The algebraic diversity framework generalizes temporal averaging over multiple observations to algebraic group action on a single observation for second-order statistical estimation. The central open problem in this framework is $\textit{group selection}$: given an $M$-dimensional observation with unknown covariance structure, find the finite group whose spectral decomposition best matches the covariance. Naive enumeration of all subgroups of the symmetric group $S_M$ requires exponential time in $M$. We prove that this combinatorial problem reduces to a generalized eigenvalue problem derived from the double commutator of the covariance matrix, yielding a polynomial-time algorithm with complexity $O(d^2M^2 + d^3)$, where $d$ is the dimension of a generator basis. The minimum eigenvector of the double-commutator matrix directly constructs the optimal group generator in closed form, with no iterative optimization. The reduction is exact: the double-commutator minimum eigenvalue is zero if and only if the optimal generator lies in the span of the basis, and its magnitude provides a certifiable optimality gap when it does not. This problem does not appear in the standard catalogs of computational complexity (Garey and Johnson, 1979) and represents a new class linking group theory, matrix analysis, and statistical estimation. We establish connections to independent component analysis (JADE), structured matrix nearness problems, and simultaneous matrix diagonalization, and we show that the double-commutator formulation is the unique approach that is simultaneously polynomial-time, closed-form, and certifiable. We extend the framework to non-Abelian symmetry recovery via a Sequential GEVP with deflation, and add two identifiability theorems characterizing the commutant-lattice ambiguity and the dichotomy on whether $\mathrm{Aut}(\mathbf{R})$ recovers a generative subgroup or only a supergroup.

翻译：代数多样性框架将基于多次观测的时间平均推广到基于单次观测的代数群作用，以用于二阶统计估计。该框架的核心未解决问题是“群选择”：给定一个具有未知协方差结构的M维观测，寻找其谱分解与协方差最匹配的有限群。对对称群S_M的所有子群进行朴素枚举需要指数时间。我们证明这一组合问题可简化为由协方差矩阵的双换位子导出的广义特征值问题，从而得到复杂度为O(d^2M^2 + d^3)的多项式时间算法，其中d是生成元基的维度。双换位子矩阵的最小特征向量直接以闭式构造最优群生成元，无需迭代优化。该化简是精确的：当且仅当最优生成元位于该基的张成空间中时，双换位子最小特征值为零；否则其大小提供了可认证的最优性间隙。该问题未出现在标准计算复杂性目录（Garey and Johnson, 1979）中，代表了一类连接群论、矩阵分析与统计估计的新类型。我们建立了与独立成分分析（JADE）、结构化矩阵逼近问题和同步矩阵对角化的联系，并证明双换位子公式是同时满足多项式时间、闭式解和可认证性的唯一方法。通过带收缩的序列广义特征值问题（Sequential GEVP），我们将该框架扩展到非阿贝尔对称性恢复，并添加了两个可辨识性定理，刻画了换位子格模糊性以及Aut(R)恢复的是生成子群还是仅为其超群的二分性。