Nima Afsharhajari, Jonathan Yu-Meng Li
arXiv 18 Apr 2026 · Finance — General
arXiv:2604.17166 · PDF · DOI · OpenAlex · Extracted main text
Sparsity or complexity? In modern high-dimensional asset pricing, these are often viewed as competing principles: richer feature spaces appear to favor complexity, while economic intuition has long favored parsimony. We show that this tension is misplaced. We distinguish capacity sparsity-the dimensionality of the candidate feature space-from factor sparsity-the parsimonious structure of priced risks-and argue that the two are complements: expanding capacity enables the discovery of factor sparsity. Revisiting the benchmark empirical design of Didisheim et al. (2025) and pushing it to higher complexity regimes, we show that nonlinear feature expansions combined with basis pursuit yield portfolios whose out-of-sample performance dominates ridgeless benchmarks beyond a critical complexity threshold. The evidence shows that the gains from complexity arise not from retaining more factors, but from enlarging the space from which a sparse structure of priced risks can be identified. The virtue of complexity in asset pricing operates through factor sparsity.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Didisheim, Antoine and Ke, Shikun Barry and Kelly, Bryan T. and Mala… (2025) APT or “AIPT”? The Surprising Dominance of Large Factor Models | 1.000 | 15 | 4 | 100% |
| 2 | Kelly, Bryan T. and Malamud, Semyon and Zhou, Kangying (2024) The Virtue of Complexity in Return Prediction | 0.737 | 3 | 2 | 100% |
| 3 | Gu, Shihao and Kelly, Bryan T. and Xiu, Dacheng (2020) Empirical Asset Pricing via Machine Learning | 0.644 | 2 | 2 | 100% |
| 4 | Jensen, Theis Ingerslev and Kelly, Bryan T. and Pedersen, Lasse Heje (2023) Is There a Replication Crisis in Finance? | 0.644 | 2 | 2 | 100% |
| 5 | Wilson, Andrew Gordon (2025) Deep Learning is Not So Mysterious or Different | 0.644 | 2 | 2 | 100% |
| 6 | Kelly, Bryan T. and Malamud, Semyon (2025) Understanding The Virtue of Complexity | 0.511 | 2 | 1 | 100% |
| 7 | Kozak, Serhiy and Nagel, Stefan and Santosh, Shrihari (2020) Shrinking the Cross-Section | 0.511 | 2 | 1 | 100% |
| 8 | Bartlett, Peter L. and Long, Philip M. and Lugosi, Gabor and Tsigler… (2020) Benign Overfitting in Linear Regression | 0.405 | 1 | 1 | 100% |
| 9 | Chen, Luyang and Pelger, Markus and Zhu, Jason (2024) Deep Learning in Asset Pricing | 0.405 | 1 | 1 | 100% |
| 10 | Chen, Andrew Y. and Zimmermann, Tom (2022) Open Source Cross-Sectional Asset Pricing | 0.405 | 1 | 1 | 100% |
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