Baptiste Huguet

POINCAR É INEQUALITIES AND INTEGRATED CURVATURE-DIMENSION CRITERION FOR GENERALISED CAUCHY AND CONVEX MEASURES

Keywords: Curvature-dimension criterion, generalised Cauchy measures, heavy tails, Poincaré inequality. 1

We obtain new sharp weighted Poincaré inequalities on Riemannian manifolds for a general class of measures. When specialised to generalised Cauchy measures, this gives a unified and simple proof of the weighted Poincaré inequality for the whole range of parameters, with the optimal spectral gap, the error term and the extremal functions.

Introduction

The family of generalised Cauchy distributions is a family of probability measures defined on R n by dµ β ∼ (1 + |x| 2 ) -β dx, for β > n/2. This family plays a significant role in a variety of mathematics issues. For example, it acts as the heat kernel for fast diffusions. In this context, it is known as the Barenblatt profile. [START_REF] Sergey | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF] pointed out that generalised Cauchy measures approach the Gaussian measure when β diverges, after rescaling. This prompted them to investigate the properties of these measures in terms of functional inequalities, specifically Poincaré inequalities. Due to heavy tails, these measures cannot satisfy the classical Poincaré inequality. Nevertheless, a weighted version of it has been proved, with weight ω(x) = 1 + |x| 2 . For all n ≥ 1 and all β ≥ n (or β > 1 if n = 1), we have 2(β -1)

1 + 2 β-1 + 2 β-1 2 Var µ β (f ) ≤ R n |df | 2 (1 + |x| 2 ) dµ β ,
for all smooth bounded functions f . Their result is sufficiently sharp to recover the classical Poincaré inequality for Gaussian measures in R n , by rescaling. However, it does not cover the entire range n/2 < β and the constant is not optimal.

Poincaré inequalities, whether classical or weighted, are equivalent to spectral properties of a suitable operator. In the case of generalised Cauchy measures, this operator is Lf = ω∆f -2(β -1)⟨df, x⟩, defined on smooth compactly supported functions f ∈ C ∞ c (R n ) and extended to its domain D(L) ∈ L 2 (µ β ). The spectrum of this operator was studied by [START_REF] Denzler | Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology[END_REF] from the perspective of fast diffusion equations, providing a thorough description of its bottom, its eigenvalues and associated eigenfunctions. Their proof is based on a decomposition along spherical harmonics. This description allows [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF] to prove weighted Poincaré inequalities with optimal constant. However, this approach is highly specific to generalised Cauchy measures.

These articles paved the way to many projects on weighted Poincaré inequalities. On the one hand, these works tackle the question towards more general measures. The article [START_REF] Sergey | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF] has studied the general broad framework of convex measures. These measures are defined using a Brunn-Minkowski-type inequality. In R n , they can be characterised as probability measures whose density is proportional to ω -β for a positive convex function ω defined on some convex set and β ≥ n. This is the context in which most of the work has been carried out. Their aim is to establish weighted Poincaré inequalities with sharp control over the optimal constant. On the other hand, they question the relevance of probabilistic arguments to spectral theory, in particular, the relevance of Markov semigroups and curvature-dimension theory. These works often lead to more general functional inequalities, such as logarithmic-Sobolev or other Φ-entropies.

Sharp weighted Poincaré inequalities were proved by [START_REF] Van Hoang Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF], for convex measures, with an additional assumption of strong convexity, that is dµ ∼ ω -β dx on R n , where ω is smooth and Hess(ω) uniformly lower bounded. His proof is inspired by Hörmander's L 2 -method. His result depends on Hess(ω) explicitly, but is only valid for β ≥ n + 1. In the case of generalised Cauchy measures, his result is optimal and associated with an eigenfunction.

The case of generalised Cauchy measures was completed by [START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF] in dimension one using the intertwining approach. A milestone was reached by [START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF] for the n-dimensional case. They obtained a general spectral gap comparison between a radial measure and its one-dimensional radial part. When applied to generalised Cauchy measures, they prove weighted Poincaré inequalities for the whole range β > n/2. Nonetheless, they cannot reach the optimal constant over a small range of β.

In addition to these major articles, several papers recovered partial results and enlightened new perspectives. Among them, [START_REF] Saumard | Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one[END_REF] used Stein kernels to recover the optimal bound in dimension 1 for β > 3/2. In [START_REF] Arnaudon | Intertwinings and generalized Brascamp-Lieb inequalities[END_REF], a new bound was obtained for β > n, using metric twisting and intertwining. Unfortunately, their bound is not optimal. In [START_REF] Huguet | Intertwining relations for diffusions in manifolds and applications to functional inequalities[END_REF], the previous method was generalised to Riemannian manifolds. Interpreting the weight as a metric, it can be applied to generalised Cauchy measures in dimension n = 2. Then it provides a better bound, even optimal for 1 < β ≤ 1+ √ 2. The optimal bound was recovered for β ≥ n+1 in a Riemannian setting by [START_REF] Bakry | Sharp Beckner-type inequalities for Cauchy and spherical distributions[END_REF], using harmonicness and curvature-dimension arguments. Their result is a geometric generalisation of [START_REF] Van Hoang Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF]. This Riemannian result was recovered by [START_REF] Gentil | A family of Beckner inequalities under various curvature-dimension conditions[END_REF], using the curvature-dimension criterion on the Laplace operator only, with applications to Poincaré and Beckner inequalities.

Each one of these articles uses different perspectives and different techniques. Together, they do not allow to properly understand the link between curvature-dimension criterion and Poincaré inequality, especially for generalised Cauchy measures. The goal of the present work is to obtain a simple, self-contained and intrinsic proof of sharp Poincaré inequalities using curvature-dimension arguments only. A novelty in our work is to find out a result beyond the condition β ≥ n + 1 and even beyond convex measure condition β ≥ n. When specialised to generalised Cauchy measures, we recover the spectral gap λ 1 (-L), that is to say the optimal constant such that

λ 1 (-L) Var µ β (f ) ≤ R n |df | 2 (1 + |x| 2 ) dµ β ,
for the whole range of parameters. Our guiding principle is to recover the following result from [START_REF] Denzler | Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology[END_REF].

Theorem 1.1. For n = 1, we have

λ 1 (-L) = (β -1/2) 2 if 1/2 < β ≤ 3/2 2(β -1) if 3/2 ≤ β.
For n ≥ 2, we have

λ 1 (-L) =    (β -n/2) 2 if n/2 < β ≤ n/2 + 2 4(β -n/2 -1) if n/2 + 2 ≤ β ≤ n + 1 2(β -1) if n + 1 ≤ β.
By doing so, we obtain some general Poincaré inequalities for a large class of measures. Also, we obtain an estimation of the error term. In the case of generalised Cauchy measures, this estimation is exact and sufficiently explicit to characterise the associated extremal functions, when they exist. Our method is inspired by the integrated curvature-dimension criterion. This criterion has been firstly introduced by [START_REF] Obata | Certain conditions for a Riemannian manifold to be isometric with a sphere[END_REF], for the Laplace operator on compact Riemannian manifolds. The generalisation of this criterion thank to the carré du champ formalism was introduced by [START_REF] Bakry | Diffusions hypercontractives[END_REF]. and developed by [START_REF] Ledoux | On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions[END_REF]. An improvement of this criterion was proposed by [START_REF] Chafaï | On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures[END_REF], with an explicit expression of the error term, or deficit, in Poincaré inequality. This improvement allows to characterise extremal functions.

Let us describe the structure of this article. In Section 2 we present the integrated curvaturedimension criterion and its link to weighted Poincaré inequalities. In Section 3, we obtain some upper bound of the spectral gap for generalised Cauchy measures by studying its eigenfunctions. In Section 4, we prove weighted Poincaré inequalities for the measures dµ ∼ ω -β , where ω is smooth, positive and strongly convex. These inequalities will be satisfied under assumptions which need to be discussed. We apply the results to generalised Cauchy measures so as to obtain the exact spectral gap and the extremal functions. Finally, in Section 5, we return to the onedimensional case, as a toy application of our method. This section can be read directly after Section 2, as it exposes most of our arguments in a simpler way.

Variance representation formula

Let M be a complete Riemannian manifold of dimension n. Let µ be a probability measure on M and L a reversible diffusion operator associated with µ. We denote by P the L 2 -semigroup generated by L. We assume that the semigroup P is ergodic. Let Γ and Γ 2 be the carré du champ and iterated carré du champ operators, respectively defined on

f ∈ C ∞ c (M ) by 2Γ(f ) = L(f 2 ) -2f Lf and 2Γ 2 (f ) = LΓ(f ) -2Γ(f, Lf ).
Under suitable conditions, discussed in [START_REF] Bakry | Analysis and Geometry of Markov Diffusion operators[END_REF], these operators can be uniquely extended to the domain D(L) ⊂ L 2 (µ). The link between these operators and functional inequalities satisfied by the measure µ has been studied since the work of [START_REF] Bakry | Diffusions hypercontractives[END_REF]. The operator L satisfies the curvaturedimension criterion CD(ρ, ∞), also called Bakry-Émery criterion, with ρ ∈ R, if

Γ 2 ≥ ρΓ.

Moreover, if ρ > 0, then under the CD(ρ, ∞) criterion, the associated measure µ satisfies a Poincaré-type inequality with constant ρ:

(2) ρ Var µ (f ) ≤ M Γ(f ) dµ.
For the operator L = ∆ -∇V • ∇ associated with dµ = e -V dx, the carré du champ operator is Γ(f ) = |df | 2 and (2) is the classical Poincaré inequality. However, for a general L, this inequality is a weighted Poincaré inequality. Actually, both Poincaré inequality and weighted Poincaré inequality are equivalent to a spectral gap property on the generator L. We denote by λ 1 (-L) this spectral gap:

λ 1 (-L) = inf M -f Lf dµ Var µ (f ) , f ∈ D(-L) .
Actually, curvature-dimension criterion implies stronger functional inequalities than Poincarétype inequalities, such as logarithmic-Sobolev inequalities. In [START_REF] Ledoux | On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions[END_REF], it was proved in R n that the classical Poincaré inequality is equivalent to an integrated version of the Bakry-Émery criterion. This result is very general and has been adapted to a larger class of diffusion and space. See [START_REF] Bakry | Analysis and Geometry of Markov Diffusion operators[END_REF] or [START_REF] Cattiaux | A journey with the integrated Γ 2 criterion and its weak forms[END_REF] for more details. Here, we recall a version of this result from [START_REF] Chafaï | On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures[END_REF], where the deficit term is characterised.

Proposition 2.1 ([11]). For all f ∈ C ∞ c (M )
and for all ρ ̸ = 0 we have

Var µ (f ) = 1 ρ M Γ(f )dµ - 2 ρ +∞ 0 M (Γ 2 -ρΓ)(P t f ) dµ dt.
In particular,

λ 1 (-L) is the largest ρ > 0 such that M (Γ 2 -ρΓ) dµ ≥ 0. Proof. Let us define h(t) = M (P t f ) 2 dµ. We have Var µ (f ) = -[h t ] +∞ 0 = -1 2ρ h ′ (0) - 1 ρ +∞ 0 1 2 h ′′ (t) + ρh ′ (t) dt.
Then, we compute the first derivatives of h.

h ′ (t) = M ∂ t (P t f ) 2 dµ = M 2P t f LP t f dµ = -2 M Γ(P t f ) dµ, h ′′ (t) = -2 M ∂ t Γ(P t f ) dµ = -4 M Γ(P t f, LP t f ) dµ = 4 M Γ 2 (P t f ) dµ.
This proves the formula. Besides, if L satisfies an integrated Bakry-Émery criterion, with constant ρ > 0, then it is clear that λ 1 (-L) ≥ ρ. Conversely, using Cauchy-Schwarz inequality, we have

M Γ(f ) dµ = M f - M f dµ (-L)f dµ ≤ Var µ (f ) 1/2 M Γ 2 (f ) dµ 1/2
. Hence, we have

λ 1 (-L) ≤ inf f ∈C ∞ c (M ) M Γ 2 (f ) dµ M Γ(f ) dµ . □
Let us remark that this formula can be extended to other Φ-entropy inequalities, such as logarithmic-Sobolev inequality, using

h(t) = M Φ(P t f ) dµ.

