On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu [Dum07]. To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP. The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference [HMMR22] and later extended to [HMMR24]. In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance. Partial results of this article have been presented as a poster at the ISSAC 2023 conference [Met23].


Introduction
We present a sums-of-squares-based algorithm to minimize a multivariate trigonometric polynomial under symmetry assumptions, which reduces computational complexity whilst preserving numerical accuracy and convergence rate.The algorithm allows exploitation of symmetry with respect to any finite group that acts on a lattice.
Compared to classical polynomials, trigonometric ones are associated to a lattice.This makes them relevant for a variety of problems in geometry and information theory, with incidence in physics and chemistry, where lattices often provide optimal configurations.For example, the hexagonal lattice is classically known to be optimal for sampling, packing, covering, and quantization in the plane [CS99,KAH05], but also proved, or conjectured, to be optimal for energy minimization problems [PS20,BF23].More recently, the E 8 lattice was proven to give an optimal solution for the sphere packing problem and a large class of energy minimization problems in dimension 8 [Via17, CKM + 22].From an approximation point of view, such lattices describe Gaussian cubature [LX10,MP11], a rare occurence on multidimensional domains, and are relevant in graphics and computational geometry, see for instance [CKW20] and references therein.
The distinguishing feature of the above lattices is their intrinsic symmetry.The latter is given by the action of a finite group on the lattice, which induces a linear action on trigonometric polynomials.For the goal of trigonometric optimization, it is this feature that is emphasized and exploited in an optimization context.
Computing the global minimum of a polynomial is algorithmically hard and can be achieved numerically with an algorithm based on sums-of-squares reinforcements [Dum07,JM18].The convergence rate of this approach for trigonometric polynomials was recently shown to be exponential [BR23].
The approach is summarized as follows.One restricts to suitable finite subsets Ω  ⊆ Ω +1 ⊆ … ⊆ Ω of the lattice, to obtain a degree bound.Then a lower bound for the minimum of a trigonometric polynomial  is obtained by computing the maximal  ∈ ℝ, such that  −  is a Hermitian sum of squares, where the summands are bounded in the degree.This means, that  −  can be represented as a positive semi-definite Hermitian Toeplitz matrix so that the problem of computing  becomes a semi-definite program (SDP).By increasing the parameter , that is, the degree bound, one improves the quality of the approximation, but the problem becomes more and more costly to solve.
It is at this point, where we exploit symmetry.If the objective function is invariant under the action of a finite group, then the Hermitian matrices in the SDP can be block diagonalized according to a symmetry adapted basis.This reduces the number of variables of the SDP significantly and thus reduces required memory for storing and computational effort for solving the problem.
The main tool in this article, the symmetry adapted basis, is computed by decomposing a representation that corresponds to the finite subset Ω  into isotypic components [Ser77].The fact that one can do so is widely used in several areas, such as combinatorics [Sta79,Ber09], optimization [GP04,Val09,ALRT13], approximation [KdK23], interpolation [RBH21,RBH22b], dynamical systems [Gat00, HL13], polynomial systems [Gat90, FR09], computational invariant theory [HL16,RBH22a], as well as their fields of application [FS92].It was exploited for the special case of the symmetric group in [KdK23] and is here extended to finite groups acting on lattices.
An alternative approach to exploiting symmetry in trigonometric optimization was explored by Hubert, Moustrou, Riener and the author in [HMMR22,HMMR24] and the thesis [Met22].The idea there was to rewrite an invariant trigonometric polynomial to a classical one and then use relaxation tools from polynomial optimization [Las01,Par03].The present paper complements this work in the sense that we first rewrite the problem to a semi-definite program and then exploit symmetry.One goal is to provide a framework for a fair comparison between polynomial and Hermitian sums of squares.This completes a recently presented poster at the ISSAC 2023 conference [Met23].
Next to to symmetry, one can also exploit sparsity in the optimization of complex polynomials at a cost of numerical accuracy and convergence, see [WM22, LFF + 20, WSKM23].A further step towards improved computational efficiency would be to exploit both sparsity and symmetry.
The article is structured as follows: In Section 2, we define trigonometric polynomials and the associated concepts of periodicity and degree.We recall how to rewrite a trigonometric polynomial in terms of a Hermitian Toeplitz matrix.The first novelty is to construct a representation of a finite group that induces the action on trigonometric polynomials up to a fixed degree.Under the assumption that the trigonometric polynomial is invariant, we show that the associated Hermitian Toeplitz matrix is equivariant with respect to the constructed representation and can thus be block diagonalized via a symmetry adapted basis according to the isotypic decomposition of the representation.
The Hermitian sums-of-squares reinforcement is recalled in Section 3. The feasible region of the arising semidefinite program can be restricted to equivariant Hermitian Toeplitz matrices.Thus, it suffices to consider block diagonal matrices.We show how to apply symmetry reduction in this context with a clarifying example.
Finally, in Section 4, we show how to compute the symmetry adapted basis for our setup and consider a larger example to discuss possible implementations in practice.
There is a MAPLE worksheet dedicated to the examples and computations in this article, which requires the MAPLE package GENERALIZEDCHEBYSHEV 1 .Beyond that, the package offers a large variety of functionalities such as an implementation of the irreducible root systems and computational aspects of multiplicative invariants.

