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arXiv2026

Coherent information deletion: Bayes' theorem and generalized Bayesian unlearning

Bayes' theorem admits an information-processing interpretation due to Zellner (1988): under the Shannon-information criterion, the posterior is the unique rule that processes prior and data information without information loss. We revisit these ideas, but from the perspective of information deletion. Given a posterior based on a complete dataset, what distribution should replace it when a subset of the data is removed? We define information deletion using the same information conservation principle as Zellner (1988), and show that the optimalpost-deletion distribution is exactly the leave-data-out posterior. We then extend the framework beyond likelihood-based inference from Bayes to the generalized Bayesian updating of Bissiri et al. (2016) based on loss functions. We introduce a sequential coherence requirement for deletion, under which, removing two pieces of information jointly is equivalent to removing them successively. The resulting coherent deletion rule exactly recovers the generalized Bayesian posterior based only on the retained data. Restricting these optimization problems to variational families yields corresponding formulations of variational Bayesian and generalized Bayesian unlearning.

stat.ME↗

arXiv2026

Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $σ^d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.

stat.ML↗

arXiv2026

Entropy-Generated Attention Beyond Softmax and Entmax: Kaniadakis and Reciprocal-Symmetric Abe Operators

We derive two attention operators from generalized statistical entropies. Kaniadakis entropy yields an exact full-support normalization whose weights and low-score sensitivities decay algebraically, rather than exponentially as in Softmax or by exact truncation as in entmax. Classical Abe entropy yields an implicit reciprocal-symmetric operator. With $q=e^ε$, the involution $q\leftrightarrow q^{-1}$ removes every odd correction about Softmax; we obtain the normalized second- and fourth-order terms, including the deformation of the normalization multiplier. These stationary laws follow from a Fisher-metric Lagrangian on the probability simplex, whose Shannon sector recovers scaled dot-product Softmax. We also give a tangent-gradient test for deciding whether changing the entropy changes the attention profile or only its scale. Rényi and two-parameter Sharma--Mittal entropies retain the Tsallis--entmax inverse-gradient shape, but their global moments make the effective temperature input dependent when the external temperature is fixed. Distinguishing profile-shape equivalence from fixed-parameter operator equivalence separates new normalization shapes from adaptive rescalings and organizes the operators by support, tail behavior, and realization complexity.

cs.LG↗

arXiv2026

Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

stat.ML↗

arXiv2026

Deciding When to Decide: Testing Operational Suboptimality Under Distributional Shift

Deployed decisions are often optimized once and retained because updates impose operational, regulatory, or switching costs. As operating conditions change, when should such decisions be re-optimized? We study this question for stochastic optimization when the objective's functional form is known but the decision maker's trade-offs are encoded by an unknown preference parameter. Standard distribution-shift tests are poorly aligned with this goal: they can flag detectable yet decision-irrelevant changes without determining whether the incumbent decision has become materially suboptimal. We propose \texttt{RADAR} (Regret-based Assessment of Decision Adequacy and Risk), a decision-focused framework that uses inverse optimization to infer latent preferences and tests the deployed decision's optimality gap under the current distribution. By targeting regret, \texttt{RADAR} ignores decision-irrelevant shifts while detecting changes that warrant re-optimization. We develop two-sample and sequential changepoint procedures and establish asymptotic guarantees for Type-I error and power. Across synthetic optimization problems, a semi-synthetic capacity allocation task, and police-zone planning, \texttt{RADAR} more reliably distinguishes harmful from harmless shifts than decision-agnostic alternatives.

