Emergent phenomena -- onset of epileptic seizures, sudden customer churn, or pandemic outbreaks -- often arise from hidden causal interactions in complex systems. We propose a machine learning method for their early detection that addresses a core challenge: unveiling and harnessing a system's latent causal structure despite the data-generating process being unknown and partially observed. The method learns an optimal feature representation from a one-parameter family of estimators -- powers of the empirical covariance or precision matrix -- offering a principled way to tune in to the underlying structure driving the emergence of critical events. A supervised learning module then classifies the learned representation. We prove structural consistency of the family and demonstrate the empirical soundness of our approach on seizure detection and churn prediction, attaining competitive results in both. Beyond prediction, and toward explainability, we ascertain that the optimal covariance power exhibits evidence of good identifiability while capturing structural signatures, thus reconciling predictive performance with interpretable statistical structure.

In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $β$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.

Transformers have revolutionized deep learning across various domains but understanding the precise token dynamics remains a theoretical challenge. Existing theories of deep Transformers with layer normalization typically predict that tokens cluster to a single point; however, these results rely on deterministic weight assumptions, which fail to capture the standard initialization scheme in Transformers. In this work, we show that accounting for the intrinsic stochasticity of random initialization alters this picture. More precisely, we analyze deep Transformers where noise arises from the random initialization of value matrices. Under diffusion scaling and token-wise RMS normalization, we prove that, as the number of Transformer layers goes to infinity, the discrete token dynamics converge to an interacting-particle system on the sphere where tokens are driven by a \emph{common} matrix-valued Brownian noise. In this limit, we show that initialization noise prevents the collapse to a single cluster predicted by deterministic models. For two tokens, we prove a phase transition governed by the interaction strength and the token dimension: unlike deterministic attention flows, antipodal configurations become attracting with positive probability. Numerical experiments confirm the predicted transition, reveal that antipodal formations persist for more than two tokens, and demonstrate that suppressing the intrinsic noise degrades accuracy.

Modern large language models (LLMs) such as GPT, Claude, and Gemini have transformed the way we learn, work, and communicate. Y et, their ability to produce highly human-like text raises serious concerns about misinformation and academic integrity, making it an urgent need for reliable algorithms to detect LLMgenerated content. In this paper, we start by presenting a geometric approach to demystify rewrite-based detection algorithms, revealing their underlying rationale and demonstrating their generalization ability. Building on this insight, we introduce a novel rewrite-based detection algorithm that adaptively learns the distance between the original and rewritten text. Theoretically, we demonstrate that employing an adaptively learned distance function is more effective for detection than using a fixed distance. Empirically, we conduct extensive experiments with over 100 settings, and find that our approach demonstrates superior performance over baseline algorithms in the majority of scenarios. In particular, it achieves relative improvements from 57.8% to 80.6% over the strongest baseline across different target LLMs (e.g., GPT, Claude, and Gemini). The past few years have witnessed the emergence and rapid development of large language models (LLMs) such as GPT (Hurst et al., 2024), DeepSeek (Liu et al., 2024), Claude (Anthropic, 2024), Gemini (Comanici et al., 2025), Grok (xAI, 2025) and Qwen (Y ang et al., 2025). Their impact is everywhere, from education, academia and software development to healthcare and everyday life (Arora & Arora, 2023; Chan & Hu, 2023; Hou et al., 2024). On one side of the coin, LLMs can support users with conversational question answering, help students learn more effectively, draft emails, write computer code, prepare presentation slides and more. On the other side, their ability to closely mimic human-written text also raises serious concerns, including the generation of biased or harmful content, the spread of misinformation in the news ecosystem, and the challenges related to authorship attribution and intellectual property (Dave et al., 2023; Fang et al., 2024; Messeri & Crockett, 2024; Mahajan et al., 2025; Laurito et al., 2025). Addressing these concerns requires effective algorithms to distinguish between human-written and LLM-generated text, which has become an active and popular research direction in recent literature (see Crothers et al., 2023; Wu et al., 2025, for reviews).

We design a variational state estimation (VSE) method that provides a closed-form Gaussian posterior of an underlying complex dynamical process from (noisy) nonlinear measurements. The complex process is model-free. That is, we do not have a suitable physics-based model characterizing the temporal evolution of the process state. The closed-form Gaussian posterior is provided by a recurrent neural network (RNN). The use of RNN is computationally simple in the inference phase. For learning the RNN, an additional RNN is used in the learning phase. Both RNNs help each other learn better based on variational inference principles. The VSE is demonstrated for a tracking application - state estimation of a stochastic Lorenz system (a benchmark process) using a 2-D camera measurement model. The VSE is shown to be competitive against a particle filter that knows the Lorenz system model and a recently proposed data-driven state estimation method that does not know the Lorenz system model.