Periodicity domain
The property "Λ-periodic" means that, for all  ∈  and  ∈ Λ, we have  ( + ) =  ().In particular, Λ acts on  as an additive group by translation and  is constant on all residue classes  + Λ of the compact torus  ∕Λ.To understand the periodicity domain of a trigonometric polynomial, we consider the Voronoï cell of Λ where ‖‖ ∶= √ ⟨, ⟩ is the induced norm.This is a compact, convex set and tiles the space by Λ-translation, that is, The interiors of the cells Vor(Λ) +  are disjoint and the intersections of two adjacent cells is an entire face of both of them [CS99, Ch. 2, §1.2].The origin 0 ∈  is contained in the cell Vor(Λ) + 0 = Vor(Λ) itself.
We conclude that the Voronoï cell Vor(Λ) is the closure of the periodicity domain.

Degree concept
We now introduce the notion of degree for trigonometric polynomials.For  ∈ ℕ, is a finite subset of the lattice, because Ω is discrete and Vor(Λ) is compact.We say that a trigonometric polynomial  ≠ 0 with coefficients   has degree deg( ) =  ∈ ℕ if  ∈ Ω  whenever  ≠ 0 and  is minimal with this property.
The following statement shows that this is a suitable notion of degree.
Lemma 2.1.We have Proof.The first and third statement are clear.Furthermore, because Ω is additively closed and Vor(Λ) is convex.□ Example 2.2 (Using the notation from Appendix A).

It may happen that
None of the lattice generators   lies in the Voronoï cell and thus Ω 1 = Ω 0 = {0}.On the other hand, we have Ω 2 = {0,  1 ,  8 }, where  is the Weyl group.Hence, 2. This definition of degree yields a filtration of the trigonometric polynomials in the sense that the product of two degree  polynomials is of degree at most 2.However, it is not a grading, as the degree might be strictly smaller than 2.A counter example is given by the hexagonal lattice in Figure 1.

Toeplitz matrix representations
Given a trigonometric polynomial  = ∑      , let  ∈ ℕ be the minimal integer, such that  ∈ Ω  + Ω  whenever   ≠ 0. We shall write  as (2.4) where † denotes the complex conjugate transpose and • ( ) is a matrix with rows and columns indexed by Ω  and independent of  ∈  .
By comparing coefficients in Equation (2.4), we obtain the equation for  ∈ Ω.Therefore, we may assume without loss of generality that the entries ( )  depend only on the the lattice element  =  − .Hence, ( ) can be chosen uniquely as a Toeplitz matrix.Then the conjugate sign symmetry The space of all Hermitian Toeplitz matrices with rows and columns indexed by Ω  is denoted by Toep  .

