Jean-Bernard Chatelain
email: jean-bernard.chatelain@univ-paris1.fr

Kirsten Ralf
email: kirsten.ralf@esce.fr.

Super-inertial interest rate rules are not solutions of Ramsey optimal monetary policy

Keywords: C61, C62, E43, E44, E47, E52, E58 New-Keynesian model, Ramsey optimal policy, Interest rate smoothing, Super-inertial rule, Inertial rule JEL classi…cation numbers: C61, C62, E43, E44, E47, E52, E58 new-Keynesian model, Ramsey optimal policy, interest rate smoothing, super-inertial rule, inertial rule

found that the equilibrium determined by commitment to a super-inertial rule (where the sum of the parameters of lags of interest rate exceed ones and does not depend on the auto-correlation of shocks) corresponds to the unique bounded solution of Ramsey optimal policy for the new-Keynesian model. By contrast, this note demonstrates that commitment to an inertial rule (where the sum of the parameters of lags of interest rate is below one and depends on the auto-correlation of shocks) corresponds to the unique bounded solution.

1 Introduction [START_REF] Giannoni | How forward-looking is optimal monetary policy?[END_REF], henceforth GW, found that Ramsey optimal monetary policy under commitment miminizing an in…nite horizon discounted loss function including the variance of the interest rates subject to the new-Keynesian model has its unique bounded equilibrium solution determined by commitment from date t onwards to a "super-inertial" interest rate rule where the sum of parameters of the lags of interest rate is strictly above one. This super-inertial policy rule does not depend on the auto-correlation of shocks [START_REF] Giannoni | Optimal interest-rate rules and in ‡ation stabilization versus price-level stabilization[END_REF], p.123).

In recent new-Keynesian dynamic stochastic general equilibrium models of research departments of central banks, GW result provides a rationale for computing super-inertial optimal simple rules without setting a restriction on the sum of parameters for lagged funds rate to be below one when using simulation grids [START_REF] Adolfson | Monetary policy trade-o¤s in an estimated open-economy DSGE model[END_REF], [START_REF] Chung | E¤ective Monetary Policy Strategies in New Keynesian Models: A Reexamination[END_REF]). By contrast, positive restrictions and upper limit restrictions are usually set on the parameters for the response of funds rate to in ‡ation and to output gap in simulation grids.

This note demonstrates that the unique bounded equilibrium solution of GW model is determined by commitment from date t onwards to an inertial policy rule, where the sum of parameters of the lags of interest rate is strictly below one and depends on the auto-correlation parameters of shocks.

GW super-inertial policy rule is a necessary …rst order Euler condition but not a su¢ cient condition to the unique bounded solution. It corresponds no only to the unique bounded solution but also to an in…nity of out-of-equilibrium unbounded paths with dynamics for the funds rate which are explosive. For those paths, the discounted variance of funds rate included into the loss function tends to in…nity when the growth rate of funds rate exceeds the households discount factor. Those paths maximizes the Central Bank loss function instead of minimizing it. GW super-inertial policy rule may not be a relevant advice to practitioners in Central Banks and in the IMF.

This super-inertial policy rule is one equation of the Hamiltonian system of Ramsey optimal policy. This Hamiltonian system has a saddle-point equilibrium which satis…es [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] determinacy condition (GW proposition 1). After proposition 1, GW stop half-way in the computation of optimal policy.

Once the determinacy condition is ful…lled (GW proposition 1), [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] and Vaughan (1970) demonstrated that the unique bounded solution and the optimal policy rule are obtained seeking the stable subspace of the Hamiltonian system using the method of undetermined coe¢ cients or, equivalently, solving a matrix Riccati equation which amounts to seek the eigenvectors of stable roots of a Jordan decomposition of the Hamiltonian matrix. This is also demonstrated by [START_REF] Anderson | Mechanics of Forming and Estimating Dynamic Linear Economies[END_REF] Section two details the Hamiltonian system of Ramsey optimal policy and GW superinertial rule. Section three presents its solution. Section four concludes.

The Hamiltonian system of Ramsey optimal policy

GW apply Ramsey optimal policy on the new-Keynesian monetary policy transmission mechanism. The new-Keynesian private sector's four-equations model is written with all variables as log-deviations of an equilibrium. In the representative household's intertemporal substitution consumption Euler equation, current output gap x t is positively correlated with expected output gap and negatively correlated with the real rate of interest, equal to the nominal rate i t minus expected in ‡ation E t t+1 . The intertemporal elasticity of substitution (IES) is a measure of the responsiveness of the growth rate of consumption to the interest rate, usually considered to be smaller than one. It is the inverse or the relative degree of resistance to intertemporal substitution of consumption (RISC, the relative ‡uctuation aversion), which measure the strength of the preference for smoothing consumption over time, usually considered to be larger than one. We keep GW's notation for the IES, although some other papers denote it 1 .

x t = E t x t+1 (i t E t t+1 ) + z x;t where > 0.

An exogenous stationary and predetermined shock z t is auto-regressive of order one (0 < z < 1) with given initial value. Disturbances " z;t are zero-mean, independently and identically normally distributed. z t = z z t 1 + " z;t where " z;t is i.i.d. N 0; s 2

x , z 0 given.

In the new-Keynesian Phillips curve, current in ‡ation is positively correlated with discounted expected in ‡ation and to current output, with a slope parameter .

t = E t t+1 + x t + u t where > 0 and > 0 (3)
An exogenous stationary and predetermined variable cost-push shock u t is autoregressive of order one ( 0 < u < 1) with given initial value. Disturbances " u;t are zero-mean, independently and identically normally distributed.

u t = u u t 1 + " u;t where " u;t is i.i.d. N 0; s 2 , u 0 given. ( 4
)
The policy maker minimizes the expectation of the present value of a discounted quadratic loss function by a discount factor . The relative weight of in ‡ation variance is set to one. The relative weight on the second policy target (output gap) is positive x 0. The relative weight on the volatility of her policy instrument (interest rate variance) is a strictly positive i > 0:

E t +1 X t=0 t 2 t 2 + x x 2 t 2 + i i 2 t 2 (5)
This loss function is subject to the private sector's new-Keynesian four equations model (equations [START_REF] Anderson | Mechanics of Forming and Estimating Dynamic Linear Economies[END_REF][START_REF] Adolfson | Monetary policy trade-o¤s in an estimated open-economy DSGE model[END_REF][START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF][START_REF] Clarida | The science of monetary policy: a new Keynesian perspective[END_REF]). This is a discounted linear quadratic regulator (LQR) augmented by two forcing variables (z t ; u t ) (Hansen and Sargent (2008), section 3.5). A quadratic loss function including a discount factor is minimized subject to a linear dynamic system with boundary conditions. Denoting Lagrangian multipliers x;t for the consumption Euler equation ( 1) and ;t for the new-Keynesian Phillips curve (3), the Lagrangian L is:

L = E 0 +1 X t=0 t 2 t 2 + x x 2 t 2 + i i 2 t 2 + x;t [x t x t+1 + (i t t+1 )] + ;t [ t t t+1 x t ] (6)
The interest rate (policy instrument) is a linear decreasing function of the Lagrange multiplier of the output gap x;t :

@L @i t = 0 ) i i t + x;t = 0 ) x;t = i i t or i t = i
x;t for t = 0; 1; 2; :::

Because in ‡ation and the output gap are forward-looking variables, they are optimally chosen at the initial date t = 1 and at the …nal date according to transversality conditions. The marginal values of the loss function with respect to output gap x;t= 1 and to in ‡ation ;t= 1 are equal to zero at the initial date:

;t= 1 = x;t= 1 = 0 and lim t!+1 ;t = lim t!+1 x;t = 0. (8)
Because the marginal value of the loss function with respect to the output gap at the initial date t = 1 is zero ( x;t= 1 = 0), the initial interest rate is set to zero: i 1 = 0.