The error term does not have an explicit expression because of the dependence on

Pf . Yet, it is sufficiently explicit to provide a necessary condition for extremal function, up to regularity considerations, or at least, a heuristic to find them. To do so, we firstly need to extend the error formula to the domain D(L). This can be done if L is essentially self-adjoint and if it satisfies any CD(ρ, ∞) criterion for ρ ∈ R (for instance, see [START_REF] Bakry | Analysis and Geometry of Markov Diffusion operators[END_REF]). Then, we also need some argument to prove the convergence of P t f as t converges to 0. Under these considerations, if L satisfies an integrated curvature-dimension criterion with ρ > 0 and if f ∈ D(L) is extremal for the associated Poincaré inequality, then we have

M (Γ 2 -ρΓ)(f ) dµ = 0.
This characterisation was used in [START_REF] Chafaï | On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures[END_REF] on the Ornstein-Uhlenbeck operator to recover the extremal functions in Poincaré and logarithmic-Sobolev inequalities for the Gaussian measures.

Afterwards, we use Proposition 2.1 to determine weighted Poincaré inequalities. In the case of generalised Cauchy measures, the expression of the error term is sufficiently explicit to characterise the extremal functions or to prove that there do not exist any extremal functions.

Spectral gap upper bound

The goal of this section is to introduce generalised Cauchy measures and recall the spectral gap upper bounds for generalised Cauchy measures These upper bounds were already obtained in [START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF] for all parameters. Nevertheless, the study of eigenfunctions associated with these bounds, or the lack of eigenfunctions, will be a very useful clue in Section 4 so as to choose the appropriate integration by parts formulae. The study of the upper bound highlights three different ranges of the parameter β, on which the spectral gap has three different behaviours.

We denote by ω the function defined on R n by ω(x) = 1 + |x| 2 . The generalised Cauchy measure µ β is the probability measure defined on R n , for β > n/2, by

dµ β = 1 Z n,β ω -β dx, with Z n,β = π n/2 Γ(β -n/2) Γ(β) .
This probability measure is associated with the symmetric operator, defined on

C ∞ c (R n ) by Lf = ω∆f -2(β -1)⟨x, df ⟩.
The carré du champ operator associated with L is then

Γ(f ) = ω|df | 2 .
The operator L is essentially self-adjoint and can be extended in a unique operator, still denoted by L defined on a domain D(L) ⊂ L 2 (µ β ). The spectrum of this operator was completely described in [START_REF] Denzler | Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology[END_REF] with a decomposition on spherical harmonics. Here, we are interested in its spectral gap and in the associated eigenfunctions only. Proposition 3.1. Let be n ≥ 1 and β > n/2 + 1. For all v ∈ R n , the linear function x → ⟨v, x⟩ belongs to L 2 (µ β ) and is an eigenfunction of -L associated with the eigenvalue 2(β -1). In particular, for all β > n/2 + 1, we have

λ(-L) ≤ 2(β -1). For β > n/2 + 2, the function x → |x| 2 - n 2β-n-2 belongs to L 2 (µ β )
and is an eigenfunction associated with the eigenvalue 4(β -n 2 -1). In particular, for all β > n/2 + 2, we have

λ(-L) ≤ 4 β - n 2 -1 .
For the smallest β, the spectral gap is not an eigenvalue any longer. In [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF], it is interpreted as the bottom of the continuous spectrum. We give a new proof of this optimality based on the one-dimensional argument from [START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF], recalled in Proposition 5.3. The sequence of functions exhibited in their proof is peculiar to the dimension n = 1 and needs some adaptation to the n-dimensional case. It leads to the following result:

Proposition 3.2. For n ≥ 2, n/2 < β ≥ n/2+2 and ε < (2β -n)/4 the functions f ε : x → ω ε (x) belongs to D(L) and we have lim ε↑(2β-n)/4 R Γ(f ε ) dµ β Var µ β (f ε ) = β - n 2 2 .
In particular, for all β > n/2, we have

λ(-L) ≤ β - n 2 2 .
Proof. Firstly, we remark that f ε ∈ L 2 (µ β ) if and only if ε < (2β -n)/4 but at the limit we still have

f (2β-n)/4 ∈ L 1 (µ β ). So, we have Var µ β (f ε ) R n f 2 ε dµ β = 1 -R n f ε dµ β 2 R n f 2 ε dµ β -→ ε→(2β-n)/4
1.

Then, for all ε < (2β -n)/4 and for all x ∈ R n , we have

Lf ε (x) = εω ε-1 (x)Lω(x) + ε(ε -1)ω ε-2 (x)Γ(ω)(x) = εω ε-1 (x) 2nω(x) -4(β -1)|x| 2 + ε(ε -1)ω ε-2 (x) 4ω(x)|x| 2 = ε [2n -4(β -ε)] f ε (x) + 4(β -ε).
Thus,

R n -f ε Lf ε dµ β R n f 2 ε dµ β = ε [4(β -ε) -2n] -4(β -ε) R n f ε (x) dµ β R n f 2 ε dµ β . So, we have lim ε↑(2β-n)/4 R Γ(f ε ) dµ β Var µ β (f ε ) = lim ε↑(2β-n)/4 R -f ε Lf ε dµ β R n f 2 ε dµ β = β - n 2 2 .
□ Hence, we have obtained three different upper bounds for the spectral gap, on different intersecting ranges. After optimising, we get the global result.

Corollary 3.3 (Cauchy upper bounds).

For n ≥ 2, we have the following spectral gap upper bound:

λ 1 (-L) ≤    (β -n/2) 2 if n/2 < β ≤ n/2 + 2 4(β -n/2 -1) if n/2 + 2 ≤ β ≤ n + 1 2(β -1) if n + 1 ≤ β.

Weighted Poincaré inequalities

Let ω be a smooth positive function on M . We assume that there exists β > 0 such that the measure with density ω -β with respect to the Riemannian volume measure, denoted µ β , is a probability measure. Up to additional assumptions on ω we will be able to determine on which range of β this measure assumption is satisfied.

The goal of this section is to obtain general weighted Poincaré inequalities for the probability µ β , under additional assumptions on ω such as strong convexity or bounded convexity. These inequalities will be sharp in the sense that for generalised Cauchy measures, the weight and the constant are optimal. The main innovations of this section is the Poincaré inequality on a range of small parameters β ≤ n + 1, beyond the criterion of convex measures. In particular, when applied to generalised Cauchy measures, the integrated curvature-dimension approach allows to recover the exact spectral gap even for n/2 < β ≤ n + 1.

The measure µ β is associated with the symmetric diffusion operator, defined on

C ∞ c (M ) by Lf = ω∆f -(β -1)⟨dω, df ⟩.
This special operator is chosen such that its carré du champ gives the weight ω in the Poincarétype inequalities. Indeed, we have the following result:

Proposition 4.1. For all f ∈ C ∞ c (M ), we have Γ(f ) =ω|df | 2 , Γ 2 (f ) =∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + 1 2 ω∆ω -(β -1)|dω| 2 |df | 2 + ⟨d|df | 2 , ωdω⟩ -⟨∆f df, ωdω⟩ + (β -1)ω Hess(ω)(∇f, ∇f ).
Proof. The computation of Γ does not present any difficulties. We will only develop the compu-

tation of Γ 2 . For f ∈ C ∞ c (M ), we have 2Γ 2 (f ) = LΓ(f ) -2Γ(f, Lf ) = ω∆(ω|df | 2 ) -(β -1)⟨dω, d(ω|df | 2 )⟩ -2ω ⟨df, d(ω∆f -(β -1)⟨dω, df ⟩)⟩ = ω∆ω|df | 2 + 2ω⟨dω, d|df | 2 ⟩ + ω 2 ∆|df | 2 -(β -1)|dω| 2 |df | 2 -(β -1)⟨d|df | 2 , ωdω⟩ -2⟨∆f df, ωdω⟩ -2ω 2 ⟨df, d|df | 2 ⟩ + 2(β -1)ω ⟨df, d⟨df, dω⟩⟩ .
Then, we use Bochner's formula for the Laplacian

(3) ∆|df | 2 -2⟨df, d∆f ⟩ = 2∥ Hess(f )∥ 2 HS + 2 Ric(∇f, ∇f ). Together with the remark that Hess(f )(∇f, •) = 1 2 d|df | 2 ,

this ends the proof. □

Let us note that, as ω is positive, then the carré du champ cancels on constant functions only. This means that the diffusion is ergodic and Proposition 2.1 can be applied.

In this general setting, it remains unclear whereas L satisfies or not a curvature-dimension criterion. Yet, in the special case of generalised Cauchy measures, we have a positive answer. First, we specify the Γ 2 operator associated with generalised Cauchy measures.

Corollary 4.2 (Cauchy carré du champ). Let

ω : x ∈ R n → 1 + |x| 2 . For all f ∈ C ∞ c (R n ) we have Γ(f ) = ω|df | 2 , Γ 2 (f ) = ∥ω Hess(f )∥ 2 HS + [nω + 2(β -1)]|df | 2 + 2⟨d|df | 2
, ωx⟩ -2⟨∆f df, ωx⟩. This formula can be factorised so as to obtain a CD(0, ∞) criterion and to prove the optimality of this criterion. This property was already proved in [START_REF] Huguet | Intertwining relations for diffusions in manifolds and applications to functional inequalities[END_REF] in dimension n = 2. Proposition 4.3 (Cauchy CD criterion). Let ω : x ∈ R n → 1 + |x| 2 . For n ≥ 1, L satisfies CD(0, ∞). Moreover, for every ρ > 0, it does not satisfy CD(ρ, ∞).

Proof. The idea of this proof is to factorise every second-order derivative terms as the squared Hilbert-Schmidt norm of a more intricate operator. For all

f ∈ C ∞ c (R n ), we have Γ 2 (f ) = ∥ω Hess(f )∥ 2 HS + 2⟨d|df | 2 , ωx⟩ -2⟨∆f df, ωx⟩ + [nω + 2(β -1)]|df | 2 = ∥ω Hess(f )∥ 2 HS + 4ω Hess(f )(∇f, x) -2ω Tr(Hess(f ))⟨df, x⟩ + [nω + 2(β -1)]|df | 2 = ∥ω Hess(f ) + x ⊗ ∇f + ∇f ⊗ x -⟨df, x⟩ id∥ 2 HS -∥x ⊗ ∇f + ∇f ⊗ x -⟨df, x⟩ id∥ 2 HS + [nω + 2(β -1)]|df | 2 = ∥ω Hess(f ) + x ⊗ ∇f + ∇f ⊗ x -⟨df, x⟩ id∥ 2 HS + (n -2) |df | 2 |x| 2 -⟨df, x⟩ 2 + (2β + n -2)|df | 2 .
Let us notice that in dimension n = 1 the second term vanishes. Then, these three terms are non-negative and so L satisfies the CD(0, ∞) criterion.

The optimality comes from the comparison with the special function f : x ∈ R n → 1/2 ln(|x| 2 )ξ(x), where ξ is a cutoff function, which is equal to 1 on a neighbourhood of x 0 ∈ R n , such that f belongs to C ∞ c (R n ). For every x on this neighbourhood, we have

Γ(f )(x) = 1 + 1 |x| 2 , Γ 2 f (x) = n |x| 4 + 2β + n -2 |x| 2 .
Choosing |x 0 | large enough, it follows that for every ρ > 0, there exists a function

f ∈ C ∞ c (R n ) and x ∈ R n such that Γ 2 (f )(x) < ρΓ(f )(x). □
Let us remark that the symmetric term in the Hilbert-Schmidt norm is necessary to obtain this optimal result. The same computation with a term ∥ω Hess(f ) + x ⊗ ∇f -⟨df, x⟩ id ∥ 2 HS , would not have been sufficient. This phenomenon will be encountered again while dealing with the lower range of β. Also, this proof highly relies on the special form of ω and its generalisation to other ω is not trivial.

