Hidden Markov Models

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.

Accurate posterior prediction need not require accurate approximation of Bayesian updates. We prove that an unbounded gap between the update maps can coexist with vanishing predictive KL for every fixed finite $K\ge2$ in a stationary symmetric Gaussian HMM. Exact Bayesian mixing and an explicit deterministic radial filter act on the same $K-1$ belief coordinates. As $q\to0^+$, their separation in centered logits in the worst case grows at least linearly in the natural confidence scale $L_K(q)$, while their categorical $D_{\mathrm{KL}}(\mathrm{exact}\|\mathrm{radial})$ vanishes at the same explicit witness. Along stationary HMM trajectories, the expected terminal KL between filtered posteriors also converges to zero at $H(q)=\lceil-\log(q)/c\rceil+1$. Typical blocks without switches drive both filters into a common confidence cone, where softmax curvature suppresses their disagreement; a single Gaussian maximal event controls adaptive noise. A sweep with equally spaced Gaussians over $K\in\{2,4,8\}$ illustrates the opposing trends, and binary controls at long horizons compare saturating and nonsaturating recurrences. The result isolates two missing links between internal update gaps and predictive cost: the contribution of separating states to expected loss and decoder sensitivity. Thus even an unbounded internal update gap does not by itself certify predictive failure. The construction is fixed in $K$ and does not provide a universal criterion for when compression is harmless or characterize when internal gaps must incur task loss.

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.

Automated solar filament detection using deep learning faces several challenges. Semantic segmentation of solar filaments is a complicated multiscale feature extraction task with long-tail distribution. Furthermore, a large-scale, highly complete, and finely detailed dataset has become mandatory for providing abundant information. To address these challenges, we present a series of machine learning approaches to develop a solar filament detection workflow that performs superbly. First, we manually annotated a small-scale solar filament dataset based on H$α$ spectra called MHAS. Next, we developed the Multiscale ORiented DENdritic (MORDEN) model, a semantic segmentation model focusing on multiscale feature extraction. We also introduced the Dense Conditional Random Field (DenseCRF) and Density-Based Spatial Clustering of Applications with Noise (DBSCAN) methods for post-processing. Using the proposed workflow, we generated a large-scale, high-quality dataset called AHAS. Experimental results demonstrate that MORDEN outperforms several existing solar filament semantic segmentation models with open access. DenseCRF has been demonstrated to effectively capture fine edge details. We also evaluated the effects of data scaling and the reliability of DBSCAN and found that both approaches yield satisfactory performance. Multiple visualization results substantiate our quantitative findings. Our work provides a foundation for maximizing the potential of deep learning models for solar filament detection.

Reliability assessment of large language models (LLMs) seeks to estimate the probability that a model produces correct responses under a specified operational profile. Conventional benchmark-based evaluation, often summarized by aggregate accuracy, provides a point estimate of performance but does not characterize the uncertainty associated with reliability claims. Currently, statistical inference methods for LLM reliability assessment are emerging. However, a key assumption underlying these models is that test outcomes can be treated as independent repeated trials. This assumption may be inappropriate in sequential settings, where later responses depend on earlier interactions through retained context, error propagation, or an evolving interaction state. We extend a hierarchical Bayesian framework for LLM reliability assessment by relaxing the assumption of independent task outcomes and introducing a Hidden Markov Model to capture sequential dependence in benchmark-constructed interaction sessions. In this formulation, outcomes are generated from a latent interaction state evolving according to a first-order Markov process, capturing changes in interaction context. Through experiments using Anthropic Claude and OpenAI on four datasets, we demonstrate the potential impact of sequential dependence on reliability assessment. The results suggest that ignoring sequential dependence may lead to overconfident reliability estimates.