The Hamiltonian system includes the Euler consumption equation ( 1), the new-Keynesian Phillips curve (3) and the two …rst order conditions with respect to output gap and to in ‡ation:

@L @x t = 0 ) x x t + x;t 1
x;t 1

;t = 0 for t = 1; 2; :::

@L @ t = 0 ) t 1

x;t 1 + ;t ;t 1 = 0 for t = 1; 2; :::

One considers two observationally equivalent representations of the Hamiltonian system using linear substitution between variables across equations. The most common representation substitutes the policy instruments by the Lagrange multipliers: the interest rate i t in the Euler consumption equation ( 1) is substituted by x;t [START_REF] Hansen | Robustness[END_REF], chapter 3). Then, the Hamiltonian system involves four variables: output gap and in ‡ation (x t ; t ) and their Lagrange multipliers ( x;t ; ;t ) and four equations, with a symplectic Hamiltonian matrix: its transpose is similar to its inverse. This matrix has an equal number of stable eigenvalues and of unstable eigenvalues, which is equal to two in this case. The number of stable eigenvalues is equal to the number of predetermined variables ( x;t ; ;t ) which corresponds to predetermined values of the policy instrument and of its …rst lag (i t ; i t 1 ). The number of unstable eigenvalues is equal to the number of forward-looking variables (x t ; t ). [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] determinacy condition is satis…ed: the number of stable eigenvalue is equal to the number of predetermined variables. [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] acknowledged that the determinacy condition is always satis…ed in the general LQR case and in optimal growth models. In addition, the non-controllable part of the economy has two stable eigenvalues ( z , u ) which are the auto-correlation parameters of two predetermined forcing variables (z t ,u t ), so that Blanchard and Kahn (1980) determinacy condition is satis…ed.

An alternative representation of the Hamiltonian system is to substitute the Lagrange multipliers by the policy instrument. If the number of instruments is smaller than the number of policy targets or the number of their Lagrange multipliers, one includes lags of the policy instrument. Linear substitutions of the Lagrange multipliers ( x;t ; ;t ) by the policy instrument and its lag (i t ; i t 1 ) leads to an equivalent Hamiltonian linear dynamic system with variables (x t ; t ; i t ; i t 1 ).

In the case of an optimal commitment for Ramsey policy that has been in force since at least period t 2 ( x;t 3 = ;t 3 = 0 and i t 3 = 0), GW reasoning is that "one can infer the value of the Lagrange multiplier x;t 1 from the value of i t 1 using Equation [START_REF] Chatelain | A Finite Set of Equilibria for the Indeterminacy of Linear Rational Expectations Models[END_REF], and similarly, the value of x;t 2 from the value of i t 2 : @L @i t = 0 ) x;t = i i t for t = 2; 1; 0; 1; :::

Then, substituting theses values into Equation ( 9) for period t 1, one can also infer the value of ;t 1 from the value of x t 1 . One can, of course, similarly solve for the period t Lagrange multipliers as functions of x t ; i t and i t 1 : @L @x t = 0 ) ;t = 1 x x t 1 i i t + 1 i i t 1 for t = 1; 0; 1; :::

Using these expressions to substitute out the Lagrange multipliers in equation ( 10), one obtains a linear relation among the endogenous variables t , x t , x t 1 , i t 1 , i t 1 and i t 2 that must hold for any period t 0, for an optimal commitment for Ramsey policy that has been in force since at least period 2:

i t = 1 + i t 1 + 1 i t 1 + i t + 4 x i
x t 4 for t = 0; 1; 2; :::

GW label "super-inertial" this representation (13) of the …rst order condition of in- ‡ation [START_REF] Chatelain | A Simple Algorithm for Solving Ramsey Optimal Policy with Exogenous Forcing Variables[END_REF]: the implied dynamics of the funds rate are explosive, because the sum of parameters for lagged funds rate exceeds one: 1 + > 1. GW proposition 1 states that commitment to this super-inertial policy rule (13) is a necessary condition which implies determinacy. They prove that their representation of the Hamiltonian system including the super-inertial policy rule (13) has a number of stable eigenvalues equal to the number of predetermined variables (GW appendix one). As mentioned above, this is usually demonstrated with the previous representation of the Hamiltonian system with variables (x t ; t ) and their Lagrange multipliers ( x;t ; ;t ) in the general LQR case.

This super-inertial rule ( 13) corresponds to the unique bounded solution and to an in…nity of out-of-equilibrium unbounded paths with dynamics for the funds rate which are explosive. At this stage, GW proved that their super-inertial policy (the …rst order condition on in ‡ation) is a necessary condition to this unique bounded solution but not a su¢ cient condition ruling out unbounded paths. GW, p.1430, just below proposition 1, state without proof that their super-inertial policy is ruling out unbounded paths: "The equilibrium determined by commitment to this rule (13) from date t=0 onwards corresponds to the unique bounded solution to equations (1,2,3,4,7,9,10) when the initial condition ((8): x; 1 = ; 1 = 0 and i t 1 = 0) is replaced by the values of x; 1 and ; 1 that would be inferred from the historical values of x 1 ; i 1 , and i 2 under the reasoning described above."

The change of initial conditions corresponds to the case of an optimal commitment for Ramsey policy that has been in force since at least period t 2 ( x;t 3 = ;t 3 = 0 and i t 3 = 0) instead of instead of period t = 0 ( x; 1 = ; 1 = 0 and i t 1 = 0). The super-inertial rule ( 13) is then valid after two periods starting at t 0. The "reasoning described above" is the substitution of the Lagrange multipliers by the interest rate and its two lags in the …rst order equation of in ‡ation. These practical informations do not rule out the unbounded paths predicted by the super-inertial rule [START_REF] Chatelain | The Indeterminacy of Determinacy with Fiscal, Macro-prudential or Taylor Rules[END_REF].

Once the determinacy condition is ful…lled (GW proposition 1), [START_REF] Blanchard | The solution of linear di¤erence models under rational expectations[END_REF] and Vaughan (1970) demonstrated that the unique bounded solution and the optimal policy rule are obtained seeking the stable subspace of the Hamiltonian system using the method of undetermined coe¢ cients or, equivalently, solving a matrix Riccati equation which amounts to seek the eigenvectors of stable roots of a Jordan decomposition of the Hamiltonian matrix.