Group symmetry
Let  ⊆ GL( ) be a finite group.We shall define a group action of  on the space of Hermitian Toeplitz matrices.
In order to do so, we assume that Ω is a -lattice, that is, (Ω) = Ω.Then a group element  ∈  acts on a trigonometric polynomial  with coefficients   by where   denotes the transpose of  with respect to ⟨⋅, ⋅⟩.We call  -invariant, if  ⋅  =  for all  ∈ .To proceed, we need to assume that (Ω  ) = Ω  .Then a group element  ∈  also acts on a matrix  ∈ Toep  with entries   by (2.7) We denote the fixed-point space of this action by Proof.Let  ∈  and  ∈ Ω.Then Therefore, we have  ⋆ ( ) = ( ) if and only if  ⋅  =  .□

Symmetry adapted bases
We show that the matrices Toep   are block diagonal for a certain choice of basis.In order to do so, we start with a representation  of , which induces the action on Toep  by Equation (2.7).
Note that a trigonometric polynomial of degree at most  can be written uniquely in terms of the   with  ∈ Ω  .The coordinates in this basis yield a vector  ∈ ℂ Ω  with  − =   , where ℂ Ω  denotes the space of complex-valued vectors indexed by Ω  .For  ∈ , let () be the permutation matrix that takes  ∈ ℂ Ω  to the vector with entries (2.8) Then we obtain the following version of Maschke's theorem.
Proposition 2.4.The space ℂ Ω  is a semi-simple -module with representation .For all  ∈ Toep  , we have because ()  = 1 when  =  −1  and 0 otherwise.□ In particular, if  ∈ Toep   , then ()  =  ().Hence,  commutes with the representation  and is therefore equivariant.Now, since the -module ℂ Ω  is semi-simple, it decomposes into simple submodules.By Schur's lemma, any -homomorphism between complex simple modules is either a scalar isomorphism or zero.Hence, ℂ Ω  has an isotypic decomposition where, for all 1 ≤  ≤ ℎ, the   1 , … ,     are isomorphic, simple -submodules with dimension   ∶= dim(   ) and multiplicity   ∈ ℕ so that According to this decomposition, there is a change of basis matrix  ∈ GL(ℂ Ω  ) that transforms any  ∈ Toep   into where each   consists of   equal blocks X of size   ×   .By applying Gram-Schmidt, we may assume additionally that  is unitary, that is,  −1 =  † .The column vectors of  form a symmetry adapted basis of ℂ Ω  .In particular, we obtain a basis for the trigonometric polynomials of degree at most  according to the decomposition.

Optimization
We present a numerical algorithm to minimize a -invariant trigonometric polynomial  with coefficients   =  − =   −1 () for  ∈  a finite group and  ∈ Ω a full-dimensional -lattice in a finite-dimensional real vector space  as in the previous Section 2. The global minimum exists and is assumed in some minimizer in the compact periodicity domain Vor(Λ), where Λ is the dual lattice of Ω.
Our strategy is to follow a known relaxation technique of this optimization problem to a hierarchy of semi-definite programs (SDP), whose solutions will provide lower bounds of increasing quality for the minimum.This technique is explained in the book of Dumitrescu for the case Ω = ℤ  [Dum07, Ch. 2, 3, 4], which is adapted to any lattice Ω with the notion of degree (= order of the hierarchy) as it was defined in the previous section in a straightforward manner.We then exploit the symmetry of  by writing the SDP matrices in block diagonal form according to the isotypic decomposition in Equation (2.10).
The fundamental difference to [HMMR24] is that we first relax to an SDP and then exploit symmetry.

