Convexity of Non-homogeneous Quadratic Functions on the Hyperbolic Space

In this paper, some concepts related to the intrinsic convexity of non-homogeneous quadratic functions on the hyperbolic space are studied. Unlike in the Euclidean space, the study of intrinsic convexity of non-homogeneous quadratic functions in the hyperbolic space is more elaborate than that of homogeneous quadratic functions. Several characterizations that allow the construction of many examples will be presented.


Introduction
The hyperbolic space was discovered due to attempts to understand Euclid's axiomatic basis for geometry dating back to the 1800s.This space is one of the most interesting models of non-Euclidean Riemannian manifold of negative constant sectional curvature, see for example [1,3,14].Since the discovery of the hyperbolic space, several efforts have been made to understand its properties and several models of it have emerged over the years, including the hyperboloid model (also called Lorentz model), the Poincaré half-plane model, the Poincaré disk model and the Klein model, see [1].The growing interest over the years in the hyperbolic space has resulted in a proven success story, helping to make impressive advances in many fields of science, one of the best known being general relativity, see [17].It is also worth mentioning that in many practical applications, the natural structure of the data is modeled in the hyperbolic space.Various topics of research use this type of modeling, see for example the papers in machine learning [13], artificial intelligence [12], neural circuits [15], low-rank approximations of hyperbolic embeddings [7,16], financial networks [8], complex networks [9,11], embeddings of data [19], strain analysis [18,20] and the references therein.
The convex quadratic functions are the most popular convex functions in the Euclidean space as well as various geometric contexts, occurring in many problems, such as eigenvalue optimization, least square approximation and linear regression.
The convex quadratic functions are the most popular convex functions in Euclidean space as well as in various geometric contexts, occurring in many problems such as eigenvalue optimization, least square approximation, and linear regression.A comprehensive study of the convexity of homogeneous quadratic functions in the context of spheres is discussed in [4].The aim of this paper is to study the convexity of non-homogeneous quadratic functions on the hyperbolic space in an intrinsic way.In particular, in this study, we will present several characterizations that allow the construction of several examples.To this end, among the aforementioned models of hyperbolic space, we choose the hyperboloid model.The study of the convexity of homogeneous quadratic functions in the hyperboloid model of the hyperbolic space was started in [5].As it is well-known, in Euclidean space, there is no conceptual difference between the convexity of homogeneous and non-homogeneous quadratic functions.However, we will see that in the hyperboloid model of the hyperbolic space, the conceptual intrinsic hyperbolic convexity of homogeneous and non-homogeneous quadratic functions are quite different, requiring much more effort than in the Euclidean scenario to understand it.The primary challenge lies in establishing the role of the linear term on the hyperbolic convexity of a non-homogeneous quadratic function.This is because introducing a linear term to a hyperbolically convex homogeneous function can result in the newly formed non-homogeneous quadratic function losing its hyperbolic convexity.
The structure of this paper is as follows.In Sect.1.1, we recall some notations and basic results.In Sect.2, we recall some notations, definitions and basic properties about the geometry of the hyperbolic space.The main results are presented in Sect.3. We conclude the paper by making some final remarks in Sect. 4.

Notation and Basics Results
For any real number α denote α + := max(α, 0) and α − := (−α) + .Let R m be the m-dimensional Euclidean space.Denote by e i is the i-th canonical unit vector in R n+1 .The Euclidean norm of u ∈ R m is denoted by u 2 := √ u u .The set of all m × n matrices with real entries is denoted by R m×n and R m ≡ R m×1 .For M ∈ R m×n , the matrix M ∈ R n×m denotes the transpose of M. The operator norm associated with Euclidean norm of a matrix M ∈ R m×m is defined by A 2 := max{ Mu 2 : u 2 = 1, u ∈ R m }.The numbers λ min (M) and λ max (M) stand for the minimum and maximum eigenvalue of the matrix M ∈ R m×m , respectively.If u ∈ R m , then diag(u) ∈ R m×m denotes a diagonal matrix with (i, i)-th entry equal to u i , i = 1, . . ., m.The matrix I denotes the m × m identity matrix.
In the following, we state a version of Finsler's lemma, see [6].A proof of it can be found, for example, in [10,Theorem 2].Lemma 1.1 Let M, N ∈ R n×n be two symmetric matrices with N = 0.If x N x = 0 implies x M x ≥ 0, then there exists λ ∈ R such that M + λN is positive semidefinite.
In order to state a special version of Lemma 1.1 in a convenient form, we take the diagonal matrix J ∈ R (n+1)×(n+1) defined by (1) By using the matrix (1), the Lorentz cone L and its boundary ∂L are defined, respectively, by A matrix M is called ∂L -copositive if z Mz ≥ 0, for all z ∈ ∂L .Then, combining Lemma 1.1 with the second equality in (2), we obtain the following special version of Lemma 1.1.
Corollary 1.1 Let M ∈ R n×n be a symmetric matrix.If M is ∂L -copositive, then there exists λ ∈ R such that M + λJ is positive semidefinite.
The dual cone of a cone K ⊂ R m is a cone defined by

