Papers

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $\Theta(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $\Theta(1/n)$ at $d=1$ and $\Theta((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.

We propose a new unit of analysis for longitudinal data: the Latent Memory Table. The scientific contribution is not the encoder. It is that table, treated as a reusable statistical object on the same footing as a matrix of principal-component scores, a table of estimated random effects, or a table of predicted probabilities. We estimate a statistical table that summarizes recent longitudinal history and is intended to be stored, queried, analysed and reused throughout the statistical workflow. A memory operator maps each masked windowed history to a finite-dimensional state; collecting those states with uncertainty yields the Latent Memory Table. Validation is organized around six properties—recoverability, personalization, temporal coherence, interpretability, stability and reusability—summarized by a composite quality index \(Q\); the Transformer, the SoccerMon case study and the simulations exist to argue that this table deserves that status. Classical exponentially weighted moving averages and related short- and long-horizon scalar summaries arise as restricted, typically univariate special cases of the same operator class. A simulation study with known memory mechanisms shows that \(Q\) and rotation-invariant recovery scores discriminate genuine multivariate or personalized memory from negative controls and from misspecified windows, whereas regime classification accuracy alone does not. SoccerMon serves as an empirical case study: a constructed Latent Memory Table attains \(Q\approx 0.73\) versus about \(0.40\) for classical and lagged principal-component baselines, with incremental held-out value for some wellness targets and Procrustes ensembles for row-wise reliability.

This work presents a novel attention-based framework for estimating the Individual Probability of Treatment Benefit (IPTB) in survival analysis contexts. The proposed model, called Surv-IPTB, directly quantifies the probability that a specific patient will experience extended survival time under treatment versus control. We reformulate IPTB estimation as a binary classification problem, leveraging pairwise patient comparisons across treatment and control cohorts. The framework incorporates a principled handling of right-censored observations through imprecise probability representations, where uncertain treatment effects are characterized by interval-valued probabilities. An attention mechanism with learnable query-key transformations enables flexible, data-driven aggregation of pairwise comparisons, while simultaneously learning soft class probabilities for censored cases. Through extensive experiments on synthetic datasets with complex nonlinear structures, including spiral, bell-shaped, and circular feature spaces, we demonstrate that our approach maintains robust performance across varying censoring rates and treatment effect strengths. The model consistently outperforms meta-learner baselines (T-learner and S-learner) equipped with random survival forests, Cox proportional hazards, and Beran estimators, particularly in challenging nonlinear scenarios where conventional methods exhibit significant degradation. The results establish the proposed attention-based framework as a scalable and statistically principled solution for personalized treatment benefit assessment in survival settings. The code implementing the model is publicly available.

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures. To handle the superlinear regime, taming techniques are employed to produce a stable, explicit scheme. We derive non-asymptotic convergence bounds in Wasserstein-2 distance, with all constants tracked explicitly in terms of dimension and inverse temperature, improving upon the currently known rates for subgradient-based Langevin algorithms. We further provide excess risk estimates for the associated optimisation problem. We verify the assumptions, with explicit constants, for the regularized pretraining potential of a LLM in the GPT-2 lineage and the boosted coordinate-wise variant of SG-TULA pretrains the former competitively against finetuned AdamW and Muon, for which no comparable non-asymptotic guarantees are presently available.

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

In overparameterised classification, training data can be linearly separable even when the underlying distribution is not. In this setting, gradient descent (GD) on the logistic loss diverges in norm while converging in direction to a max-margin interpolating classifier, whose implicit bias can be statistically suboptimal. In this work, we show that early stopping can overcome this suboptimality: in a Gaussian mixture model with label-flipping noise, GD stopped at an appropriate oracle time achieves minimax-optimal excess zero-one risk for covariance spectra with fast and continuous decay, including polynomial and exponential spectral decays. Our analysis combines a sharp upper bound for the early-stopped iterate with a matching statistical lower bound over arbitrary classifiers, yielding optimal rates that are validated by experiments. A central technical contribution is a new calibration result that converts excess logistic risk into excess zero-one risk; it handles the model misspecification induced by the label-flipping noise, and removes the square-root rate in standard bounds. We also establish a lower bound for linear interpolators, showing that interpolation can require exponentially more samples than early stopping to achieve the same excess risk.

Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an $O(h^\beta)$ localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.

Probabilistic Regression Trees (PRTrees) are a smooth and consistent alternative to classical regression trees, producing continuous predictions through probabilistic split assignments. This paper extends the PRTree framework to accommodate missing predictor values directly during tree construction, eliminating the need for prior imputation. Three strategies are proposed, each exploiting the available information differently: a uniform-probability approach, a partial-observation approach, and a dimension-reduced smoothing approach. These modifications are defined to preserve the fundamental probabilistic properties of the original methodology, including probability conservation and marginal compatibility, under arbitrary patterns of missing covariate values. The proposed methods are evaluated on several real-world datasets exhibiting different levels of missingness and are compared with classical regression trees. The results show that the effectiveness of probabilistic tree construction depends strongly on the treatment of missing observations. Across the considered datasets, the fill strategy emerged as the dominant modeling component, often exerting a larger influence on predictive performance than either the smoothing distribution or the proxy-selection criterion. In datasets where a substantial proportion of observations contained missing predictor values, the proposed methods frequently outperformed CART, while maintaining the interpretability and flexibility of tree-based models.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon–Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert–Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

The experience sampling method (ESM) is a longitudinal research design where participants report their thoughts, emotional states and behaviours multiple times a day. Our work is motivated by such data collected by the GrowIt! app, which was released to investigate daily emotions among adolescents during the COVID-19 pandemic. Current procedures to analyse ESM data face various challenges. While standard statistical techniques may not scale well to a high-dimensional setting, machine learning procedures can give biased results due to selection bias introduced by missingness. In our motivating dataset, adolescents dropped out due to previous strong feelings of negative emotions. Hence, the implied missing data are of the missing-at-random type that standard machine learning procedures cannot accommodate. We develop a novel neural network architecture that generalises mixed effects models to deep learning to overcome these challenges. It allows semi-parametric and flexible modelling of data's mean and correlation structure through fixed and random effects. For estimation, we use an adaptation of variational auto-encoders and a Bayesian data augmentation algorithm. Through this approach, the model can accommodate longitudinal outcomes following generic distributions, scale well to high-dimensional settings and provide valid inference when data are missing-at-random. We applied the Deep Generalised Mixed Model to the GrowIt! study and various simulations. The results show potential for the Deep Generalised Mixed Model, yet suboptimal performance due to model instability.

We introduce a fuzzy network jump model for clustering time-varying observations indexed by the nodes of a weighted graph. The framework allows flexible graph representations with spatial and temporal regularization promoting smooth soft cluster assignments across connected nodes and consecutive time points. Estimation is performed through an efficient alternating optimization scheme that exploits the quadratic structure of the regularization terms. A simulation study covering different levels of spatial dependence and cluster overlap shows that the proposed method accurately recovers the true membership probabilities and outperforms competing clustering methods. An application to traffic-network data for the city of San Francisco identifies interpretable traffic regimes and reveals their evolution over time and across connected road segments.

Bayesian model selection couples a discrete model indicator with a model-specific continuous parameter space. We introduce structured dimension-matched variational transdimensional inference (SM-VTI) for finite enumerable model spaces. A rooted construction graph expresses a model as a sequence of local stop/child decisions. Each typed edge compiles a declared scientific parent-child edit into an exact native-coordinate dimension-matching lifting; an edge-conditioned flow then learns the residual continuous transport. The resulting local policy and conditional flow define one direct joint variational distribution, without embedding every model in a saturated maximum-dimensional surrogate. We derive its exact path density and optimize the joint reverse-KL objective. On a controlled 15-model target, SM-VTI-Joint recovers terminal masses, local actions, and nonlinear conditional geometry. On a 128-model misspecified robust variable-selection problem, a 10-data-set nearly parameter-matched affine comparison with AVTI shows stronger early model-mass recovery and competitive final joint accuracy under the same target-evaluation budget.

Autonomous agents now carry out entire data analyses, selecting cohorts, joining tables, and fitting models with little step-by-step supervision. When such an analysis turns out to be wrong, someone must determine which operation caused it. A recent approach does this without any labelled mistakes, learning instead from analyses known to be sound and flagging operations that depart from what that model predicts; how reliable such audits are has not been studied. This paper supplies that analysis. The choice of score determines whether an error can be localized at all. If each operation is scored by how surprising it is given the operation immediately preceding it, then operations that merely inherit an earlier error are indistinguishable from correct ones, so one mistake produces one flag; scores computed against a longer reconstruction of the intended analysis instead spread a single mistake across many operations. We quantify how far they spread, and how to choose the comparison length when an error accumulates gradually rather than at once. We then give procedures that control the proportion of falsely flagged operations within a single audited analysis, requiring only that sound analyses be exchangeable rather than that the fitted model be correct, and we quantify how much the guarantees weaken when the model is imperfect or when the analysis was selected for review in a way that depends on its content. Finally we establish a limit on what any such audit can report: errors below a certain magnitude cannot be attributed at all, being indistinguishable from ordinary variation among sound analyses. This limit falls so slowly as more sound analyses are collected that at the representation sizes now in use a hundredfold increase reduces it by under two percent, so the dimension of the representation rather than the volume of training data is the binding constraint.

Knowledge distillation (KD) is widely used to transfer the capabilities of large language models (LLMs) to smaller students, but existing objectives often struggle to balance faithful imitation and robust generation. In particular, existing methods mainly combine FKL and RKL, overlooking that RKL itself provides a mechanism for adjusting the student's imitation strength. Motivated by this, we revisit on-policy Reverse Kullback-Leibler (RKL) distillation and decompose its objective into a teacher-fitting term and a student-entropy term, without introducing an explicit FKL branch. We show theoretically that the token-level optimal student distribution corresponds to a tempered variant of the teacher distribution, where the adaptive weight controls the trade-off between mode-seeking and uncertainty preservation. Guided by this insight, we propose Adaptive Entropy Distillation (AED), which uses the teacher's entropy to dynamically calibrate token-level imitation strength. Experiments on instruction-following and mathematical reasoning benchmarks demonstrate that AED achieves superior overall performance and generally improves teacher–student distributional and entropy alignment.

Real-world time series inherently intertwine global periodic structures with localized non-stationary variations. Existing approaches process these heterogeneous dynamics within a single computational domain, incurring fundamental limitations: time-domain models suffer from periodic misalignment over long horizons, while frequency-domain models over-smooth transient spikes. We argue that the optimal computational domain is not a property of the model, but of the data itself. Based on this principle, we propose RouteTS, a unified forecasting framework that partitions the frequency spectrum via amplitude routing and delegates components to their mathematically optimal domains. Dominant frequencies are processed by a complex-valued linear predictor in the frequency domain to preserve periodic structure, while residual spectral energy is reverted to the time domain and modeled by a lightweight MLP for local variations. Extensive experiments demonstrate that RouteTS achieves competitive prediction accuracy across diverse real-world datasets, with routing decisions guided by the underlying spectral signature. Furthermore, the lightweight design of RouteTS provides significant computational efficiency advantages, offering a principled solution to the longstanding dilemma between global periodicity and local transience.

Multi-agent LLM systems route among model-backed advisors, yet a deployer rarely knows before shipping whether routing will help at all. Prevailing routers optimize a gate's AUC and presume that advisor complementarity suffices. We show that neither determines the deployable gain. We introduce RouteGuard, a deployment-certification framework. Routing gain decomposes as $G = \pi \Delta_E$, and the achievable gain is governed by a conditional-regret functional $\Phi$, not by AUC. A finite-sample certification bracket comes with a matching Le Cam lower bound, constant-sharp over the fixed-activity class, and a robustness phase transition. On two benchmarks the framework acts as a guardrail. On RouterBench (11 cross-family models) the verdict depends on the sampling unit: the protocol certifies a gain over GPT-4 under prompt-level sampling and withholds it under workload-cluster resampling, because the gain rests on 3 of 86 workload cells. On OpenRCA (three Gemini advisors) the advisors are statistically redundant: the realized oracle sits at or below the independence baseline in all pools we tested (221 RouterBench pools and three OpenRCA distributions), so the protocol correctly refuses to certify. A pre-registered semi-synthetic control confirms calibration: the protocol certifies a genuine gain once $m \ge m^\star$ and does not certify a true null. Code and frozen artifacts will be released with the published version.

Probabilistic prediction heads in neural networks typically output either a Gaussian mixture or a single conformal region. Neither separates the distinct sources of uncertainty often present in real prediction tasks: a discrete choice among modes, bounded systematic drift within the chosen mode, and irreducible stochastic noise. We introduce the Hybrid Probabilistic Zonotope (HProbZ), an output head that represents these three sources as binary, bounded, and stochastic generators of a zonotope, and admits a closed-form likelihood by convolution. Sharing the bounded generator across prediction steps couples future predictions algebraically, so observing one step refines the predictive distribution at every remaining step in a single forward pass. We establish that the three generators are identifiable from the likelihood up to permutation, and that an HProbZ density is representationally distinct from any finite Gaussian mixture. The same shared structure provides analytic per-mode risk and distribution-free multi-modal conformal sets at inference time. Empirical analysis on representative prediction benchmarks supports the effectiveness of the design relative to same-encoder mixture baselines, while offering structural properties that mixture or convex-conformal predictors do not jointly provide.

Sequential clinical decision-making often involves more than maximizing average efficacy. Clinicians may need to simultaneously optimize clinically relevant tails of the outcome distribution, control treatment-related risk, and choose among multiple treatment options. Existing quantile dynamic treatment regime (DTR) methods capture distributional features of treatment outcomes but remain largely restricted to efficacy-only objectives and binary treatments. To address these limitations, we propose Risk-Aware Quantile Dynamic Treatment Regimes (RQDTR), a unified framework that optimizes a prespecified quantile of the cumulative potential outcome while explicitly incorporating treatment-related risk. We also develop an angle-based formulation for jointly learning decision rules across multiple treatment categories. Our framework includes three interpretable subclasses: efficacy-only quantile learning, constraint-based learning with population-level risk control, and utility-based learning through a composite benefit-risk utility. Theoretically, we establish identification and oracle equivalence, Fisher consistency of the smoothed surrogate, consistency of the estimated regime, and finite-sample performance error rates. Extensive simulation studies and applications to All of Us major depressive disorder and MIMIC-III sepsis data demonstrate that RQDTR improves tail-oriented efficacy and achieves more favorable benefit-risk trade-offs than existing quantile DTR methods.

We developed a unified covariate-adjusted causal inference framework for estimating the desirability of outcome ranking (DOOR) probability for benefit-risk evaluation in randomized trials and observational studies. The framework expresses the DOOR probability as a bilinear functional of the marginal ordinal outcome distributions under the two treatment strategies, estimates conditional ordinal distributions through sequential risk-set hazards, and derives the efficient influence function (EIF) of the DOOR probability. The point-estimation simulations compared G-computation, normalized inverse probability weighting (IPW), augmented IPW (AIPW), and targeted maximum likelihood estimation (TMLE), with nuisance functions estimated using generalized linear models or Super Learner (SL). TMLE-SL showed the strongest and most consistent point-estimation performance, with AIPW-SL ranking second. EIF-based inference was then evaluated for AIPW-SL and TMLE-SL, with and without cross-fitting, across settings varying in overlap, treatment-effect heterogeneity, and treatment allocation. CVTMLE-SL showed the strongest overall performance across DOOR-scale bias, recovery of the underlying ordinal distributions, standard-error accuracy, and confidence-interval coverage. We illustrate the methodology using data from the multidrug-resistant organism network of the Antibacterial Resistance Leadership Group.

Factorized generative models commonly regularize a latent style variable z_s by matching its marginal distribution to a fixed Gaussian prior and interpret this as evidence that the style representation is independent of class information. We show that this interpretation is incorrect. Matching only the marginal distribution places no constraint on the class-conditional distributions, allowing the latent style to remain highly predictive of the label despite appearing perfectly Gaussian in aggregate. We derive an exact decomposition showing that this mismatch is one of four conditions required for factorized sampling, and demonstrate that eliminating it is necessary but not sufficient to obtain the intended factorization. Empirically, our case-study model and four representative latent baselines achieve near-zero global MMD while still allowing a linear probe to recover class labels with 74%–100% accuracy (10% chance level). Our model reaches 99.15% clustering accuracy, whereas externally evaluated class-conditional generation succeeds only 16% of the time. This leakage remains under six independent perturbations involving model capacity, curriculum, prior geometry, and supervision across two datasets. Four mitigation strategies reduce probe accuracy to 21%–46%, although they leave within-class dependence largely unchanged. A post-hoc conditional prior improves externally evaluated class generation to 0.97 on MNIST without retraining but reaches only 0.41 on CIFAR-10, while an empirical style bank achieves 0.88 on CIFAR-10. These results demonstrate that no divergence computed solely on the marginal distribution of the style latent can certify independence from class labels, and that reporting marginal statistics alone does not verify the property commonly claimed in factorized generative models.