Sums of squares
To begin, let  be a trigonometric polynomial and let  ∈ ℕ be the minimal integer so that  ∈ Ω  + Ω  whenever   ≠ 0. The number  can be seen as the starting index of a hierarchy of lower bounds, which is non-decreasing and converges to  * .We shall see that exploiting symmetry in this setup does not effect the quality of the bound, but reduces the computational effort.

Symmetry reduction
Now that we have formulated the hierarchy of SDPs, let us assume that  is -invariant.Proof.If  ∈ Toep   is feasible for  , , then it is especially feasible for   .Hence, we have   ≤  , .For the converse, let  ∈ Toep  be feasible for   and define We claim that X is feasible for  , .Indeed, for  ⪰ 0,  ∈  and  ∈ ℂ Ω  , we have because ()  = () † .Thus,  ⋆  is Hermitian positive semi-definite.Furthermore, if Trace() = 1, then Thus,  ⋆  is also feasible for   .Since the feasibility region of   is convex, it also contains the convex combination X.In particular, X is feasible for  , with Proof.Since the trace does not depend on the basis, we have Note that  ⪰ 0 if and only if  †   ⪰ 0, that is, for all 1 ≤  ≤ ℎ, we have X ⪰ 0. Thus,  , =  block , and the statement follows from Proposition 3.

Computing symmetry adapted bases
We have seen the advantage of using a symmetry adapted basis in trigonometric optimization in the previous Section 3. Using the computer algebra systems OSCAR2 and MAPLE3 , we explain how to obtain said basis in practice with the algorithm from [Ser77, Ch. 2].

Projection onto isotypic components
The semi-simple -module ℂ Ω  has an isotypic decomposition where, for each , the  ()  are isomorphic, simple -submodules with multiplicity   and dimensions   = dim( ()  ).The characters   of the representations associated to the  ()   are the complex irreducible characters of .

Computing representing matrices
Next, we need the representing matrices associated to the irreducible characters.The representations associated to the characters  1 and  2 are 1-dimensional.Hence, the representing matrices are For  2 , we discuss three possible approaches.

Over ℤ with MAPLE
This strategy works, because  is a Weyl group (see Appendix A): It acts by permutation of coordinates on  = ℝ 3 ∕⟨[1, 1, 1]  ⟩, which is a simple -module with character  2 .The root system A 2 from Figure 2 is a root system for  and the weight lattice Ω = ℤ  1 ⊕ ℤ  2 is hexagonal.The reflection associated to the root   takes   to It is the output of the following MAPLE command.
By reordering the columns, we obtain a matrix , so that, for  ∈ Toep   , the matrices  †   have the block diagonal structure , that is,  2 = 2 equal blocks of size  2 = 2 and  3 = 1 equal block of size  3 = 3.

Conclusion
For large ,  and ||, solving the SDP via Theorem 3.3 is vastly more efficient than without the symmetry reduction via Equation (3.1), because (i) the group symmetry reduces the number of variables and (ii) the matrices are blockdiagonal and thus the matrix size is reduced from |Ω  | 2 to  2 1 , … ,  2 ℎ (which simplifies the certification of positive semi-definiteness).
The bottleneck in the presented approach is the computation of the representing matrices  ()    () for the irreducible representations.In the examples, we have used computer algebra systems, such as MAPLE and OSCAR.Current work in progress revolves around the implementation of the presented approach to the point where the input consists of an invariant trigonometric polynomial  , a finite group  and an order of the relaxation  and the output of the matrices defining the SDP from Theorem 3.3.
The dense approach is in any case the least efficient.

Open problems
How to decide a priori which approach gives a better numerical result is unkown to the author.By Theorem 3.3, we already know that the approach via symmetry adapted basis gives the same solution as the one for the dense approach.
The exponential convergence rate is proven in [BR23].The Chebyshev approach on the other hand uses polynomial instead of trigonometric optimization techniques.In this case, the convergence rate is given in [BM23].This article now provides the means to compare the two approaches, which the author intends to address in future work.