Phonetic forced alignment is a key technique in phonetic research, yet existing alignment systems lack specialized models for low-resource language varieties. We address this by training text-dependent and text-independent aligners for Chengdu Mandarin using a 17-hour corpus and a custom G2P dictionary. We trained a text-dependent GMM-HMM model (Chengdu-MFA) and fine-tuned a pretrained audio encoder on frame classification with Chengdu-MFA's pseudo label for text-independent alignment (Chengdu-FC). Evaluation on an expert-annotated test set show that both methods significantly outperform Standard Mandarin baselines. Chengdu-MFA reduced average phone boundary differences by 31.8%, while Chengdu-FC achieved a 61.2% reduction. This work establishes a practical bootstrapping pipeline for developing accurate aligners for under-resourced varieties without labor- and time-intensive manual annotation.

Sampling from discrete Markov random fields (MRFs) is a hard problem. We study amplitude-encoded i.i.d. sampling for small MRFs where $2^n$ target probabilities are precomputed classically. This removes quantum exponential speedup but allows a clean comparison against classical MCMC based on independent circuit samples ($τ\approx 1$). Across 60 instances spanning five graph families (1k-step burn-in, 3k retained samples), the mean ESS ratios of Quantum to Single-Site Gibbs, Block Gibbs, Tuned-Block, and Parallel Tempering are $16.35$, $7.29$, $1.82$, and $1.79$, showing modern classical samplers substantially close this gap. Amortizing $O(2^n)$ preprocessing into wall-clock time, exact inverse-CDF sampling yields $17.7\text{M}$ ESS/s versus $488\text{K}$ ESS/s for the quantum sampler ($36\times$ mean rate, $153\times$ per-instance), confirming no wall-clock advantage. We characterize MCMC autocorrelation costs and benchmark amplitude-encoded state preparation at $n \in \{8,10,12\}$. An MPS scaling study ($n \le 40$) shows bond dimension $χ=32$ achieves $F=0.721\pm0.059$ at $n=40$. Finally, a matched-budget VQC vs. MPS comparison at $n \in \{8,10,12\}$ shows VQC fidelities fall far below MPS: $(F_{\mathrm{VQC}}, F_{\mathrm{MPS}}) = (0.31, 0.99), (0.21, 0.96), (0.17, 0.88)$ at compressions $10.7\times$, $34.1\times$, and $113.8\times$.

A common method for the representation and analysis of time-series data is the hidden Markov model (HMM), where each observation is associated with a hidden state that evolves over time. However, many real-world systems are influenced by multiple independent factors, which are more naturally represented by factorial hidden Markov models (fHMM), where several hidden Markov chains jointly generate the observed data. Although an fHMM provides a richer and more realistic representation of many real-world systems, it can be reformulated as an equivalent HMM, but with a significantly larger state-space, leading to a severe increase in computational cost. In particular, the forward filtering algorithm, which is central to evaluation, decoding, and estimation tasks, becomes prohibitively expensive even for small systems. This work focuses on developing scalable methods for time-series analysis using tensor algebra to exploit the multidimensional structure of fHMM directly, without constructing intermediate HMM representations. Our novel filtering approach significantly improves computational performance and enables the efficient analysis of large systems and datasets, extending the scope of fHMM and providing a practical framework for data intensive applications.

Federated learning typically aggregates client updates using fixed or heuristic weighting rules, which can be suboptimal when clients have heterogeneous data and varying contributions to the global model. We propose a framework that refines client aggregation weights using Conditional Random Fields (CRFs). Our method defines unary potentials for individual clients and pairwise potentials for all client pairs, allowing the server to model both client-specific reliability and interactions between clients. The resulting CRF inference produces aggregation weights that enable better convergence of the global training objective. Experiments show that, under non-IID heterogeneity, our approach consistently improves performance over well-established federated learning baselines.