3 Optimal inertial policy rules 3.1 Clarida Gali and Gertler (1999) case.

Proposition 1 Clarida, [START_REF] Clarida | The science of monetary policy: a new Keynesian perspective[END_REF]. The equilibrium determined by commitment from date t=0 onwards to an inertial output gap rule, where the auto-correlation of the output gap is below one, corresponds to the unique bounded solution of Ramsey optimal policy for the new-Keynesian Phillips curve (3) and the cost-push shock (4)

Proof. In Clarida [START_REF] Clarida | The science of monetary policy: a new Keynesian perspective[END_REF] and Gali (2015, chapter 5) optimization of Ramsey optimal policy, the intertemporal substitution equation of consumption (1) and its auto-correlated shock equation ( 2) are not taken into account, so that there is no interest rate smoothing: i t = 0, i = 0, x = 0 and z t = 0. The output gap x t plays the role of the policy instrument in a "targeting rule": x > 0. In ‡ation is the only policy target. The Hamiltonian system includes the new-Keynesian Phillips curve and the noncontrollable dynamics of the cost-push shock. The marginal equation with respect to the single policy instrument (output gap) implies that it is a linear increasing function of the Lagrange multiplier of in ‡ation: @L @x t = 0 ) x x t ;t = 0 for t = 0; 1; 2; :::

) x t = x ;t or ;t = x x t (14)
The initial natural boundary initial sets to zero the initial value of the policy instrument at the initial date 1:

; 1 = 0 ) x 1 = x ; 1 = 0
The …rst order condition on in ‡ation is: @L @ t = 0 ) t + ;t ;t 1 = 0 for t = 0; 1; 2; :::

One substitutes the Lagrange multiplier of in ‡ation ;t by the policy instrument x t : @L @ t = 0 ) x t = x t 1

x t for t = 0; 1; 2; 3:::

This equation corresponds to GW super-inertial policy rule. The policy instrument (the output gap) has unit root non-stationary dynamics. When time goes to in…nity, this non-stationary policy rule allows an in…nity of unbounded out-of-equilibrium paths with an in…nite variance of the policy instrument and a unique bounded stationary solution. Clarida, [START_REF] Clarida | The science of monetary policy: a new Keynesian perspective[END_REF] and [START_REF] Gali | Monetary Policy, In ‡ation and the Business Cycle[END_REF] seek the unique stable eigenvalue of the Hamiltonian system:

0 < = 1 2 1 + 1 + 2 x s 1 4 1 + 1 + 2 x 2
1 < 1

The Hamiltonian system satis…es Blanchard and Kahn (1980) determinacy condition (as GW proposition one): two stable eigenvalues and u for two predetermined variables ;t and u t . By contrast to GW, Clarida, Gali and Gertler (1999) and Gali (2015) go one step further. They …nd the optimal stationary inertial policy rule using the method of undetermined coe¢ cients:

x t = x t 1 1 u x
u t for t = 1; 2; ::: and x 0 = 1

u x u 0 ( 16
)
The representation of the inertial optimal policy rule for the output gap has an autoregressive component which is strictly lower than one. It also depends on the auto-correlation of the cost push shock u and on the shock u t . One can …nd another representation of this inertial policy rule after substitution of the cost-push shock u t by in ‡ation t (Chatelain and Ralf 2017a)).

Inertial optimal policy rule

Proposition 2 The equilibrium determined by commitment from date t=0 onwards to an inertial interest rate rule, where the sum of the parameters of lags of interest rate is below one and depends on the auto-correlation of shocks, corresponds to the unique bounded solution of Ramsey optimal policy for the new-Keynesian model (equations 1,2,3,4).

Proof.

i t = F x x t + F t + F z z t + F u u t (17)
There is no closed form solution for these parameters, only numerical solutions. Doing linear substitutions within the system of equations including the policy rule [START_REF] Giannoni | How forward-looking is optimal monetary policy?[END_REF] and the private sector equations (1,2,3,4) leads to an equivalent system with observational equivalence. Applying the operator p

(L) = (1 u L) (1 z L)
, where L is the lag operator, on [START_REF] Hansen | Robustness[END_REF] optimal rule (17) obtains an observationally equivalent representation [START_REF] Hansen | Robustness[END_REF] of the optimal policy rule [START_REF] Giannoni | How forward-looking is optimal monetary policy?[END_REF], where the two auto-regressive forcing variables z t and u t are substituted by the two …rst lags of the policy instrument: i t 1 and i t 2 : p (L) i t = F x p (L) x t + F p (L) t + F z (1 u L) " z;t + F u (1 z L) " u;t [START_REF] Hansen | Robustness[END_REF] The sum of the parameters of the two lags of the dependent variables is the sum S less the product P of the two auto-correlation coe¢ cients of shocks:

i t = ( z + u ) i t 1 ( z u ) i t 2 + :::

Because the two auto-correlation coe¢ cients are between zero and one and are the roots of the polynomial p, p(1) > 0. Hence, the sum of the two parameters of the lags of the interest rate is strictly below one:

p(1) = 1 S + P > 0 ) S P < 1

Hence, the observationally equivalent representation (18) of the optimal policy rule ( 17) is inertial. It depends on the auto-correlation of shocks. [START_REF] Simon | Dynamic programming under uncertainty with a quadratic criterion function[END_REF] proved that the optimal rule parameters of the LQR are independent of the disturbances of the shocks (" z;t ; " u;t ), in particular of their variance, and of the initial conditions of private sectors variables (x 0 ; 0 ; z 0 ; u 0 ). Using the optimal initial condition (x 0 ; 0 ) or ad hoc initial jumps for forward-looking variables (x 0 ; 0 ) such as timeless perspective or predetermined values for (x 0 ; 0 ) do not change the optimal policy rule parameters. One …nds the optimal initial conditions (x 0 ; 0 ) of Ramsey optimal policy using Ljungqvist and Sargent (2012) chapter 19 and Chatelain and Ralf (2017b) algorithm.

section 5 ,

 5 by Hansen and Sargent (2008) section 3.5, by Ljungqvist and Sargent (2012) chapter 19. Closer to GW model, this computation is done by Clarida, Gali, Gertler (1999) and by Gali (2015) chapter 5 on a smaller Hamiltonian system included into GW Hamiltonian system.

  [START_REF] Hansen | Robustness[END_REF] section 3.5 and Anderson et al. (1996) solution of this discounted LQR augmented with forcing variables (z t ; u t ) solves a discrete algebraic Riccati equation for rule parameters (F x ; F ) which depend on parameters ( ; ; ) and preferences ( x ; i ) but do not depend on the auto-correlation of shocks and solves a Sylvester equation for rule parameters (F z ; F u ) which depends on the auto-correlation of shocks (

z ; u ):

Conclusion

Ramsey optimal policy for the new-Keynesian model implies a stationary optimal policy rule which is not super-inertial and which depends on the auto-correlation of shocks.


Thibault Groueix

Matthew Fisher

Vladimir G Kim

Bryan C Russell

Mathieu Aubry

3D-CODED : 3D Correspondences by Deep Deformation

Keywords: 3D deep learning, computational geometry, shape matching (a) Input Shape

We present a new deep learning approach for matching deformable shapes by introducing Shape Deformation Networks which jointly encode 3D shapes and correspondences. This is achieved by factoring the surface representation into (i) a template, that parameterizes the surface, and (ii) a learnt global feature vector that parameterizes the transformation of the template into the input surface. By predicting this feature for a new shape, we implicitly predict correspondences between this shape and the template. We show that these correspondences can be improved by an additional step which improves the shape feature by minimizing the Chamfer distance between the input and transformed template. We demonstrate that our simple approach improves on stateof-the-art results on the difficult FAUST-inter challenge, with an average correspondence error of 2.88cm. We show, on the TOSCA dataset, that our method is robust to many types of perturbations, and generalizes to non-human shapes. This robustness allows it to perform well on real unclean, meshes from the the SCAPE dataset.

