Large and Deep Factor Models

Predicting cross-sectional stock returns is challenging due to low signal-to-noise ratios and evolving market regimes. Classical factor models offer interpretability but limited flexibility, while deep learning models achieve strong performance yet often underutilize financial priors. We address this gap with PRISM-VQ (PRior-Informed Stock Model with Vector Quantization), a dynamic factor framework that integrates expert prior factors, vector-quantized discrete latent factors learned from cross-sectional structure, and a structure-conditioned Mixture-of-Experts to generate time-varying factor loadings. Vector quantization acts as an information bottleneck that suppresses noise while capturing robust market structure, with discrete codes serving both as latent factors and as routing signals for temporal expert specialization. Experiments on CSI 300 and S&P 500 show consistent improvements in cross-sectional return prediction and portfolio performance over strong baselines while preserving interpretability. Our code is available at https://github.com/finxlab/PRISM-VQ.

Sparsity or complexity? In modern high-dimensional asset pricing, these are often viewed as competing principles: recent empirical evidence favors richer models, while economic intuition has long favored parsimony. We reconcile this tension by distinguishing capacity sparsity-restrictions on effective model capacity-from factor sparsity-the parsimonious structure of priced risks. Revisiting the benchmark empirical design of Didisheim et al. (2025), we combine nonlinear feature expansions with basis pursuit, using column generation and GPU acceleration to scale estimation to 432 million candidate factors. Reaching this scale reveals a reversal in out-of-sample performance: sparse portfolios trail dense ridgeless benchmarks at lower complexity but achieve a higher Sharpe ratio and lower pricing error at the largest candidate set. Capacity expansion and factor sparsity are therefore complements: enlarging the candidate space allows a parsimonious pricing kernel to outperform its dense counterpart.

Sparse tangent portfolio optimization aims to learn an interpretable, low-cardinality portfolio in the tangency direction of the mean-variance frontier. However, the associated cardinality-constrained formulation is NP-hard, and standard predict-then-optimize pipelines often misalign forecasting accuracy with downstream portfolio quality. We propose an end-to-end decision-focused learning framework that reformulates Sharpe ratio maximization as a Disciplined Parametrized Programming (DPP)-compliant convex programming layer and replaces discrete selection with a smooth top-$k$ operator enforcing an exact cardinality $k$. This enables gradient flow through prediction, asset selection, and re-optimization, allowing the predictive model to directly optimize portfolio performance. Across four major equity markets, our method achieves competitive and often superior out-of-sample Sharpe ratios compared with historical and prediction-focused baselines, with particularly strong gains in larger asset universes. Our \href{https://github.com/feuerwerksh/Diffble-card-SR}{code} is publicly available.