Gaussian Processes (GPs) are a powerful tool for Bayesian time-series modeling, yet their cubic computational cost remains a severe barrier for application to long, high-cadence datasets in astronomy. While specialized scalable solvers like Celerite elegantly reduce this scaling to linear time, repeatedly evaluating the exact likelihood during iterative Bayesian sampling is a bottleneck for developing more complex models, like hierarchical or additive models in which Celerite is only one component. To make this inference computationally tractable, we introduce a generative surrogate framework. By utilizing a Variational Autoencoder (VAE) to learn a compressed representation of the Celerite prior, we map highly correlated stochastic dependencies into a low-dimensional, isotropic manifold. This transition completely bypasses exact covariance operations, shifting the computational burden to a rapid neural network forward pass. Through an extensive simulation study, we show that the generative surrogate accurately reproduces the structural fidelity of exact physical kernels like Celerite. Finally, we demonstrate embedding our VAE approximation into an additive model that combines Celerite and a hidden Markov model (HMM) for stellar flare detection in time series data of stars. We evaluate the joint VAE+HMM architecture against the exact Celerite+HMM framework on empirical astrophysical time series and demonstrate that the proposed methodology achieves significant reductions in computational time, enabling the rigorous, large-scale characterization of stellar flares across massive data archives.

Human Action Recognition (HAR) is frequently investigated in Human-Robot Collaboration (HRC) research to understand what actions have been performed and hence the state of a collaborative task. Accurately tracking an assembly state from HAR is however not fully investigated, and in realistic scenarios is not a trivial task. This research systematically investigates and compares methods for tracking assembly state using action recognition inputs. Investigations using two diverse datasets and five state tracking approaches, including logic-based, Hidden Markov Model (HMM), and neural network (NN) methods, show that optimal approaches are not uniform across different tasks and that different methods fail under different circumstances. Testing is performed using both simulated inputs with varying noise levels and realistic inputs from a HAR model. Results show NN and HMM methods can perform well in tasks with limited variability, but for other scenarios logic-based approaches can be more robust. Methods which model expected action duration are also important for tasks with repeated actions where no additional sensing is provided.

The challenge associated with recognizing dysarthric speech primarily arises from pronounced acoustic variability attributed to impaired articulatory precision. Past research has demonstrated improved recognition through the use of hybrid DNN/HMM sequence discriminative training. This paper presents a comprehensive investigation of various combinations of acoustic features tailored to different Acoustic Models, offering suitable feature selections for each. The incorporation of Pitch features notably improved recognition performance, especially for sentence recognition tasks involving dysarthric speech. Through a systematic examination of the TORGO database, we have demonstrated the potential to enhance the performance of the state-of-the-art Factorized Time Delay Neural Network (F-TDNN) model for recognizing dysarthric speech. Our methods, implemented with the F-TDNN model, resulted in a 4.65% relative improvement in isolated word recognition and a 4.63% relative improvement in sentence recognition for dysarthric speech, compared to previous research. This improvement effectively compensates for speech variability, attributable to our deliberate selection of the number of overlapping frames between consecutive training example chunks.

The challenge associated with recognizing dysarthric speech primarily arises from pronounced acoustic variability attributed to impaired articulatory precision. Past research has demonstrated improved recognition through the use of hybrid DNN/HMM sequence discriminative training. This paper presents a comprehensive investigation of various combinations of acoustic features tailored to different Acoustic Models, offering suitable feature selections for each. The incorporation of Pitch features notably improved recognition performance, especially for sentence recognition tasks involving dysarthric speech. Through a systematic examination of the TORGO database, we have demonstrated the potential to enhance the performance of the state-of-the-art Factorized Time Delay Neural Network (F-TDNN) model for recognizing dysarthric speech. Our methods, implemented with the F-TDNN model, resulted in a 4.65% relative improvement in isolated word recognition and a 4.63% relative improvement in sentence recognition for dysarthric speech, compared to previous research. This improvement effectively compensates for speech variability, attributable to our deliberate selection of the number of overlapping frames between consecutive training example chunks.

Sequence labelling, a core task of Natural Language Processing (NLP), consists in assigning each token of an input sentence a label. From a Machine Learning point of view, sequence labelling is often cast as a Linear-Chain Conditional Random Field (CRF) parametrised by a neural network. While this approach gives good empirical results, CRFs assume a finite decision span (eg label bigrams) which can limit their expressivity and hurt performance when long-range dependencies are required. We show we can leverage diffusion to train a CRF conditioned on an entire label sequence, with the caveat that the condition is on a noisy version of labels. We show experimentally that this method, in conjunction with approximate CRF inference, improves label accuracy with a 16.5% error reduction for POS-tagging.

The family of linear recurrent neural networks has shown strong performance as recurrent memory units in partially observable reinforcement learning. We provide a theoretical justification for their empirical effectiveness by constructing and studying two linear filters: (i) the first exactly reproduces the pre-softmax logits of the belief vector in a hidden Markov model (HMM) under a deterministic transition matrix, thereby serving as a sufficient statistic for optimal policy learning, (ii) the second achieves vanishing state-decoding error under a nearly deterministic transition matrix, thus reducing state ambiguity to near zero. The results extend to action-controlled HMMs, where the corresponding linear filters become time-varying with action-dependent dynamics. We illustrate our main results through numerical experiments and further show that the constructed linear filter serves as a strong feature extractor in a small reinforcement learning game.

We describe libhmm, a C++20 library for Hidden Markov Model parameter estimation, sequence decoding, and model selection. libhmm addresses two gaps in existing software: the absence of a well-maintained, zero-dependency C++ HMM library suitable for embedding in production systems, and the widespread use of method-of-moments (MOM) approximations in the emission distribution M-step of the Baum-Welch algorithm. The library implements correct maximum likelihood estimators for sixteen scalar emission distributions, including an ECME algorithm for the location-scale Student-t distribution, Newton-Raphson maximization for Gamma, Beta, Weibull, and Negative Binomial distributions, and the von Mises distribution for circular data. All forward-backward and Viterbi calculations operate in full log-space. SIMD acceleration is provided for AVX-512, AVX2, SSE2, and ARM NEON via compile-time dispatch with scalar fallback. Version 4 adds multivariate observation support via the BasicHmm<Obs> template, with three multivariate emission families (diagonal Gaussian, full-covariance Gaussian, and independent components) each with correct weighted MLE M-steps. Python bindings are available via the companion package pylibhmm. We compare libhmm against established C and C++ HMM libraries and against published R reference packages on seven real-data benchmarks, and discuss the architectural tradeoffs made in the design.

MedER refers to the identification of medical entities. It is crucial for extracting structured clinical information from unstructured medical text. Many existing systems rely on transformer-based models, which are computationally expensive and difficult to deploy in resource-constrained environments. Furthermore, earlier works often use relaxed evaluation metrics that artificially inflate performance by rewarding correct prediction of dominant "Outside" (O) tokens. In this paper, we propose a lightweight Medical Entity Recognition (MedER) framework for the Bangla language. We establish a rigorous baseline using a 12-layer BanglaBERT model combined with a Conditional Random Field (CRF) layer for exact-boundary entity detection. To address deployment constraints, we compress this teacher model into a 4-layer student network through Knowledge Distillation (KD), where the student learns from the teacher's pre-CRF soft emission logits. Finally, we apply INT8 dynamic quantization to further reduce model size and inference cost. Our final quantized student achieves an 8.6x CPU speedup while requiring nearly 48 percent less storage than the CRF teacher model.