Basics Results About the Hyperbolic Space
In this section, we recall some notations, definitions and basic properties about the geometry of the hyperbolic space used throughout the paper.They can be found in many introductory books on Riemannian and differential geometry, for example in [1,14], see also [2].Let •, • be the Lorentzian inner product of x := (x 1 , . . ., x n , x n+1 ) and y := (y 1 , . . ., y n , y n+1 ) on R n+1 defined as follows: x, y := x 1 y 1 + • • • + x n y n − x n+1 y n+1 . (3)

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For each x ∈ R n+1 , the Lorentzian norm (length) of x is defined to be the complex number Here, x is either positive, zero, or positive imaginary.By using (1), the Lorentz inner product (3) can be stated equivalently as follows: x, y := x Jy, ∀x, y ∈ R n+1 . ( Throughout the paper, the n-dimensional hyperbolic space and its tangent hyperplane at a point p are denoted by respectively.It is worth noting that the Lorentzian inner product defined in (3) is not positive definite in the entire space R n+1 .However, one can show that its restriction to the tangent spaces of H n is positive definite; see [2,Section 7.6].Consequently, v > 0 for all v ∈ T p H n and all p ∈ H n with v = 0. Therefore, •, • and • are in fact a positive inner product and the associated norm in T p H n , for all p ∈ H n .Moreover, for all p, q ∈ H n , p, q ≤ −1 and p, q = −1 if and only if p = q.Therefore, (3) actually defines a Riemannian metric on H n , see [3, pp. 67].The Lorentzian projection onto the tangent hyperplane T p H n is the linear mapping defined by The Lorentzian projection (7) is self-adjoint with respect to the Lorentzian inner product (3), i.e., (I + pp J)u, v = u, (I + pp J)v , for all u, v ∈ R n+1 and all p ∈ H n .Moreover, we also have The intrinsic distance on the hyperbolic space between two points p, q ∈ H n is defined by It can be shown that (H n , d) is a complete metric space, so that d( p, q) ≥ 0 for all p, q ∈ H n , and d( p, q) = 0 if and only if p = q.Moreover, (H n , d) has the same topology as R n .The intersection curve of a plane though the origin of R n+1 with H n is called a geodesic.If p, q ∈ H n and q = p, then the unique geodesic segment from p to q is Let f : H n → R be a twice differentiable function.The Hessian on the hyperbolic space of f at a point p ∈ H n is the mapping Hess f ( p) : where

Hyperbolically Quadratic Convex Functions
Our aim is to study the hyperbolic convexity of the non-homogeneous quadratic function f : where For that we first recall some general characterizations for a non-homogeneous quadratic convex function.We begin with the general definition of a convex function on the hyperbolic space.
Definition 3.1 A function f : H n → R is said to be hyperbolically convex (respectively, strictly hyperbolically convex) if for any geodesic segment γ , the composition f • γ is convex (respectively, strictly convex) in the usual sense.
In the following, we recall the general second-order characterization for hyperbolically convex functions on hyperbolic spaces, for a proof see [5,Proposition 5.4].

Proposition 3.1 Let f : H n → R be a twice differentiable function. The function f is hyperbolically convex if and only if the Hessian
Hess f on the hyperbolic space satisfies the inequality Hess f ( p)v, v ≥ 0, for all p ∈ H n and all v ∈ T p H n , or equivalently, where D2 f ( p) is the usual Hessian and D f ( p) is the usual gradient of f at a point p ∈ H n .If the above inequalities are strict, then f is strictly hyperbolically convex.
In the following, we present a general characterization for convexity of the function (10), which is an immediate consequence of Proposition 3.1.