Thus, for generalised Cauchy measures at least, we cannot obtain any CD(ρ, ∞) criterion for positive ρ. This is a reason why we are interested in integrated criterion. Before going further, here are the following integration by parts formulae:

Lemme 4.4. For all f ∈ C ∞ c (M ), we have (4)
M ⟨d|df | 2 , ωdω⟩ dµ β = M -ω∆ω + (β -1)|dω| 2 |df | 2 dµ β , M ⟨∆f df, ωdω⟩ dµ β = M - 1 2 ⟨d|df | 2 , ωdω⟩ -ω Hess(ω)(∇f, ∇f ) + (β -1)⟨df, dω⟩ 2 dµ β . (5)
Furthermore, if β ̸ = 2, we have

(6) M ⟨d|df | 2 , ωdω⟩ dµ β = 1 β -2 M ω 2 ∆|df | 2 dµ β , (7)
M ⟨∆f df, ωdω⟩ dµ β = 1 β -2 M ω 2 ⟨df, d∆f ⟩ + (ω∆f ) 2 dµ β .
Proof. For the first two formulae, we lower the differentiation degree on f . For all f ∈ C ∞ c (M ), we have

M ⟨d|df | 2 , ωdω⟩ dµ β = M ⟨d|df | 2 , ω 1-β dω⟩ d vol = - M |df | 2 Div(ω 1-β dω) d vol = - M |df | 2 ω 1-β Div(dω) + (1 -β)ω -β |dω| 2 d vol = M -ω∆ω + (β -1)|dω| 2 |df | 2 dµ β . M ⟨∆f df, ωdω⟩ dµ β = M ∆f ⟨df, ω 1-β dω⟩ d vol = - M df, d⟨df, ω 1-β dω⟩ d vol = - M ω 1-β Hess(f )(∇f, ∇ω) + ω 1-β Hess(ω)(∇f, ∇f ) + (1 -β)ω -β ⟨df, dω⟩ 2 d vol = M - 1 2 ⟨d|df | 2 , ωdω⟩ -ω Hess(ω)(∇f, ∇f ) + (β -1)⟨df, dω⟩ 2 dµ β .
Now, for β ̸ = 2, we can also increase the differentiation degree on f . Then, we have

M ⟨d|df | 2 , ωdω⟩ dµ β = M ⟨d|df | 2 , ω 1-β dω⟩ d vol = M ⟨d|df | 2 , dω 2-β 2 -β ⟩ d vol = - M ∆|df | 2 ω 2-β 2 -β ⟩ d vol = 1 β -2 M ω 2 ∆|df | 2 dµ β . M ⟨∆f df, ωdω⟩ dµ β = M ⟨∆f df, ω 1-β dω⟩ d vol = M ⟨∆f df, dω 2-β 2 -β ⟩ d vol = - M Div(∆f df ) dω 2-β 2 -β d vol = - M [⟨d∆f, df ⟩ + ∆f Div(df )] dω 2-β 2 -β d vol = 1 β -2 M ω 2 ⟨df, d∆f ⟩ + (ω∆f ) 2 dµ β .

□

These formulae allow us to establish a first simplification.

Lemme 4.5. For all f ∈ C ∞ c (M ), we have M Γ 2 (f ) dµ β = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + (β -1)ω Hess(ω)(∇f, ∇f ) + 1 2 ⟨d|df | 2 , ωdω⟩ -⟨∆f df, ωdω⟩ dµ β . Proof. M Γ 2 (f ) dµ β = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + (β -1)ω Hess(ω)(∇f, ∇f ) + 1 2 ω∆ω -(β -1)|dω| 2 |df | 2 + 1 2 ⟨d|df | 2 , ωdω⟩ (i) + 1 2 ⟨d|df | 2 , ωdω⟩ -⟨∆f df, ωdω⟩ dµ β .
The result is achieved by using the formula (4) on the term (i). □

This last formula for M Γ 2 dµ β can be divided into four parts. The first one is a squared norm of the Hessian and is non-negative. The second one, gathering the Ricci tensor and the Hess(ω), is a kind of Bakry-Émery curvature, upon which, we will probably need some additional assumption. The last two terms are more problematical. In the sequel, we will deal with these terms using appropriate combinations of our integration by parts formulae, depending on the parameter β.

4.1. Strong convexity. Firstly, we recover a general weighted Poincaré inequality. It was proved in the Euclidean space R n in [START_REF] Van Hoang Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF] (Theorem 12) and in Riemannian manifolds with nonnegative Ricci curvature in [START_REF] Gentil | A family of Beckner inequalities under various curvature-dimension conditions[END_REF]. This classical result is only valid for convex measures with β ≥ n + 1 and under strong convexity property. Here, we complete this result and provide a criterion for general Riemannian manifolds.

Proposition 4.6. For all β ̸ = 2 and f ∈ C ∞ c (M ), we have M Γ 2 (f ) dµ β = M β -(n + 1) β -2 ∥ω Hess(f )∥ 2 HS + n β -2 ∥ω Hess(f )∥ 2 HS - 1 n (ω∆f ) 2 + (β -1)ω 1 β -2 ω Ric + Hess(ω) (∇f, ∇f ) dµ β .
Proof. From Lemma 4.5, we have

M Γ 2 (f ) dµ β = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + 1 2 ⟨d|df | 2 , ωdω⟩ (i)
-⟨∆f df, ωdω⟩

+(β -1)ω Hess(ω)(∇f, ∇f ) = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + (β -1)ω Hess(ω)(∇f, ∇f ) + ω 2 2(β -2) (∆|df | 2 -2⟨df, d∆|df | 2 ⟩) - ω 2 β -2 (∆f ) 2 , (ii)
where we used the equalities ( 6) and ( 7) on (i) and (ii) respectively. Using again Bochner's formula (3), we obtain the result. □

Let us note that this expression of M Γ 2 (f ) dµ can be found in [START_REF] Van Hoang Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF], without the use of the carré du champ formalism.

The previous proposition suggests some conditions under which we can obtain a spectral gap inequality. The more important point to understand is the condition upon the Bakry-Émery like-curvature. A first assumption can be set as follows. Let us assume that β ̸ = 2 and that there exists a c > 0 such that

(H1) ω β -2 Ric + Hess(ω) ≥ c id .
This assumption is not very convenient because it depends on β and because this dependence is not explicit. Yet, in the Euclidean space, the assumption is equivalent to strong convexity on ω: it exists ρ -> 0 such that (8) Hess(ω) ≥ ρ -id .

Let us remark that under this strong convexity assumption, the measure dµ β is a probability measure for every β > n/2. The hypothesis ( 8) is also sufficient in a Riemannian manifold with non-negative Ricci curvature. We can now state our first result.

Theorem 4.7. For n ≥ 2 and β ≥ n + 1, under assumption (H1), we have λ 1 (-L) ≥ c(β -1).

For n = 1 and β > 1, under assumption (H1), we have λ 1 (-L) ≥ c(β -1).

Proof. Let us begin with the general case n ≥ 2. We must remark that for n ≥ 2 and β ≥ n + 1 the assumption β ̸ = 2 is redundant as we have β > 2. Then, under the assumption (H1) we have proved that

M Γ 2 (f ) -c(β -1)Γ(f ) dµ β ≥ M β -(n + 1) β -2 ∥ω Hess(f )∥ 2 HS + n β -2 ∥ω Hess(f )∥ 2 HS - 1 n (ω∆f ) 2 dµ β .
The numerical coefficients are non-negative and the Cauchy-Schwarz inequality implies that for all linear operator A acting on the tangent bundle T M , we have

∥A∥ 2 HS ≥ 1 n (Tr A) 2 .
Applied to A = ω Hess(f ), it proves an integrated curvature-dimension criterion. We conclude with Proposition 2.1.

In the particular case n = 1, the two problematic terms from Lemma 4.5 offset each other. Under assumption (H1), we get

M Γ 2 (f ) -c(β -1)Γ(f ) dµ β ≥ M ∥ω Hess(f )∥ 2
HS dµ β .

□

In this general case, we only obtain an upper bound of the error term although a lower bound would have been more interesting. In the case of the generalised Cauchy measures, the potential ω satisfies Hess(ω) = 2 id. Then we can apply our theorem to get a spectral gap lower bounds together with an exact expression of the error term.

∈ C ∞ c (R n ), we have 2(β -1) Var µ β (f ) - R n |df | 2 (1 + |x| 2 ) dµ β = -2 +∞ 0 R n β -(n + 1) β -2 ∥ω Hess(P t f )∥ 2 HS + n β -2 ∥ Hess(ωP t f )∥ 2 HS - 1 n (ω∆P t f ) 2 dµ β dt.
As explained in Section 2, our expression of the error term characterises the extremal function. For β > n + 1, an extremal function f must satisfy ∥ Hess(f )∥ HS = 0.

Then it is an affine function. This is coherent as, in this range, linear functions are eigenfunctions associated with 2(β -1). In the particular case β = n + 1, an extremal function f must satisfy

| Hess(f )∥ 2 HS - 1 n (∆f ) 2 .
Then in this case, f is quadratic. This is coherent as, for this parameter, linear functions and

x ∈ R n → |x| 2 - n 2β-n-
M Γ 2 (f ) dµ β = M n n -1 ∥ω Hess(f )∥ 2 HS - 1 n (ω∆f ) 2 + (n + 1 -β)(β -1) n -1 |df | 2 |dω| 2 -⟨df, dω⟩ 2 + ω n n -1 ω Ric +(β -1) Hess(ω) (∇f, ∇f ) + n + 1 -β n -1 ω [Hess(ω) -∆ω id] (∇f, ∇f ) dµ β .
Proof. We begin with the generic case β ̸ = 2. Let λ ∈ R be a parameter to be optimised later. From Lemma 4.5, for all f ∈ C ∞ c (M ), we have

M Γ 2 (f ) dµ β = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) + (β -1)ω Hess(ω)(∇f, ∇f ) + λ 2 ⟨d|df | 2 , ωdω⟩ -λ⟨∆f df, ωdω⟩ + 1 -λ 2 ⟨d|df | 2 , ωdω⟩ (i) -(1 -λ) ⟨∆f df, ωdω⟩ (ii)
dµ β = M ∥ω Hess(f )∥ 2 HS + 1 -λ β -2 ∥ω Hess(f )∥ 2 HS -(ω∆f ) 2 + 1 + 1 -λ β -2 ω 2 Ric(∇f, ∇f ) + (β -1)ω Hess(ω)(∇f, ∇f ) + λ 2 ⟨d|df | 2 , ωdω⟩ (iv)
-λ ⟨∆f df, ωdω⟩

dµ β .

Here, we have used the formulae ( 6) and ( 7) for (i) and (ii) respectively and together with Bochner's formula. Then, we choose λ in order to exactly compensate the term ∥ Hess(f )∥ 2 HS with 1 n (∆f ) 2 . It results that

λ = n + 1 -β n -1 .
To conclude, we use the formulae ( 4) and ( 5) to treat the terms (iv) and (v).

In the specific case, β = 2, we do not need the forbidden formulae ( 6) and [START_REF] Sergey | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF]. Indeed, thanks to Bochner's formula, we have

M (ω∆f ) 2 dµ 2 = M ∆f ∆f d vol = - M ⟨df, d∆f ⟩ d vol = - M ⟨df, d∆f ⟩ - 1 2 ∆|df | 2 d vol = M ∥ Hess(f )∥ 2 HS + Ric(∇f, ∇f ) d vol = M ∥ω Hess(f )∥ 2 HS + ω 2 Ric(∇f, ∇f ) dµ 2 .
It follows that

M ∥ω Hess(f )∥ 2 HS dµ 2 = M n n -1 ∥ω Hess(f )∥ 2 HS - 1 n (ω∆f ) 2 + 1 n -1 Ric(∇f, ∇f ) dµ 2 .
The proof ends as in the generic case. □

In order to derive some spectral gap inequality, for β < n + 1, the assumption (H1) is not sufficient any more. In this case, we make the following assumption: it exists c > 0 such that

(H2) n n -1 ω Ric +(β -1) Hess(ω) + n + 1 -β n -1 ω[Hess(ω) -∆ω id] ≥ c id .
In a general Riemannian manifold, it seems difficult to obtain an handier condition. However, in the Euclidean space R n , this condition is equivalent to boundedness of the Hessian operator: it exists 0 < ρ -≤ ρ + such that (9) ρ -id ≤ Hess(ω) ≤ ρ + id .