stat.ML↗

arXiv2026

TaskGuard: Task-Conditioned Restoration Utility for Risk-Aware Object Detection

Image restoration is commonly applied before object detection under adverse conditions, yet a visually improved image need not improve the downstream task. We study this mismatch as restoration utility prediction: given a degraded image and its candidate restoration, should the restoration be used or should the original observation be preserved? We introduce TaskGuard, a post-hoc controller for frozen restoration and detection pipelines. TaskGuard characterizes the realized restoration residual through its interaction with detector sensitivity and predicts whether the intervention is task-beneficial. Exact regional counterfactuals reveal substantial within-image utility heterogeneity, while a deployable pseudo-gradient preserves statistically reliable directional information. Feature-group ablation further shows that task-conditioned evidence contributes information beyond detector-response and residual statistics. The TaskGuard utility predictor is trained only on Gaussian degradation and frozen before final evaluation, then transferred to unseen motion blur, rain, and defocus. Across these unseen families, TaskGuard reduces lossnegative interventions by 54.2% (family macro) and practical per-image detection deteriorations by 37.0% (pooled), while preserving 98.8% of the Always-Restore COCO AP. On natural-rain DAWN, it reduces loss-negative interventions by 97.9% while retaining 77.8% of the AP improvement obtained by deraining. These results support restoration utility as a task-conditioned property of the specific intervention rather than image appearance alone.

cs.CV↗

arXiv2026

First-Order Efficiency for Probabilistic Value Estimation via A Statistical Viewpoint

Probabilistic values, including Shapley values and semivalues, provide a model-agnostic framework to attribute the behavior of a black-box model to data points or features, with a wide range of applications including explainable artificial intelligence and data valuation. However, their exact computation requires utility evaluations over exponentially many coalitions, making Monte Carlo approximation essential in modern machine learning applications. Existing estimators are often developed through different representation strategies, including weighted averages, self-normalized weighting, regression adjustment, and weighted least squares. Our key observation is that these seemingly distinct constructions share a common first-order expansion, in which the leading term is determined by the sampling law and a working surrogate function. This first-order representation yields an explicit expression for the leading mean squared error (MSE), which characterizes how the sampling law and the surrogate jointly determine statistical efficiency. Guided by this criterion, we propose an Efficiency-Aware Surrogate-adjusted Estimator (EASE) that directly chooses the sampling law and surrogate to minimize the first-order MSE. We demonstrate that EASE consistently outperforms existing estimators for various probabilistic values.

cs.AI↗

arXiv2026

PQMass: Probabilistic Assessment of the Quality of Generative Models using Probability Mass Estimation

We propose a likelihood-free method for comparing two distributions given samples from each, with the goal of assessing the quality of generative models. The proposed approach, PQMass, provides a statistically rigorous method for assessing the performance of a single generative model or the comparison of multiple competing models. PQMass divides the sample space into non-overlapping regions and applies chi-squared tests to the number of data samples that fall within each region, giving a p-value that measures the probability that the bin counts derived from two sets of samples are drawn from the same multinomial distribution. PQMass does not depend on assumptions regarding the density of the true distribution, nor does it rely on training or fitting any auxiliary models. We evaluate PQMass on data of various modalities and dimensions, demonstrating its effectiveness in assessing the quality, novelty, and diversity of generated samples. We further show that PQMass scales well to moderately high-dimensional data and thus obviates the need for feature extraction in practical applications.

stat.ML↗

arXiv2026

Stereo 4D Radar for 3D Object Detection: Integrating Geometric Alignment and Absolute Velocity Estimation

Four-dimensional (4D) Radar is a powerful sensing modality capable of detecting surrounding three-dimensional (3D) objects under diverse weather conditions and providing Doppler-based motion information. However, raw 4D Radar signals contain significant clutter from road surfaces, guardrails, and surrounding vehicles, along with multipath-induced ghost reflections and the receiver's inherent noise floor. Consequently, preprocessing algorithms designed to remove such invalid measurements often make the Radar data excessively sparse. Moreover, the Doppler measurements provided by 4D Radar describe only the radial component of an object's velocity, limiting their ability to recover the full motion state. In this paper, we introduce a stereo 4D Radar-based 3D object detection framework that exploits the geometric disparity between left and right Radars to estimate the absolute velocity of objects and achieve more robust perception through the fusion of their complementary features. The effectiveness of the proposed framework is validated on our in-house stereo 4D Radar dataset, demonstrating performance gains of 8.82 points in AP 3D and 9.0 points in AP BEV over state-of-the-art mono 4D Radar baselines. These results demonstrate that absolute velocity estimation combined with stereo geometry-aware feature fusion leads to substantial improvements in 3D object detection.