The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard approaches typically require high-fidelity training data. In many practical settings, the data are indirectly observed through noisy and nonlinear measurement. The challenge lies not only in approximating the coefficients of the SDEs, but in simultaneously inferring the posterior updates given the observations. In this work, we present a neural path estimation approach to solve stochastic dynamical systems based on variational inference. We first derive a stochastic control problem that solve filtering posterior path measure corresponding to a pathwise Zakai equation. We then construct a generative model that maps the prior path measure to posterior measure through the controlled diffusion and the associated Randon-Nykodym derivative. Through an amortization of sample paths of the observation process, the control is learned by an embedding of the noisy observation paths. Thus, we learn the unknown prior SDE and the control can recover the conditional path measure given the observation sample paths and we learn an associated SDE which induces the same path measure. In the end, we perform experiments on nonlinear dynamical systems, demonstrating the model's ability to learn multimodal, chaotic, or high dimensional systems.

We introduce a model-agnostic forward diffusion process for time-series forecasting that decomposes signals into spectral components, preserving structured temporal patterns such as seasonality more effectively than standard diffusion. Unlike prior work that modifies the network architecture or diffuses directly in the frequency domain, our proposed method alters only the diffusion process itself, making it compatible with existing diffusion backbones (e.g., DiffWave, TimeGrad, CSDI). By staging noise injection according to component energy, it maintains high signal-to-noise ratios for dominant frequencies throughout the diffusion trajectory, thereby improving the recoverability of long-term patterns. This strategy enables the model to maintain the signal structure for a longer period in the forward process, leading to improved forecast quality. Across standard forecasting benchmarks, we show that applying spectral decomposition strategies, such as the Fourier or Wavelet transform, consistently improves upon diffusion models using the baseline forward process, with negligible computational overhead. The code for this paper is available at https://anonymous.4open.science/r/D-FDP-4A29.

We introduce ECSEL, an explainable classification method that learns formal expressions in the form of signomial equations, motivated by the observation that many symbolic regression benchmarks admit compact signomial structure. ECSEL directly constructs a structural, closed-form expression that serves as both a classifier and an explanation. On standard symbolic regression benchmarks, our method recovers a larger fraction of target equations than competing state-of-the-art approaches while requiring substantially less computation. Leveraging this efficiency, ECSEL achieves classification accuracy competitive with established machine learning models without sacrificing interpretability. Further, we show that ECSEL satisfies some desirable properties regarding global feature behavior, decision-boundary analysis, and local feature attributions. Experiments on benchmark datasets and two real-world case studies i.e., e-commerce and fraud detection, demonstrate that the learned equations expose dataset biases, support counterfactual reasoning, and yield actionable insights.

We introduce Statsformer, a principled framework for integrating large language model (LLM)-derived knowledge into supervised statistical learning. Existing approaches are limited in adaptability and scope: they either inject LLM guidance as an unvalidated heuristic, which is sensitive to LLM hallucination, or embed semantic information within a single fixed learner. Statsformer overcomes both limitations through a guardrailed ensemble architecture. We embed LLM-derived feature priors within an ensemble of linear and nonlinear learners, adaptively calibrating their influence via cross-validation. This design yields a flexible system with an oracle-style guarantee that it performs no worse than any convex combination of its in-library base learners, up to statistical error. Empirically, informative priors yield consistent performance improvements, while uninformative or misspecified LLM guidance is automatically downweighted, mitigating the impact of hallucinations across a diverse range of prediction tasks.

Randomized subspace methods reduce per-iteration cost; however, in nonconvex optimization, most analyses are expectation-based, and high-probability bounds remain scarce even under sub-Gaussian noise. We first prove that randomized subspace SGD (RS-SGD) admits a high-probability convergence bound under sub-Gaussian noise, achieving the same order of oracle complexity as prior in-expectation results. Motivated by the prevalence of heavy-tailed gradients in modern machine learning, we then propose randomized subspace normalized SGD (RS-NSGD), which integrates direction normalization into subspace updates. Assuming the noise has bounded $p$-th moments, we establish both in-expectation and high-probability convergence guarantees, and show that RS-NSGD can achieve better oracle complexity than full-dimensional normalized SGD.