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arXiv2026

The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal scheduling

We study a class of product-reference diffusion algorithms for sampling from a discrete distribution. We show that their sampling performance can be characterized using a path-based measure of data geometry that we call the interaction growth complexity (IGC). We show that a bivariate IGC kernel gives an exact representation of both the KL discretization error and a simple one-step upper bound. The simpler univariate IGC density can be used to study the effect of stepsize choices on the iteration complexity required to obtain $ε$-accurate samples in KL divergence. Samplers that traverse the path with equi-spaced steps in log-squared-reliability-odds have performance that depends on the aggregate IGC mass, whereas refined choices of stepsizes have a lower complexity depending on a square-root functional. In the fine-grid limit, both of these characterizations become sharp. We also allow general product reference distributions and show that the reference law can substantially reshape the IGC profile and the resulting sampling complexity; in particular, references far from both the uniform and the data marginals can yield dimension-dependent improvements. Finally, the aggregate IGC mass admits bounds in terms of total correlation and dual total correlation, thereby connecting the pathwise geometry to classical measures of multivariate dependence.

stat.ML↗
arXiv2026

Understanding Deep Learning via Notions of Rank

Despite the extreme popularity of deep learning in science and industry, its formal understanding is limited. This thesis puts forth notions of rank as key for developing a theory of deep learning, focusing on the fundamental aspects of generalization and expressiveness. In particular, we establish that gradient-based training can induce an implicit regularization towards low rank for several neural network architectures, and demonstrate empirically that this phenomenon may facilitate an explanation of generalization over natural data (e.g., audio, images, and text). Then, we characterize the ability of graph neural networks to model interactions via a notion of rank, which is commonly used for quantifying entanglement in quantum physics. A central tool underlying these results is a connection between neural networks and tensor factorizations. Practical implications of our theory for designing explicit regularization schemes and data preprocessing algorithms are presented.

cs.LG↗
arXiv2026

AI-Generated Measurements for Identification and Inference with Missing Data: A Weak Shadow Variable Approach

Across business and social science applications, outcomes are often missing in ways that depend on the unobserved outcomes themselves. In service systems, for example, whether a customer submits a rating depends on the rating they would have provided. Such missing-not-at-random (MNAR) mechanisms make population quantities difficult to identify without strong assumptions on the observation process. Meanwhile, rich unstructured data, such as customer interaction histories, are increasingly available and can be used to construct structured measurements using tools such as large language models (LLMs). In this work, we develop an assumption-lean partial identification framework that uses such measurements as weak shadow variables, defined as outcome-informative proxies that are conditionally independent of missingness given the true outcome and observed covariates. Importantly, they need not accurately predict missing outcomes or satisfy the completeness requirement in the classical shadow variable literature. For identification, we characterize sharp bounds on population quantities through a pair of linear programs. For estimation and inference, we propose a localized penalized estimator that remains feasible under sampling error, and a subsampling algorithm for constructing confidence intervals. In semi-synthetic experiments using real customer-service dialogues, weak-shadow-variable intervals are about 89\% narrower than those without auxiliary information, while their midpoints have around 41\% lower estimation error than classical MNAR methods.

stat.ML↗
arXiv2026

Learning PDE Time-Stepping with Neural Cellular Automata

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.

cs.LG↗
arXiv2026

Simulation-Based Evaluation of Energy-Constrained Quantum-Classical Competition

This paper develops a simulation-based framework for evaluating the energy implications of quantum and classical computing firms competing in a market with limited energy resources. We model providers as differentiated Cournot competitors whose feasible service capacity is induced by technology-specific energy scaling laws: polylogarithmic for quantum algorithms that achieve an equivalent computational target and polynomial for classical emulation. For symmetric groups of quantum and classical firms, the equilibrium reduces to a tractable two-equation system that supports large scenario sweeps over market size, technology mix, and hardware coefficients. We characterize the capacity-constrained Nash equilibrium, prove the existence of a demand scale beyond which quantum service becomes more energy efficient, and report numerical experiments calibrated to trapped-ion and Rydberg platforms. The results identify when quantum energy advantage is only asymptotic and when it becomes operationally relevant.

quant-ph↗
arXiv2026

Improved off-policy training of diffusion samplers

We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at https://github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

cs.LG↗
arXiv2026

Stein's method for marginals on large graphical models

Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel $δ$-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.

