Papers

We formulate generative modeling as a path integral in which flow-based, diffusion-based, variational, and adversarial models arise as different evaluation principles for a single master action. Its Martin-Siggia-Rose-Janssen-de~Dominicis (MSRJD) form separates free from interacting probability flows and opens them to diagrammatic perturbation theory. The expansion yields a one-loop correction to deterministic samplers at no stochastic-sampling cost, which we validate on solvable and nonlinear drifts, where it reduces a 53 % tree-level error to 1.6 %. Imperfect learned scores enter as insertions and yield a response-weighted score-matching objective, and symmetry-equivariant drift design becomes an operator expansion with EFT power counting.

Two free satellite signals carry real information about glacial-lake outburst risk in the Nepal Himalaya: radar interferometry sees a moraine dam slowly sagging, and satellite weather marks the weeks when a primed lake is under stress. A companion feasibility study found that deformation indicates which lake is destabilizing and weather indicates when it is at risk, but proposed no predictive model. To address this gap, we propose and evaluate models that predict which site is susceptible and when a trigger arrives. We test three related hazards on free data alone: large moraine- and ice-dammed bursts, rainfall-triggered landslides, and smaller floods from ponds on and around a glacier. Each hazard gets two questions, never blended. Using 589 dated outbursts from HMAGLOFDB and several thousand catalogued landslides, we match each event against similar but unfailed sites, and hold every model to a strong simple baseline under spatial cross-validation that withholds whole map tiles, so no model succeeds by recognising a trained-on neighbourhood. Antecedent weather times the trigger at ROC 0.73 for big bursts, 0.83 for landslides, and 0.82 for small floods. Terrain ranks susceptibility only in part: scored naively it appears near 0.9, largely because catalogued failures cluster in wetter ranges; matched against comparable nearby sites the honest figures are 0.76, 0.71, and 0.54 (no better than chance). The burst signal holds within single regions, reaching 0.89 in Nepal alone. Five deep-learning models do not decisively beat a simple gradient-boosted baseline. Three score marginally higher on landslides, a hint too small to confirm. For the lake hazards the baseline wins outright, reproduced by a three-rule decision tree on ruggedness and monsoon rainfall. We close with a ranked Nepal watchlist, a prioritisation aid, not a prediction, and note where free data reaches its limits.

We develop a probabilistic cavity-contiguity framework for proving replica-symmetric convergence of local marginals in mean-field Gibbs systems. The approach is based on cavity decompositions of the Hamiltonian together with direct comparison of probability measures through Radon-Nikodym derivatives and Hellinger-type estimates. At a conceptual level, the method separates the concentration of the relevant order parameters from the identification of the asymptotic cavity model and the comparison of the associated Gibbs measures. In contrast with interpolation-based approaches, the argument relies only weakly on the Gaussianity of the disorder and naturally accommodates concentration tools such as Poincaré and log-Sobolev inequalities. Rather than pursuing maximal generality, with the aim of making the method transparent, we implement the framework in a canonical example: the high-temperature Sherrington-Kirkpatrick model. In this setting, we prove that the marginal law of a fixed spin converges in total variation toward the effective one-dimensional cavity measure predicted by the replica method. Beyond the specific result for the Sherrington-Kirkpatrick model, the paper illustrates a broader cavity-contiguity methodology which is expected to extend naturally to other mean-field Gibbs systems, particularly Bayesian inference models with or without mismatch.

What a city makes reachable depends less on what it contains than on who can still afford to move when travel costs rise. We leverage the 2026 US-Iran oil shock as a natural experiment, applying a hierarchical panel regression discontinuity design to 1.7 trillion point-of-interest visits across 122,000 neighbourhoods in China and the United States. Mobility range declined in nearly three-quarters of neighbourhoods, but responses varied systematically with pre-shock urban conditions. Exposure to energy-intensive travel explained the largest share of modelled heterogeneity in both countries, while adaptive capacity and activity composition further shaped how travel was reorganized. Longer baseline travel intensified contraction, whereas greater car dependence constrained adjustment. Crucially, similar mobility outcomes arose from different processes: some neighbourhoods maintained travel by absorbing higher costs, whereas others appeared structurally locked into travel they could not reorganize. Fuel-price shocks, therefore, act as urban stress tests, revealing which neighbourhoods a city keeps connected.

Let $\widehat R$ be the Pearson sample correlation matrix formed from $n$ independent Gaussian observations in $p$ dimensions, and write $m=n-1\ge p$. Under the null correlation $R=I_p$, the classical independent beta product, exact cumulants, and full Fourier inversion yield, along every sequence $p\to\infty$ with $m\ge p$, a uniform first Edgeworth expansion for $\log\det\widehat R$, centered by its exact mean and scaled by its exact standard deviation. The expansion identifies the exact finite dimensional skewness correction and gives the sharp Kolmogorov equivalent $A_{m,p}/\{6\sqrt{2\pi}V_{m,p}^{3/2}\}$, where $V_{m,p}$ is the exact variance and $A_{m,p}$ is the absolute third cumulant. This equivalent unifies the square, fixed gap, growing gap, proportional, and dilute regimes; in the square regime the error has order $(\log p)^{-3/2}$ with an exact constant. For every positive definite population correlation matrix $R$, we prove a uniform finite sample Berry-Esseen bound that explicitly tracks population dependence. All theoretical results have exact or proved equivalent Lean 4 formulations whose declarations and dependencies are kernel checked.

Recovering a low-dimensional latent subspace from nonlinear observations of Gaussian covariates in high dimensions is a fundamental problem in feature learning. Here, we consider Gaussian multi-index models in which the covariates $\boldsymbol{x}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0,\boldsymbol{I}_d)$ and the responses $\boldsymbol{y}_i$ depend on $\boldsymbol{x}_i$ only through its projection onto an unknown $r$-dimensional subspace. Earlier work based on approximate message passing (AMP) identified a sharp threshold for weak recovery [Troiani et al., 2025], raising the question of whether it can be attained, without side information, by a spectral method. We answer this affirmatively and develop a general random matrix theory for matrix-valued spectral estimators of the form \[\boldsymbol{D}_n=\frac{1}{n}\sum_{i=1}^n\boldsymbol{T}(\boldsymbol{y}_i)\otimes\boldsymbol{x}_i\boldsymbol{x}_i^\top,\] where $\boldsymbol{T}$ is an arbitrary bounded symmetric matrix-valued preprocessing map of fixed dimension. As $n,d \to \infty$ with $n/d\to\alpha$, we prove that the empirical spectral measure of $\boldsymbol{D}_n$ converges almost surely to a deterministic compactly supported distribution characterized by a matrix-valued self-consistent equation. We then establish a spectral phase transition for the largest eigenvalue: below threshold it sticks to the bulk edge, while above threshold an outlier emerges. We characterize the outlier location through a finite-dimensional deterministic equation and show that the associated spectral estimator achieves weak recovery of the latent subspace. Finally, we prove that the AMP-derived preprocessing of [Defilippis et al., 2025] is optimal among all bounded matrix-valued preprocessing maps of any fixed dimension. Its transition coincides with the AMP weak-recovery threshold, proving the general spectral conjecture of [Defilippis et al., 2025].

Let $\boldsymbol{X}_i=(X_{1i},X_{2i})^\top$ be independent and identically distributed observations with mean $\boldsymbol{\mu}=(\mu_1,\mu_2)^\top$ constrained by $\mu_1\leq\mu_2$. We study empirical-likelihood inference for a fixed mean vector and distinguish it from the previously known test of equality against an ordered alternative. At a fixed interior point, the constrained empirical likelihood ratio has the usual $\chi^2_2$ limit. At a fixed boundary point $(m,m)^\top$, its limit is the chi-bar-square distribution $\tfrac12\chi^2_1+\tfrac12\chi^2_2$. By contrast, profiling the unknown common mean in the equality-versus-order test yields $\tfrac12\chi^2_0+\tfrac12\chi^2_1$, the $k=2$ ordered-mean case of El Barmi (1996). We give an exact reduction of the latter statistic to the empirical likelihood of the paired differences, establish the localization step needed for the fixed-boundary expansion, and derive a local-to-boundary limit showing that interior calibration is not uniform over $n^{-1/2}$-neighborhoods of the boundary. Monte Carlo experiments under Gaussian, Student $t_5$, and shifted log-normal sampling examine fixed, boundary, and local regimes with explicit numerical-failure accounting. Illustrative paired-data analyses show the practical distinction between fixed-candidate confidence regions, directional equality tests, and ordinary scalar empirical-likelihood intervals truncated to the nonnegative parameter space.

In high-stakes applications, reliable confidence estimates are as important as the predictions themselves. Confidence calibration ensures that predicted probabilities reflect the likelihood of correctness, making it essential for safe deployment of deep learning models. However, existing methods typically assume access to clean validation data, which is often unrealistic due to label noise and domain shifts. This thesis develops methods for improving calibration under these conditions. First, we address calibration under label noise. Standard methods can produce misleading confidence estimates when labels are unreliable. We propose a framework that uses an estimated noise model to reconstruct noise-free confidence estimates by modeling the relationship between noisy and clean label distributions. We extend this approach to Conformal Prediction (CP), which provides set-valued predictions with guaranteed coverage. Our noise-aware CP method estimates clean conformity scores despite label noise, enabling reliable uncertainty quantification. Next, we study calibration in unsupervised domain adaptation, where a model trained on a labeled source domain is adapted to an unlabeled target domain. Since labeled target data are unavailable, we estimate target-domain accuracy from source performance and domain discrepancies, enabling calibration without target labels. We also consider privacy-preserving settings in which user labels and model outputs must remain protected. We propose a locally differentially private conformal prediction framework that provides valid uncertainty quantification while maintaining privacy guarantees and balancing privacy, computational feasibility, and prediction reliability. Our results bridge calibration theory and practical deployment in safety-critical applications, contributing to reliable, privacy-preserving, and noise-resilient neural network predictions.

