,(t) is bounded by a constant. follows from Corollary the metrics (t) converge uniformly in the norm induced by g(t) to smooth metric gT)). Based on Theorem 2 is possible to find a solution to the Ricci flow on t e [o,T) because the smooth metric g(T) is uniformly equivalent to initial metric g()).

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Hence, extend the solution of the Ricci flow after the time point T, which is the result for continuous derivatives at t = T. This tell us that the time of existence of the Ricci flow has not been maximal, which contradicts our assumption. In other words, (t) is unbounded. O
As approaching the singular time T', the Riemann curvature Rmm (g(e) becomes no longer convergent and tends to explode.
EVOLUTION OF LINEARLY NEARLY EUCLIDEAN METRICS
Next, this paper will focus on linearly nearly Euclidean metrics, proving that them can have good performance in terms of stability, i.c., the convergence of Ricei-DeTurck flow g(t) to linearly nearly Euclidean metric g(xc). Before that, we have to construct a family Do of linearly nearly Euclidean reference metrics such that Be go = O((g(t) - (t)))) Let
2
F {g(() E M
(i)(((()) + L Cesth g() }}
be the set of stationary points under the Ricci-DeTuree flow. We are able establish a manifold
===Nu
(6)
where 4 is an L'-neighbourhood of integral go-
Before starting the discussion about long time stability of linearly nearly Euclidean metrics, we need some prior knowledge about short time existence (Appendix [ and Appendix Eb. In particular, we yield
Lemma Let g(t) be a Ricci-DeTurck flow on maximal time interval t (0,7) an neigh- bourhood of gu- We have the following estimate such that:
- do(() Di
i
sc c310 (d(t) - do) (d())
(()} be family of L"-orthonormal bases of the kernel kerga such that 37 (() depends linearly on of dd((t). For an isomorphism orthogonal projection
Proof. Let e((), &
Tg. F - keT by the Hardy inequality (Minerbe 2009), one has similar proofs by referring the details (Deruelle & Kröncke 2021).
Theorem Let (M", 3o) be linearly nearly Euclidean n-manifold which is linearly stable and integrable. Furthermore, there exists a constant ago such that
(Ad(t) + Rm((0)) * d(t), S -ago /
for all g(t) E F which as in 6
Proof. The similar proofs can be found in Devyver 2014) with some minor modifications. Due
the linear stability requirement of linearly nearly Euclidean manifolds in Definition X -gg non-negative. Then there exists positive constant Oggo such that
O'jo - Ad()) d(()) < (-Ad(() Rm(gn) d((), da)))
By Taylor expansion, one repeatedly uses elliptic regularity and Sobolev embedding (Pacini 2010 to obtain the estimate.
Corollary Let (M", 3o) be linearly nearly Euclidean n-manifold which is integrable. For Ricci-DeTurek flow g(t) on maximal time interval [0, T], if it satisfies l[ğ(t) - go Soll where e > O, then there exists constant C. < 00 for t E [0, T] such that the evolution inequality satisfies
[|((t) - -doll 2 cc/ | (d(t) do) do)1(3 dt.
12
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Proof. Based on Equation 16), know
① at
(d(t) do) (d((t) - do) + Rm (d((t) - do)
+ Fa- - (d(t) - do) 8 (d(t) - do)
V 80 (Grog)) * (d(t) - do) * 9 (d(t) do)).
Followed by Lemma and Theorem 5 we further obtain
θ
öt [d(()) doll2> =2 (d((() - do) + Rm * (d(() do), do)
(Fg-1 (d(t) do) (d(() da), d(t) do) 12
(v (Gr(w)) (d(1)) do) v - do)) ,d(l) - da) 1.2
/ (d(t) do) (d(t) - do) do do(t)du
8
+ d(t) - do. 0t
do(t)
+
122
S 20 V (d(t) do)
12
C ((d(())) - do) | (d(t) - do) do>l
+
do(t)
lld(t) - doll
0t
122
<-~20g. + +c (d(t) - do) '.
Let c be small enough that -20g0 + C. < 0 holds, we can find
ld(t) dollis S -C|| (d(t) do>)l 01
holds.
Theorem Let (M" .0)) be a linearly nearly Euclidean n-manifold which is linearly stable and integrable. For every E1 > 0, there exists 62 >> O satisfying: For any metric g(t) E Bra e2), there is a complete Ricci-DeTurek flow (M", ,g()) starting from
from g(t) converging to a linearly nearly
Euclidean metric g(oc) Br (1), Note that Bg= <) is the e-ball with respect to the I?2-Nomm induced by go and centred at go.