stat.ML↗
arXiv2026

Trajectory balance: Improved credit assignment in GFlowNets

Generative flow networks (GFlowNets) are a method for learning a stochastic policy for generating compositional objects, such as graphs or strings, from a given unnormalized density by sequences of actions, where many possible action sequences may lead to the same object. We find previously proposed learning objectives for GFlowNets, flow matching and detailed balance, which are analogous to temporal difference learning, to be prone to inefficient credit propagation across long action sequences. We thus propose a new learning objective for GFlowNets, trajectory balance, as a more efficient alternative to previously used objectives. We prove that any global minimizer of the trajectory balance objective can define a policy that samples exactly from the target distribution. In experiments on four distinct domains, we empirically demonstrate the benefits of the trajectory balance objective for GFlowNet convergence, diversity of generated samples, and robustness to long action sequences and large action spaces.

cs.LG↗
arXiv2026

Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample

We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit $(n-1)^{-2}$ bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.

stat.ML↗
arXiv2023

Constraint-Free Structure Learning with Smooth Acyclic Orientations

The structure learning problem consists of fitting data generated by a Directed Acyclic Graph (DAG) to correctly reconstruct its arcs. In this context, differentiable approaches constrain or regularize the optimization problem using a continuous relaxation of the acyclicity property. The computational cost of evaluating graph acyclicity is cubic on the number of nodes and significantly affects scalability. In this paper we introduce COSMO, a constraint-free continuous optimization scheme for acyclic structure learning. At the core of our method, we define a differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous work, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite the absence of explicit constraints, we prove that COSMO always converges to an acyclic solution. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performance on graph reconstruction compares favorably with competing structure learning methods.

cs.LG↗
arXiv2026

Robust Assortment Optimization from Observational Data

Assortment optimization is a fundamental challenge in modern retail and recommendation systems, where the goal is to select a subset of products that maximizes expected revenue under complex customer choice behaviors. While recent advances in data-driven methods have leveraged historical data to learn and optimize assortments, these approaches typically rely on strong assumptions -- namely, the stability of customer preferences and the correctness of the underlying choice models. However, such assumptions frequently break in real-world scenarios due to preference shifts and model misspecification, leading to poor generalization and revenue loss. Motivated by this limitation, we propose a robust framework for data-driven assortment optimization that accounts for potential distributional shifts in customer choice behavior. Our approach models potential preference shift from a nominal choice model that generates data and seeks to maximize worst-case expected revenue. We first establish the computational tractability of robust assortment planning when the nominal model is known, then advance to the data-driven setting, where we design statistically optimal algorithms that minimize the data requirements while maintaining robustness. Our theoretical analysis provides both upper bounds and matching lower bounds on the sample complexity, offering theoretical guarantees for robust generalization. Notably, we uncover and identify the notion of ``robust item-wise coverage'' as the minimal data requirement to enable sample-efficient robust assortment learning. Our work bridges the gap between robustness and statistical efficiency in assortment learning, contributing new insights and tools for reliable assortment optimization under uncertainty.

stat.ML↗
arXiv2026

Seq2Synth: Benchmarking Temporal Fidelity in Synthetic Sequential Tabular Data

Synthetic sequential tabular data are increasingly used for privacy-preserving data sharing and research, yet conventional tabular metrics often overlook temporal structure. Existing single-table and relational evaluation protocols largely collapse records into static distributions, leaving key temporal properties insufficiently evaluated. We introduce Seq2Synth, a unified benchmark for assessing these properties. Its taxonomy characterizes temporal and schema properties to determine applicable evaluations, covering timestamp, cross-sectional, longitudinal, and structural fidelity, alongside trajectory-aware utility and privacy. Across seven core datasets from a 13-dataset benchmark and eight generators, models with near-perfect static fidelity still violate basic temporal constraints, producing duplicate timestamps, irregular intervals, and incomplete observation grids. Moreover, static and temporal-aware rankings diverge substantially, showing that temporal fidelity must be evaluated directly rather than inferred from static or relational scores. Project page and online appendices are available at: https://seq2synth.github.io/.

cs.LG↗
arXiv2026

Which Metrics Save the Most Human Annotation? Prediction-Powered Evaluation and Meta-Evaluation

Across various non-verifiable tasks, human evaluation is reliable but expensive, while automatic metrics are more scalable but often biased. Building on prediction-powered inference (PPI), we propose prediction-powered evaluation, a framework that combines limited human judgments with large-scale automatic scores to obtain data-efficient system comparisons that are provably unbiased. We develop parametric and non-parametric procedures, analyze the efficiency trade-off between paired and unpaired designs, and validate the framework on six WMT datasets. We further introduce the Prediction-Powered Saving Ratio (PPSR), a meta-metric that measures how much human annotation an automatic metric can save when used within prediction-powered evaluation. PPSR directly targets metric utility for prediction-powered evaluation and yields more discriminative and stable metric rankings than existing system-level meta-metrics. Overall, our new paradigm reframes automatic metrics as tools for reducing human annotation cost rather than replacing human judgment, and applies broadly to non-verifiable tasks.

