Approximation to uniform distribution in SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SO}(3)$$\end{document}

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SO}(3)$$\end{document}, with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SO}(3)$$\end{document}. The variance of rotation matrices sampled by a certain determinantal point process is estimated, and formulas for the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest.


Introduction and results
In this paper we study properties of a finite collection of randomly generated points in SO(3), the rotation group of 3-dimensional Euclidean space, sampled bya certain determinantal point process (dpp). It turns out that these points tend to be well distributed, a property that is important for discretization, integration and approximation. Our goal is not to compute actual collections of evenly distributed rotation matrices, but rather to provide a comparison tool that allows one to decide the effectiveness of any given method.
If one is given an algorithm to generate finite (but arbitrarily large) collections of matrices, common methods to measure how well distributed these are include either calculating some discrete energy of them or looking at the speed of convergence of the counting measure towards uniform measure. Most work in this direction has been done on spheres of various dimensions, see the monography [8] for a very complete survey of the state of the art of this question; the particular question of finding collections of spherical points with small energy was posed by Shub and Smale in [21] and is nowadays known as Smale's 7th problem [22].
In order to extend part of the work done on spheres to the context of rotation matrices, we will obtain bounds on various energies for points generated through a certain dpp (technically speaking, a dpp is a counting measure where one identifies the measure with its set of atoms). Briefly, such a process is obtained by taking a Hilbert space H(X ) (usually H(X ) = L 2 (X )) of an underlying measure space (X , μ) and an N -dimensional subspace H ⊂ H(X ), with projection kernel K onto H. Then, under mild conditions on X , one is guaranteed almost surely the existence of such a process with N distinct points in X associated to K.
The theory of those processes has been summarized in [7]; there one also finds a pseudo-code which samples points from any given dpp. A main feature of the underlying points is that they tend to repel each other, and hence have become the theoretical basis of construction of well-distributed points on various symmetric spaces, see for instance [2,5,6,19].
Since one can sometimes compute the expected value of the energy of points coming from these processes with high precision, they have been used as a tool to understand the asymptotic properties of the discrete energy in that context; and in particular, for even dimensional spheres, with the exception of the usual 2-sphere, the best known bounds for some energies have been proved using this approach.
We will employ the same method for SO (3), considering first the (discrete) Riesz s-energy for A = {α 1 , . . . , α N } ⊂ SO (3): with α j being thought of as rotation matrices, · F being the Frobenius or L 2 -norm, and s ∈ (0, ∞). In contrast to this, the continuous Riesz s-energy is given by replacing the double sum by the double integral over SO (3). We further set The investigation of these sums is very popular and results usually describe the behavior of the first leading terms. This seems particularly interesting in case s equals the dimension, where we have following result.
for L ∈ N, then the Riesz 3-energy satisfies where γ is the Euler-Mascheroni constant.
The right-hand side is the expected value of the Riesz 3-energy with underlying points generated by a certain dpp. Now, given any particular method of generating finite point sets in SO (3) Here β 3 is the volume of the unit ball in R 3 and V ol(SO (3)) is the volume, i.e., the Haursdoff measure, of SO(3) as a subset of R 3×3 ≡ R 9 ; see [15] for a computation of that volume. We can thus see that random points from our dpp give the correct order of the asymptotic. We now turn our attention to the Green energy, where we obtain bounds with the continuous Green energy as coefficient of the factor N 2 (zero in this case), and narrow the domain of the leading coefficient of the second term.
To recap, a Green function G L for a linear differential operator L is an integral kernel to produce solutions for inhomogeneous differential equations and is unique modulo Ker(L). In our case, we deal with the Laplace-Beltrami operator g , and note that Ker( g ) is the set of harmonic functions-which are just constants on a compact Riemannian manifold (M, g). We will construct G = G g in such a way that it integrates to zero and speak of the Green function.
The (discrete) Green energy for A = {α 1 , . . . , α N } ⊂ SO(3) will be given by and we let It is noteworthy that G(α, β)d(α, β) ≈ 1 for α close to β in geodesic distance d(·, ·), and a set of points with small Green energy is hence expected to be well-distributed, which is indeed the main result in [4]. We know that if {α 1 , . . . , α N } attains the minimal possible energy, then the associated discrete measure approaches the uniform distribution in SO(3) as N → ∞. A set of points with small Green energy is also expected to be well-separated, see [9]. Now, G(·, β) is for any β ∈ SO(3) a zero mean function by definition, and if {α 1 , . . . , α N } were simply chosen uniformly and independently in SO(3), then the expected value of the Green energy would equal 0, so in particular we have E G (N ) ≤ 0. In this note we prove the following much stronger result.
The right-hand side is the expected value of the Green energy with underlying points generated by a dpp, and that is where we have the restriction for N , as the process is related to subspaces H that we can project onto. The lower bound is valid for all N .
As mentioned above, another classical measure of the distribution properties of {α 1 , . . . , α N } is the speed of convergence to uniform measure, which can be understood by choosing some range sets {A j } j∈I measurable with respect to the Haar measure μ and investigating the behavior of as N grows large. We will tackle this problem probabilistically, where we turn the count of points in A j into a random variable.
In analogy to spherical caps on spheres, the range sets for SO(3) will be the balls B(α, 2ε) := {β ∈ SO(3) : ω(α −1 β) < 2ε} for ε ∈ 0, π 2 and ω(·) being the rotation angle distance introduced in the following sections. For given random points {α 1 , . . . , α N } and fixed α ∈ SO(3), we define random variables via characteristic functions Now, for a collection of random uniform points chosen independently in SO(3), denoting by 1 the identity matrix in SO(3), we have 2ε)), and the variance can also be computed from the independence of the points: We are able to bound the variance of this quantity for our dpp, proving that it is much smaller than in the previous case.
for L ∈ N, and ε ∈ 0, π 2 be fixed. Then the points generated by the dpp given in Lemma 2.3 satisfy and moreover From Theorem 1.4 and for any fixed ε, we then have by Chebyshev's inequality for example, letting T = N 1/3 log(N ) and with some little arithmetic we obtain sup α∈SO(3) In other words, for large N the counting and Haar measures are very similar with high probability.