In this case, we will denote by κ the corresponding condition number:

κ = ρ + ρ - ≥ 1.
As previously, in a Riemannian manifold with non-negative Ricci curvature, we can also use the simpler boundedness assumption [START_REF] Bonnefont | Spectral gap for spherically symmetric log-concave probability measures, and beyond[END_REF]. Even if it is more restrictive than (H2), it is explicit and independent of β.

Let us remark that if the Hessian is uniformly bounded, then µ is a probability on R n if and only if β > n/2. Theorem 4.10. For n ≥ 2 and β ≤ n + 1, under assumption (H2), we have λ 1 (-L) ≥ c. Furthermore, if Ric ≥ 0, then, under assumption (9), we have

λ 1 (-L) ≥ ρ -β -1 - n + 1 -β n -1 (nκ -1) , ∀ β ∈ n(n + 1)κ -2 n(κ + 1) -2 , n + 1 .
Proof. The first general result is clear, using Cauchy-Schwarz inequality. Then, we remark that β -1 + λ ≥ 0 and λ ≥ 0. Under assumption (9), we have

(β -1 + λ) Hess(ω) -λ∆ω id ≥ [ρ -(β -1 + λ) -λnρ + ] id ≥ ρ -[β -1 -λ(nκ -1)] id . The term ρ -[β -1 -λ(nκ -1)] is positive if and only if n(n+1)κ-2 n(κ+1)-2 < β. □
The assumption (9) implies the assumption (8), then it is interesting to remark that the associated spectral gap lower bounds are continuous at β = n + 1. As the condition number κ is always greater than For n/2 + 2 < β < n + 1, an extremal function must satisfy |df | 2 |x| 2 = ⟨df, x⟩ 2 . Then there exist a, b ∈ R such that for all x ∈ R n , f (x) = a|x| 2 + b. For β = n + 1, we recover the same error term as in the previous subsection and so, the same extremal function. The parameter β = n + 1 marks a transition between affine extremal functions and quadratic ones.

This result for mid-range β improves the previous results from the literature as it is the first one which obtains the optimal spectral gap with curvature-dimension argument. 4.3. Lower range. It seems difficult to obtain any result for the smallest β and for unspecified ω. Yet, in the particular case of generalised Cauchy measures, we will be able to conclude. As for the one-dimensional case, the appropriate formula for the lower range is obtained by factorising the second-order derivative, in the way we proved the curvature-dimension criterion in Proposition 4.3. However, we need to factorise a fraction of it only and use integration parts formulae on the remaining second-order terms. The result is achieved by optimising on these fractions.

In the previous subsection, we have obtained the following formula

R n Γ 2 (f ) dµ β = R n n n -1 ∥ω Hess(f )∥ 2 HS - 1 n (ω∆f ) 2 + 4(β -1) n + 1 -β n -1 |df | 2 |x| 2 -⟨df, x⟩ 2 + 4(β - n 2 -1)Γ(f ) dµ β . (10)
Let ε be the optimisation parameter. We want to force the following factorisation ω Hess f + ε ∇f ⊗ x + x ⊗ ∇f 2 Our computation only provides a lower bound of the spectral gap. On this range, there is no eigenfunction, as in the n-dimensional case. The article [START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF] proved the optimality using a sequence of functions. While looking for an extremal function f associated with (β -1/2) 2 , we saw that, up to regularity, if it does exist, this function would satisfy the equation

ωf ′′ + 3 2 -β f ′ = 0.
The solutions of this equation are primitive functions of ω (2β-3)/4 . This explains the choice of f ε . Heuristically, the sequence f ε converges towards a function close to cancel the Hessian part but which is not in L 2 (µ β ).

Corollary 4 . 8 (

 48 Cauchy -Upper range). Let ω : x ∈ R n → 1 + |x| 2 . For all n ≥ 2 and β ≥ n + 1, we have λ 1 (-L) ≥ 2(β -1), and for all f

2 HS=

 2 ∥ω Hess f ∥ 2 HS + ε⟨d|df | 2 , xω⟩ + ε 2 2 [|df | 2 |x| 2 + ⟨df, x⟩ 2 ] (ω∆f + ε⟨df, x⟩) 2 = (ω∆f ) 2 + 2ε⟨∆f df, xω⟩ + ε 2 ⟨df, x⟩ 2 .

Proposition 5 . 3 (- 1 2 2 .

 532 [START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF]). For n = 1, 1/2 < β ≤ 3/2 and ε < (2β -3)/4, the function f ε : x → xω ε (x) belongs to L 2 (µ) and we havelim ε↑(2β-3)/4 R Γ(f ε ) dµ β Var µ β (f ε ) = β

  2 are eigenfunctions associated with 2n. 4.2. Upper bounded convexity. To obtain Proposition 4.6, we kept an Hessian term. This term is cancelled by affine functions which are extremal functions for generalised Cauchy measures for β ≥ n + 1. However, we have seen that the extremal functions for β ≤ n + 1 are no longer linear. This suggests a formula for M Γ 2 dµ without the Hessian term. On the other hand, the term, specific to the dimension n ≥ 2 is a very good candidate as it is cancelled by the extremal functions for generalised Cauchy measures on this range. This idea provides a Poincaré inequality for general ω and β ≤ n + 1. This result for non-convex measures is an important innovation of our work. Nonetheless, it requires a stronger assumption than the strong convexity.

	∥ω Hess(f )∥ 2 HS -	1 n	(ω∆f ) 2 ,
	Proposition 4.9. For all n ≥ 2, β > n/2 and f ∈ C ∞ c (M ), we have

  Hence, our two spectral gap lower bounds do not cover the entire range ]n/2, +∞[ admissible under assumption[START_REF] Bonnefont | A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions[END_REF]. The equality κ = 1 is attained for generalised Cauchy measures. Then we have the following result. Corollary 4.11 (Cauchy -Intermediate range). Let ω : x ∈ R n → 1 + |x| 2 . For all n ≥ 2 and n/2 + 1 < β ≤ n + 1, we have λ 1 (-L) ≥ 4(β -n/2 -1), and for all f ∈ C ∞ c (R n ), we have 4(β -n/2 -1) Var µ β (f ) -

		1, we have
						n(n + 1)κ -2 n(κ + 1) -2	≥	n 2	+ 1.
						|df | 2 (1 + |x| 2 ) dµ β
						R n
	= -2	0	+∞	R n	n n -1	∥ω Hess(P t f )∥ 2 HS -	1 n	(ω∆P t f ) 2

+ 4(β -1) n + 1 -β n -1 |dP t f | 2 |x| 2 -⟨dP t f, x⟩ 2 dµ β dt.

Acknowledgements

We would like to thank François Bolley for his encouragements and for our many useful discussions. We also thank Geneviève Ropars for her helpful suggestions. This research is partially supported by the Centre Henri Lebesgue (ANR-11-LABX-0020-0) and the ANR project RAGE "Analyse Réelle et Géométrie" (ANR-18-CE40-0012).

We inject these twisted terms in equation [START_REF] Cattiaux | A journey with the integrated Γ 2 criterion and its weak forms[END_REF] and we use the integration by parts formulae [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF] and then [START_REF] Bakry | Sharp Beckner-type inequalities for Cauchy and spherical distributions[END_REF]. We have

where B, C and D are defined by

Note that the particular form of ω has been used in several simplifications, as in this case |dω| 2 and ω Hess(ω) are easily comparable to ω. To optimise this expression, we are looking for the maximum of D over the parameters ε such that B and C are non-negative. This optimum is reached for ε 0 = n/2 + 2 -β, and we have

Then, for all n ≥ 2 and n/2 < β, we have the following formula:

Remark that the optimisation parameter ε 0 was expected, as we recover the expression (10) at the transition β = n/2 + 2. For β ≥ n/2 + 2, we can prove that the optimal is ε 0 = 0.

To conclude, for all n ≥ 2 and n/2 < β ≤ n/2 + 2 we have B ε0 ≥ 0, C ε0 ≥ and D ε0 > 0. Hence, we have proved the following weighted Poincaré inequality on the lower range.

Corollary 4.12 (Cauchy -Lower range). For all n ≥ 2 and n/2 < β ≤ n/2+2, we have

where ε 0 = β -n 2 -2. Note that the error term is positive for all n/2 < β < n/2 + 2. This implies that there cannot exist any extremal function.

When all has been said and done, we have obtained a global lower bound for all n/2 < β. Comparing to the upper bounds from Corollary 3.3, this bound is optimal and we have proved the n-dimensional part of Theorem 1.1.

Corollary 4.13. For n ≥ 2, we have

It is noticeable that the obtained expressions are continuous in β. Let us remark that in dimension n = 2, the intermediate range n/2 + 2 ≤ β ≤ n + 1 is reduced to β = 3. Our method shows that curvature-dimension arguments are sufficient to obtain the spectral gap for generalised Cauchy measures. In this work, the error term has been used only to recover extremal functions. It should be interesting to use these formulae in order to get some explicit lower bounds of the deficit in terms of the distance to extremal function in some topology, as it can be done for Gross' logarithmic Sobolev inequality in [START_REF] Sergey | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF].

The one-dimensional case

In this section, we end the proof of Theorem 1.1 with the dimension n = 1. For the sake of completeness, we recover this result with our method and make some further remarks, especially on the error term and on extremal functions.

We still denote by ω the function ω : x ∈ R → 1 + x 2 . As shown in Section 4, we have

Firstly, we remark that, the operator L satisfies a nice curvature-dimension criterion. Indeed, in the one-dimensional case, the factorisation from Proposition 4.3 brings

This factorisation formula enlightens one of the two options we have. The first one is the integration by parts, as used in our general method. The second one, which appears here, is the factorisation. A mix between these two approaches allows to recover the spectral gap for the whole range β ∈]1/2, +∞[. Let ε ∈ R be an optimisation parameter. We have

Now, using the integration by parts formula (4), we get

We optimise to obtain the maximum over ε of min{A ε , B ε }. Two cases appear: small β and large β.

If β ≥ 3/2 then the optimum is reached for ε = 0, as a direct application of Theorem 4.7. We obtain

Then, according to Proposition 2.1, we have the following result.

Furthermore, if β > 3/2 then λ(-L) = 2(β -1) as linear functions are eigenfunctions associated with 2(β -1).

Actually, affine functions are the only extremal functions, up to regularity, as any extremal function, f , satisfies f ′′2 = 0. Remark that we cannot conclude yet when β = 3/2 as affine functions do not belong to L 2 (µ 3/2 ).

If 1/2 < β ≤ 3/2, the maximum over ε is obtained for ε = 3/2 -β and we have

Then, we obtain the final result.


Yuka Suzuki
email: yuka.suzuki@pasteur.fr

Hervé Ménager

Bryan Brancotte

Raphaël Vernet

Cyril Nerin

Christophe Boetto

Antoine Auvergne

Christophe Linhard

Rachel Torchet

Pierre Lechat

Lucie Troubat

Michael H Cho

Emmanuelle Bouzigon

Hugues Aschard
email: hugues.aschard@pasteur.fr

Hanna Julienne
email: hanna.julienne@pasteur.fr

Trait selection strategy in multi-trait GWAS: Boosting SNPs discoverability

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 increased detection of variants as compared to univariate screening. Our analyses identified multiple factors associated with the gain in the association detection in multi-trait tests. Together, these factors of the analysed sets are predictive of the gain of the multi-trait test (Pearson's 𝜌 equal to 0.43 between the observed and predicted gain, P < 1.6 x 10 -60 ). Applying an alternative multi-trait approach (MTAG, multi-trait analysis of GWAS), we found that in most scenarios but particularly those with larger numbers of traits, JASS outperformed MTAG. Finally, we benchmark several strategies to select set of traits including the prevalent strategy of selecting clinically similar traits, which systematically underperformed selecting clinically heterogenous traits or selecting sets that issued from our datadriven models. This work provides a unique picture of the determinant of multi-trait GWAS statistical power and outline practical strategies for multi-trait testing.