Proof. By Lemma 4 can find small c2 such that d(t) E (0, holds. By Lemma and Corollary 2 we have
[ sc
dt 12
SC S/ 1| (d(t) - do(t)))l dt
s <Cld((1) do (1)1l s clld(1))li SC- (
Furthermore, we obtain
[ - do((T) do))) S [ld(1) do(1)llz -2.
By the triangle inequality, we get
[ld S C- 2 2 +
Followed by Corollary and Lemma
2 such T should be pushed further outward, because
i (1)
lim sup 6-+00
S lim sup | 5o (d(1) do()))l 0. -
Øt
L7
Thus, as f approaches to +00 g(t) converges to g(oo) Do do (oe). By the Euclidean Sobolev
inequality (Minerbe 2009). ) d(t) (t) converges to as f goes to +00,
lim [d()) do do(t)llz= < lim C|| (d(t) do 1. = 0.
+0++00
+++
We now conclude a result for linearly nearly Euclidean manifolds under the Dicci-DeTurek flow, which ensures infinite time existence.
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4 GRADIENT FLOW WITH LINEARLY NEARLY EUCLIDEAN METRICS
Consequently, we have clarified the convergence of linearly nearly Euclidean manifolds under the Ricci-DeTuree 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 pack-propagation in the linearly nearly Euclidean manifold. Empirically, introduce information geometry (Amari & Nagaoka, 2000; Amari (016) and mirror descent algorithm (Bubeck et al. (013) to construct the gradient flow with the help of divergences.
4.1 LINEARLY NEARLY EUCLIDEAN DIVERGENCE
From the perspective of information geometry and mirror descent algorithm, the metric ğ can be deduced by the divergence that needs satisfy certain criteria (Basseville) 2013). We now consider two nearby points P and Q in manifold M, where these two points are expressed in coordinates as Ep and Eq. where & column vector. Moreover, the divergence is defined as half the square of an infinitesimal distance between two sufficiently close points in Definition
Definition D[P: Q) is called divergence when satisfies the following criteria:
(1) D[P: Q) 2 0, (2) D[P : Q] 0 when and only when P = Q, (3) When P and Q are sufficiently close, by denoting their coordinates by Ep and EQ Ep + dg, the Taylor expansion of D) written
D'gr: : Er + de) = Σ 9is (Er) + 0(
and metric Isj is positive-defiitt, depending on &p.
In order construct linearly nearly Euclidean metric, according to Definition 2 one can con- struct the divergence to obtain the expression of the metric indirectly. And the advantage is that constructed divergence metries. Therefore, with the assist of Definition we introduce a symmetrized convex function to construct the needed divergence:
be used to calculate the gradient flow under linearly nearly Euclidean
o(E) = Σ log 2 (exp(TÉ:) + exp(-rEE))
Σ
log (cosh(TE.))
where T constant parameter.
Definition 3 The Bregman divergence (Bregman 1967) D E E'] is defined as the difference
berween convex function ΦfE) and its tangent hyperplane ((E)) -''S-'''''', depending on the Taylor expansion at the point E':
DELE E'l = ((E) o(E') -E =')VV(')))
Theorem For convex function & defined by Equation 77 the linearly nearly Euclidean diver- gence between two points & and g' is
cosh(rE!)
DLNEIE' E] Σ
log
(g) - {x) -x) tanh(TTi)
(8)
72
cosh(Tēt)
where the Riemannian metric is
Dig (E(t)) = 5zz [tanh(TT) tanh(rE)
(tn)(T(:(t)) (Tn)(
tanh(75:(1)) tanh (rnn(r
(9)
tanh(TE.. (t)) (t))
1 tanh(rT,, tan
Proof. The proofs be found in Appendix
Based on Theorem the form of the metrics g(t) constructed by the linearly nearly Euclidean divergence is consistent with the definition of linearly nearly Euclidean metries, long adjust
parameter 7 to satisfy Definition/) 77] Moreover, we have also proven that the linearly nearly Euclidean divergence satisfies the criteria of divergence followed by Definition 2

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4.2 WEAK APPROXIMATION OF THE GRADIENT FLOW
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 manifold endowed with linearly nearly Euclidean metries.