A predictive model can fit its data even when its information set is insufficient; fit alone cannot establish sufficiency. This Perspective introduces Level II-A, a new design-based inference framework to test this distinction, illustrated in anticipatory EEG using contingent negative variation. A pre-event endpoint is committed before the delay to the imperative event is randomised. That later-assigned delay thereby becomes a negative-control probe of whether past-adapted information was sufficient for an already committed result. Under the past-adapted factorisation, accounts using only pre-commitment information cannot systematically order the endpoint by that delay. Leakage-safe preprocessing, a frozen label-blind comparator and retained-sample qualifications carry the exclusion to the confirmatory residual. A qualified material negative ordering supports conditional insufficiency without identifying a mechanism; an adequately sensitive null supports a bounded affirmative conclusion calibrated by the pipeline's false-adequacy rate. A non-compensatory rule separates these from diagnostic failure, selection-limited, opposite-direction and inconclusive outcomes. No human EEG data are analysed. In the synthetic benchmark, grid-based false-adequacy boundaries are $15\,\mu\mathrm{V\,s^{-1}}$ for assignment isolation and $30\,\mu\mathrm{V\,s^{-1}}$ for the sequential e-value route, in both directions. The design transfers wherever endpoint commitment precedes an exogenous label, probing sufficiency only where a declared alternative predicts ordering by it. It turns "the past explains it" from a working explanatory assumption into a magnitude-qualified, testable claim.

We prove new deviation inequalities for infinite-order U-processes, that is, U-processes whose order may increase with the sample size. Our goal is to obtain the sharpest possible dependence of the tail bounds on the order of the process, as well as to characterize the largest range in which the process exhibits sub-Gaussian deviations. We derive closed-form inequalities for processes indexed by Euclidean classes, including VC-subgraph classes of functions. We then illustrate the statistical applications of our results to multivariate depth. In particular, we introduce a multivariate analogue of the Hodges-Lehmann estimator based on Tukey's median, establish sub-Gaussian concentration inequalities with explicit constants for its deviations from the mean, and identify an interesting phenomenon concerning its asymptotic distribution.

The unit square $I=[0,1]^2$ is divided into two rectangles by the vertical line $x=p$, where $p\in(0,1)$. Consider $N$ independent uniformly distributed points on $I$, which we interpret as a population of individuals, with the line $x=p$ representing a community boundary that separates the population into two communities. Whether a pair of individuals share a connection depends on their locations in $I$ and on whether they belong to the same community. Specifically, two individuals in the same community are connected if they are within distance $R_N$ of each other, while two individuals in different communities are connected if they are within distance $R_N'$ of each other, giving rise to a geometric variant of the so-called stochastic block model. A statistician observes the adjacency matrix of the resulting graph together with the geometric locations of the individuals and is tasked with estimating the boundary location $p$. Depending on how $R_N$ and $R_N'$ scale as $N\to\infty$, we establish necessary and sufficient conditions for consistent estimation of $p$. Whenever consistent estimation is possible, we devise an estimator that converges to $p$ as $N\to\infty$ and provide explicit bounds on its estimation error.

The presence of unmeasured confounding factors during the collection of observational data may lead to biased estimates of the effect of an exposure on an outcome. Consequently, a central problem in causal inference based on observational data is sensitivity analysis with respect to unmeasured confounding. Existing sensitivity analyses generally focus on worst-case bounds. We propose an exact method for quantifying the significance of confounding, defined here in terms of the complete range of analytical estimates of the stratification-based Risk Ratio over the set of all joint distributions compatible with the observed characteristics. We refer to the proposed method as Inverse Confounding Analysis (ICA). The proposed ICA method extends the widely used E-value approach but, in contrast to it, does not restrict the analysis to a worst-case lower bound. Instead, it provides exact estimates over the entire set of admissible configurations. This requires several additional input parameters, namely the frequencies of the exposure, the confounder, and the outcome. The ICA method is based on an inverse problem: reconstructing the set of admissible joint distributions from specified frequencies and pairwise associations. We formulate this reconstruction problem as a system of nonlinear equations and obtain an analytical solution. Surprisingly, the complete solution set can be parameterized linearly by a single free parameter. The corresponding stratification-based Risk Ratio is then represented as a fractional-linear function of this parameter. This representation makes it possible to derive exact analytical measures of the significance of confounding over the entire set of admissible statistical configurations.