Fig. 1: Our approach predicts shape correspondences by learning a consistent mesh parameterization with a shared template. Colors show correspondences.

Introduction

There is a growing demand for techniques that make use of the large amount of 3D content generated by modern sensor technology. An essential task is to establish reliable 3D shape correspondences between scans from raw sensor data or between scans and a template 3D shape. This process is challenging due to low sensor resolution and high sensor noise, especially for articulated shapes, such as humans or animals, that exhibit significant non-rigid deformations and shape variations.

Traditional approaches to estimating shape correspondences for articulated objects typically rely on intrinsic surface analysis either optimizing for an isometric map or leveraging intrinsic point descriptors [START_REF] Sun | A concise and provably informative multiscale signature-based on heat diffusion[END_REF]. To improve correspondence quality, these methods have been extended to take advantage of category-specific data priors [START_REF] Boscaini | Anisotropic diffusion descriptors[END_REF]. Effective human-specific templates and registration techniques have been developed over the last decade [START_REF] Zuffi | The stitched puppet: A graphical model of 3d human shape and pose[END_REF], but these methods require significant effort and domain-specific knowledge to design the parametric deformable template, create an objective function that ensures alignment of salient regions and is not prone to being stuck in local minima, and develop an optimization strategy that effectively combines a global search for a good heuristic initialization and a local refinement procedure.

In this work, we propose Shape Deformation Networks, a comprehensive, all-in-one solution to template-driven shape matching. A Shape Deformation Network learns to deform a template shape to align with an input observed shape. Given two input shapes, we align the template to both inputs and obtain the final map between the inputs by reading off the correspondences from the template.

We train our Shape Deformation Network as part of an encoder-decoder architecture, which jointly learns an encoder network that takes a target shape as input and generates a global feature representation, and a decoder Shape Deformation Network that takes as input the global feature and deform the template into the target shape. At test time, we improve our template-input shape alignment by optimizing locally the Chamfer distance between target and generated shape over the global feature representation which is passed in as input to the Shape Deformation Network. Critical to the success of our Shape Deformation Network is the ability to learn to deform a template shape to targets with varied appearances and articulation. We achieve this ability by training our network on a very large corpus of shapes.

In contrast to previous work [START_REF] Zuffi | The stitched puppet: A graphical model of 3d human shape and pose[END_REF], our method does not require a manually designed deformable template; the deformation parameters and degrees of freedom are implicitly learned by the encoder. Furthermore, while our network can take advantage of known correspondences between the template and the example shapes, which are typically available when they have been generated using some parametric model [START_REF] Bogo | FAUST: Dataset and evaluation for 3D mesh registration[END_REF][START_REF] Varol | Learning from synthetic humans[END_REF], we show it can also be trained without correspondence supervision. This ability allows the network to learn from a large collection of shapes lacking explicit correspondences.

We demonstrate that with sufficient training data this simple approach achieves state-of-the-art results and outperforms techniques that require complex multiterm objective functions instead of the simple reconstruction loss used by our method.

Related work

Registration of non-rigid geometries with pose and shape variations is a long standing problem with extensive prior work. We first provide a brief overview of generic correspondence techniques. We then focus on category specific and template matching methods developed for human bodies, which are more closely related to our approach. Finally, we present an overview of deep learning approaches that have been developed for shape matching and more generally for working with 3D data.

Generic shape matching. To estimate correspondence between articulated objects, it is common to assume that their intrinsic structure (e.g., geodesic distances) remains relatively consistent across all poses [START_REF] Mémoli | A theoretical and computational framework for isometry invariant recognition of point cloud data[END_REF]. Finding point-to-point correspondences that minimize metric distortion is a non-convex optimization problem, referred to as generalized multi-dimensional scaling [START_REF] Bronstein | Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching[END_REF]. This optimization is typically sensitive to an initial guess [START_REF] Bronstein | Efficient computation of isometryinvariant distances between surfaces[END_REF], and thus existing techniques rely on local feature point descriptors such as HKS [START_REF] Sun | A concise and provably informative multiscale signature-based on heat diffusion[END_REF] and WKS [START_REF] Aubry | The wave kernel signature: A quantum mechanical approach to shape analysis[END_REF], and use hierarchical optimization strategies [START_REF] Sahillioglu | Coarse-to-fine combinatorial matching for dense isometric shape correspondence[END_REF][START_REF] Raviv | Hierarchical framework for shape correspondence[END_REF]. Some relaxations of this problem have been proposed such as: formulating it as Markov random field and using linear programming relaxation [START_REF] Chen | Robust nonrigid registration by convex optimization[END_REF], optimizing for soft correspondence [START_REF] Solomon | Soft maps between surfaces[END_REF][START_REF] Kim | Exploring Collections of 3D Models using Fuzzy Correspondences[END_REF][START_REF] Solomon | Entropic metric alignment for correspondence problems[END_REF], restricting correspondence space to conformal maps [START_REF] Lipman | Mobius voting for surface correspondence[END_REF][START_REF] Kim | Blended Intrinsic Maps[END_REF], heat kernel maps [START_REF] Ovsjanikov | One point isometric matching with the heat kernel[END_REF], and aligning functional bases [START_REF] Ovsjanikov | Functional maps: A flexible representation of maps between shapes[END_REF].

While these techniques are powerful generic tools, some common categories, such as humans, can benefit from a plethora of existing data [START_REF] Bogo | FAUST: Dataset and evaluation for 3D mesh registration[END_REF] to leverage stronger class-specific priors.

Template-based shape matching. A natural way to leverage class-specific knowledge is through the explicit use of a shape model. While such templatebased techniques provide the best correspondence results they require a careful parameterization of the template, which took more than a decade of research to reach the current level of maturity [START_REF] Allen | Articulated body deformation from range scan data[END_REF][START_REF] Allen | The space of human body shapes: reconstruction and parameterization from range scans[END_REF][START_REF] Allen | Learning a correlated model of identity and pose-dependent body shape variation for real-time synthesis[END_REF][START_REF] Loper | Smpl: A skinned multi-person linear model[END_REF][START_REF] Zuffi | The stitched puppet: A graphical model of 3d human shape and pose[END_REF]. For all of these techniques, fitting this representation to an input 3D shape requires also designing an objective function that is typically non-convex and involves multiple terms to guide the optimization to the right global minima. In contrast, our method only relies on a single template 3D mesh and surface reconstruction loss. It leverages a neural network to learn how to parameterize the human body while optimizing for the best reconstruction.

Deep learning for shape matching. Another way to leverage priors and training data is to learn better point-wise shape descriptors using human models with ground truth correspondence. Several neural network based methods have recently been developed to this end to analyze meshes [START_REF] Rodola | Dense nonrigid shape correspondence using random forests[END_REF][START_REF] Masci | Geodesic convolutional neural networks on riemannian manifolds[END_REF][START_REF] Boscaini | Learning shape correspondence with anisotropic convolutional neural networks[END_REF][START_REF] Monti | Geometric deep learning on graphs and manifolds using mixture model cnns[END_REF] or depth maps [START_REF] Wei | Dense human body correspondences using convolutional networks[END_REF]. One can further improve these results by leveraging global context, for example, by estimating an inter-surface functional map [START_REF] Litany | Deep functional maps: Structured prediction for dense shape correspondence[END_REF]. These methods still rely on hand-crafted point-wise descriptors [START_REF] Tombari | Unique signatures of histograms for local surface description[END_REF] as input and use neural networks to improve results. The resulting functional maps only align basis functions and additional optimization is required to extract consistent point-to-point correspondences [START_REF] Ovsjanikov | Functional maps: A flexible representation of maps between shapes[END_REF]. One would also need to optimize for template deformation to use these matching techniques for surface reconstruction. In contrast our method does not rely on hand-crafted features (it only takes point coordinates as input) and implicitly learns a human body representation. It also directly outputs a template deformation.