Many sequence computations are easier to study as movement through internal states than as isolated local circuits. We introduce Markovian Circuit Tracing (MCT), a diagnostic pipeline for testing whether transformer activations contain coarse state-transition structure. The benchmark uses synthetic Hidden Markov Model (HMM) tasks where latent states, transition matrices, Bayesian belief vectors, Bayes-optimal predictions, and forced-state counterfactual targets are known exactly. Across six HMM families and three seeds per family, tiny causal transformers learn near-Bayes next-token predictors, with mean excess loss over Bayes of 0.0138. Residual activations contain partial Bayesian belief information in this controlled synthetic benchmark. State abstractions extracted from these activations recover coarse transition signal, strongest in persistent and lower-state regimes, and weaker in ambiguous-emission and six-state regimes. The clearest result comes from state forcing. Patching a recovered-state centroid reduces KL to the exact HMM counterfactual target from 0.1957 in the unpatched model to 0.0532 on average, beating wrong-state, mean-activation, random-activation, and shuffled-label controls. The contribution is a controlled benchmark and evaluation framework for transformer state-dynamics interpretability, with MCT as a simple reference pipeline

LLM-powered coding agents spend the majority of their token budget reading repository files, yet much of the retrieved code is irrelevant to the task at hand. Existing learned pruners compress this context with a single-objective sequence labeler, collapsing all facets of code relevance into one score and one transition matrix. We show that this formulation creates a modeling bottleneck: a single CRF transition prior must serve heterogeneous retention patterns, including contiguous semantic spans and sparse structural support lines. We propose LaMR (Latent Multi-Rubric), a structured pruning framework that decomposes code relevance into two interpretable quality dimensions, semantic evidence and dependency support, each modeled by a dedicated CRF with dimension-specific transition dynamics. A mixture-of-experts gating network dynamically weights the per-rubric emissions conditioned on the query, and a final CRF layer on the fused emissions produces the aggregate keep-or-prune decision. To supervise each dimension without additional annotation cost, we derive multi-rubric labels from the existing training corpus via AST-based program analysis, simultaneously denoising the teacher's binary labels. By effectively filtering distracting noise, LaMR frequently matches or even outperforms unpruned full-context baselines. Experiments on four benchmarks (SWE-Bench Verified, SWE-QA, LCC, LongCodeQA) show that LaMR wins 12 of 16 head-to-head multi-turn comparisons. It saves up to 31% more tokens on multi-turn agent tasks and improves Exact Match by up to +3.5 on single-turn tasks, while performance is frequently enhanced by denoising the context, and any remaining drops are marginal.

Forecasting multivariate hidden Markov processes is challenging due to nonlinear and nonstationary observations, latent state transitions, and cross-sequence dependencies. While deep learning methods achieve strong predictive accuracy, they typically lack explicit state modeling, whereas Hidden Markov Models (HMMs) provide interpretable latent states but struggle with complex nonlinear emissions and scalability. To address these limitations, we propose DRL-STAF, a Deep Reinforcement Learning based STate-Aware Forecasting framework that jointly predicts next-step observations and estimates the corresponding hidden states for complex multivariate hidden Markov processes. Specifically, DRL-STAF models complex nonlinear emissions using deep neural networks and estimates discrete hidden states using reinforcement learning, reducing the reliance on predefined transition structures and enabling flexible adaptation to diverse temporal dynamics. In particular, DRL-STAF mitigates the state-space explosion encountered by typical multivariate HMM-based methods. Extensive experiments demonstrate that DRL-STAF outperforms HMM variants, standalone deep learning models, and existing DL-HMM hybrids in most cases, while also providing reliable hidden-state estimates.

We study positive-data learning of bounded-fan-out linear multiple context-free grammars under a fixed explicit finite monoid homomorphism $h$. The main obstacle beyond the context-free case is that an MCFG nonterminal derives a tuple whose components may be placed in a surrounding sentence in different orders. We introduce sentence-interface types as finite external control objects for such tuple occurrences. A type records the permutation of tuple components in the final sentence together with the $h$-values of the boundary intervals between them. For reduced working binary linear nondeleting MCFG presentations whose string languages satisfy $(f,h)$-tuple substitutability, we build a typed refinement, a finite characteristic sample, and a canonical positive-data learner. Once the sample contains this characteristic sample and remains contained in the target language, the learner reconstructs the language exactly. Consequently, for fixed fan-out bound $f$ and fixed explicit $h$, the resulting class is identifiable in the limit from positive data. Moreover, the hypothesis associated with any given finite sample is constructible in polynomial time for fixed $f$ and fixed $h$, including output size. Thus sentence-interface control is the finite mechanism that lifts fixed-$h$ distributional reconstruction from context-free grammars to bounded-fan-out linear MCFGs.