Introduction

Despite a continuous increase of the sample size of Genome-Wide Association Studies (GWAS), many genetic variants underlying human complex traits and diseases remain undetected. To increase statistical power and detection of associations at low cost, investigators have developed various multitrait approaches based on GWAS summary statistics [START_REF] Bhattacharjee | A subset-based approach improves power and interpretation for the combined analysis of genetic association studies of heterogeneous traits[END_REF][START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF][START_REF] Qi | Heritability informed power optimization (HIPO) leads to enhanced detection of genetic associations across multiple traits[END_REF][START_REF] Turley | Multi-trait analysis of genome-wide association summary statistics using MTAG[END_REF][START_REF] Wang | sCLC: a Clustering Linear Combination (CLC) Method for Multiple Phenotype Association Studies Based on GWAS Summary Statistics[END_REF][START_REF] Zhu | Meta-analysis of correlated traits via summary statistics from GWASs with an application in hypertension[END_REF] . Although other factors might also be involved, few studies investigated how the choice of phenotypes impacts the gain in the power of multi-trait approaches. In the standard univariate GWAS, statistical power mostly depends on minor allele frequency, sample size, the size of genetic effect, and polygenicity (the number of causal variants) [START_REF] Visscher | 10 years of GWAS discovery: biology, function, and translation[END_REF] . In the multi-trait GWAS, it additionally depends on complex characteristics of the set of traits: their shared aetiology, their genetic correlation, and the number of traits in the set. As previously described, the increased power of multi-trait test partly comes from adjusting for the correlation across GWAS studies due to sample overlaps and genetic relationships across phenotypes [START_REF] Turley | Multi-trait analysis of genome-wide association summary statistics using MTAG[END_REF][START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Porter | Multivariate simulation framework reveals performance of multi-trait GWAS methods[END_REF] . Previous works explored this question using simulated data [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Porter | Multivariate simulation framework reveals performance of multi-trait GWAS methods[END_REF] . However, simulations are limited in their scope, and more studies are needed to better characterise scenarios increasing the gain of multi-trait tests.

Here we empirically examined how trait characteristics impact the statistical power of multi-trait GWAS. We performed our analyses on 72 curated GWAS summary statistics and analysed the impact of 11 genetic features describing both individual and collective characteristics of sets of GWAS. Among available multi-trait methods, we used a standard 𝑘-degree of freedom joint test (omnibus test) of 𝑘 GWAS implemented in JASS [START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF] as our primary analysis. The JASS package and its associated tools solve all practical issues commonly encountered in multi-trait GWAS analyses, including missing data and computational efficiency, allowing for a large-scale power analysis on real data. We inspected the association between these genetic features and the gain in the association detection in multi-trait GWAS over univariate GWAS. We then assessed how well these genetic features predict the gain. We further compared JASS and MTAG, an alternative multi-trait approach, in terms of the impact of the trait selection strategy on the gain of multi-trait GWAS.

Results

Study Overview

We conducted a series of analyses to identify features that influence the gain in association detection of multi-trait GWAS relative to univariate GWAS. The key principles and main steps of the study are depicted in Figure 1. We used 72 curated GWAS summary statistics spanning a range of clinical domains (Table S1) and considered 11 features describing the univariate and multivariate genetic architecture of the phenotypes. Five of them characterise single GWAS: mean genetic effect size (MES), polygenicity, effective sample size (𝑁 !"" ), linear additive heritability of common variants

(ℎ #$%& '
), the proportion of uncaptured linear additive heritability of common variants (%ℎ ( ' ). The remaining six features characterise sets of GWAS. It includes the number of GWAS analysed jointly, and five metrics related to the genetic (Σ ) , Table S2 and Fig. S1) and residual (Σ * , Table S3 and Fig. S2) covariance matrices, where the latter represents the covariance between the Z-score statistics of two GWAS due to sample overlap and phenotypic correlation. Those five features are the mean of the off-diagonal terms denoted as Σ ' ) and Σ ' * ; the conditional numbers of Σ ) and Σ * denoted as 𝜅 ) and 𝜅 * , which we used as a measure of multi-collinearity; and the average difference between the two matrices Δ + . Δ + is expected to be a key driver of the power of multi-trait test [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Porter | Multivariate simulation framework reveals performance of multi-trait GWAS methods[END_REF] (Supplementary note 1, 2, Fig. S3).

Given the 72 traits, there are over 4.7 x 10 21 possible sets of two to 72 traits. In this study, we used 19,266 unique random sets of two to 12 GWAS drawn using a stratified sampling conditional on mean effect size, heritability and the number of traits to maximize the range of the genetic features studied (see Material and methods, Supplementary note 3). For each set, we derived the average of the single GWAS feature (polygenicity, MES, Neff, h 2 GWAS, and %h 2 u) and the six set features (Σ ' ) , Σ ' * , 𝜅 ) , 𝜅 * , Δ + , and the number of GWAS selected). Note that we used the log of polygenicity, MES, Σ ' ) , Σ ' * , 𝜅 ) , 𝜅 * , and Δ + when investigating their association with other variables due to their skewed distributions (Fig. S4). Then, we compared multi-trait and univariate GWAS results based on the minimum P-value of the multi-trait test across all variants within LD-independent loci (𝑃 ,%&& ) and the minimum P-value of univariate tests across all variants and all GWAS within the same loci (𝑃 (-. ). We used two metrics: i) the proportion of significant loci found associated at genome-wide level (P < 5 x 10 -8 ) by JASS and missed by the univariate GWAS, and ii) the fraction of loci where 𝑃 ,%&& was smaller than 𝑃 (-.

(𝑓 ,%&&/(-.0 ).

We first describe the distribution of the genetic features and their correlation. Second, we applied JASS to each set and compared multi-trait and univariate GWAS association results (Fig. 1B). Third, we build a predictive model of the association gain using a five-fold cross-validation approach (Fig. 1C).

For each cross-validation, we pulled a training set of 1,980 out of the 19,266 unique random sets and conducted a regression analysis to estimate the contribution of a subset of genetic features of trait sets using a joint model of the features. We measured the correlation between observed and predicted gain in an independent validation dataset. Fourth, we compared the association gain from JASS against MTAG 4 , a popular multi-trait approach that leverages genetic correlation across traits to boost statistical power. Finally, we compare common trait selection strategies-e.g. choosing clinically homogenous or heterogeneous traits-in their impacts on the association gain to evaluate our model prediction and provide a practical strategy in multi-trait test (Fig. 1D).

Distribution of the genetic features across GWAS trait sets

The individual genetic features of the 72 studied traits were distributed as follows. The sample sizes ranged from 5,318 to 697,828 with a median of 85,559. Heritability (ℎ #$%& '

) ranged from 1% to 48%

with a median at 10% (Fig. 2A) and was consistent across the software used for its estimation (Fig. S5).

Polygenicity and MES were highly variable: Polygenicity (i.e. the estimated number of causal variants) ranged from 6.9 (Estimated Glomerular

Multi-trait versus univariate GWAS across 19,266 random sets

We applied JASS to all 19,266 sets and quantified the gain in association detection of the multitrait against univariate test. On average, in a set, JASS detected 26 new loci while 285 were previously associated with the univariate tests, i.e. a 1.1-fold increase in the total number of loci detected (Fig. S7A). JASS gain was maximal for sets with a relatively low number (< 300 loci) of previously detected loci (Fig. S7B). JASS detected at least one new association in 98% of the trait sets (18,787 sets, Fig. S7C), and 508,829 new associations in total (note that these can be overlapping loci). These numbers are obtained when applying JASS on variants with beta coefficients available for all traits in the set.

When analysing all variants including those with missing values as allowed by JASS (see Material and methods), about 1.4 times more new associations were detected (693,382 new associations in total by JASS including variants with missing values, Fig. S7D).

We assessed the marginal relationship between genetic features and the number of loci detected by the univariate and multi-trait GWAS tests, and the association gain of multi-trait test. Figure 3 presents the correlation between each feature and the number of univariate and multi-trait GWAS associated loci. As expected, the number of univariate associated loci was positively correlated with the number of traits, mean Neff, the mean h can lead to a lack of robustness of the omnibus test [START_REF] Troubat | Multi-trait GWAS for diverse ancestries: Mapping the knowledge gap[END_REF] . We checked how the condition number 𝜅 * was related to JASS gain (Fig. 3B, Fig. S8). The condition number stayed in a reasonable range for 99% of sets (min=1, max=11). The rest (1%) of sets were flagged for having singular residual matrices.

Furthermore, note that the trait sets contained overlapping traits and are therefore not fully independent from each other. To robustly assess the impacts of genetic features on the multi-trait gain by addressing this issue, below we conducted regression analyses with a cross validation scheme establishing independence between training and validation data (Fig. S9).

Predicting multi-trait test gain from genetic features

We investigated the predictive power of the association gain (𝑓 ,%&&/(-.0 ) from a joint modelling of the genetic features (Fig. 4). Note that some features are almost linear combinations of other ones (e.g. Δ + is proportional to the difference between Σ ' ) and Σ ' * ). To avoid extreme collinearity and ensure parsimony, we selected six moderately correlated features: the number of traits, log 12 Δ + , log 12 Σ ' ) , mean Neff, mean h²GWAS, and mean %ℎ ( ' (see Material and methods). We favoured mean %ℎ ( ' over mean MES and mean polygenicity since mean %ℎ ( ' captures both log10(MES) and log10(polygenicity) (Fig. 2). We assessed performances using a five-fold cross validation (CV). For each CV the model parameters were derived on a training data, and the prediction accuracy was derived in an independent data (Material and methods, Fig. S9). All six features were highly associated to the multi-trait gain (Table 1). Mean %ℎ ( ' was overall positively associated to multi-trait gain, whereas mean h²GWAS was negatively associated. Overall, the association gain of multi-trait test, more specifically the omnibus test, seems driven by genetic correlation, polygenicity, and the number of traits. The predicted gain was significantly correlated with the observed gain on validation data (median Pearson 𝜌=0.43, P < 1.6 x 10 -60 , Fig. S10).

We conducted a series of sensitivity analyses to explore further performances. First, to interpret this model behaviour from the perspective of polygenicity and MES, we fitted two models replacing mean %ℎ ( ' with mean log10(polygenicity) and mean log10(MES) in turn (Fig. S11). In these models, mean log10(polygenicity) had a positive contribution to the gain, whereas mean log10(MES) had a negative contribution. In other words, multi-trait tests likely detect new associations in settings where univariate tests perform poorly, confirming the correlation analysis (Fig. 3). Additionally, the model suggested the number of traits and log 12 Σ ' ) further enhance multi-trait test gain. Contrary to previous observation on simulation studies, log 12 Δ + likely diminishes the multi-trait test gain (see Supplementary note 4 for a hypothetical explanation). Second, we also considered two nonlinear models for comparison purposes, support vector regression and random forest regression. As showed in Figure S12, these two models appear to outperform the multivariate linear model on the training datasets. However, they performed similarly or worse on the test dataset, suggesting a strong overfitting in the training data. Overall, we did not find any benefit in using these more complex models.

Comparison of JASS versus MTAG

We repeated the association screening and the prediction analysis using MTAG (Multi-Trait Analysis of GWAS), a popular multi-trait approach leveraging genetic correlation among closely related traits to inform GWAS screening. Regarding the association screening, MTAG detected 153,061 new association regions in total across the sets (30% of the number of new associations detected by JASS).