Deep Learning for 3D data. Following the success of deep learning approaches for image analysis, many techniques have been developed for processing 3D data, going beyond local descriptor learning to improve classification, segmentation, and reconstruction tasks. Existing networks operate on various shape representations, such as volumetric grids [START_REF] Girdhar | Learning a predictable and generative vector representation for objects[END_REF][START_REF] Wu | 3d shapenets: A deep representation for volumetric shapes[END_REF], point clouds [START_REF] Qi | PointNet: Deep learning on point sets for 3D classification and segmentation[END_REF][START_REF] Qi | PointNet++: Deep hierarchical feature learning on point sets in a metric space[END_REF][START_REF] Fan | A point set generation network for 3D object reconstruction from a single image[END_REF], geometry images [START_REF] Sinha | Deep learning 3d shape surfaces using geometry images[END_REF][START_REF] Sinha | Surfnet: Generating 3d shape surfaces using deep residual networks[END_REF], seamlessly parameterized surfaces [START_REF] Maron | Convolutional neural networks on surfaces via seamless toric covers[END_REF], by aligning a shape to a grid via distance-preserving maps [START_REF] Ezuz | Gwcnn: A metric alignment layer for deep shape analysis[END_REF], by folding a surface [START_REF] Yang | Foldingnet: Point cloud auto-encoder via deep grid deformation[END_REF] or by predicting chart representations [START_REF] Groueix | AtlasNet: A Papier-Mâché Approach to Learning 3D Surface Generation[END_REF]. We build on these works in several ways. First, we process the point clouds representing the input shapes using an architecture similar to [START_REF] Qi | PointNet: Deep learning on point sets for 3D classification and segmentation[END_REF]. Second, similar to [START_REF] Sinha | Surfnet: Generating 3d shape surfaces using deep residual networks[END_REF], we learn a surface representation. However, we do not explicitly encode correspondences in the output of a convolution network, but implicitly learn them by optimizing for parameters of the generation network as we optimize for reconstruction losss.

Method

Our goal is, given a reference shape S r and a target shape S t , to return a set of point correspondences C between the shapes. We do so using two key ideas. First, we learn to predict a transformation between the shapes instead of directly learning the correspondences. This transformation, from 3D to 3D can indeed be represented by a neural network more easily than the association between variable and large number of points. The second idea is to learn transformations only from one template A to any shape. Indeed, the large variety of possible poses of humans makes considering all pairs of possible poses intractable during training. We instead decouple the correspondence problem into finding two sets of correspondences to a common template shape. We can then form our final correspondences between the input shapes via indexing through the template shape. An added benefit is during training we simply need to vary the pose for a single shape and use the known correspondences to the template shape as the supervisory signal. ) by performing a regression step over the latent variable x, minimizing the Chamfer distance between D (A, x) and S. (c) Finally, given two point clouds S r and S t , to match a point q r on S r to a point q t on S t , we look for the nearest neighbor p r of q r in D (A, x r ), which is by design in correspondence with p t ; and look for the nearest neighbor q t of p t on S t . Red indicates what is being optimised.

Our approach has three main steps which are visualized figure 2. First, a feed-forward pass through our encoder network generates an initial global shape descriptor (Section 3.1). Second, we use gradient descent through our decoder Shape Deformation Network to refine this shape descriptor to improve the reconstruction quality (Section 3.2). We can then use the template to match points between any two input shapes (Section 3.3).

Learning 3D shape reconstruction by template deformation

To put an input shape S in correspondence with a template A, our first goal is to design a neural network that will take S as input and predict transformation parameters. We do so by training an encoder-decoder architecture. The encoder E φ defined by its parameters φ takes as input 3D points, and is a simplified version of the network presented in [START_REF] Qi | PointNet: Deep learning on point sets for 3D classification and segmentation[END_REF]. It applies to each input 3D point coordinate a multi-layer perceptron with hidden feature size of 64, 128 and 1024, then maxpooling over the resulting features over all points followed by a linear layer, leading to feature of size 1024 E φ (S). This feature, together with the 3D coordinates of a point on the template p ∈ A, are taken as input to the decoder D θ with parameters θ, which is trained to predict the position q of the corresponding point in the input shape. This decoder Shape Deformation Network is a multi-layer perceptron with hidden layers of size 1024, 512, 254 and 128, followed by a hyperbolic tangent. This architecture maps any points from the template domain to the reconstructed surface. By sampling the template more or less densely, we can generate an arbitrary number of output points by sequentially applying the decoder over sampled template points.

This encoder-decoder architecture is trained end-to-end. We assume that we are given as input a training set of N shapes S (i) N i=1 with each shape having a set of P vertices {q j } P j=1 . We consider two training scenarios: one where the correspondences between the template and the training shapes are known (supervised case) and one where they are unknown (unsupervised case). Supervision is typically available if the training shapes are generated by deforming a parametrized template, but real object scans are typically obtained without correspondences.

Supervised loss. In the supervised case, we assume that for each point q j on a training shape we know the correspondence p j ↔ q j to a point p j ∈ A on the template A. Given these training correspondences, we learn the encoder E φ and decoder D θ by simply optimizing the following reconstruction losses,

L sup (θ, φ) = N i=1 P j=1 |D θ p j ; E φ S (i) -q (i) j | 2 (1)
where the sums are over all P vertices of all N example shapes.

Unsupervised loss. In the case where correspondences between the exemplar shapes and the template are not available, we also optimize the reconstructions, but also regularize the deformations toward isometries. For reconstruction, we use the Chamfer distance L CD between the inputs S i and reconstructed point clouds D θ A; E φ S (i) . For regularization, we use two different terms. The first term L Lap encourages the Laplacian operator defined on the template and the deformed template to be the same (which is the case for isometric deformations of the surface). The second term L edges encourages the ratio between edges length in the template and its deformed version to be close to 1. More details on these different losses are given in supplementary material. The final loss we optimize is:

L unsup = L CD + λ Lap L Lap + λ edges L edges (2)
where λ Lap and λ edges control the influence of regularizations against the data term L CD . They are both set to 5.10 -3 in our experiments.

We optimize the loss using the Adam solver, with a learning rate of 10 -3 for 25 epochs then 10 -4 for 2 epochs, batches of 32 shapes, and 6890 points per shape.

One interesting aspect of our approach is that it learns jointly a parameterization of the input shapes via the decoder and to perdict the parameters E φ (S) for this parameterization via the encoder. However, the predicted parameters E φ (S) for an input shape S are not necessarily optimal, because of the limited power of the encoder. Optimizing these parameters turns out to be important for the final results, and is the focus of the second step of our pipeline.