Introductory Concepts
In this section we collect some definitions and previous results that we will use and that are intended to make this manuscript reasonably self-contained. Definitions of Chebyshev polynomials and alike are postponed to Sect. 2.4.

Structure, distances and integration in SO(3)
The special orthogonal group SO(3) is the compact Lie group of 3 by 3 orthogonal matrices over R that represent rotations in R 3 ; i.e., with determinant equal to one. It is a 3 dimensional manifold and since it is naturally included in R 9 it is customary to let it inherit its Riemannian submanifold structure.
Following [14], using Euler angles (ϕ 1 , θ, are rotations around the z-axis and x-axis respectively. The normalized Haar measure (i.e., the unique left and right invariant probability measure in SO (3)) is given by dμ(R) = 1 8π 2 sin(θ )dϕ 1 dθ dϕ 2 , and it corresponds to the inherited Riemannian submanifold structure of SO(3) up to the normalizing constant.
The Riemannian distance associated to the structure of SO (3) is certainly a natural and useful concept, but for us it will be more convenient to use the so called rotation angle distance defined as follows: for α, β ∈ SO (3), Its convenience stems from the following fact, see for example, [14, page 173]: By the monotone convergence theorem (1) is also valid if f ,f are just assumed to be non-negative and measurable.

Laplace-Beltrami operator and Green function in SO(3)
The Laplace-Beltrami operator g is defined on any Riemannian manifold (M, g) in terms of the Levi-Civita connection. Following [10], if γ 1 (t), . . . , γ n (t) is a set of geodesics in an n-dimensional manifold such that γ j (0) = p ∈ M for all 1 ≤ j ≤ n, and such that {γ j (0)} form an orthonormal basis of the tangent space T p M (geodesic normal coordinates), then the action of g on C 2 -functions f at p is given by Note the convention given by the minus sign in front of the sum, which sometimes leads to confusion given the Laplacian in R n . The convention we use here is widely accepted, see for example [17]. A Green function G = G g is a distributional solution to where μ dV (M) is the Riemannian volume form in M. Defined this way it is unique modulo Ker( g ) and it is common practice to add a constant in such a way that for all y ∈ M the function G(·, y) has zero mean, see [3]. We use this convention and simply refer to G as the Green function. It further follows from classical Fredholm theory that where 0 = λ 0 < λ 1 ≤ λ 2 ≤ · · · is the sequence of eigenvalues for g and {φ j }, j ≥ 1 is a complete orthonormal set of associated eigenfunctions. Hence, this is true locally on any smooth manifold.
In the case M = SO (3), we obtain a Green function which is independent of any particular chart, thus valid globally. The eigenvalues and eigenfunctions of g are known from the classical theory of continuous groups and have been intensively studied, see [14,16], [25, §15]:

then the dimension of H is (2 + 1) 2 and an orthonormal basis of H is given by
The actual form of the Wigner D-functions will not be important for us, since we will only use the fact that they constitute an orthogonal basis and the following summation formula: (3) where U 2 (x) is the Chebyshev polynomial of second kind and degree 2 , which will be briefly introduced in Sect. 2.4. For more on formula (3)

Determinantal point processes
We point the reader to the excellent monograph [7] for an introduction to point processes, and we briefly summarize part of this material below. As in [5,6], we will use only a fraction of the theory.
A simple point process on a locally compact Polish space with reference measure μ is a random, integer-valued positive Radon measure η, that almost surely assigns at most measure 1 to singletons-we shall think of it as a counting measure The joint intensities of η with respect to μ, if they exist, are functions ρ k : k → [0, ∞) for k > 0, such that for pairwise disjoint sets {D s } k s=1 ⊂ , the expected value of the product of number of points falling into D s is given by and ρ k (y 1 , . . . , y k ) = 0 in case y j = y s for some j = s.
A simple point process is determinantal with kernel K if and only if for every k ∈ N and all y j 's (4) where we write dμ(x, y) as an abbreviation for dμ(x) dμ(y) and • E g(x 1 , . . . , x N ) means expected value of some function defined from M ×· · ·× M (N copies of M) to [0, ∞], when x 1 , . . . , x N are chosen from the point process associated to H; the orthogonal projection of f onto H can be computed via: Note that if ϕ 1 , . . . , ϕ N is an orthonormal basis of H, then we can write and clearly Coming back to the case of interest and following ideas in [6], we choose as subspace H the span of the first eigenspaces of g . Recall the definition of classical Gegenbauer polynomials C (λ) n (t), λ > −1/2, a sequence of degree n polynomials orthogonal with respect to the weight (1 − t 2 ) λ−1/2 in [−1, 1], normalized in such a way that An equivalent definition of these polynomials is given by the formal power series (3)) be the span of the union of eigenspaces for eigenvalues λ 0 , . . . , λ L of g . Then, we define Moreover , the projection kernel is: We then consider the dpp associated to H L .

Proofs of the basic lemmas
The degree n+1 Chebyshev polynomials of first and second kind satisfy the recurrence relation with T 0 ≡ 1, T 1 (x) = x and U −1 ≡ 0, U 0 (x) ≡ 1 in their respective notation. With this said, using (2), (3), and (5), we obtain Further we list some equations for later reference and the reader's convenience.  , then by (7) and (9) The formula for the dimension of H L can be proved as follows. The eigenspace associated to λ = ( + 1) has dimension (2 + 1) 2 since this is the number of elements of its basis D m,n . Thus dim(H L ) is given by L =0 (2 + 1) 2 = 2L+3 3 .

Proof of Lemma 2.2
In (8) we apply the equality and argue, under the assumption w := ω(α −1 β)∈ (0, π], as follows where we used the well known fact that the power series for log(1 − x) at 1 converges at the boundary of its disc of convergence (except for x = 1) and equals the logarithm at these values: and similarly Further, by 1 − e −iw = 2ie −i w 2 sin( w 2 ), we conclude where we used a property of the principal branch of the complex logarithm: log(re iϕ ) = log(r ) + iϕ.

Riesz s-energy: Proof of Theorem 1.1
Recall that if A is a real matrix, we have A 2 F := Trace(A t A). We set throughout N = N (L) = C (2) 2L (1) for L ∈ N.
Proof We abbreviate w = ω(α −1 β), and use the half-angle formula for sine: Recall the definition of Euler's Beta function B(a, b) : We are now ready to state our first proposition.
The next line is, apart from the factor 4 8 s/2 π , the continuous Riesz s-energy: On the other hand, for 0 < s < 3 we have where we have used that |C The case s = 1 is Lemma 6.2; the case s = 2 follows from Lemma 6.4: where c u,u = c (2) u,u (2L) with notation as in Lemma 6.4.
In the next proof we use (1) and the digamma function ψ, see Sect. 6.
Proof of Theorem 1. 1 We proceed as in the previous proof and use Lemma 6.4. In particular, we use the notation of that lemma for c j,k = c (2) j,k (2L): We use (14) and obtain By c r +u,u = c and hence, by well known formulas for the sum of powers of integers: Invoking Lemma 3.3 yields (2L) 6 6 ψ(2L) − (2L) 6 proving the claim when multiplied by 4 8 3/2 π . as the sum can be bounded from above and below by the same integral, apart from integration boundaries, where we obtain the error term. We finish by integrating:

Green energy: Proof of Theorem 1.3
We prove the lower and upper bound separately in the following two sections.

Estimate of the Green energy: lower bound
We follow an exposition due to N. Elkies, found in [18,. The results in [18] are stated in detail for Riemann surfaces, i.e., one-dimensional complex manifolds, although it is mentioned that the argument can be extended to more general manifolds. Here we work out the details for SO (3).
The idea is to find a function with nice properties smaller than G, and to bound its energy from below. For α, β ∈ SO(3) and t > 0, we define: Quantitative estimates depend on asymptotics for this function. The following is the version of Elkies' result for SO(3).

Lemma 4.1 For all t > 0 and α = β we have
Proof Using uniform convergence, we differentiate term by term and define Given a smooth test function φ, with uniformly converging representation as ∞ =0 φ , where φ = m,n ϕ m,n D m,n √ 2 + 1, we set where we interchanged integration and summation by uniform convergence and used that {D m,n √ 2 + 1} is an orthonormal basis. Now we have uniformly For t > 0 fixed, we can interchange differentiation and integration yielding By the strong maximum principle (Theorem A.2), we have for every t > 0: The same PDE and estimates hold for v(α, t) = u(α, t) + φ 0 .
If φ ≥ 0, then so is v(α, t) for all t > 0 by the maximum principle as v(α, 0) = φ(α). Hence We further set where we interchanged sum and integral again. Differentiating term-wise for t > 0 yields Finally, for fixed α let t > > 0, then by the fundamental theorem of calculus: and thus, for all non-negative test functions φ Since the integrand is continuous, this proves that for t > and for any fixed α, β with α = β taking the limit as → 0 proves the result. Now by Lemma 4.1, we have for some t > 0 which will be determined later, and any collection of distinct points {α 1 , . . . , α N } ⊂ SO (3): Thus our remaining task is to find an asymptotic for G t (α, α) in t. First we note that For 0 < t 1 we then obtain proving the lower bound in Theorem 1.3.

Estimate of the Green energy: upper bound
According to (4), we have to estimate the integral which by Lemmas 2.2 and 2.3 and by invariance of Haar measure equals 2L cos ω(α) The integrand is in L 1 (SO (3)) since the singularity of the cotangent is removed by the zero of the difference of Gegenbauer polynomials, thus being a continuous function on a compact set. We can then apply (1) getting: We simplify by noticing that where we used that odd functions integrate to zero over symmetric intervals. But by the following equality, valid for ν > 1 2 and found in [13,Eq. 7.314]: We have then proved that Next we use Lemmas 6.1 and 6.2 in and obtain so that, by (10), Hence where the equality follows from Lemma 3.1. Note that by rotation invariance it suffices to study the variance of the random variable where {α 1 , . . . , α N } are generated by our dpp. The expected value of η A satisfies E[η A ] = μ(A)N , and the variance of η A is, by definition (using χ A (α k ) 2 = χ A (α k )), The expected value of the right-hand side equals, by (4), In other words, we have and therefore, using invariance of Haar measure, (1), and (10) All in one we have proved the variance version of [20, Eq. 28]:

Lemma 6.2
The Gegenbauer polynomials C (2) n−2 (x) satisfy For the proofs, we need a result from [11], showing the following recursive formula for squares of Gegenbauer polynomials: which, for λ = 1, i.e., Chebyshev polynomials of 2nd kind [11, Corollary 6.2], is Proof of Lemma 6. 1 We will use a well known identity for m ≤ n: which follows by induction on m, starting and re-applying the recurrence (6). Using (13) with m = n in (12) and integrating yields 1 0 (x 2 − 1) C (2) n (x) 2 dx where we used (9) and that T 2n+1 (x) is odd. By (11), we state for later use: