Large Signal Libraries: Equal-Weight Limits and the Divergent Spectra of Signals and PnL

Battle Count: 75

Highly relevant to quantitative trading. Directly addresses the behavior of large signal libraries (3,000+ signals) used in systematic trading, the relationship between equal-weight ensembles and principal components, the distinction between signal PCA and PnL PCA, information coefficient dynamics, and the geometric conditions under which signal averages align with dominant return factors (e.g., reversal). Provides practical diagnostics (residual-and-gap bounds, equal-centrality tests, moving-frame decomposition) for evaluating signal libraries. The paper clarifies that signal count alone does not explain factor alignment, which is critical for practitioners managing large alpha libraries.

The theoretical framework requires advanced mathematics (Hilbert spaces, operator theory, kernel methods, random matrix theory). However, the practical diagnostics (Section 12) are implementable: computing row-degree CVs, residual-and-gap bounds, moving-frame decomposition, and spectral convergence checks. The synthetic example is reproducible with standard numerical libraries. The main complexity lies in correctly specifying the sampling distribution, identifying research families, and calibrating null hypotheses for empirical validation.

The paper provides a reproducible synthetic numerical example (Figure 1) with a fixed random seed (20260911), K=3000, q=19. Supplementary code, threshold.csv, validation.json, and build instructions are available as arXiv ancillary files. However, no market-data empirical results are presented, limiting full empirical reproducibility. The theoretical results are fully specified with proofs.