A Lattices and Weyl groups associated to crystallographic root systems
We recall the definition for the lattices and groups, which appeared in the examples throughout the article and which are of interest for the applications mentioned in the introduction.Such lattices arise from root systems, which appear, for example, in the representation theory of semi-simple complex Lie algebras, see [Bou68,Ser01].
Let  be a finite-dimensional real vector space with inner product ⟨⋅, ⋅⟩.A subset R ⊆  is called a (crystallographic, reduced) root system in  , if the following conditions5 hold.R1 R is finite, spans  as a vector space and does not contain 0. A weight of R is an element  ∈  , such that, for all  ∈ R, we have ⟨,  ∨ ⟩ ∈ ℤ.By the "crystallographic" property R2, every root is a weight.For a base B = { 1 , … ,   } of the root system, the fundamental weights are the elements { 1 , … ,   }, where, for 1 ≤ ,  ≤ , we have ⟨  ,  ∨  ⟩ =  , .The set Ω of all weights  is a full-dimensional -lattice in  and called the weight lattice of R. By definition, it is the dual lattice of the coroot lattice, that is, Ω * = Λ.Assume that  =  (1) ⊕ … ⊕  () is the direct sum of proper orthogonal subspaces and that, for each 1 ≤  ≤ , there is a root system R () in  () .Then R ∶= R (1) ∪ … ∪ R () is a root system in  called the direct sum of the R () .If a root system is not the direct sum of at least two root systems, then it is called irreducible, see [Bou68, Ch.VI, §1.2].
Example A.1.For  = 1, a root system in ℝ must be of the form R = {±} for some  ∈ ℝ >0 , which is the only base element and the highest root.It admits the reflection at the origin   = −1 and so the Weyl group is  = {±1}.The fundamental Weyl chamber is Λ Λ = ℝ >0 and the coroot is  ∨ = 2 ∕⟨, ⟩ = 2∕.For  ∈ ℝ to be the fundamental weight, we require 1 = ⟨,  ∨ ⟩ = 2 ∕, that is,  = ∕2.When we choose  = 2, then R admits the self-dual lattice Ω = Λ = ℤ.For  = 2, the irreducible root systems are depicted in Figures 2 to 5. The roots are depicted in green, the base in red and the fundamental weights in blue.The gray shaded region is a Voronoï cell of the coroot lattice Λ ∶ we have two squares (C 2 and B 2 ) and two hexagons (A 2 and G 2 ).The blue shaded triangle is a fundamental domain of the semi-direct product  ⋉ Λ.

Figure 1 :
Figure 1: The subsets {0} = Ω 0 ⊆ Ω 1 ⊆ … ⊆ Ω 4 of the hexagonal lattice in the plane (specifically the weight lattice of the root system A 2 ) are contained in scaled copies of the Voronoï cell of the dual lattice.

Table 1 :
The multiplicities   of the irreducible representations of  ≅  3 occurring in the representation  by which  acts on the trigonometric polynomials up to degree .Those are the  1 : sign-,  2 : reflection-and  3 : trivial representation.
R4 For  ∈ R and  ∈ ℝ, we have  ∈ R if and only if  = ±1.The elements of R are called roots and the  ∨ are called coroots.The lattice Λ spanned by all coroots  ∨ is fulldimensional in  and called the coroot lattice.The Weyl group  of R is the group generated by the reflections   for  ∈ R.This is a finite subgroup of GL( ) and orthogonal with respect to the inner product ⟨⋅, ⋅⟩.The coroot lattice is a -lattice and the group product of  by Λ is semi-direct.Every root system contains a base, that is, a subset B = { 1 , … ,   } of R satisfying the following conditions [Bou68, Ch.VI, §1, Thm.3].B1 B is a vector space basis of  .B2 Every root  ∈ R can be written as  =  1  1 + … +     or  = − 1  1 − … −     for some  ∈ ℕ  .