The daily allocation of the finite 24-hour time budget is strongly associated with physical, mental, and cognitive health. While predictive models can estimate the relationship between time-use compositions and health outcomes such as body mass index, life satisfaction, and cognition, most optimization approaches focus only on maximizing expected benefit and do not consider the uncertainty inherent in data-driven prediction. Ignoring uncertainty in health-related decisions can lead to unrealistic time-use recommendations. To address this gap, we introduce an uncertainty quantification Quality Diversity (QD) framework for a more reliable time-use recommendation. Objective functions are derived using compositional data analysis using a large child cohort dataset n > 1000, to capture the relationship between daily activity compositions and multiple health indicators. We develop a new approach that incorporates predictive uncertainty into QD processes and produces more reliable recommendations that balance the expected health benefits with the confidence of the model. We explore the solution space through variable-based and objective-based behavioral representations, revealing diverse high-quality time-use composition and explicit relationships between health outcomes under uncertainty. By embedding uncertainty directly into optimization, our framework shifts the time-use recommendations toward regions of lower uncertainty while preserving high-quality structures for more reliable decision-making in behavioral health.

Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Achieving local differential privacy in distributed optimization while maintaining low communication cost remains challenging. Existing vector quantization methods, such as vqSGD, rely on high-dimensional geometric constructions but incur unfavorable dimension-dependent variance. In this work, we propose Subsampled Stochastic TurboQuant (SSTQ), a framework that combines a bounded Kashin representation, data-independent coordinate subsampling, and privacy-aware one-dimensional quantization. SSTQ includes two variants: (1) a Flat Randomized Response variant that is unbiased and, for a fixed codebook bit-width, frame redundancy, and dimension-independent Kashin level, achieves reconstruction MSE that scales linearly with the ambient dimension $d$, while using only $\lceil \log_2 N \rceil + b$ bits per message. Here, $N = \Theta(d)$ denotes the frame size in the Kashin transform and $b$ is the codebook bit-width; and (2) a metric-aware truncated-Laplace variant that removes the exponential dependence on bit-width at the cost of a non-vanishing bias. We also derive a convex uniform-surrogate codebook objective whose worst-case codebook-dependent upper bound improves from $O(4^b)$ to $O(2^b)$. Experiments on synthetic regression, Fashion-MNIST, and CIFAR-10 compare the per-message privacy-utility and uplink-communication trade-offs of SSTQ with those of established baselines, demonstrating favorable utility and communication efficiency.

The Subspace Constrained Mean Shift (SCMS) algorithm is a popular nonparametric method for extracting density ridges, which serve as a low-dimensional representation of high-dimensional data. It is a widely held belief in the literature that SCMS trajectories converge to the classical density ridge, which we call the "static ridge", defined via the density gradient and the eigenvalues and eigenvectors of the density's Hessian. In this paper, we demonstrate that this assumption does not hold in general, as the static definition fails to account for the rotation of the trailing eigenspace along the continuous flow of the algorithm's underlying vector field. To resolve this, we propose a paradigm shift by introducing the "stable ridge", a novel geometric structure defined through the lens of dynamical systems and the Jacobian of the projected density gradient. We prove that this stable ridge is the true theoretical target of the SCMS algorithm. Building upon this foundation, we develop a generalized SCMS framework utilizing a constant step size, establishing its uniform R-linear convergence and topological surjectivity onto the stable ridge. We further derive the rates of convergence for estimating the stable ridge in terms of the Hausdorff distance. Finally, we expose that the original SCMS algorithm suffers from polynomial-time computational complexity, which is caused by implicitly coupling the step size to the smoothing bandwidth via the Mean Shift operator, and demonstrate how our generalized framework provides a statistically consistent and more efficient solution.

Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth $\Omega(N)$ in the worst case. We further show that measurement-based quantum computation (MBQC) provides a natural implementation of the required shared classical randomness through suitable adaptation of the random measurement outcomes. Numerical experiments on MBQC-based generative models support the analytical results.