Optimizing shape reconstruction

We now assume that we are given a shape S as well as learned weights for the encoder E φ and decoder D θ networks. To find correspondences between the template shape and the input shape, we will use a nearest neighbor search to find correspondences between that input shape and its reconstruction. For this step to work, we need the reconstruction to be accurate. The reconstruction given by the parameters E φ (S) is only approximate and can be improved. Since we do not know correspondences between the input and the generated shape, we cannot minimize the loss given in equation [START_REF] Allen | Articulated body deformation from range scan data[END_REF], which requires correspondences. Instead, we minimize with respect to the global feature x the Chamfer distance between the reconstructed shape and the input:

L CD (x; S) = p∈A min q∈S |D θ (p; x) -q| 2 + q∈S min p∈A |D θ (p; x) -q| 2 . (3)
Starting from the parameters predicted by our first step x = E φ (S), we optimize this loss using the Adam solver for 3,000 iterations with learning rate 5 * 10 -4 . Note that the good initialization given by our first step is key since Equation( 3) corresponds to a highly non-convex problem, as shown in Figure 5.

Finding 3D shape correspondences

To recover correspondences between two 3D shapes S r and S t , we first compute the parameters to deform the template to these shapes, x r and x t , using the two steps outlined in section 3.1 and 3.2. Next, given a 3D point q r on the reference shape S r , we first find the point p on the template A such that its transformation with parameters x r , D θ (p; x r ) is closest to q r . Finally we find the 3D point q t on the target shape S t that is the closest to the transformation of p with parameters x t , D θ (p; x t ). Our algorithm is summarized in Algorithm 1 and illustrated in Figure 2.

Results

Datasets

Synthetic training data. To train our algorithm, we require a large set of shapes. We thus rely on synthetic data for training our model.

For human shapes, we use SMPL [START_REF] Bogo | FAUST: Dataset and evaluation for 3D mesh registration[END_REF], a state-of-the-art generative model for synthetic humans. To obtain realistic human body shape and poses from the SMPL model, we sampled 2.10 5 parameters estimated in the SURREAL dataset [START_REF] Varol | Learning from synthetic humans[END_REF]. One limitation of the SURREAL dataset is it does not include any humans bent over. Without adapted training data, our algorithm generalized poorly to these poses. To overcome this limitation, we generated an extension of the dataset. We first manually estimated 7 key-joint parameters (among 23 joints in the SMPL skeletons) to generate bent humans. We then sampled randomly the 7 parameters around these values, and used parameters from the SURREAL dataset for the other pose and body shape parameters. Note that not all meshes generated with this strategy are realistic as shown in figure 3. They however allow us to better cover the space of possible poses, and we added 3 • 10 4 shapes generated with this method to our dataset. Our final dataset thus has 2.3 • 10 5 human meshes with a large variety of realistic poses and body shapes.

For animal shapes, we use the SMAL [START_REF] Zuffi | 3D menagerie: Modeling the 3D shape and pose of animals[END_REF] model, which provides the equivalent of SMPL for several animals. Recent papers estimate model parameters from images, but no large-scale parameter set is yet available. For training we thus generated models from SMAL with random parameters (drawn from a Gaussian distribution of ad-hoc variance 0.2). This approach works for the 5 categories available in SMAL. In SMALR [START_REF] Zuffi | Lions and tigers and bears: Capturing nonrigid, 3D, articulated shape from images[END_REF], Zuffi et al. showed that the SMAL model could be generalized to other animals using only an image dataset as input, demonstrating it on 17 additional categories. Note that since the templates for two animals are in correspondences, our method can be used to get inter-category correspondences for animals. We qualitatively demonstrate this on hippopotamus/horses in the appendix [START_REF] Groueix | Supplementary material[END_REF].

Testing data. We evaluate our algorithm on the FAUST [START_REF] Bogo | FAUST: Dataset and evaluation for 3D mesh registration[END_REF], TOSCA [START_REF] Bronstein | Numerical geometry of non-rigid shapes[END_REF] and SCAPE [START_REF] Anguelov | Scape: shape completion and animation of people[END_REF] datasets.

The FAUST dataset consists of 100 training and 200 testing scans of approximately 170,000 vertices. They may include noise and have holes, typically missing part of the feet. In this paper, we never used the training set, except for a single baseline experiment, and we focus on the test set. Two challenges are available, focusing on intra-and inter-subject correspondences. The error is the average Euclidean distance between the estimated projection and the groundtruth projection. We evaluated our method through the online server and are the best public results on the 'inter' challenge at the time of submission 3 .

The SCAPE [START_REF] Anguelov | Scape: shape completion and animation of people[END_REF] dataset has two sets of 71 meshes : the first set consists of real scans with holes and occlusions and the second set are registered meshes aligned to the first set. The poses are different from both our training dataset and FAUST.

TOSCA is a dataset produced by deforming 3 template meshes (human, dog, and horse). Each mesh is deformed into multiple poses, and might have various additional perturbations such as random holes in the surface, local and global scale variations, noise in vertex positions, varying sampling density, and changes in topology.

Shape normalization. To be processed and reconstructed by our network, the training and testing shapes must be normalized in a similar way. Since the vertical direction is usually known, we used synthetic shapes with approximately the same vertical axis. We also kept a fixed orientation around this vertical axis, and at test time selected the one out of 50 different orientations which leads to the smaller reconstruction error in term of Chamfer distance. Finally, we centered all meshes according to the center of their bounding box and, for the training data only, added a random translation in each direction sampled uniformly between -3cm and 3cm to increase robustness.

Experiments

In this part, we analyze the key components of our pipeline. More results are available in the appendix [START_REF] Groueix | Supplementary material[END_REF].

Results on FAUST. The method presented above leads to the best results to date on the FAUST-inter dataset: 2.878 cm : an improvement of 8% over state of the art, 3.12cm for [START_REF] Zuffi | The stitched puppet: A graphical model of 3d human shape and pose[END_REF] and 4.82cm for [START_REF] Litany | Deep functional maps: Structured prediction for dense shape correspondence[END_REF]. Although it cannot take advantage of the fact that two meshes represent the same person, our method is also the second best performing (average error of 1.99 cm) on FAUST-intra challenge.

(a) SCAPE [START_REF] Anguelov | Scape: shape completion and animation of people[END_REF] (b) TOSCA [START_REF] Bronstein | Numerical geometry of non-rigid shapes[END_REF] (c) TOSCA animals [START_REF] Bronstein | Numerical geometry of non-rigid shapes[END_REF] Fig

Geodesic error % Correspondences

Ours FMNet [START_REF] Litany | Deep functional maps: Structured prediction for dense shape correspondence[END_REF] GCNN [START_REF] Masci | Geodesic convolutional neural networks on riemannian manifolds[END_REF] LSCNN [START_REF] Boscaini | Learning class-specific descriptors for deformable shapes using localized spectral convolutional networks[END_REF] ADD3 [START_REF] Boscaini | Anisotropic diffusion descriptors[END_REF] Results on SCAPE : real and partial data. The SCAPE dataset provides meshes aligned to real scans and includes poses different from our train- . We outperform all methods except for Deep Functional Maps [START_REF] Litany | Deep functional maps: Structured prediction for dense shape correspondence[END_REF]. SCAPE also allows evaluation on real partial scans. Quantitatively, the error on these partial meshes is 4.04cm, similar to the performance on the full meshes.

Qualitative results are shown in Fig 4a.