On average, MTAG detected eight new association loci per set and at least one new association locus on 63.3% of sets (12,195 out of 19,266) compared to 98% of sets for JASS. In 93% of all the trait sets, MTAG detected fewer associations than JASS did (Fig. 5A). The performance difference further increased when applying JASS also on variants with missing values (in 96% of trait sets MTAG detected fewer associations). Despite these discrepancies in the number of association loci, the number of new association loci in MTAG and JASS were strongly correlated (Pearson 𝜌 = 0.72, P < 2.2 x 10 -308 ), and as well as the P-values of MTAG and JASS for the same loci (Pearson 𝜌 =0.75, P < 2.2 x 10 -308 ). This concordance suggests common determinants for statistical power between the two methods. We next fitted a multivariate linear model to predict MTAG gain from the same six genetic features and training data used for JASS (Fig. S13). The most notable difference from JASS was that the number of traits in the set was, consistently across cross-validation folds, negatively associated with the gain of MTAG.

Indeed, JASS particularly outperformed MTAG on larger set of traits (Fig. 5B).

To test whether the pronounced gain of JASS compared with MTAG on large sets was mostly due to multiple testing correction-as MTAG performs one test by trait present in the set of traits, multiple testing correction is applied to MTAG P-values (Material and methods)-we compared the number of associations detected by JASS and by MTAG without correction (Fig. S14). In the absence of multiple testing correction, MTAG type 1 error expectedly increased with the number of traits in the set (e.g. genomic inflation factor l was 1.3, 1.66, 1.72, and 1.77 for four cases with 2, 5, 9, and 12 traits).

Despite the increased P-value inflation for large sets, MTAG detected fewer associations overall (Fig.

S14A

) than JASS on most of the sets (in 71% and 83% of sets when excluding and including variants with partly unavailable summary statistics, respectively). The performance of JASS and MTAG were 10 most similar in small trait sets (Fig. S14B), while JASS was particularly advantageous when analysing large trait set especially when allowing for variants with summary statistics partly unavailable.

An informed strategy for trait set selection in multi-trait GWAS

A common and seemingly sound strategy when conducting multi-trait analyse is to use closely related traits. This choice is partly driven by investigators' interest in delineating the shared genetic aetiology between a disease and closely related phenotypes. This might also arise from the intuitive idea that closely related phenotypes share a fair amount of genetic aetiology that the multi-test could leverage. However, its impact on statistical power has not been evaluated. To advise investigators on the best strategy to compose sets, we compared multi-trait gain and the number of new association loci obtained using this clinically-driven strategy (referred as "homogenous", see Material and methods) to three alternatives: i) including GWAS from two to four clinical groups (noted "low heterogeneity"), ii) including GWAS from five clinical groups or more (noted "high heterogeneity"), and iii) a data-driven approach based on the linear regression predicted gain (Material and methods).

The data-driven strategy had a higher gain and larger number of new association loci by JASS compared to the other strategies (Fig. 6A and6B). The gain and the number of new association loci increased systematically with the clinical heterogeneity of the traits, and the increases were statistically significant for most pairs of trait selection strategies compared, especially when comparing the data-driven approach with the other approaches (P < 4.4 x 10 -8 for gain and P < 6.7 x 10 -7 for the number of new association loci). The average number of new association loci detected in validation data equalled 14, 20, 32, and 61 for homogeneous, low heterogeneity, high heterogeneity, and datadriven sets, respectively.

We further decomposed the gain of the homogenous sets by clinical groups (Fig. 6C,6D). Two groups consistently yielded more new association loci than others, namely: 'Circulatory and Respiratory Physiological Phenomena' and 'Physiological Phenomena.' The first group was composed of spirometry traits and asthma, characterised by a large sample size, substantial genetic correlation, and moderate heritability: a favourable setting for JASS. The second group contained anthropometric traits (Height, BMI, Hip circumference, waist circumference, and Waist hip ratio). We next investigated whether multi-trait gain was associated with specific traits and derived the fold enrichment for traits between the top 10% yielding sets (corresponding to >169 new associated loci by JASS) and the least 10% yielding sets. BMI and hip circumference were amongst the top traits (Fig. S15). Notably, we found that high yielding trait sets (i.e., those that yielded 169 or more new association regions by JASS) all contained BMI. In contrast, out of the remaining trait sets, only 9% of the trait sets contained BMI.

Other traits enriched in top sets included spirometry and to a lesser extend mental disorders, arterial pressure, and sleep patterns. BMI genetics is increasingly recognized to be a complex entanglement of metabolic and behavioural factors [START_REF] Simon | Impaired cross-talk between mesolimbic food reward processing and metabolic signaling predicts body mass index[END_REF] , and suggests that complex traits that reflect multiple biologic processes may benefit more from multi-trait analysis.

We additionally compared the multi-trait gain and the number of new association loci by MTAG across the four groups of trait sets. The results were rather opposite to what we observed above with JASS: MTAG gain increased as the trait sets became more homogenous (Fig. S17A, P < 1.4 x 10 -7 ), and the number of new association loci was greater for trait sets of low heterogeneity than high heterogeneity (Fig. S17B, P = 2.7 x 10 -6 ). We then tested whether MTAG outperforms JASS on homogenous trait sets (Fig. S17C,D) and found that there was little difference between the two (P=0.084 for gain, P=0.063 for the number of new association loci). We further tested whether MTAG outperforms JASS on homogenous trait sets of certain clinical groups (Fig. S17E,F). JASS significantly outperformed on 'Immune System Diseases' and 'Psychological Phenomena' ('Immune System Diseases': P = 8.2 x 10 -4 for gain and P = 6.7 x 10 -7 for the number of new association loci, 'Psychological Phenomena': P = 2.5 x 10 -5 for the number of new association loci). In contrast, MTAG outperformed on 'Musculoskeletal and Neural Physiological Phenomena' (which contains only one trait set).

While JASS type 1 error in JASS is properly calibrated, this large number of new associations could be spurious, and irrelevant to biology. To ensure that new associations detected by JASS are relevant, we focussed on BMI and tested if we could predict novel associations observed in a larger study (sample size= 683,365 [START_REF] Yengo | Meta-analysis of genome-wide association studies for height and body mass index in∼ 700000 individuals of European ancestry[END_REF] ) from multi-trait GWAS applied on the BMI study in the present analysis (sample size= 339,224 [START_REF] Locke | Genetic studies of body mass index yield new insights for obesity biology[END_REF] , Table S1). Across the 1,776 sets containing BMI, JASS detected 1,167 new associations (after Bonferroni correction, Material and methods) of which 537 corresponded to a new association in the larger BMI GWAS. 86 associations of the larger GWAS were missed by JASS. Hence, JASS was able to flag loci with a high recall (0.86, probability of new association detected in the larger GWAS to be detected by JASS) but a moderate specificity (0.46, probability of a detected loci to be associated in the larger GWAS). This can be explained by the generality of the null hypothesis used in JASS which requires only one trait in the set to be significant (not necessarily BMI). To improve specificity, we fitted a logistic regression predicting if a locus would be associated with BMI in the larger GWAS by combining the number of sets where the locus was associated by JASS and the minimum P-value across sets (Fig. S16). This model reached an AUC of 0.75, an accuracy 0.74, a recall of 46% and a specificity of 75% when applying a standard probability threshold of 0.5. Based on JASS results, we were able to infer a substantial number of loci associated in a GWAS with twice as many samples as the one used in the present study.

Discussion

This study investigated the genetic features associated with the statistical power of multi-trait GWAS. On average, the power increase relative to the univariate GWAS was substantial: JASS detected new association loci in 98% of 19,266 sets, with an average of 26 new association loci. This power increase appears to be highly associated with the genetic features of trait sets. More specifically, multi-trait gain tends to be higher for sets with 1) a moderate mean heritability (mean ℎ #$%& '

), 2) a smaller mean MES, 3) a larger mean polygenicity, 4) a larger genetic covariance across traits, and 5) a smaller distance between the residual and genetic covariance. Although some features, such as increased distance between genetic and residual correlation, have been previously found associated with the gain in association detection, this analysis suggests a negative or negligible gain in real data.

This might be explained by the complex correlation across features, highlighting the importance of considering multiple features jointly or by variations of genetic architecture being narrower in real data than assumed by previous simulation studies. We also found that selecting specific traits such as BMI, and more clinically heterogenous sets, specifically for the omnibus test, can strongly outperform approaches that select clinically homogenous sets. Finally, we investigated the predictive power of a multivariate linear model that could predict trait sets that most likely benefit from the multi-trait test (median Pearson's r=0.43, P < 1.6 x 10 -60 ), which can be used to astutely select traits to be tested jointly. Our findings provide an approach that can increase the identification of genetic associations using existing GWAS data, with relevance to traits in which genetic signal is scarce. We summarize in

Figure S18

a guideline for selecting the best suited methods according to the feature of their data.

Selecting clinically homogenous traits is the strategy most commonly used 4,8,10,14-19 . In our previous large-scale analysis 8 , an heterogeneous set yielded the largest number of new association as compared to clinically homogenous sets. In this work we further showed that a data-driven strategy is expected to outperform other strategies based on clinical insights. Such sets might capture highly pleiotropic signals hard to detect using univariate GWAS and recommend that investigators compose a heterogeneous set or use our predictive model to build a set of traits.

We compared the results from the standard omnibus test (implemented in JASS) against MTAG, a popular multi-trait GWAS method [START_REF] Rahmioglu | The genetic basis of endometriosis and comorbidity with other pain and inflammatory conditions[END_REF][START_REF] Sveinbjornsson | Multiomics study of nonalcoholic fatty liver disease[END_REF][START_REF] Wang | Challenges and opportunities for developing more generalizable polygenic risk scores[END_REF][START_REF] Wang | Genome-wide association analyses of physical activity and sedentary behavior provide insights into underlying mechanisms and roles in disease prevention[END_REF] . For the data we used, the omnibus almost systematically outperformed MTAG, with an overall three-fold increase in the number of loci detected. The difference was particularly striking on larger trait sets. MTAG is built on the hypothesis of homogeneous genetic correlation across genetic variants, and therefore is expected to have maximum power when this assumption is valid. Indeed, we observed that more homogenous trait sets yielded larger MTAG gain than heterogenous trait sets (Fig. S17A,B). Yet, the omnibus generally performed equally well or better than MTAG even on the homogenous trait sets (Fig. S17C-F). In contrast, by construction, the omnibus test allows for substantial heterogeneity, although at the cost of an increase in the degree of freedom. In line with previous work suggesting that genetic correlation might be fairly heterogenous across the genome [START_REF] Werme | An integrated framework for local genetic correlation analysis[END_REF] , this cost appears to be outweighed by the additional flexibility in capturing heterogeneous multi-trait genetic patterns 8 (Supplementary notes 1, 2). Thus, the omnibus test is recommended for general identification of variants that impact phenotypes, while MTAG is suitable for identifying variants associated with specific traits (Fig. S18).

Our study has some limitations. First, we focused on commonly measured genetic features, but based on these results, a number of other refined metrics could be used. These include effect size distribution as measured by the alpha parameters [START_REF] Miao | Dissecting the high-resolution genetic architecture of complex phenotypes by accurately estimating gene-based conditional heritability[END_REF] . Second, we considered GWAS derived from common diseases and anthropometric traits. Future studies might explore performances using a wider variety of molecular traits, for which GWAS summary statistics are becoming increasingly available.

Third, the estimation of the features might be also refined. Here, we used MiXeR [START_REF] Frei | Bivariate causal mixture model quantifies polygenic overlap between complex traits beyond genetic correlation[END_REF][START_REF] Holland | Beyond SNP heritability: Polygenicity and discoverability of phenotypes estimated with a univariate Gaussian mixture model[END_REF] to estimate most features. However, we observed a dependency of MES and polygenicity on Neff. Improving these metrics could improve the overall analysis and interpretations of the results. Fourth, we focused on European ancestry summary statistics. This decision was motivated by the availability of large GWAS and using one ancestry for linkage disequilibrium; however, by doing so, we disposed of many traits with which we could have had a wider variety of genetic features, which might have improved the performance of the predictive model. This focus should not lead the reader to think that multi-trait GWAS is useful only on large sample studies of European ancestry. We actually recently updated the JASS pipeline to run a Multi-ancestry Multi-trait GWAS, which was able to detect 367 new association loci, despite the modest sample size of the non-European cohorts used [START_REF] Troubat | Multi-trait GWAS for diverse ancestries: Mapping the knowledge gap[END_REF] . Future work might leverage non-European existing [START_REF] Ishigaki | Large-scale genome-wide association study in a Japanese population identifies novel susceptibility loci across different diseases[END_REF] and upcoming biobanks [START_REF]The "All of Us" Research Program[END_REF] to investigate the validity of our results for non-European ancestries.

