libhmm: A Modern C++20 Library for Hidden Markov Models with Correct MLE Emission M-Steps

Hidden Semi-Markov Models (HSMMs) are fundamental probabilistic models widely adopted across diverse domains, from computational biology to finance and signal processing. The Viterbi algorithm decodes the most likely state sequence given an HSMM and can be applied iteratively for ab initio model learning. However, existing Viterbi implementations remain sequential, and GPU-accelerated solutions are entirely absent, making HSMM decoding impractical for large-scale workloads. We present a tensor-based formulation of the Viterbi algorithm for HSMMs, restructuring the inner loops into tensor operations that naturally map onto SIMD units and massively parallel architectures. Building on this formulation, we provide optimized implementations spanning single- and multi-core CPUs, and, for the first time, GPU. Experimental evaluation demonstrates speedups of up to 14x on a single core, over 200x with multi-core, and over 570x on GPU over the state-of-the-art sequential baseline, establishing a new performance baseline for large-scale HSMM decoding.

Hidden Markov models (HMMs) are widely used probabilistic models for discrete sequential data but can be limited when hidden dynamics are complex. Hidden quantum Markov models (HQMMs) generalize HMMs by replacing probability vectors with density matrices and stochastic transitions with quantum operations, enabling richer latent representations. However, existing HQMM learning methods have not consistently outperformed Expectation--Maximization (EM)-trained HMMs on data not generated by quantum processes, limiting their practical applicability. We introduce NS-RIS, Newton--Schulz Retraction-based Inference on the Stiefel manifold, a scalable algorithm for learning trace-preserving HQMMs. NS-RIS uses Newton--Schulz orthogonalization to compute a polar-factor search direction while preserving Stiefel-manifold feasibility, avoiding costly matrix decompositions. We further establish a finite-time stationarity guarantee under standard assumptions on smoothness, stochastic gradients, and finite Newton--Schulz accuracy. Empirically, NS-RIS provides the first benchmark evidence that an HQMM can significantly outperform an EM-trained HMM on data not generated by a quantum model. On synthetic HMM-generated benchmarks, NS-RIS outperforms both EM and the state-of-the-art HQMM method COSM, improving the evaluation metric by an average of 38.5% and by up to 50.6%. On a synthetic HQMM benchmark, it improves the test metric over COSM by 18.9% while reducing runtime by 12.0%. On the real-world Splice classification benchmark, NS-RIS also surpasses both EM and COSM in higher-dimensional latent regimes, reducing mean classification error by 17.9% for latent dimension 6 and 14.9% for latent dimension 8 relative to COSM. These results move HQMMs beyond a theoretical generalization of HMMs and establish them as practical and expressive models for scientific sequence data.

Softmax Mixture Models (SMMs) are discrete $K$-component mixture models for the probabilities of selecting one of $p$ candidate feature vectors $X_1,\ldots,X_p\in\mathbb{R}^L$ in heterogeneous populations and are widely used in econometrics and scientific applications. Related softmax mixture mechanisms also appear in modern LLM architectures. We provide a theoretical and methodological study of SMMs, focusing on the Expectation-Maximization (EM) algorithm and the Method of Moments (MoM). We show that EM recovers the mixture atoms at the parametric rate, up to logarithmic factors, after $\mathcal{O}(\log N)$ iterations, provided atom separation is at least of order $\log K$. This improves on separation conditions in existing analyses of EM for high-dimensional Gaussian mixtures. We also develop MoM procedures for parameter and subspace estimation. Although MoM parameter estimates converge more slowly than EM and can deteriorate with $K$, they provide provable warm starts for EM and are useful for small $K$. For general $K$, we estimate the atom subspace via MoM and recommend running EM from multiple random initializations within this subspace. Finally, as $p\to\infty$, we show that SMMs approximate mixtures of exponential tilts of the feature distribution, yielding asymptotic identifiability.