Lemma The steepesi descent gradient flow measured by the linearly nearly Euclidean divergence is defined as
0 g = [
dg.
(10)
Proof. The proofs be found in Appendix C.2
3 involves inversion, which greatly consumes computing resources. In partieu- lar, we propose two methods for approximating the gradient flow: weak approximation and strong approximation respectively.
However, Lemma
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 also a strictly diagonally dominant iidgonally-dommnant
matrix based on Corollary
Corollary 3 The weak approximation of the gradient flow measured by the linearly nearly Eu- clidean divergence defined as
de [6., + tanh(rT(()) tanh(rE(t)) 0E
(11)
the metric satisfies strictly diagonally dominnnt.
Proof. The proofs can be found in Appendix C.3
4.3 STRONG APPROXIMATION WITH NEURAL NETWORKS
Bypassing the requirement of weak approximation in Corollary 3 our goal to approximate the gradient flow, ğ- (t)8g in Lemma 3 from the assist of multi-layer perceptron (MLP) neural net- work. Using the neural network, we can easily yield the strong approximation because neural network with single hidden layer and a finite number of neurons can be used to approximate a continuous function on compact subsets (Jeijala et al. 2020), which is stated by the universal approximation theorem (Cybenko 1989. Hornik 1991).
the gradient flow
As an n X n symmetric matrix, g(l) can be decomposed terms the combination of entries and A, where P is the entries made up of the elements of the lower triangular matrix that contains (n 1)/2 real parameters and is the entries made up of the elements of the diagonal matrix that contains real parameters. During the training in Figure] 1 (()) be used strong approximation of g (f) in the gradient flow.
(t)
P
g(t)
MLP
g(t)
(t)
A
Ä
:
Loss function
5n (t)
Figure 1: Flow chart of strong approximation of the gradient flow. The new entries P and Á pro- duced by neural network get combined into new metric ğ(t) that is used minimize the loss function by combining with the metric g(t), where the loss function defined by Equation 12
L 2()6(11)
(12)

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EXPERIMENT
1.5
1.0
0.5
ResNet20
ResNet32
B(t)
0.0
ResNet44
ResNet56
ResNet110
-0.5
-1.0
-1.5
50
100
150
200
epoch
(a) CIFARIO
1.5
1.0
0.5
ResNet20
ResNet32
B(t)
0.0
ResNet44
ResNet56
ResNet110
-0.5
-1.0
11.5
0
50
100
150
200
epoch
(b) CIFAR100
Figure 2: The evolution of metrics g(t) by the radius of ball with the epoch of training process. Note that we use radius as the calibration of a linearly nearly Fuclidean metric.
CIFAR datasets. The two CIFAR datasets Krizhevsky et al. (0099) consist of natural color images with 32x32 pixels, respectively 50,000 training and 10,000 test images, and hold out 5,000
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training images validation set from the training set. CIFARIO consists of images organized into 10 classes and CIFAR 100 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 200 where the learning strategy is lowered by 10 times 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 CIFARIO and CIFAR 100 datasets, we apply ResNet20, ResNet32, ResNet44, ResNet56 and ResNetl 10 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 Specifically, we define a ball whose radius is equal to |d>2|:
|ds²| v ES. Ptg (t)dE,dE, to intuitively reflect the change of metrics.
(t)
Σ Iis (E)
(13)
Details. We embed the linearly nearly Euclidean manifold into neural network, which means that a neural networks uses Corollary 3 for back- poe-ppropagation No other parts of the neural network need 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, metries gt) on CIFARIO converge about 100 epochs and metrics g(t) on CIFAR 100 converge in about 150 epochs. For CIFARIO, 10, metrics ğ(t) ResNet32 and ResNet44 seem to converge the fastest. For CIFARIO0, metries g()) in ResNetl 10 seem converge the fastest. In general, experiments show that all metrics in neural manifolds converges to a linearly nearly Euclidean metric. is consistent with the evolution of Ricci-Der Brcci-DeTurck flow in Section 3
For a neural network that is specified by connection weights, the set of all such weighs forms manifold. When we use the gradient flow to learn neural network (E(t) is composed of weights), we observe the evolution of its metric is consistent with the micro-surgery process under the Ricei-DeTurck flow. Consequently, the training of neural manifold is also a surgery, i.e., the manifold is gradually regular, whose process is stable and eventually converges.
Remark.
