Given a generalist model, learning a task-relevant specialist representation is fundamental for downstream applications. Identifiability, the asymptotic guarantee of recovering the ground-truth representation, is critical because it sets the ultimate limit of any model, even with infinite data and computation. We study this problem in a completely nonparametric setting, without relying on interventions, parametric forms, or structural constraints. We first prove that the structure between time steps and tasks is identifiable in a fully unsupervised manner, even when sequences lack strict temporal dependence and may exhibit disconnections, and task assignments can follow arbitrarily complex and interleaving structures. We then prove that, within each time step, the task-relevant latent representation can be disentangled from the irrelevant part under a simple sparsity regularization, without any additional information or parametric constraints. Together, these results establish a hierarchical foundation: task structure is identifiable across time steps, and task-relevant latent representations are identifiable within each step. To our knowledge, each result provides a first general nonparametric identifiability guarantee, and together they mark a step toward provably moving from generalist to specialist models.

Large language models (LLMs) are pretrained by minimizing the cross-entropy loss for next-token prediction. In this paper, we study whether this optimization strategy can induce geometric structure in the learned model weights and context embeddings. We approach this problem by analyzing a constrained layer-peeled optimization program, which serves as a mathematically tractable surrogate for LLMs by treating the output projection matrix and last-layer context embeddings as optimization variables. Our analysis of this nonconvex optimization program demonstrates that symmetries in the target next-token distributions are transferred to the global minimizers of the layer-peeled model in a precise group-theoretic sense. Specifically, we prove that when the target tokens exhibit a cyclic-shift symmetry (such as the seven days of the week or the twelve months of the year), the optimal logit matrix is exactly circulant, and the Gram matrices of both the output projections and the context embeddings form circulant geometries as well. Next, for exchangeable target distributions invariant under the symmetric group and, more generally, under two-transitive group actions, we show that the global optimal output projection matrix forms a simplex equiangular tight frame, while the optimal logit matrix and context embeddings inherit the permutation symmetries present in the input data. A key technical step is to reduce the constrained nonconvex factorized problem to an explicit logit-level convex characterization for cyclic symmetry and to a symmetry-based lower bound for permutation symmetry, together with a sharp characterization of the optimal factorization. Finally, we empirically demonstrate that open-source LLMs naturally exhibit symmetries consistent with our theoretical predictions, despite being trained without any explicit regularization promoting such geometric structure.

Open time-series forecasting (TSF) benchmarks cover retail, energy, weather, and traffic, but supply-chain logistics remains underserved. We introduce ISOMORPH, the first public digital twin of a multi-echelon logistics network with fully interpretable, user-configurable parameters and modular topology, demand process, and control rules. The simulator advances a directed routing graph in discrete time: demand arrives at the destination, is served from stock or recorded as backlog, and triggers replenishment through the network. The state vector tracks per-node on-hand inventory with outstanding orders, in-transit shipments, and a smoothed demand estimate, so the dynamics close as a Markov chain on a tractable state space whose transition kernel acts linearly on the empirical distribution of the state. The released data reproduces the bullwhip effect at empirically consistent magnitudes, and three conservation laws encoded in the Markov chain serve as verification tools when users extend the simulator. We release datasets at two catalogue scales ($C=50$ and $C=200$) with six scenario sweeps producing 30 additional rollouts and 20 Latin-hypercube perturbations, exhibiting dynamics absent from fixed TSF benchmarks: variance amplification, cascading bottlenecks, regime shifts, and cross-channel coupling through shared macro shocks. Zero-shot evaluation of four foundation models (Chronos, Moirai, TimesFM, Lag-Llama) shows MASE values exceeding public GIFT-Eval references at low-to-moderate horizons, supporting incorporation into existing benchmarks. The same pairing produces forecast confidence bands via Latin-hypercube perturbation of demand-side knobs, forward UQ from parameter uncertainty unavailable on standard TSF datasets, demonstrating that foundation models can serve as fast surrogates for the digital twin's forward UQ. Code (MIT): https://github.com/tuhinsahai/ISOMORPH.

Single-arm trials are an important study design for evaluating drug efficacy and safety without enrolling patients into a control arm. Although they do not provide the gold-standard evidence of randomized controlled trials, they are increasingly used in clinical development as they offer an efficient, ethical, and practical alternative. A wide variety of approaches can be used to construct control comparators and estimate treatment effects, from fixed comparators informed by clinical knowledge to data-based and model-based patient-level comparators, also known as synthetic controls. Powerful and flexible machine learning models can allow outcome-model-based synthetic controls to overcome key limitations of direct data-based approaches, yield more robust estimates of treatment effects, and provide a principled way to incorporate corrections or encode additional assumptions when external data are not directly comparable. In this work, we argue that outcome-model-based synthetic control arms are an important tool for single-arm trials. We focus on digital twins, personalized predictions of disease progression generated from machine learning models trained on historical datasets, which naturally leverage these flexible approaches. We review doubly robust estimators, present power and sample size formulas, and discuss trade-offs in selecting historical data for training and analysis. We also outline practical considerations for deploying digital twins within the framework of recent FDA draft guidance on the use of artificial intelligence in drug development. Finally, we reanalyze data from trials in amyotrophic lateral sclerosis and Huntington's disease to demonstrate the proposed methods.

