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 the lead detailed example, with the Constant Modulus Algorithm's residual phase ambiguity predicted analytically and matched within $1.6^\circ$ 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：我们提出代数多样性（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的群论信号处理方法，是对传统时空多样性方法的补充。信号统计量保持不变的变换构成一个匹配群；该群决定了分析所需的自然变换，而对估计量在群作用上求平均可减少方差，无需额外快照。本文从五个方向扩展了配套论文中单次观测测量的视角。秩提升使得AD适用于标量数据流，并将大数定律识别为$(G,L)$连续统中群轨道平均与样本计数结合的平凡群特例。特征张量层次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不同支点。成本对称性匹配原理将AD从测量领域推广至盲均衡和自适应信号处理领域；以盲均衡作为主要详细示例，解析预测了常模算法的残余相位模糊性，并在3GPP TDL多径信道上以1.6°精度实现匹配，同时将信号处理中的其他盲问题映射至该框架。四个定理形式化定义了结构容量$\kappa$——类似于香农和冯·诺依曼Rényi-1熵的Rényi-2模拟量，用于量化信号信息的组织方式而非信息含量。AD是对包括不变估计、极小极大鲁棒估计、代数信号处理和压缩感知在内的前期代数方法的补充。