6 CONCLUSION
In this paper, we have analysed the evolution of linearly nearly Euclidean metrics under the Ricci- DeTurck 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 spring- board 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 micro- surgery. We hope that this paper will open exciting future direction which will use Ricci flow to assist neural network training in a manifold.
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A
DIFFERENTIAL GEOMETRY
Riemann curvature tensor (Rm) a (1,3)-tensor defined for a I-form wt
Rijaar V.V jink
V Viik
where the covariant derivative of satisfies
V Ppliin ig..de
Σ
phin 19 tr in pi."
pq
s=1
In particular, coordinate form of the Riemann curvature tensor is:
R = a jk -
0,
+
TP
jk
P ik
2. Christoffel symbol in terms of an ordinary derivative operator is:
1 kl (0.9,) + 0ggu 0rghg).
r 2
3. Ricci curvature tensor (Ric) is a (0,2)-tensor,
Rig pij'
4. Scalar curvature is the trace of the Ricci curvature tensor:
R= 9¹ Rag.
Lie derivative of F' in the direction de(t), at
d dt
Cast L
d1
1=
where cptt: : M + M for E (-c,c) is a time-dependent diffemoopphin M to M.
B PROOF FOR THE RICCI FLOW
B. PROOF FOR LEMMA 1
Definition 4 The linearization of the Ricci curvature tensor is given by
(b)i 59 (V + V.Vgl V V. V. Vylip).
Proof. Based on Appendix have
V V.hjp V V.V hij
Ragghtep Ry him- him-
Combining with Definition we can obtain the deformation equation because of Vagij = 0,
V s - Vgli50 + v, Vellp - Volisp + O(tig)
D[-2Ric =
=
V,V, + O(hr)).
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B.2 DESCRIPTION OF THE DETURCK TRICK
Using time-dependent diffeomorphism pp(), we express the pullback metrics g()):
g(l) & (1)(()
solution of the Ricci flow. Based the chain rule for the Lie derivative in Appendix A calculate
)()) = a (t)g())
Ot O
+ (1+)))))
ŪT
T=0
(*(()) 05(l 05(1+ +
(1 +)o(())
ØT
T=0
r=0
(() + 4 (1) 10C
0t
With the help of Equation
1. for the reparameterized metric, we have
o (t) Ot (() + 40 aega 5(t) = 2 Ric (t)g(t)) -24
(t) (t) Ric(g(t)).
The diffeomorphism invariance of the Ricci curvature tensor is used in the last step. The above equation is equivalent to
0 क. )00 - Ric(g(()) Conto g(L).
B.3 PROOF FOR THEORFM
Considering any E M, E [0,7), V e T M,M, we have
log 9z (z(())(V
to 0
log (V, V)] dt
(0)(VVV
(g)
dt
(t)(VV
V
0r 9z (1)
(lace)
(Vlcos
dl
1o
a
<
Un (1)
dt
s(t)
C.
By exponentiating both sides of the above inequality, we have
ga (0)VV, V) (lo)(VV) e (0)(V)
This inequality can rewritten as
g0 gz (to)(V S E (O)(VV
because it holds for any V. Thus, the metrics g are uniformly equivalent to g0).
Now, we have the well-defined integral:
Ia (T') (0) = L T
gz (f)dt.
We say that this integral is well-defined because of two reasons. Firstly, long as the metrics are smooth, the integral exists. Secondly, the integral is absolutely integrable. Based on the norm inequality induced by (0), one has
19a (n(())())) < ( dt.
9(0)
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For each x M,, the above integral will approach to zero as approaches T. Because M is compact, the metrics gt)) converge uniformly to continuous metric g(T) which is uniformly equivalent to g(0) on M. Moreover, show that
gx S gz (T) S e go(
C PROOF FOR THE INFORMATION GEOMETRY
C.I PROOF FOR THEOREM 7
The linearly nearly Euclidean divergence can be defined between two nearby points & and g'. where the first derivative of the linearly nearly Euclidean divergence w.r.t. g' is:
0g. Dr F:.'' [
Σ
ag
1
log cosh(TE)) ag 72
log cosh(TE)) - ag (5i - E.) trn)((FE))
=
72
- De - log cosh(TT)) 1 Lanh(TT).