The Probabilistic Transformer (PT) establishes that the Transformer's self-attention plus its feed-forward block is mathematically equivalent to Mean-Field Variational Inference (MFVI) on a Conditional Random Field (CRF). Under this equivalence the Transformer ceases to be a black-box neural network and becomes a programmable factor graph: graph topology, factor potentials, and the message-passing schedule are all explicit and inspectable primitives that can be engineered. PT was originally developed for natural language and in this report we investigate its potential for time series. We first lift PT into the Spatial-Temporal Probabilistic Transformer (ST-PT) to repair PT's missing channel axis and weak per-step semantics, and adopt ST-PT as a shared cornerstone backbone. We then identify three distinct properties that PT/ST-PT offers as a factor-graph model and derive three Research Questions, one per property, that probe how each property can be exploited in time series: RQ1. The graph topology and potentials are direct programmable primitives. Can this be used to inject symbolic time-series priors into ST-PT through structural graph modifications, especially under data scarcity and noise? RQ2. The CRF's factor matrices are the operator's potentials. Can an external condition program these factor matrices on a per-sample basis, so that conditional generation becomes structural rather than feature-level modulation of a fixed one? RQ3. Each MFVI iteration is a Bayesian posterior update on the factor graph. Can this turn the latent transition of latent-space AutoRegressive (AR) forecasting from an opaque MLP into a principled posterior update, and can a CRF teacher distill its latents into the AR student to counter cumulative error? We give one empirical study per question. Together, these three studies position ST-PT as a programmable framework for time-series modeling.

We address Human Activity Recognition (HAR) utilizing Wi-Fi Channel State Information (CSI) under the joint requirements of causal interpretability, symbolic controllability, and direct operation on high-dimensional raw signals. Deep neural models achieve strong predictive performance on CSI-based HAR (CHAR), yet rely on continuous latent representations that are opaque and difficult to modify; purely symbolic approaches, in contrast, cannot process raw CSI streams. We propose a fully automatic and strictly decoupled pipeline in which CSI magnitude windows are compressed by a categorical variational autoencoder with Gumbel-Softmax latent variables under a capacity-controlled objective, yielding a compact discrete representation. The encoder is then frozen and used as a deterministic mapping to one-hot latent trajectories. Causal discovery is performed on these trajectories to estimate class-conditional temporal dependency graphs. Statistically supported lagged dependencies are translated into Linear Temporal Logic (LTL) rules, producing a fully symbolic and deterministic classifier based solely on rule evaluation and aggregation, without any learned discriminative head. Because rules are defined over discrete latent variables, antenna-specific rule sets can in principle be combined at the symbolic level, enabling structured multi-antenna fusion without retraining the encoder. Results from CHAR Latent Temporal Rule Extraction (CHARL-TRE) indicate competitive performance while preserving explicit temporal and causal structure, showing that deterministic symbolic classification grounded in unsupervised discrete latent representations constitutes a viable alternative to end-to-end black-box models for wireless HAR.

Semi-Markov Conditional Random Fields (semi-CRFs) assign labels to segments of a sequence rather than to individual positions, enabling exact inference over segment-level features and principled uncertainty estimates at their boundaries. However, existing implementations must materialize a large edge potential tensor whose size grows with sequence length, maximum segment length, and label count, becoming prohibitive for speech-scale state spaces and intractable at genomic scales where sequences can exceed 100,000 positions. This memory bottleneck has limited the adoption of exact segment-level inference for long sequences and large label sets. We identify that the core inefficiency is materializing edge potentials that can instead be evaluated on-the-fly from a compact prefix-sum array, and make several improvements. First, replacing the stored edge tensor with prefix-sum lookup reduces the memory footprint by a factor proportional to the product of segment length and label count. Second, a streaming forward-backward pass with checkpoint-boundary normalization keeps working memory sublinear in sequence length while preserving exact gradients. Third, zero-centered cumulative scores control numerical drift and induce an adaptive duration prior under label imbalance. We integrate these ideas into Flash-SemiCRF, a fused Triton kernel that enables exact semi-CRF inference on previously intractable problem sizes. Available at https://github.com/biobenkj/flash-semicrf.

