We present principles of algebraic diversity (AD), a group-theoretic approach to signal processing exploiting signal symmetry to extract more information per observation, complementing classical methods that use temporal and spatial diversity. The transformations under which a signal's statistics are invariant form a matched group; this group determines the natural transform for analysis, and averaging an estimator over the group action reduces variance without requiring additional snapshots. The viewpoint is broadened in five directions beyond the single-observation measurement of a companion paper. Rank promotion admits AD on scalar data streams and identifies the law of large numbers as the trivial-group case of a $(G, L)$ continuum combining sample-count with group-orbit averaging. An eigentensor hierarchy handles signals with nested symmetry. A blind group-matching methodology identifies the matched group from data via a polynomial-time generalized eigenvalue problem on the unitary Lie algebra, placing the DFT, DCT, and Karhunen--Loève transforms as distinguished points on a transform manifold. A cost-symmetry matching principle then extends AD from measurement to blind and adaptive signal processing generally; blind equalization is given as a detailed example, with the Constant Modulus Algorithm's residual phase ambiguity predicted analytically and matched within two degrees on 3GPP TDL multipath channels, and other blind problems in signal processing are mapped into the framework. Four theorems formalize a structural capacity $κ$, the Rényi-2 analog of Shannon and von Neumann's Rényi-1 entropies, quantifying how a signal's information is organized rather than how much information it contains. AD relationship to prior algebraic approaches including invariant estimation, minimax robust estimation, algebraic signal processing, and compressed sensing.

翻译：我们提出代数多样性（AD）的原理，这是一种基于群论的信号处理方法，通过利用信号对称性从每次观测中提取更多信息，补充了利用时间与空间多样性的经典方法。使信号统计量保持不变的变换构成一个匹配群；该群决定了信号分析的自然变换，并且通过在群作用上对估计量进行平均，可在无需额外快照的情况下降低方差。与配套论文中基于单次观测的测量相比，本视角在五个方向上得到拓展。秩提升使得AD可应用于标量数据流，并将大数定律识别为样本计数与群轨道平均的$(G, L)$连续谱中的平凡群特例。特征张量层级结构用于处理具有嵌套对称性的信号。盲群匹配方法通过酉李代数上的多项式时间广义特征值问题从数据中识别匹配群，将DFT、DCT及Karhunen–Loève变换定位为变换流形上的不同点。随后，代价对称性匹配原则将AD从测量扩展至通用的盲自适应信号处理；以盲均衡为详细示例，理论预测了恒模算法的残余相位模糊性，在3GPP TDL多径信道上与匹配结果相差不超过两度，并将信号处理中的其他盲问题映射至该框架中。四个定理形式化定义了结构容量$\kappa$，即香农与冯·诺依曼Rényi-1熵的Rényi-2类比，该量刻画信号信息的组织方式而非所含信息量。最后，讨论了AD与既有代数方法（包括不变估计、极小极大稳健估计、代数信号处理及压缩感知）之间的关系。