Results on TOSCA : robustness to perturbations. The TOSCA dataset provides several versions of the same synthetic mesh with different perturbations. We found that our method, still trained only on SMPL or SMAL data, is robust to all perturbations (isometry, noise, shotnoise, holes, micro-holes, topology changes, and sampling), except scale, which can be trivially fixed by normalizing all meshes to have consistent surface area. Reconstruction optimization. Because the nearest neighbors used in the matching step are sensitive to small errors in alignment, the second step of our pipeline which finds the optimal features for reconstruction, is crucial to obtain high quality results. This optimization however converges to a good optimum only if it is initialized with a reasonable reconstruction, as visualized in Figure 5. Since we optimize using Chamfer distance, and not correspondences, we also rely on the fact that the network was trained to generate humans in correspondence and we expect the optimized shape to still be meaningful. Table 1 reports the associated quantitative results on FAUST-inter. We can see that: (i) optimizing the latent feature to minimize the Chamfer distance between input and output provides a strong boost; (ii) using a better (more uniform) sampling of the shapes when training our network provided a better initialization; (iii) using a high resolution sampling of the template (∼200k vertices) for the nearest-neighbor step provide an additional small boost in performance. Adding synthetic data reduce the error by a factor of 3, showing its importance. The difference in performance between the basic synthetic dataset and its augmented version is mostly due to failure on specific poses, as in Figure 3 .

Necessary

REAL shapes, and 2.3 × 10 5 , 10 4 and 10 3 shapes from our augmented dataset.

The quantitative results are reported Table 2 and qualitative results can be seen in Figure 6. The FAUST training set only include 10 different poses and is too small to train our network to generalize. Training on many synthetic shapes from the SURREAL dataset [START_REF] Varol | Learning from synthetic humans[END_REF] helps overcome this generalization problem. However, if the synthetic dataset does not include any pose close to test poses (such as bent-over humans), the method will fail on these poses (4 test pairs of shapes out of 40). Augmenting the dataset as described in section 4.1 overcomes this limitation. As expected the performance decreases with the number of training shapes, respectively to 5.76cm and 4.70cm average error on FAUST-inter.

Unsupervised correspondences. We investigate whether our method could be trained without correspondence supervision. We started by simply using the reconstruction loss described in Equation ( 3). One could indeed expect that an optimal way to deform the template into training shapes would respect correspondences. However, we found that the resulting network did not respect correspondences between the template and the input shape, as visualized figure 7. However, these results improve with adequate regularization such as the one presented in Equarion (2), encouraging regularity of the mapping between the template and the reconstruction. We trained such a network with the same training data as in the supervised case but without any correspondence supervision and obtained a 4.88cm of error on the FAUST-inter data, i.e. similar to Deep Functional Map [START_REF] Litany | Deep functional maps: Structured prediction for dense shape correspondence[END_REF] which had an error of 4.83 cm. This demonstrates that our method can be efficient even without correspondence supervision. Further details on regularization losses are given in the appendix [START_REF] Groueix | Supplementary material[END_REF].

Rotation invariance

We handled rotation invariance by rotating the shape and selecting the orientation for which the reconstruction is optimal. As an alternative, we tried to learn a network directly invariant to rotations around the vertical axis. It turned out the performances were slightly worse on FAUSTinter (3.10cm), but still better than the state of the art. We believe this is due to the limited capacity of the network and should be tried with a larger network. However, interestingly, this rotation invariant network seems to have increased robustness and provided slightly better results on SCAPE. (c,e). Note that without regularization, because of the strong distortion, the meshes appear to barely match to the input, while the point clouds are reasonable. On the other hand surface regularization creates reasonable meshes.

Conclusion

We have demonstrated an encoder-decoder deep network architecture that can generate human shape correspondences competitive with state-of-the-art approaches and that uses only simple reconstruction and correspondence losses. Our key insight is to factor the problem into an encoder network that produces a global shape descriptor, and a decoder Shape Deformation Network that uses this global descriptor to map points on a template back to the original geometry. A straightforward regression step uses gradient descent through the Shape Deformation Network to significantly improve the final correspondence quality.

Supplementary

Choice of template

The template is a critical element for our method. We experimented with three different templates: (i) a "FAUST" template associated with SMPL parameters fitted to a body in a neutral pose in the FAUST training set, (ii) a "zero" template corresponding to the "zero" shape of SMPL, and (iii) a "separated" template in which this "zero" shape is modified to have the legs better separated and the arms higher. In this experiment, the points are not sampled regularly on the surface, and a low resolution template is used. Figure 8 shows the different templates, while table 4 shows quantitative results using the different templates. Interestingly, the best results were obtained with the more "natural" template, selected in the "FAUST" training dataset, rather than with the templates from simple SMPL parameters, where points from different body parts seem easier to separate.

(a) "FAUST" template (b) "Zero" template (c) "Separated" template Fig. 8: Shapes for template study. We evaluate three different template shapes used in our model.

Quantitative results for perturbations on TOSCA

We quantitatively evaluate the robustness of our method to perturbation on the TOSCA dataset. This dataset consists of several versions of the same synthetic mesh with different perturbations, specifically: noise, shotnoise, sampling, scale, template 0 Faust error (cm) "FAUST" template 3.255 "Zero" template 3.385 "Separated" template 3.314 Table 5: Quantitative results for perturbations on TOSCA for the horse category local scale, topology, holes, microholes, and isometry. We experimented on the horse model. In Table 5 we report quantitative results for each perturbation (with a gradual strength from 1 to 5) and show qualitative reconstruction with correspondences suggested by colors for each category with maximum strength in Figure 9. We found that we are robust to all categories of noise under study, except for strong variation in sampling (964 points instead of 19948) Surprisingly, adding noise can enhance the quantitative error.

Cross-category correspondances on animals

SMAL synthetic are in correspondences across categories. Hence the template for two different categories are in correspondence and our approach can be trivially extended to get correspondences for animals from different species. Qualitative evidence of this is show in Figure 10.

Regularization for the unsupervised case

In the unsupervised case of equation 2, if the autoencoder is trained using the Chamfer distance alone, it falls into a bad local minimum with high distorsion of the template to reconstruct the input shape. For example the left foot is propagated on left hand in Figure 7. This distortion is consistent across shapes, so correspondences are still possible, and perform reasonably well with an average error of 8.727cm on the FAUST-inter challenge. However, we expect that by minimizing distorsion in the generated shape, the Shape Deformation Network will learn to map an arm to an arm, and a foot to a foot, which will naturally encourage correspondences. We added two regularization losses to achieve this: an edge loss L edges and a laplacian loss L Lap . Correspondences are suggested by color. Notice the overall robustness to all perturbations, with small errors on the ears, tail or legs.

Edge loss L edges Let (V, E) be the graph of the template and V r the reconstructed vertices.

L edges (V r ) = 1 #E • (i,j)∈E | V r i -V r j V i -V j -1 | (4)
This enforces edges to keep the same length in the template and the generated mesh. We use L edges = 0.005. For instance, if the length of an edge doubles the contribution to the loss is L edges • 1.0 = 0.005 which is equivalent (in terms of contribution to the loss function) to a error of placement of 7.1cm. In other words, in terms on loss for the network, it is equivalent to double an edge's length or to misplace a point by 3.2cm. Laplacian loss L Lap Similar to Kanazawa et. al. [START_REF] Kanazawa | Learning category-specific mesh reconstruction from image collections[END_REF], we use the Laplacian regularization. The Laplacian matrix L is defined as :

L i,j =    d i if i = j -1 if (i, j) ∈ E 0 otherwize (5) [LV ] i = (i,j)∈E V i -V j = d i • (V i - (i,j)∈E V j d i ) (6)
This is an approximation of the following integral as explained in [START_REF] Sorkine | Differential representations for mesh processing[END_REF].

lim γ-→0 1 | γ | v∈γ (v i -v)dl(v) = -H(v i ) • n i (7)
where:

-H(v i ) is the mean curvature n i is the surface normal

We follow [START_REF] Meyer | Discrete differential-geometry operators for triangulated 2-manifolds[END_REF] and use cotangent weights in the Laplacian which have been shown to have better geometric discretization.