The second derivative of the linearly nearly Euclidean divergence w.r.t. g' is:
0g DINELE' E]
Σ
0% 72 log cosh(
We deduce the Taylor expansion of the linearly nearly Euclidean divergence E' E:
DLNE : E DLNEEE: : E +
Σ
Oe' 72
log coshi(T))) (anh(TE)
dg
E'=E
+ +2 dE
0g. 73
log cosh((T(()
de
c'=E
(cosh(TE)
0 + 0 0+
de
a
dc & cosh" (Tē)
2r2
cosh(TE)
cosh(TT)
cosh(TE)cosh(((E)
dE
de
27
82 coshi(TE)
_- [T))
[sinh(TT)
dET
de (oosh(TE) cosh(rE)
27
cosh(TT)
dij
tanh(rT) "), dgadEs.
2
C.2 PROOF FOR LEMMA
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:
dg dE &
arg min
L(E dE)
where is the constant. The loss function descends along the manifold with constant speed, regard- less the curvature.
Now, we write the minimization Lagrangian form. Combined with Theorem 7 the linearly nearly Euclidean divergence can be approximated by its second order Taylor expansion. Approximating L(E + de) with first order Taylor expansion, we get:
dg*
arg min LLE d + (DLNEEE : dE) c)
de
arg min L(E) (E) de + > TT go((t)d ch.
de
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To solve this minimization, set its derivative w.r.t dg zero:
Ô (L) + (5) dg
0 = Odg
2 ds [o.j tanh(rE(()) tanh(rg(t)) dg cd
0gL(E) + |6. - tanh(( tanb((((tt) dg
1
de - 16. - tanh((E((t)) (an)(((()))))) dgL((E)
where a constant factor 1/> can be absorbed into learning rate. Up to now, we the optimal descent direction, i.e., the opposite direction of gradient which takes into account the local curvature defined by [6.r - tanh(r tanh(TE(t))
C.3 PROOF FOR COROLLARY 3
Definition 5 For A E R=x", A is called as the strictly diagonally-dominen matrix when it satisfies
n
Layl: i= 1.2,....,7.
j=1jfi
Definition 6 A E Ruxn is a strictly diagonally-dominan matrix, then A is nonsingular matrix together.
Subsequently, we can consider the inverse matrix of the metric g(t). Due to the strictly dingonally-dominant feature in Definition | and Definition g we can approximate (6g (I)) (an)( As for we can also ignore the fourth-order small quantity Σ ((popoppep)) because of the characteristic of the strictly diagonally-dominaan that
[0. tanh(rE(t)) tanb(r
[6 + tann((F((t) tanh(rE(tt)
PIP
PP1P2
1 + PIPI
P1P2
P2P1
P2202
P2P1
+ P2P2
:
:
1 Σ (paPoPoPa))
P1P2 PIP2 E O(aphppea))
P2P1 - P2P1 - (PPPpppppa))
1 1- (PPPPoPePd)
= I.
Note that the Euclidean metric dig is equal to the identity matrix I.
D ANALYSIS ON LINEARLY NEARLY EUCLIDEAN METRICS
Let us give the definition of linearly nearly Euclidean metrics without further explanation:
Definition 7 A complete Riemannian n-manifold (M", .9)) is said to be linearly nearly Euclidean
with one end of order t > 0 if there exists compact set C M, a radius r, a point x in M and diffeomorphism satisfying M.K + ((x,)B())))))) (O(n) a finite group acting freely on I2" (00, then
where B(x, r) is the ball and
(6+ vo) O(r-T-k) wk 2 0
(14)
holds on (RR)B((,,))) r)))SO(m). go can be linearly decomposed into a form containing the Eu- clidean metric d;
(t) = 5+ vott)
(15)
In this paper, consider the linear stability and integrability of the initial metric 9u- Fortunately, similar to the proof process of (Koiso) 1983] linearly stable.
Besse 2007). we can proceed that 9o is integral and
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Definition 8 A complete linearly nearly Euclidean n-manifold (M", .9)) is said be linearly stable if the L spectrum of the Lichnerowicz operator Lgo := Ago + Rm{gu)** is in (-o0, 0] where A go the Laplacian, when Lgo acting dij satisfies
go (d) gg + Rm((go)= * d
g. d + Rm(go) (ay km In
90
Definition A n-manifold (M" .0) ) structure.