In conclusion, this study provides a first overview of what to expect when applying multi-trait tests to a variety of data and how to maximise new discoveries. These insights can be leveraged to discover genetic variants associated with human complex traits and diseases missed by univariate analysis at no cost. Beyond mapping, JASS used on clinically heterogeneous trait sets might offer a way to understand a shared genetic aetiology among unexpected traits [START_REF] Hackinger | Statistical methods to detect pleiotropy in human complex traits[END_REF] and contribute to deeper understanding of pleiotropy.

Estimated and derived genetic features

We investigated the effect of both single and multi-trait features. Single GWAS features include the effective sample size (𝑁 !"" ), linear additive common variants heritability (ℎ #$%& '

), polygenicity, MES, and the proportion of uncaptured linear additive common variants heritability (%h²u). Multi-trait GWAS features include the number of traits in a trait set (k), the average of the off-diagonal of genetic covariance and the residual covariance ( Σ ' ) and Σ ' * , respectively), condition numbers of genetic covariance matrix and residual covariance matrix (κ ) and κ < , respectively) across traits, and average distance between the genetic and residual correlation matrices (Δ + ). All parameters were aggregated to form a vector of 11 features per trait set. For the single GWAS parameters, MES, polygenicity, 𝑁 !"" ,

ℎ #$%& '
, and %h²u, we calculated mean values across each set of traits.

Polygenicity and heritability (ℎ #$%& '

) were estimated using MiXeR [START_REF] Frei | Bivariate causal mixture model quantifies polygenic overlap between complex traits beyond genetic correlation[END_REF][START_REF] Holland | Beyond SNP heritability: Polygenicity and discoverability of phenotypes estimated with a univariate Gaussian mixture model[END_REF] , with the 1000 Genomes Phase3 reference panel provided along the MiXeR package containing approximately 10 million common variants [START_REF]A global reference for human genetic variation[END_REF] . Following the authors recommendation, we defined the parameter for effective sample size as 𝑁 !"" = 1 (1 𝑁 =>?! ⁄ + 1 𝑁 =5-7*59? ⁄ ) ⁄

. For comparison purposes, we also estimated h 2 GWAS using the LDscore regression [START_REF] Bulik-Sullivan | An atlas of genetic correlations across human diseases and traits[END_REF] , and the two metrics were consistent (Pearson 𝜌=0.86, Fig. S5).

The estimated polygenicity by MiXeR showed a dependency on the GWAS sample size, with about a 10-fold increase of the polygenicity for an increase of 500,000 of the sample size (Fig. S19). We therefore adjusted polygenicity by taking the residuals of linear regression between log10 polygenicity and 𝑁 !"" : log 12 𝑝𝑜𝑙𝑦𝑔𝑒𝑛𝑖𝑐𝑖𝑡𝑦 (𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑) = log 12 𝑝𝑜𝑙𝑦𝑔𝑒𝑛𝑖𝑐𝑖𝑡𝑦 (𝑚𝑖𝑥𝑒𝑟) -𝛼𝑁 !"" , where 𝛼 was estimated by a linear regression log 12 𝑝𝑜𝑙𝑦𝑔𝑒𝑛𝑖𝑐𝑖𝑡𝑦 (𝑚𝑖𝑥𝑒𝑟) ~ 𝛼𝑁 !"" + 𝜀. We obtained 'adjusted polygenicity' as 10 [ABC !" D59E)!-.=.7E (>HI(?7!H)] . We computed MES as

L #$%& ' D59E)!-.=.7E (>HI(?7!H) .
The proportion of uncaptured linear additive common variants heritability (%ℎ M ' ) was derived as

(ℎ #$%& ' -ℎ #$%&NL.7? ' )/ℎ #$%& '
, where ℎ #$%&NL.7?

' denotes the heritability accounted by the univariate GWAS association loci. It was derived using the lead variants from each locus reaching genome-wide significance (P < 5 x 10 -8 ): ℎ #$%&NL.7? ' = ∑ 𝛽 .

' .∈P

, where 𝛽 . = 𝑧 . / l 𝑁 !"" . We excluded loci with lead variant whose 𝑎𝑏𝑠(𝛽) > 0.19, because ℎ #$%&NL.

Assessment of features associated with multi-trait association gain

The contribution of features to multi-trait gain was estimated using a five-fold cross validation. For each round of cross validation, the 72 GWAS were randomly split into a training and validation data, each including 36 GWAS. Within each cross validation, we generated 1,980 unique random trait sets, containing 2 to 12 traits, for each of the training and validation data: random sampling out of the 36 traits (660 sets generated), random sampling out of traits with common SNP heritability below median (330 sets) and above median (330 sets); random sampling out of traits with MES below median (330 sets) and above median (330 sets). For this stratification of traits by the median of MES, we used polygenicity / ℎ #$%& ' estimated by MiXeR without adjusting for the effective sample size. For each set, we ran JASS and derived the 11 features of interest (𝐹 . , 𝑖 = 1, … , 11) and the multi-trait gain. We used 19,793 trait sets out of the total 19,800 (=1,980 x 2 x 5) sampled for which the whole analysis process completed without error. Errors include cases where there was no association detected by both the univariate and joint tests.

Moving to the multivariate regression analysis, we selected six out of the 11 features based on collinearity analysis (as described below). The six features and the multi-trait gain were standardised into a range between 0 and 1 using MinMaxScaler in scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] . This standardisation was applied at once on the entire dataset including training and validation data across the five-fold cross validation sets.

We used the training data to estimate the joint effect 𝛿 y . of each feature i from a multiple regression:

𝑔𝑎𝑖𝑛 7*>.-~ ∑ 𝛿 . 𝐹 7*>.-,.
.

. This was conducted using the OLS function in statsmodels in Python [START_REF] Seabold | Statsmodels: Econometric and statistical modeling with python[END_REF] . We report Pearson's correlation coefficient as a metric of predictive power.

Collinearity and selection of features

We observed collinearity among some of the 11 features of traits across 19,266 unique trait sets.

Log10 𝜅 * , log10 𝜅 ) , log10 Σ ' * , and the number of traits were highly correlated (Pearson 𝜌 > 0.65).

Likewise, mean %h 2 u, mean log10MES, and mean log10polygenicity were highly correlated (abs(𝜌) > 0.8). These correlated features capture redundant characteristics of traits. Thus, we selected one out of each correlated features: the number of traits and mean %h 2 u. We chose the number of traits because it had the smallest P-value in a multivariate linear regression with all the features included, where inf values in condition number were replaced with their non-inf max value. We chose mean(%h 2 u) because it captures both mean(log10MES) (𝜌=-0.90) and mean(log10polygenicity) (𝜌=0.84).

The estimation of mean(%h 2 u) is also more straightforward than mean(log10MES) and mean(log10polygenicity). After this pre-selection, we built models with the remaining six features as described in the previous section.

Non-linear models

We considered two alternative non-linear models for prediction purposes: support vector regression (SVR), and random forest regression (RFR). SVR and RFR are regression approaches that allow for non-linear relationships. SVR's goal is to find a hyperplane (or line, in the case of twodimensional data) that best fits the data. It is effective at handling non-linear and complex data by using the kernel trick-mapping data with a kernel function into a higher-dimensional space where it is easier to find the best-fit hyperplane. SVR also penalises the complexity of the model and gives the flexibility in how much error is acceptable. Random forest regression performs a regression using decision trees. It generates multiple trees, fits each to a random subset of training data, and averages the predictions across trees as the final prediction. While the random sampling and averaging supposedly makes the model robust to outliers, RFR's performance relies on a high quality of training data; the training data needs to cover a wide range as RFR does not work well for extrapolation, and RFR leads to biased predictions when the training data is sampled in a biased way [START_REF] Hengl | Random forest as a generic framework for predictive modeling of spatial and spatio-temporal variables[END_REF] . In contrast, SVR is suggested to be capable of extrapolation [START_REF] Balabin | Interpolation and extrapolation problems of multivariate regression in analytical chemistry: benchmarking the robustness on near-infrared (NIR) spectroscopy data[END_REF] . We used the scikit-learn 37 python implementation of RFR and of the SVR. We fitted SVR and RFR models to the 𝑔𝑎𝑖𝑛 7*>.-including hyperparameters.

Hyperparameters were tuned using RandomizedSearchCV in scikit-lean across the following range:

SVR's kernel=[linear, rbf, sigmoid, poly], C= [1,10,50,100], epsilon=[10 -3 ,10 -2 ,10

MTAG analyses

For comparison purposes, we repeated the multi-trait test and the prediction analyses with the MTAG approach [START_REF] Turley | Multi-trait analysis of genome-wide association summary statistics using MTAG[END_REF] . MTAG uses a weighted sum of Z-score (Supplementary note 1). Trait weights are derived using the generalised method of moments, as (𝛽 y I 𝛽 y I R -𝛺 -𝛴 I ) = 0 , where 𝛺 is the genetic covariance matrix to be estimated for the weights, and 𝛴 I is the genetic covariance matrix estimated using the LD score regressions by Bulik-Sullivan et al [START_REF] Bulik-Sullivan | An atlas of genetic correlations across human diseases and traits[END_REF] . The model assumes that the genetic covariance matrix is homogenous across variants. We used MTAG with its default setting, which considered only complete cases. We ran MTAG for the same trait sets used for the analysis with JASS (all data used in the five-fold cross validation). MTAG outputs P-values for each trait in each trait set, whereas JASS gives a single P-value for a trait set. To account for the number of tests run by MTAG for one set, we obtained minimum P-values by MTAG across traits and variants in each locus and multiplied the minimum P-values by the number of traits in the set. For the comparison of the association gain between MTAG and JASS, we used both the minimum P-values with and without multiple test correction (Fig. 5, and Fig. S14).

Comparison of strategies for trait selection.

To compare the performances of trait selection strategies, we classified the 19,266 unique sets into clinically homogenous, low heterogeneity, high heterogeneity, or high predicted gain according to our predictive model. To assess clinical homogeneity, we first classified the 72 traits into clinical groups using the broadest categories in the MeSH Tree Structures 41 . The grouping was further refined based on clinician's insights (Table S1). We labeled each set of traits as 'homogeneous' if all the traits are in the same clinical group, 'low heterogeneity' if trait belonged to two to four clinical groups, 'high heterogeneity' if traits spanned five or more clinical groups. For the 'data-driven' method, we selected 100 sets of traits by CV fold that had the largest gains predicted by the linear model with aggregated coefficients across CV folds (Table 1). We used the Welch's t-test to evaluate the differences in gain and the number of new associated loci. We used a two-sided Welch's t-tests to determine whether the data driven method achieves a greater association gain than other methods, and whether jointly analysing clinically heterogenous traits achieves a greater association gain than jointly analysing clinically homogenous traits. For this test, we used a pair of training data and validation data that are mutually exclusive to ensure independence.

Evaluation of the relevance of new association

To evaluate the relevance of new associations detected by JASS (i.e. if most of them were true positive), we attempted to predict loci discovered in a recent large meta-analysis on BMI (sample size of 683,365 on average across ~2.3 million variants [START_REF] Yengo | Meta-analysis of genome-wide association studies for height and body mass index in∼ 700000 individuals of European ancestry[END_REF] ) from the results of multi-trait GWAS applied on 1,776 trait sets containing a smaller study of BMI (sample size of 339,224, Table S1). First, we compared loci detected by JASS (after a Bonferroni correction to account for the number of sets) and in the larger GWAS using the standard genome wide significance threshold of 5 x 10 -8 . Second, we fitted a logistic regression to predict associated loci in the larger GWAS by combining JASS P-value and the number of sets where the loci was considered associated with JASS. We use odd number chromosomes to fit the logistic regression and evaluated its performances on even number chromosomes.