cs.CV↗

arXiv2026

Clustering and Pruning in Causal Data Fusion

Data fusion, the process of combining observational and experimental data, can enable the identification of causal effects that would otherwise remain non-identifiable. Although identification algorithms have been developed for specific scenarios, do-calculus remains the only general-purpose tool for causal data fusion, particularly when variables are present in some data sources but not others. However, approaches based on do-calculus may encounter computational challenges as the number of variables increases and the causal graph grows in complexity. Consequently, there exists a need to reduce the size of such models while preserving the essential features. For this purpose, we propose pruning (removing unnecessary variables) and clustering (combining variables) as preprocessing operations for causal data fusion. We generalize earlier results on a single data source and derive conditions for applying pruning and clustering in the case of multiple data sources. We give sufficient conditions for inferring the identifiability or non-identifiability of a causal effect in a larger graph based on a smaller graph and show how to obtain the corresponding identifying functional for identifiable causal effects. Examples from epidemiology and social science demonstrate the use of the results.

stat.ML↗

arXiv2026

Provable Pluralistic Alignment: Multi-Party RLHF under Offline Human Feedback

Pluralistic alignment requires learning from feedback that reflects persistent and potentially conflicting stakeholder preferences while ultimately selecting a single collective policy. We study this problem in offline reinforcement learning from human feedback (RLHF), where the party associated with each comparison is observed. Under a shared low-rank linear reward model, we jointly estimate party-specific rewards and perform pessimistic policy optimization under Nash, Utilitarian, and Egalitarian social-welfare objectives. We establish nonasymptotic bounds for party-specific reward estimation and the resulting policy suboptimality under offline coverage conditions. We further consider general pairwise preferences that need not admit a scalar reward representation and may exhibit cycles. In this setting, we construct a pessimistic von Neumann winner policy and derive corresponding performance guarantees. Under these models, our results provide a unified finite-sample solution to a central challenge in pluralistic alignment: learning from limited, heterogeneous, and potentially cyclic feedback, and producing a single policy with explicit collective-welfare guarantees. Our framework thereby makes preference aggregation an explicit and statistically analyzable design choice rather than an implicit consequence of pooling human feedback.

cs.LG↗

arXiv2026

Residual-augmented flow matching operators for probabilistic partial differential equations

Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural operators fail to characterize uncertainty, while generative approaches require large ensembles of high-fidelity solution operator simulations and often sacrifice resolution generalizability. In this work, we propose a residual-augmented probabilistic operator learning framework that casts flow-matching-based generative modeling in infinite-dimensional function spaces while leveraging inexpensive low-fidelity solution operators as an inductive bias. Rather than learning the full high-fidelity stochastic solution operator directly, the proposed framework learns probabilistic residual operators that characterize the discrepancy between low- and high-fidelity solutions. By parameterizing the vector field in flow matching using neural operators conditioned on both the known system input and low-fidelity solution, the framework amortizes probabilistic inference across input conditions while enabling uncertainty-aware and resolution-generalizable predictions across spatial discretizations. Numerical experiments on stochastic advection, Burgers', and Darcy flow systems demonstrate that the residual-augmented formulation improves predictive accuracy under the same high-fidelity data budget, while the probabilistic operator learning formulation enables accurate characterization of uncertainty in low-data regimes compared to learning high-fidelity stochastic operators directly from data.

stat.CO↗

arXiv2026

A computational approach to maximum likelihood thresholds for colored Gaussian graphical models

Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.

stat.ML↗

arXiv2026

Causal DAG Identification for Count Data via Poisson Thinning Structural Equation Models

Count-valued variables arise in many scientific and applied settings, yet explicit structural models that allow full identification of causal DAGs from observational data remain limited. The Poisson branching structural causal model (PB-SCM) provides a count-valued analogue of linear structural equation models using binomial thinning and independent Poisson exogenous variables, but its causal DAG is generally only partially identifiable. Building on this framework, we propose the Poisson thinning structural equation model (PT-SEM), which replaces binomial thinning in PB-SCM with Poisson thinning and allows node-wise exogenous distributions from diverse count-distribution families. Under node-wise regularity conditions, we establish identifiability of the causal DAG, the thinning coefficients, and the node-wise exogenous distributions. The same identification analysis extends to binomial thinning, yielding full identifiability whenever every nonsink has non-Poisson exogenous noise. We further develop a structure learning algorithm that optimizes, via dynamic programming, a BIC score based on local likelihoods evaluated at plug-in moment estimates, and establish its consistency for DAG selection. Simulations demonstrate favorable performance in DAG recovery and thinning-coefficient estimation, and a real-data application illustrates the practical utility of PT-SEM.