Joint Energy-Based Models (JEM) unify classification and generation within a single network and support out-of-distribution (OOD) detection. Canonical JEM training relies on stochastic gradient Langevin dynamics (SGLD); a theoretically motivated alternative, the Predictor-Corrector (PC) sampler, has not previously undergone a systematic replication test on the canonical model. We reproduce canonical JEM on WideResNet-28-10 without normalisation layers on two independent runs and test a fixed-noise PC adaptation - with the degenerate annealed-noise predictor replaced by a deterministic gradient step - across three protocols: the adapted sampler replacing SGLD throughout the full training trajectories (115-132 epochs); cold-start generation (FID); and refinement-style multi-OOD detection (AUROC). The reconstruction reaches 92.88% test accuracy and buffer-FID 44.46 (canonical: 92.90% and 38.40). We document two failure modes: catastrophic late-training divergence with the signature of the canonical outlier-buffer mechanism (all four runs), and run-dependent SVHN OOD-discrimination dynamics. No consistent method-level advantage of the adaptation over SGLD is observed on any protocol: refinement AUROC differences stay below 0.007 across ten checkpoint-OOD pairs; seeded cold-start generation favours SGLD by about five FID points; on the training protocol a hierarchical seed-by-image bootstrap gives a 95% confidence interval on the macro-averaged AUROC difference that contains zero, while a seed-level equivalence test with two runs per method cannot establish formal equivalence. The training-protocol data are consistent both with equivalence and with a small directional effect. This outcome is consistent with theory: the guarantees of the annealed-noise PC framework do not transfer to the constant-noise regime of canonical JEM.

Every SVAR result is conditional on two choices: the restrictions that identify the shock and the variables on which they operate. The literature disciplines the first; the second is chosen by hand. We develop a Bayesian methodology that constructs information sets, uses an out-of-sample criterion, and retains the largest system it admits. Under recursive identification, output rises with housing production rather than household credit alone. For monetary policy, an anchor-free joint Bayesian proxy SVAR with multiple instruments strengthens the credit spread channel. A core system augmented with the selected corporate spread identifies expected default risk as a potent transmission margin.

Metamodels are instrumental in reducing the computational burden associated with nested reliability analyses and optimization loops in Performance-Based Risk Optimization (PBRO) of structures under stochastic loads. In this context, stochastic emulators are particularly useful because they approximate response distributions while accounting for the intrinsic stochasticity of the simulator. Among these methods, Stochastic Polynomial Chaos Expansion (SPCE) is especially attractive because it does not require replications of nonlinear analyses at fixed input conditions. However, SPCE may present limitations in accurately representing extreme responses in the tails of structural response distributions. To address this limitation, this study proposes a framework that combines Generalized Stratified Sampling (GSS) with SPCE. The GSS scheme partitions the input space into strata according to the intensity of the hazard, improving the representation of extreme responses, while independent SPCE emulators are trained within each stratum. The conditional exceedance probabilities estimated in each stratum are then recombined using the total probability theorem to evaluate the probabilistic constraints. The proposed GSS-SPCE framework is applied to the optimal design of buckling-restrained brace cross-sectional areas in a two-story steel building. The objective is to minimize the initial construction cost while satisfying prescribed probabilistic performance constraints. Results show that the proposed framework accurately estimates structural response distributions, including their tail regions, while substantially reducing the number of nonlinear model evaluations required for PBRO.

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave–convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

This paper develops procedures for nonparametric goodness-of-fit testing under covariate shift, where labelled data are drawn from a source population but goodness-of-fit is evaluated for a target population. The distribution mismatch is quantified by either a bounded moment condition or a sub-exponential tail condition on the target-to-source density ratio. Our method combines truncated importance-weighting kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. The truncation stabilizes the importance- weighting kernel ridge regression as well as the bootstrap calibration, making our approach applicable even when the density ratio has heavy tails. We prove nonasymptotic validity and sharpness of the resulting confidence sets under suitable operator compatibility conditions, and establish explicit error rates for coverage probability under specific conditions on the target- to-source density ratio and on the spectral decay of the kernel integral operator. Numerical experiments corroborate our theoretical findings.