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 complements prior algebraic approaches including invariant estimation, minimax robust estimation, algebraic signal processing, and compressed sensing.

翻译：本文提出代数多样性（AD）的原理，这是一种利用信号对称性从每次观测中提取更多信息、补充利用时间与空间多样性的经典方法的群论信号处理方法。使信号统计量保持不变的变换构成一个匹配群；该群决定分析的自然变换，对群作用下的估计量进行平均化可在无需额外快照的情况下降低方差。本文从五个方向拓展了姊妹篇中单次观测测量的视角：秩提升使得标量数据流上的AD成为可能，并将大数定律识别为$(G, L)$连续体中结合样本计数与群轨道平均化的平凡群情形；特征张量层级结构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和Karhunen–Loève变换置于变换流形上的特征点；成本对称匹配原理随后将AD从测量拓展至盲及自适应信号处理的一般情形；以盲均衡为详细实例，解析预测恒模算法的残余相位模糊，并在3GPP TDL多径信道上实现两度以内的匹配，同时将信号处理中的其他盲问题映射至该框架。四个定理形式化了结构容量$\kappa$——相当于香农和冯·诺依曼的Rényi-1熵的Rényi-2模拟——量化信号信息的组织方式而非所含信息量。AD补充了包括不变估计、极小极大鲁棒估计、代数信号处理和压缩感知在内的先验代数方法。