Tables and Figures

Figure 1. Study overview

We conducted a power analysis on real data to understand in which setting a standard multi-trait test-the omnibus test-outperforms univariate GWAS. A) To assemble our real data, we curated 72 GWAS summary statistics and formed about 20k sets of traits by random sampling. Each set of traits was characterised by assessing key genetic features such as polygenicity, mean effect size (MES), and heritability. (B) For each set of traits, we ran omnibus test using JASS and computed the association gain compared to the univariate test. We defined this association gain as the number of LDindependent loci where the omnibus yields a smaller P-value than the univariate test. We repeated the analysis using MTAG, a popular multi-trait approach. (C) To investigate which genetic features (highlighted in A) explain the JASS (omnibus) association gain, we applied statistical models to predict the gain as a function of genetic features. Several models were benchmarked to optimise prediction performances. (D) To suggest a practical strategy for selecting traits that yield a large association gain, we compared the performance of JASS on trait sets that are clinically similar, clinically heterogenous, and that were predicted to have a large gain by the predictive model highlighted in (C). A) Illustration of the three metrics to assess univariate and multi-trait GWAS outcomes. On a quadrant plot representing the P-value of the multi-trait test with respect to the P-value of the univariate test, the following areas represent regions where: (green) only the univariate test is significant, (pink) only the multi-trait test is significant, (purple) the multi-trait test is more significant than the univariate test. B) Heatmap of the Pearson correlation between the number of univariate association loci, the number of new association loci detected by JASS, the association gain of JASS (fJASS<univ) and the 11 genetic features across 19,266 trait sets.

7 ?'R

 7 tended to become larger than ℎ #$%& ' .The mean genetic covariance and mean residual covariance were defined as the mean of the absolute value of the upper off-diagonal elements of the genetic and residual covariance matrices (𝚺 𝒈 and 𝚺 𝒓 ), i.e. 𝜎 ' ) ;./I , respectively, where 𝜎 ).I and 𝜎 *.I are 𝑖𝑗 elements of 𝚺 𝒈 and 𝚺 𝒓 , and k is the number of traits. The condition numbers of genetic and residual covariance matrices were computed as: 𝜅 ) = omaxs𝜆 ),. u /min (𝜆 ),. ), and 𝜅 * = omaxs𝜆 *,. u /min (𝜆 *,. ) where 𝜆 ),. and 𝜆 *,. are the eigenvalues of the genetic and residual covariance matrices. We used numpy.linalg.eig[START_REF] Harris | Array programming with NumPy[END_REF] for the eigen decomposition and assigned an infinite value to 𝜅 when the minimum eigenvalue was negative or close to zero. The average distance Δ + was defined as the mean over the absolute values of pairwise difference between the corresponding upper off, where 𝜌 ).I and 𝜌 *.I indicate the 𝑖𝑗 elements in the genetic and residual correlation matrices.

1 .

 1 univ,i = min(p 1, p 2, p 3, p 4 ) Association gain = f multi-trait<univ (p univ, p multi-trait ) Multi-trait and univariate tests comparison B o Linear model o Support vector regression o Random forest regression C Predict multi-trait test gain from genetic features Predict Gain = f (Genetic features) Mean effect size (mean MES) 2. # of traits (k) 3. Mean polygenicity 4. Mean effective sample size (mean N eff ) 5. Mean h 2 GWAS 6. Mean %h 2 u 7. Mean genetic-residual distance (∆ ! ) 8. Mean genetic covariance ( " Σ " ) 9. Mean residual covariance ( " Σ # ) 10.Condition no. of genetic cov. ($ " ) 11.Condition no. of residual cov. (! ! )

Figure 2 .Figure 3 .

 23 Figure 2. Genetic features characteristics derived from 72 traits and 19,266 random trait

!

  !"##$%&'( # of univariate associated loci # of multi-trait associated loci A B # of univariate associated loci # of multi-trait associated loci ! !"##$%&'(

Figure 4 .

 4 Figure 4. Model prediction power and feature contributions. (A) Boxplots of the prediction

Figure 5 .

 5 Figure 5. Comparison of MTAG with JASS. A) The number of new association loci found by

Figure 6 .

 6 Figure 6. Comparison between clinical and data-driven trait sampling methods. (A, B)

  a median of 1,110, and MES ranged from 4.48 x 10 -8 (variability sleep duration) to 3.3 x 10 -3 (Fasting proinsulin) with a median of 1.0 x 10 -4 . The distribution of the average these parameters across the 19,266 random sets is presented in Figure2Balong with the GWAS set features. The latter metrics ranged (in 25-75 percentiles) as -2.04 to -1.79 for log 12 Σ ' ) , -1.96 to -1.44 for log 12 Σ ' * , 0.36 to 0.78 for log 12 𝜅 ) , 0.05 to 0.32 for log 12 𝜅 * , and -1.11 to -0.89 for log 12 Δ + . In particular, the variability in log 12 Δ + was limited (given the theoretical upper bound of log 12 Δ + is log 12 2 ≅ 0.3). ) with log 12 Σ ' * and log 12 Δ + , log 12 𝜅 ) with log 12 𝜅 * , and log 12 Σ ' * with log 12 𝜅 * and log 12 𝜅 ) , which further correlated with the number of traits.

	We measured the correlation across features at the trait and set levels (Fig. 2C, 2D). The
	proportion of uncaptured linear additive heritability of common variants ( %ℎ ( ' ) was strongly
	associated with log 12 polygenicity (𝜌 = 0.76, P = 8.8 x 10 -15 ) and with log 12 MES (𝜌 = -0.8, P = 3.5 x
	10 -17 ) (Fig. 2C, S6), in agreement with previous reports showing that univariate GWAS performs better
	for traits with a larger MES and a smaller polygenicity 7 . As expected, the means of individual trait
	feature were correlated in the same way across sets as across traits (e.g. mean %ℎ ( ' and mean
	polygenicity are highly correlated as are %ℎ ( ' and polygenicity, Fig. 2D). Metrics of genetic and
	covariance matrices were positively correlated to each other, i.e. log 12 Σ '

Filtration Rate from Cystatin C) to 570,102 (variability sleep 6 duration) with

Table 1 . Coefficients of the multivariate linear regression models from five-fold cross validations.

 1 Mean genetic residual distance and mean genetic covariance were log10 transformed. All the features were scaled using MinMaxScaler in scikit-learn[START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] .

				CV1	CV2	CV3	CV4	CV5	Mean
	# of traits		0.096	0.077	0.082	0.049	0.083	0.077
				(p= 3.447x10 -44 )	(p= 1.592x10 -17 )	(p= 1.638x10 -44 )	(p= 6.148x10 -8 )	(p= 3.071x10 -19 )	(std=0.017)
	Mean log10 ∆ !	-0.787	-0.536	-0.393	-1.174	-0.610	-0.700
				(p= 8.338x10 -29 )	(p= 8.338x10 -29 )	(p= 2.242x10 -10 )	(p= 1.086x10 -26 )	(p= 1.947x10 -09 )	(std=0.301)
	Mean log10 Σ # "	0.767	0.789	0.355	1.149	0.675	0.747
				(p= 3.813x10 -45 )	(p= 2.055x10 -20 )	(p= 5.990x10 -15 )	(p= 1.604x10 -56 )	(p= 5.636x10 -21 )	(std=0.284)
	Mean Neff		0.206	-0.016	-0.071	0.260	-0.015	0.073
				(p= 4.342x10 -15 )	(p= 0.554)	(p= 0.003)	(p= 1.099x10 -10 )	(p= 0.598)	(std=0.149)
	Mean h 2	GWAS	-0.509	-0.437	-0.518	-0.690	-0.429	-0.516
				(p= 2.937x10 -96 )	(p= 1.455x10 -34 )	(p= 2.216x10 -111 )	(p= 6.670x10 -79 )	(p= 2.225x10 -26 )	(std=0.105)
	Mean %h 2	u	0.112	0.370	0.091	0.277	-0.064	0.157
				(p= 5.158x10 -6 )	(p= 4.353x10 -35 )	(p= 3.877x10 -10 )	(p= 3.022x10 -33 )	(p= 0.002)	(std=0.169)

Acknowledgments

This research was supported by the Agence Nationale pour la Recherche (GenCAST, ANR-20-CE36-0009). This work has been conducted as part of the INCEPTION program (Investissement d'Avenir grant ANR-16-CONV-0005). MHC was supported by R01HL162813, R01HL153248, R01HL149861, and R01HL147148.

Data and Code availability

https://gitlab.pasteur.fr/statistical-genetics/jass_suite_pipeline https://gitlab.pasteur.fr/statistical-genetics/jass https://gitlab.pasteur.fr/statistical-genetics/multitrait_power_traitselection

Author contributions

Declaration of interests

Material and Methods

Database of curated summary statistics

We assembled a database of 72 genome-wide GWAS summary statistics of quantitative traits and diseases conducted in European ancestry population pulled from the GWAS catalogue 31 and a variety of publicly available meta-analyses. We cleaned, harmonised and imputed each study using our previously developed pipeline [START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF] . In brief, the process includes the following steps: 1) alignment of each GWAS to the 1000G GRCh37 reference panel [START_REF]A global reference for human genetic variation[END_REF] , 2) imputation of missing summary statistics using RAISS [START_REF] Julienne | RAISS: robust and accurate imputation from summary statistics[END_REF] , 3) computation of the heritability, genetic and residual covariance matrices, referred further as ℎ #$%& ' , 𝚺 𝒈 and 𝚺 𝒓 , using LD-score regression [START_REF] Bulik-Sullivan | An atlas of genetic correlations across human diseases and traits[END_REF] , 4) aggregation of curated GWAS in a unique entry file used as input for JASS. We filtered all GWAS with negative heritability, resulting in a total of 72 traits (Table S1). Curated GWAS summary statistics used in the analysis are available on the JASS webserver https://jass.pasteur.fr/.

Joint test and association gain

Multi-trait analyses were conducted using the omnibus test implemented in the JASS package [START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF][START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF] .

For a set of 𝑘 GWAS, the omnibus statistics is defined as 𝑇 56-. = 𝐳 7 𝚺 𝒓 𝒛 where 𝐳 is the vector of Zscores across traits 𝐳 = (𝑧 1 … 𝑧 8 ) and 𝚺 𝒓 is the residual Z-score covariance derived using the LD-score regression. Under the null hypothesis of no association with any of the 𝑘 phenotypes, 𝑇 56-. follows a 𝜒 ' distribution with 𝑘 degree of freedom. To maximize data usage, the default setting of JASS uses all variants even those with missing association statistics. In this case, JASS returns association P-value based on the subset of Z-scores available. In contrast, the -remove-nans option removes variants with incomplete data. Here, we used -remove-nans option as the primary analysis for a better characterisation of trait sets and for a fair comparison with the default setting of MTAG 4 that do not allow for missing statistics, whereas we also provide some results from the default setting of JASS as an additional information.

All power comparisons were conducted at a locus level. The entire genome was split into a total of 1,703 quasi-independent loci defined based on linkage disequilibrium (LD)-independent blocks, as proposed by Berisa and Pickrell [START_REF] Berisa | Approximately independent linkage disequilibrium blocks in human populations[END_REF] . For both multi-trait and univariate analyses, we obtained the minimum P-value across variants in each locus. The gain of the multi-trait test was derived as the fraction of loci whose P-values were smaller than corresponding P-values in univariate GWAS corrected for the number of traits jointly analysed: 𝑔𝑎𝑖𝑛 = 𝑓 6(97./(-.0 = N (𝑃 6(97... < 𝑃 (-... • 𝑘)

where 𝑅 is the number of loci (R=1,703), 𝑘 is the number of traits (the number of GWAS studies) jointly analysed, 𝑃 6(97... is the minimum P-value of the multi-trait test in region 𝑖, and 𝑃 (-.0,. is the minimum P-value of the univariate tests across all GWAS analysed in locus 𝑖.