Abdulrahman Alswaidan

Abdulrahman Alswaidan, Jeffrey D. Varner

Attention heads retrieve: given a query, they return a weighted average of stored values. We showed that this computation is one step of gradient descent on the modern Hopfield energy, and that Langevin sampling from the corresponding Boltzmann distribution yielded stochastic attention, a training-free sampler controlled by a single temperature parameter. Lowering the temperature gave exact retrieval; raising it gave open-ended generation. Because the energy gradient equals the attention map, no score network, training loop, or learned model was required, making the approach particularly suited to the low-data regime where learned generative models are starved of training signal. We derived an entropy inflection condition that identified the retrieval-to-generation transition temperature for any memory geometry and validated the sampler on five domains spanning two orders of magnitude in dimension. A single Boolean mask on the attention softmax, identical to the causal mask used in transformers but applied along the memory axis rather than the sequence axis, turned the sampler into a zero-shot class-conditional generator on Olivetti faces with no retraining and no learned classifier. On MNIST digit images, stochastic attention produced samples that were markedly more novel and more diverse than the best learned baseline while matching a Metropolis-corrected gold standard. On protein sequences from a small Pfam family, the generation regime preserved amino acid composition far more faithfully than a variational autoencoder at matched novelty, indicating that the training-free score function retained family-level fidelity that learned models lost. A denoising diffusion baseline failed across all memory sizes tested, producing samples indistinguishable from isotropic noise. The approach required no architectural changes to the underlying attention mechanism.

Abdulrahman Alswaidan, Jeffrey D. Varner

Generating synthetic financial time series that preserve statistical properties of real market data is essential for stress testing, risk model validation, and scenario design. Existing approaches, from parametric models to deep generative networks, struggle to simultaneously reproduce heavy-tailed distributions, negligible linear autocorrelation, and persistent volatility clustering. We propose a hybrid hidden Markov framework that discretizes continuous excess growth rates into Laplace quantile-defined market states and augments regime switching with a Poisson-driven jump-duration mechanism to enforce realistic tail-state dwell times. Parameters are estimated by direct transition counting, bypassing the Baum-Welch EM algorithm. Synthetic data quality is evaluated using Kolmogorov-Smirnov and Anderson-Darling pass rates for distributional fidelity, and ACF mean absolute error for temporal structure. Applied to ten years of SPY data across 1,000 simulated paths, the framework achieves KS and AD pass rates exceeding 97% and 91% in-sample and 94% out-of-sample (calendar year 2025), partially reproducing the ARCH effect that standard regime-switching models miss. No single model dominates all quality dimensions: GARCH(1,1) reproduces volatility clustering more accurately but fails distributional tests (5.5% KS pass rate), while the standard HMM without jumps achieves higher distributional fidelity but cannot generate persistent high-volatility regimes. The proposed framework offers the best joint quality profile across distributional, temporal, and tail-coverage metrics. A Single-Index Model extension propagates the SPY factor path to a 424-asset universe, enabling scalable correlated synthetic path generation while preserving cross-sectional correlation structure.