said to be integrable if neighbourhood of go has smooth
E SHORT TIME CONVERGENCE IN THE L2-NORM
For convenience, rewrite the Ricci-DeTurek flow 33 in terms of the difference d(t) g(t) 9u. such that
0 Ot d(t) =
Lavco g(t) Ot g()) = - Ric( + Ric( + L Dutto 9o
Bt
= Ad(t) + Rm #d(t) + Fg * 9 d(t) 9 d(t) + d (Gr(s)) d(t) * V
(16)
where the tensors F' and G' depend on / and (') Note that go is linearly nearly Euclidean metric which satisfies the above formula, where (t) = To (t) Gu. so that d(t) (o = g()
(t) holds. Note that || 1/ || || denotes the L -Norm _NomN with respect to the metric To-
Lemma Let (M" ) be complete linearly nearly Euclidean n-manifold. If g(0) is metric satisfying (l)(0) goll 1. where e 0, then there exists constant C < ao and unique Ricci-DeTurck flow gt) that satisfies
l0((1) - Zo < <C]lg(0) - Boll <.
a Ricci-DeTurck flow in Brn (go, e) for t 1, there exist constants such that
(g(1)- <(k)e, VkEN.
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).
Lemma Let (M"
(M',9)) 90) be linearly nearly Euclidean n-manifold. For Ricci-DeTurre flow g(t) on maximal time interval E [0, T), if it satisfies (1ğ(0) Do Jol|n. < e where e > 0, then there exists a constant C < 30 for E (0, T) such that
((ī(() Bollz> <C
(17)
Proof. Based on Lemma we can consider about [|ğ(t) goll Let x be a function such that [2x r), M B(r, 2r) and [< 2/r where M and radius
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Followed by Equation (6). obtain
/ [d(t) du 22 / (Ad(t), +u+ / [d(t)[px(dd
01
M
+ +c|d(t))a= / [d(tt) / (V(Gr dx Vd). d) 13 dx
< 5 [dd()PPPuu+C /. Vd(())d(()
+C(70) /,. |d(0)322p CId(O1IL= J. [vd(1)
S(-2 C(+2+) /. vd(t))3
C (0)) / ld(t))2<2dp
+ Ci Imw |d(O)PVA((dk
s <(C(Go) + C,r² B(a,2r)
2
dd
Note that we can always find a suitable C1 to make the above formula true. By integration in time t, we can further obtain
Jm [d(t)) s / [d(0))2xdpp + (C(O)) + Cir²
2
So B(w.2r)
(d(s)
Consequently, we can find a finite ball that satisfies this estimate. Corollary Based on Lemma
we further have
sup / M
xd()) < cc.
(18)
Proof. We obtain
sup [d(t) (xtt) S sup / m (d(0))2x2dp
2
% sup / (x2s))x"ddudds
(go) + Cir²
where each ball of radius 2r on M can be covered by N balls of radius r because (M" go) is linearly nearly Euclidean. By the Gronwall inequality, we have
sup [d(t))x2dpp S exp (N (CG)) + c )1) sup / xd
For the T.2-Norm, the Ricci-DeTTurek flow in linearly nearly Euclidean manifolds has solution for a short time.
17


Reviews:
[{"review_id": "mAHMAkPFryY", "reviewer": "Reviewer_HjEu", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper presents seemingly new theoretical results and sets the stage for the analysis of gradient flow used for training neural networks. For me, the most interesting finding is the remark in Sec 4.2 stating that \"When we use the gradient flow to learn a neural network ($\\xi(t)$ is composed of weights), we observe the evolution of its metric is consistent with the micro-surgery process under\nthe Ricci-DeTurck flow\". Unfortunately, this argument is not well supported in the paper, and only a small empirical test is shown in Appendix C.4.\n\nConsidering the venue (ICLR), I believe the paper is currently unbalanced. On the one hand, this is a theoretical paper presenting new analysis, which is interesting and is a contribution by itself. On the other hand, too much focus is put on the math, and I am sure how much it is of interest to the ML community. In contrast, the utility of the analysis for training neural networks using gradient flow is of major importance to the community, but it is not well treated in the paper. If the authors wish to publish their work in such ML conferences, I would recommend moving much of the analysis into the appendix and focusing instead on the utility, by describing in more detail the setting and the connection between the analysis of the evolution of this linearly nearly Eulicdean metric under the Ricci flow and the gradient flow and/or by extending the experimental study significantly.", "summary_of_the_paper": "This paper investigates linearly nearly Euclidean metrics on Riemannian manifolds under the Ricci flow. Analysis, focusing on the stability and convergence of the evolution of such metrics, is presented. In addition, the utility of the analysis for measuring the approximation of gradient flow in the context of training neural networks is demonstrated.", "summary_of_the_review": "Interesting new theoretical results, but the connection to gradient flow is lacking", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "fLe3Pz8LJm", "reviewer": "Reviewer_ibvS", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is heavily theoretic and expects a lot of background knowledge in differential geometry from the reader. I do have some background but I was not able to follow every detail in the derivations given the short review period. I am especially not familiar with the term micro-surgery for Ricci flows, there is no citation for this concept and any explanation I was able to find did not make sense for how it is used in this paper for me. The main idea of linearizing the Ricci flow to increase the robustness sounds meaningful to me, and