Corollary 3.1 Let
Thus, it follows from Proposition 3.1 that the function f is hyperbolically convex in H n if and only if 2v Av + 2 p Ap + b p ≥ 0, for all p ∈ H n , all v ∈ T p H n with v Jv = 1.Considering that p ∈ H n and v ∈ T p H n with v Jv = 1 if and only if p J p = −1, v Jv = 1 and p Jv = 0, the result follows.
Next, we relate the hyperbolic convexity of f with the boundary of the Lorentz cone ∂L .

Lemma 3.1 Let
The following three conditions are equivalent: Proof First, we prove the equivalence between (i) and (ii).For that, it is convenient first to consider the following invertible transformations where x, y, p, v ∈ R n+1 .By using the first two equalities in (11), after some calculations, we have On the other hand, by using the last two inequalities in (11), we obtain the following three equalities Moreover, the equalities in (11) also imply that First we prove (i) implies (ii).Take x, y ∈ ∂L and x Jy = −1, and consider the transformation (11).Thus, by using ( 12), we conclude that p J p = −1, v Jv = 1 and p Jv = 0. Hence, item (i) together with Corollary 3.1 implies that 2v Av+2 p Ap+ b p ≥ 0. Therefore, by using ( 11) and ( 14), we conclude that 4x Ax + 4y Ay + √ 2 b (x + y) ≥ 0 and item (ii) holds.Next, we prove that (ii) implies (i).Assume that the item (ii) holds, and take p, v ∈ R n+1 with p J p = −1, v Jv = 1 and p Jv = 0, and consider (11).Hence, by using (13), we have x (or − x) ∈ ∂L , y (or − y) ∈ ∂L and x Jy = −1, and item (ii) implies that 4x Ax + 4y Ay + √ 2 b (x + y) ≥ 0. Thus, (11) and ( 14) implies that 2v Av + 2 p Ap + b p ≥ 0, which implies that item (i) holds.
We proceed to prove the equivalence between (ii) and (iii).Assume that item (ii) holds and take z, w ∈ ∂L and z Jw < 0. Since z Jw < 0, we define Thus, considering that z, w ∈ ∂L and z Jw < 0, some calculations show that x, y ∈ ∂L and x Jy = −1.Therefore, using (15) together item (ii), we conclude that z Az + w Aw = −z Jw x Ax + y Ay ≥ 0, and the item (iii) holds.Finally, (iii) implies (ii) is immediate, which concludes the proof.

Proposition 3.2 Let
where for all z, w ∈ ∂L with z Jw < 0 and all k ∈ N with k = 0. Hence, by tending with k to infinity, we obtain that 4z Az ≥ 0, for all z ∈ ∂L .Therefore, A is ∂L -copositive, and by using Corollary 1.1, the result follows.
Since the condition x Jy ≤ 0, for any x, y ∈ ∂L , is equivalent to the n-dimensional Cauchy inequality, it all holds.Then, the equivalence between items (i) and (iii) of Lemma 3.1 can be stated equivalently in the following form.

Proposition 3.3 Let
In next corollary, we present a characterization for a linear function to be hyperbolically convex.In the Euclidean context, the linear term of a quadratic function has no influence on the convexity of the function.As we will see in the next corollary, this is not the case in the hyperbolic setting.

Proposition 3.4 Let
is a positive semidefinite matrix.

Corollary 3.5 Let
Proof The proof follows from Proposition 3.4 by using the items (i) and (iii) of [5,Theorem 5.1].
for all x, y ∈ ∂L and k ∈ N .Multiplying the latest inequality by k 3 , then tending with k to infinity and finally dividing by −2x Jy , we obtain b x ≥ 0, for all x ∈ ∂L .Hence, b ∈ (∂L ) * = L .Therefore, item (v) holds and the proof is completed.
For simplifying the statement and proof of the next results, it is convenient to introduce the following notations.For a given A ∈ R (n+1)×(n+1) , consider the following decomposition: Denote Ī ∈ R n×n the identity matrix.In addition, for a given vector z ∈ R n+1 , consider the following decomposition: By using the above decompositions, we have the following result: Proof First note that, by using ( 2) and ( 19), we conclude that all x, y ∈ ∂L can be written Hence, using ( 18), ( 19) and ( 20), we obtain that By using equations ( 21), ( 22), ( 23) and ( 24), we conclude that Therefore, by using the last equality together with Proposition 3.3, the desired result follows.