stat.ME↗

arXiv2026

Differentially Private Model-X Knockoffs via Johnson-Lindenstrauss Transform

We introduce a novel privatization framework for high-dimensional controlled variable selection. Our framework enables rigorous False Discovery Rate (FDR) control under differential privacy constraints. While the Model-X knockoff procedure provides FDR guarantees by constructing provably exchangeable ``negative control" features, existing privacy mechanisms like Gaussian noise injection disrupt its core exchangeability conditions. In this work we consider privatizing the data knockoff matrix through Johnson--Lindenstrauss Transform (JLT), a dimension reduction technique that simultaneously preserves covariate relationships through approximate isometry for $(ε,δ)$-differential privacy. We theoretically characterize both FDR and the power of the proposed private variable selection procedure asymptotically. Our theoretical analysis characterizes the role of different factors, such as the privacy parameters, sample size, and feature dimension, in shaping the privacy-power trade-off. Our analysis is based on a novel `debiasing technique' for high-dimensional private knockoff procedure. We further establish sufficient conditions under which the power of the proposed procedure converges to one. This work bridges two critical paradigms---knockoff-based FDR control and private data release. Our analysis demonstrates that structural privacy preservation through random projections outperforms the classical noise addition mechanism, maintaining statistical power even under strict privacy budgets.

stat.ML↗

arXiv2026

Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing

The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.

math.ST↗

arXiv2026

Statistical Inference for Privatized Data with Unknown Sample Size

We develop both theory and algorithms to analyze privatized data in unbounded differential privacy (DP), where even the sample size is considered a sensitive quantity that requires privacy protection. We show that the distance between the sampling distributions under unbounded DP and bounded DP goes to zero as the sample size $n$ goes to infinity, provided that the noise used to privatize $n$ is at an appropriate rate; we also establish that Approximate Bayesian Computation (ABC)-type posterior distributions converge under similar assumptions. We further give asymptotic results in regimes where the privacy budgets vary, establishing similarity of sampling distributions as well as showing that the MLE in the unbounded setting converges to the bounded-DP MLE. To facilitate valid, finite-sample Bayesian inference on privatized data under unbounded DP, we propose a reversible jump MCMC algorithm which extends the data augmentation MCMC of Ju et al. (2022). We also propose a Monte Carlo EM algorithm to compute the MLE from privatized data in both bounded and unbounded DP. We apply our methodology to analyze a linear regression model as well as a 2019 American Time Use Survey Microdata File which we model using a Dirichlet distribution.

math.ST↗

arXiv2026

The Price of Sparsity: Sufficient Conditions for Sparse Recovery using Sparse and Sparsified Measurements

We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrices we identify sufficient conditions on the minimal sample size for maximum-likelihood recovery in the high-SNR regime $ds/p \to \infty$, where $p$ denotes the signal dimension, $s$ the number of non-zero components of the signal, and $d$ the expected number of non-zero components per row of measurement. Combined with known lower bounds, this yields an information-theoretic threshold of order $s\log(p/s) / \log(ds/p)$, making explicit the price of measurement sparsity. In particular, we highlight a regime where the sample-complexity loss from measurement sparsity is logarithmic while the computational gain is nearly linear. Second, we study recovery after sparsifying an originally dense Gaussian design: the observations are generated from the dense design, while estimation uses an independently sparsified design and a rescaled response. In the proportional regime $s=αp$, $d=ψp$, we prove that, for every fixed target error level $δ$ and every slack $\varepsilon>0$, a sample size of order $p/ψ^2$ is sufficient for support recovery for arbitrarily small $ψ$.

stat.ML↗