as far as I can tell the math behind it looks convincing (though I did not check all details). \n\nThe paper does not describe the context in which it exists (as in there is no related work section, and what related work is mentioned is about properties of manifelds). It is not clear to me what the method has to do with the topics of ICLR, there is no mention of which papers use Ricci flow for anything, and the results are also not clear in how this method is used on CIFAR exactly. This is my major critisism of this paper, as someone who is not an expert in this topic, it was impossible for me to find a starting point to judge the contributions and novelty. \n\nThe main paper does not present any experiments, and the experiments in the appendix on CIFAR are neither properly explained, nor was I able to interpret them (also due to the lack of description of the experiment setup). The authors only claim that \"Actually, the change in metric is not so obvious\" which I must interpret as it did not really work. I see that this is a very basic theoretical work, and I would have been fine with experiments on artifical and simplistic data but the complete lack of any positive results does not raise faith. ", "summary_of_the_paper": "The paper proposes a linerization method for ricci flow that is used for manifold surgery which makes the process more robust. ", "summary_of_the_review": "First of all, this is not my area of expertise, and the review period was too short to go into all the mathematical details. Therefore, it might as well be possible that this is quite genius and I was just not able to see it. However, the paper does not explain the impact of its contribution or its applications well, does not have a proper related work section, and the results are half hidden in the appendix and their positive aspects not explained. \nI have rated reject because putting contributions into context through the related work (while often annoying and seemingly just taking away space) is an essential part of a paper and skipping over this leaves the paper incomprehensible for many people.\nMy ratings on correctness and novelty are only approximations, and I would not oppose acceptance if the theory has strong supporters in the other reviewers. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "5vIakQ0xxRB", "reviewer": "Reviewer_g1ct", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Ricci flows and the techniques involved in the study of Ricci flows such as surgery are no doubt a very exciting topic. The authors survey existing results on the Ricci flow and the occurrence of singularities. They continue to define nearly Euclidean metrics and prove various properties of these. I have not carefully checked the correctness of the survey and the presented results.\n\nThe sections on divergences, gradient flows and approximations with neural networks are largely unreadable. I did not succeed in understanding what the authors which to demonstrate here and what is the application. What do the authors mean by sentences such as \"we dynamically consider the gradient flow followed with the optimal descent direction on this manifold\" and \"Therefore, this gradient flow is a weak approximation under the manifold micro-surgery.\"? What is the underlying manifold and metric in section 4.3 on approximations with neural networks?\n\nUnfortunately, the point of the last sections is unclear to me. Because of this, it is not clear what is the contribution and point of the paper besides a survey and discussion of some very exciting mathematical topics.", "summary_of_the_paper": "The paper concerns the Ricci flow and surgery to handle singularities occuring during the Ricci flow. The authors propose linearly nearly Euclidean metrics that they prove are stable under the Ricii-DeTurck flow. The authors use this to approximate steepest descent gradient flows in information geometry and state that they obtain a new method for geometric optimization on manifolds.", "summary_of_the_review": "Unfortunately, while I very much appreciate the topic and the underlying mathematics, I find the paper and presentation in its present form to be too unclear that I can recommend acceptance.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: Ricci flow is a central topic in geometric analysis. It has had stunning applications in mathematics, most notably the proof of the Poincare conjecture. The major issue is that it, while it can be used to make a manifold more well-behaved, it frequently develops singularities. The main contribution fo this paper is in introducing linearly nearly Euclidean metrics. They give a proof of convergence in both short and infinite time, under the Ricci-DeTurk flow, and exploit connections to information geometric and mirror descent to develop methods for approximating the gradient flow. The paper is confusingly written (compounded by poor organizational structure and many grammatical mistakes). Perhaps the biggest issue is that it does not have any clear relevance to machine learning. Some sections mention connections to neural networks, but the reviewers found these sections to be indecipherable.
Decision: Reject