The Virtue of Sparsity in Complexity

TL;DR

This paper demonstrates that increasing feature space complexity in asset pricing enhances the discovery of sparse, well-performing risk factors, challenging the notion that simplicity always prevails.

Contribution

It clarifies that capacity sparsity and factor sparsity are complementary, showing that higher complexity enables better identification of sparse risk structures.

Findings

Nonlinear feature expansions with basis pursuit outperform simple benchmarks.

Performance gains are due to larger feature spaces, not more factors.

Complexity facilitates the discovery of sparse, effective risk factors.

Abstract

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.