Large language models are increasingly deployed as protocols: structured multi-call procedures that spend additional computation to transform a baseline answer into a final one. These protocols are evaluated only by end-to-end accuracy, giving limited insight into when they help, when they hurt, and whether their behavior transfers under distribution shift or composition. We propose a paired-outcome measurement interface for auditing a single protocol step on exact-match tasks. For each instance, the interface records a baseline correctness bit $E_0\in\{0,1\}$ and a post-step correctness bit $E_1\in\{0,1\}$, separating correction ($E_0=0\to E_1=1$) from corruption ($E_0=1\to E_1=0$) through two rates: $c=\Pr(E_1=1\mid E_0=0)$ and $γ=\Pr(E_1=0\mid E_0=1)$. These rates predict accuracy changes and define a reusable empirical interface testable across seeds, mixtures, and pipelines. We identify three failure mechanisms. Under mixture shift, pooled estimates of $(c,γ)$ become biased when calibration and deployment mixtures differ; conditioning on a difficulty proxy restores stability without additional model calls. Under presentation contamination, selection protocols alter the interface through stable presentation artifacts when candidate content is fixed. Under state insufficiency, the correctness bit may not carry enough history for multi-step pipelines to compose predictably; a Markov factorization test identifies when composition is valid and where additional state is needed. When a protocol step passes these diagnostics, it becomes an auditable module: gated by estimated gain, conditioned on a difficulty proxy to correct mixture bias, and composed into multi-step pipelines with predictable accuracy. We demonstrate these ideas on synthetic mathematical tasks and on GSM8K, where the calibrated interface correctly predicts when protocol steps should be activated or suppressed.

This paper presents a method to efficiently classify the gastroenterologic section of images derived from Video Capsule Endoscopy (VCE) studies by exploring the combination of a Convolutional Neural Network (CNN) for classification with the time-series analysis properties of a Hidden Markov Model (HMM). It is demonstrated that successive time-series analysis identifies and corrects errors in the CNN output. Our approach achieves an accuracy of $98.04\%$ on the Rhode Island (RI) Gastroenterology dataset. This allows for precise localization within the gastrointestinal (GI) tract while requiring only approximately 1M parameters and thus, provides a method suitable for low power devices

Depth completion aims to predict a dense depth map from a color image with sparse depth measurements. Although deep learning methods have achieved state-of-the-art (SOTA), effectively handling the sparse and irregular nature of input depth data in deep networks remains a significant challenge, often limiting performance, especially under high sparsity. To overcome this limitation, we introduce the Gaussian Belief Propagation Network (GBPN), a novel hybrid framework synergistically integrating deep learning with probabilistic graphical models for end-to-end depth completion. Specifically, a scene-specific Markov Random Field (MRF) is dynamically constructed by the Graphical Model Construction Network (GMCN), and then inferred via Gaussian Belief Propagation (GBP) to yield the dense depth distribution. Crucially, the GMCN learns to construct not only the data-dependent potentials of MRF but also its structure by predicting adaptive non-local edges, enabling the capture of complex, long-range spatial dependencies. Furthermore, we enhance GBP with a serial & parallel message passing scheme, designed for effective information propagation, particularly from sparse measurements. Extensive experiments demonstrate that GBPN achieves SOTA performance on the NYUv2 and KITTI benchmarks. Evaluations across varying sparsity levels, sparsity patterns, and datasets highlight GBPN's superior performance, notable robustness, and generalizable capability.