cs.CL↗
arXiv2026

Representation Learning with Quantum Signal Processing

Representation learning begins when training changes the features that define similarity between data. A frozen-kernel model only reweights a fixed geometry. We establish quantum signal processing (QSP) as a solvable quantum model of the representation-learning regime. At arbitrary depth, we compute the exact mean and variance of its quantum neural tangent kernel, revealing an input-dependent angular geometry whose diagonal remains non-self-averaging even when the underlying unitary approaches Haar randomness. We also prove a sparse-data guarantee for the full nonlinear gradient flow without freezing or ensemble-averaging the kernel: the realized dynamics converges to an integrable scalar flow with a time-dependent kernel closure and explicit convergence times. A finite-depth speed limit holds for every data set and trajectory. At higher data density, numerical results show coupled evolution beyond both the scalar and frozen-kernel descriptions. These results give a controlled theory of learned quantum data geometry with provable training dynamics beyond the frozen limit.

quant-ph↗
arXiv2026

A Deep Latent Variable Framework for Jointly Modeling Missingness, Measurement Error, and Heterogeneity

Missing data, measurement error, and population heterogeneity are pervasive challenges in analyzing data arising from modern observational studies and machine learning applications. Although these problems frequently coexist and interact, they are often treated separately in existing works. We propose a unified probabilistic framework that jointly addresses these issues utilizing deep latent variable representation. The proposed method integrates a novel hierarchical tree-routed variational autoencoder with pattern-aware latent representations and calibration-based denoising. The framework accommodates missing data mechanisms, including MCAR, MAR, and MNAR, while simultaneously learning subgroup-specific and globally shared latent structure. The introduced reconvergent routing mechanism enables selective parameters to be shared across related subpopulations, which offers flexibility as well as improved statistical efficiency. Simulation studies demonstrate substantial improvements over existing deep generative imputation approaches under complex heterogeneous missingness and measurement-error settings. The proposed framework provides a principled approach for learning from noisy and incomplete data in modern healthcare and other high-dimensional applications.

stat.ML↗
arXiv2026

Fairness in multi-class multi-group classification problems via contextial coherent risk measures

We propose a new design of fair classifiers for multi-class classification problems in the presence of vector-valued sensitive attributes. In that scenario each sensitive attribute has multiple values and forms several groups relevant to the fairness consideration. Naturally those groups are overlapping and one should also analyze the interaction of factors. Additionally, the decision makers aided by the classification should not violate individual rights at the expense of satisfying fairness metrics at the group level. We propose an approach using the theory and methods of coherent measures of risk aiming at resolving the fairness challenges. Further, we propose a specialized numerical method for solving the resulting optimization problem. The method scales well with the increase of the number of observations. Additionally, we note that the obtained classifier is robust with respect to corrupted data or to situation when data is scarce. We demonstrate the advantages of the proposed framework in comparison to the support-vector machine framework and other methods handling fairness.

stat.ML↗
arXiv2026

Inverting Foundation Models of Brain Function with Simulation-Based Inference

Foundation models of brain activity promise a new frontier for in silico neuroscience by emulating neural responses to complex stimuli across tasks and modalities. A natural next step is to ask whether these models can also be used in reverse. Can we recover a stimulus or its properties from synthetic brain activity? We study this question in a proof-of-concept setting using TRIBEv2. We pair the brain emulator with large language models (LLMs) that generate news headlines from linguistic parameters such as valence, arousal, and dominance. We then use simulation-based inference to learn a probabilistic mapping from brain maps to latent stimulus parameters. Our results show that these parameters can be recovered from predicted brain maps, demonstrating that the emulator's synthetic neural encodings preserve information about the controlled stimulus dimensions. They also show that LLMs can serve as controllable stimulus generators for simulated experiments. Together, these findings provide a step toward decoding and inverse design with foundation brain models.

cs.LG↗
arXiv2026

Online Learning-to-Defer with Varying Experts

Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. Real-world deployments require handling streaming data, changing expert availability, shifting expert reliability, and feedback observed only for the selected action. We introduce an online multiclass L2D algorithm that combines queried-action bandit feedback with a dynamically varying pool of experts. Let $N=n+n_e$, let $B$ bound the Frobenius norm of the linear score matrix, and let $ρ$ bound the augmented input norm. Assuming linear calibration and zero surrogate minimizability gap for the projected comparator class, our method achieves expected true-deferral regret $O((BN^{3/2}ρ+1)T^{2/3})$, improving to $O(BN^{3/2}ρ\sqrt T+B^2N^3ρ^2)$ under a concentrated-score condition. The analysis combines an online $\mathcal H$-consistency transfer bound with projected online convex optimization. Experiments on synthetic and real-world datasets demonstrate selective routing under varying expert availability and reliability.

stat.ML↗