Remark 3.2 Let
To see that set ȳ = 0 in Proposition 3.6.

Corollary 3.6 Let
Proof First note that by letting ȳ = − x in Proposition 3.6, we obtain that if f is hyperbolically convex then 8 x Ā + σ Ī x + 4b n+1 x 2 2 ≥ 0, for all x ∈ R n .Considering that and f is hyperbolically convex, the result follows.

Proposition 3.7 Let
Then, there exists a λ > 0 such that the function f λ : Proof Consider the following matrix The eigenvalues of the matrix (25) are the form β + σ + 1 2 (b n+1 − λ), where β is an eigenvalue of Ā.Thus, the matrix (25) have negative eigenvalue by taking λ > 0 sufficiently large.Consequently, the matrix (25) will not positive semidefinite for λ > 0 sufficiently large.Therefore, by applying Corollary 3.6 for f = f λ the result follows.

Theorem 3.1 Let
R be defined by f ( p) = p Ap + b p + c and g : H n → R be defined by g( p) = p T p + b p + c.Then, the following statements hold: (ii) If there exists a μ ∈ R and a λ ≥ 0 such that the matrix A − λI + μJ is positive semidefinite and b for any x, y ∈ R n+1 with x, y ∈ ∂L .Divide (27) by A 2 and use (26) to obtain 4x x + 4y y + −2x Jy b A (x + y) ≥ 0, for any x, y ∈ R n+1 with x, y ∈ ∂L .Thus, applying Proposition 3.3 with f = h, it follows that h is hyperbolically convex.Proof of item (ii): First note that by using Cauchy and triangular inequalities, we have Taking into account that b + 4λe n+1 ∈ L , we obtain that b n+1 − b 2 ≥ −4λ.Hence, we have On the other hand, some calculations show that If x and ȳ are not parallel, then Cauchy inequality implies that 4 x 2 ȳ 2 ≥ 2 x 2 ȳ 2 − 2 x ȳ > 0. Thus, it follows from the last inequality that Combining the previous equality with (28), we obtain that which, after some algebraic manipulations, can be rewritten equivalently as follows: The last inequality holds for any x, ȳ ∈ R n , since it is also true for x and ȳ parallel or if any of x and y is zero.To proceed, note that by using ( 2) and ( 19), we conclude that all x, y ∈ ∂L can be written In addition, for any x, y ∈ ∂L , we have (32) Applying Corollary 3.6 with f = h, it follows that is positive semidefinite, which implies that 2 + (1/2)b A n+1 ≥ 0. Therefore, it follows from (32) that 2 + (1/2)(b n+1 / A 2 ) ≥ 0, which is equivalent to the required inequality.for all x, ȳ ∈ R n .The first statement follows after dividing the last inequality by the quantity

Corollary 3.9 Let
and then using the definition of the infimum.The second statement is a particular instance of the first one.

Corollary 3.10 Let
Then, the following statements hold: Proof Proof of item (i): By using the notations of Theorem 2 and triangular and Cauchy inequality, we have Considering that λ min ( Ā) + σ ≥ 2 ā 2 , we obtain from the last inequality that Taking into account that inf or equivalently that b n+1 + ϕ b, A ≥ 0. Hence, applying Theorem 3.2, we obtain that f is hyperbolically convex and the first statement of item (i) is proved.The second statement is an immediate consequence of the first one.

Final Remarks
This paper is a natural continuation of [5], where the study of convexity of homogeneous quadratic functions in hyperbolic spaces was started.In contrast to the Euclidean space, the study of non-homogeneous quadratic functions in the hyperbolic space is more elaborate than the study of homogeneous quadratic functions.In [4], the study of the convexity of quadratic functions on the sphere was conducted, with a specific focus on homogeneous functions.An intriguing investigation that merits further exploration involves characterizing non-homogeneous quadratic functions on the sphere, complementing the findings presented in the paper [4].It is worth noting that when defining a quadratic function on the sphere and in the hyperbolic space, we rely on the fact that these manifolds are subsets of the Euclidean space.In fact, this kind of study could potentially be extended to manifolds that are subsets of matrix spaces.However, conducting a comprehensive study of convex quadratic functions, or more generally, engaging in convex analysis within a general Riemannian manifold, remains a challenging endeavor.
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