[L c V ] i = 1 Ω i i∼j 1 2 (cotα ij + cotβ ij )(V i -V j ) (8)
where :

-Ω i is the size of the Voronoi cell of i α ij and β ij denote the two angles opposite of edge (i, j)

Our Laplacian loss is thus written :

L Lap (V g ) = 1 t • L c • (V template -V r ) (9)
We use λ laplace = 0.005. In practice we notice that using Laplacian regularization constrains the network to keep sound surfaces. It may still suffer from error in symmetry and can still invert right and left, and front and back. Figure 11 illustrates that optimizing an asymmetric Chamfer distance can in some cases, especially when the 3D scans have holes, produce qualitatively better results. However, Table 6 shows that the symmetric version of the Chamfer distance performs better. Investigating how other losses behave, such the Earth-Mover distance loss (also known as Wasserstein loss) behave is left to future work.

Failure cases

Figure 12 shows the two main sources of error our algorithm faces: Table 6: Analysis on the Chamfer distance. We compare the latent feature search with Chamfer Distance against latent feature searches with asymmetric Chamfer distances. On average, the Chamfer distance (symmetric) performs better (no regular sampling on the surface, low-resolution template).

-Nearest-neighbor step in overlapping regions failure: a point is matched with the closest point in Euclidean distance but the match is very far in geodesic distance. This could be addressed by adding some regularity in the matches found by the nearest neighbor step. We leave this to future work. -Failures in reconstruction: in such cases, the initial guess of the autoencoder is just too far away from the input, and the regression step fails. The average in centimeters over each vertex of (a), of the Euclidean distance between its projection and the ground truth, is reported (d). Red vertices have an error higher than 20cm, blue ones lower than 2cm. The largest error are observed in places where the Euclidean distance is small, while the geodesic distance is high, such as touching skin (zoom in on the leg). In such region, the nearest neighbors step is match a vertex in mesh A in a distant (in terms of geodesic distance) vertex in mesh A's reconstruction. High error can also come from a bad reconstruction, such as the head of the second example.

  Network training (b) Local optimization of feature x (c) Correspondences

Fig. 2 :

 2 Fig. 2: Method overview. (a) A feed-forward pass in our autoencoder encodes input point cloud S to latent code E (S) and reconstruct S using E (S) to deform the template A. (b) We refine the reconstruction D (A, E (S)) by performing a regression step over the latent variable x, minimizing the Chamfer distance between D (A, x) and S.(c) Finally, given two point clouds S r and S t , to match a point q r on S r to a point q t on S t , we look for the nearest neighbor p r of q r in D (A, x r ), which is by design in correspondence with p t ; and look for the nearest neighbor q t of p t on S t . Red indicates what is being optimised.

Algorithm 1 : 9 CFig. 3 :

 193 Fig. 3: Examples of the different datasets used in the paper.

. 4 :

 4 Fig. 4: Other datasets. Left images show the input, right images the reconstruction with colors showing correspondences. Our method works with real incomplete scans (a), strong synthetic perturbations (b), and on non-human shapes (c).

Fig. 5 :

 5 Fig. 5: Reconstruction optimization. The quality of the initialization (i.e. the first step of our algorithm) is crucial for the deformation optimization. For a given target shape (a) and for different initializations (left of (b), (c) and (d)) the figure shows the results of the optimization. If the initialization is random (b) or incorrect (c), the optimization converges to bad local minima. With a reasonable initialization (d) it converges to a shape very close to the target ((d), right).

Fig. 6 :

 6 Fig. 6: Importance of the training data. For a given target shape (a) reconstructed shapes when the network is trained on FAUST training set (b) and on our augmented synthetic training set (c), before (left) and after (right) the optimization step.

  amount of training data. Training on a large and representative dataset is also crucial for our method. To analyze the effect of training data, we ran our method without re-sampling FAUST points regularly and with a low resolution template for different training sets: FAUST training set, 2 × 10 5 SURaugmented synthetic data, 2.3 × 10 5 shapes 3.26

Fig. 7 :

 7 Fig.7: Unsupervised correspondences. We visualize for different inputs (a), the point clouds (P.C.) predicted by our approach (b,d) and the corresponding meshes (c,e). Note that without regularization, because of the strong distortion, the meshes appear to barely match to the input, while the point clouds are reasonable. On the other hand surface regularization creates reasonable meshes.

Fig. 9 :

 9 Fig. 9: Robustness to perturbations on TOSCA for the horse category. Correspondences are suggested by color. Notice the overall robustness to all perturbations, with small errors on the ears, tail or legs.

Fig. 10 :

 10 Fig. 10: Inter-class correspondences on animals. Correspondences are suggested by color.

6. 5 Fig. 11 :

 511 Fig.11: Asymmetric Chamfer loss in reconstruction optimization. Given an input scan, with holes (a), our network outputs a reconstruction result (b), that can be improved by an optimization step. When the scan has holes, it is better to only consider a loss where the scan attracts the reconstruction (d), rather than using a loss where reconstruction attracts the scan (c), or the Chamfer distance where they attract each other (e).

Fig. 12 :

 12 Fig.12: Error visualization Given the input mesh (a), our autoencoder makes an initial reconstruction (b), optimized by a regression step (c). The average in centimeters over each vertex of (a), of the Euclidean distance between its projection and the ground truth, is reported (d). Red vertices have an error higher than 20cm, blue ones lower than 2cm. The largest error are observed in places where the Euclidean distance is small, while the geodesic distance is high, such as touching skin (zoom in on the leg). In such region, the nearest neighbors step is match a vertex in mesh A in a distant (in terms of geodesic distance) vertex in mesh A's reconstruction. High error can also come from a bad reconstruction, such as the head of the second example.

Table 1 :

 1 Examples of representative qualitative results are shown Fig 4b and quantitative results are reported in appendix [19]. Importance

	Method

of the reconstruction optimization step. Optimizing the latent feature is key to our results. Regular point sampling for training and high resolution for the nearest neighbor step provide an additional boost.

Table 2 :

 2 FAUST-inter results when training on different datasets.

Table 3 :

 3 Results with and without supervised correspondences. Adding regularization helps the network find a better local minimum in terms of correspondences.

Table 4 :

 4 Comparison of different template shapes. We compare different choices for the template shape shown in Figure8. Notice that the neutral "FAUST" template performs best out of the three tested shapes.

	Perturbation Error (cm) Perturbation Error (cm) Perturbation Error (cm)
	1 4.58	1 4.73	1 4.71
	Noise	Scale	Holes

http://faust.is.tue.mpg.de/challenge/Inter-subject_challenge

Acknowledgments. This work was partly supported by ANR project EnHerit ANR-17-CE23-0008, Labex Bézout, and gifts from Adobe to École des Ponts. We thank Gül Varol, Angjoo Kanazawa, and Renaud Marlet for fruitful discussions.