Experimental design has emerged as a powerful approach for improving the sample efficiency of A/B testing, yet existing designs rely critically on correctly specified models. We study robust sequential experimental design under model misspecification and develop a unified framework that covers both contextual bandit and dynamic settings. Theoretically, we prove that our design bounds the worst-case mean squared error of the estimated treatment effect. Empirically, we demonstrate the effectiveness of the proposed approach using synthetic and real-world datasets from a leading technology company.

Weak-to-strong (W2S) generalization, in which a strong model is fine-tuned on outputs of a weaker, task-specialized model, has been proposed as an approach to aligning superhuman AI systems. Existing theoretical analyses either fix the student's representations or operate in restricted settings. Whether multi-step SGD can succeed in feature learning while preserving diverse pre-trained capabilities remains open. We study W2S in the setting of reward-model learning with two-layer neural networks. The strong model has pre-trained representations organized into low-dimensional subspaces $V_k$, and is fine-tuned under the supervision of a weak model specialized on task $κ$. We prove that the strong model efficiently learns task $κ$, eliciting its pre-trained knowledge while retaining general capabilities. This establishes W2S generalization in the feature-learning regime, in the sense that the strong model acquires the target feature direction through W2S training, rather than having it given a priori. Moreover, W2S preserves pre-trained off-target features, whereas standard supervised fine-tuning causes catastrophic forgetting when off-target feature directions are correlated with the target's. Numerical experiments on synthetic data confirm our theoretical results.

Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

Density estimation in high-dimensional settings is an important and challenging statistical problem.Traditional methods based on kernel smoothing are inefficient in high dimensions due to the difficulties in specifying appropriate location-adaptive kernels. In this work, we introduce pre-training, a key idea behind many cutting-edge AI technologies, to the context of non-parametric density estimation. By establishing a pre-trained neural network that can recommend an appropriate location-adaptive kernel for each sample point, efficient density estimation with adaptive kernels is achieved in high dimensions. A wide range of numerical experiments show that this strategy is highly effective for improving density-estimation accuracy, when the target distribution is close to the distribution family for pre-training. When the target distribution is substantially different from the pre-training distribution family, the benefit from the proposed pre-training strategy may be diluted, but can be reactivated by an additional fine-tuning procedure.

This paper introduces an algorithm-agnostic approach to feature-based time series clustering via amortized neural inference. By training neural networks to approximate the optimal partitioning rule from simulated data, the proposed framework reduces reliance on conventional clustering methods, such as $K$-means, $K$-medoids, or hierarchical clustering, and their associated objective functions and heuristics. Leveraging statistical features, such as autocorrelations and quantile autocorrelations, the approach learns a data-driven affinity structure from which clustering partitions can be recovered, without requiring explicit prior specification of cluster shapes or structures. In addition, one version of the method can automatically determine the number of clusters, avoiding ad-hoc selection procedures. Comprehensive empirical studies show that the proposed framework achieves competitive or superior clustering accuracy relative to traditional methods, even in challenging scenarios where competing techniques are provided with the true number of clusters. An application to financial time series of stock returns illustrates its practical utility. By reducing the need for algorithm selection and calibration, the proposed framework opens new possibilities for automated, adaptive, and data-driven clustering of temporal data across scientific and industrial domains.

While deep learning has revolutionised inverse problems, its safe deployment is hindered by three primary reliability concerns: hallucinations, instabilities, and performance volatility [48]. Hallucinations manifest as high-fidelity features that are factually false; instabilities reflect heightened sensitivity to measurement noise; and performance volatility refers to significant fluctuations in reconstruction quality across the data, yielding high-fidelity results for some samples while failing on seemingly similar images. In many applications, the risk of generating realistic but unfaithful content can impede the safe deployment of AI methods for inverse problems. The choice of "hallucinate" as the Cambridge Dictionary's word of the year in 2023 illustrates this open problem [53]. The problem of AI hallucinations persists, as the Financial Times [44] highlighted that, "AI hallucinations haunt users more than job losses." A first step toward training AI methods that do not suffer from hallucinations is the assessment and identification of hallucinated outputs. Consider the inverse problem of recovering xfrom noisy measurements y " Fpx,eq, x PM1 ĂX, e PEĂY, (1.1)