Structured potential outcomes such as microscopy images may be recorded after an unknown, unit-specific transformation. If that transformation can depend on treatment, covariates or the intrinsic outcome, raw-coordinate analyses may mix biological effects with acquisition geometry. We study the unrestricted observation model X = {\Gamma} . Y(A) and characterize its observable information: a target is uniformly recoverable exactly when it is constant on group orbits, while a Borel maximal invariant retains every measurable invariant target. We then distinguish observability from statistical losslessness. A quotient-faithful reconstruction theorem shows that quotient reduction is sufficient for the full transformed experiment exactly when the conditional law of the raw observation given treatment, covariates and the quotient has a parameter-free version. Conditional Haar contamination on a compact group yields Blackwell equivalence as a special case; it is not imposed in the main model. We also separate independent site-specific product actions from shared diagonal actions and show why componentwise canonicalization can discard relative cross-site information. Under explicit metric and kernel regularity, an approximate-contamination theorem bounds quotient-law Wasserstein error and the induced perturbation of population maximum mean discrepancy. For finite-support multichannel lattice images, we construct a maximal invariant under integer translations and quarter turns, combine its characteristic Gaussian kernel with a complete paired-swap test, and retain the original simulations and RxRx1 HUVEC study. Under the sharp null, the quotient test rejected in 0.052 of simulation replicates; at unit effect strength its power was 0.992. The primary RxRx1 contrast had an enumerated paired-swap p-value of 0.0078.

The efficient randomized-adaptive design attains the minimal possible variance of the allocation proportion for any pre-specified target and has been extended from two to several treatment arms, but both versions target a single response. Confirmatory trials increasingly rely on several prioritized endpoints, typically efficacy followed by safety, that cannot be reduced to one summary without losing information. The efficient randomized-adaptive design is extended here to targets driven by the net treatment benefit of generalized pairwise comparisons, allowing any number of arms and endpoints of mixed type, including censored time-to-event outcomes. Strong consistency, a law of the iterated logarithm, and asymptotic normality of the allocation proportions are established by combining a sequential Hájek projection for the net benefit with the martingale technique used for the original design, and the design is shown to attain the semiparametric efficiency bound within the class of designs that track a smooth function of the net benefit. In simulations calibrated to a three-arm phase III melanoma trial, the proposed design concentrates allocation on the superior treatment more efficiently than designs driven by a single endpoint, while preserving type I error and power. A redesign of the trial illustrates the practical benefit, and the approach is discussed in relation to current regulatory thinking on adaptive designs for confirmatory trials.

Induction is the process by which empirical evidence is transformed to knowledge. Hume famously argued—and Popper and others agree—that there can be no logical justification for induction. A weaker form of induction, due to Bayes, expresses the aforementioned knowledge in terms of probabilities, and we review some well-known and not-so-well-known criticisms of the Bayesian solution. We then investigate the relatively new possibilistic inferential model (IM) framework, showing that, in addition to the IM's strong, statistical reliability guarantees that it uniquely enjoys, it is safe from those criticisms that damage the Bayesian foundations. For illustration, we reconsider the classical sunrise problem and compare our proposed solution with Laplace's famous rule of succession.

Efficient urban logistics requires a comprehensive understanding of traffic dynamics, particularly as it pertains to kinetic parameters influencing energy consumption and trip duration estimations. While real-time traffic information is increasingly accessible, current high-precision forecasting services embedded in route planning often function as opaque 'black boxes' for users. These services, typically relying on AI-processed counting data, fall short in accommodating open design parameters essential for management studies, notably within Supply Chain Management. This work revisits the modelling of traffic conditions in the context of city logistics, emphasizing its significance from the user's point of view, with two focuses. Firstly, the focus is not on the vehicle flow but on the vehicles themselves and the impact of the traffic conditions on their driving behaviour. This means opening the range of studied indicators, beyond vehicle speed, to describe extensively the kinetic and dynamic aspects of the driving behaviour. To achieve this, we leverage the Art.Kinema parameters designed to characterizing driving cycles. Secondly, this study examines how the driving context (i.e., exogenous factors to the traffic flow) determine the mentioned driving behaviour. Specifically, we explore how accurately the kinetic behaviour of a vehicle can be predicted based on a limited set of exogenous factors, such as time, day, road type, orientation, slope, and weather conditions? To answer this question, statistical analysis was conducted on real-world driving data, which include high-frequency measurements of vehicle speed. A Factor Analysis and a Generalized Linear Model have been established to link kinetic parameters with independent categorical contextual variables. The results include an assessment of the adjustment quality and of the robustness of the models, as well as an overview of the models' outputs.

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

The Global Carbon Budget (GCB), the community reference dataset for the carbon cycle, is reissued annually. The 2025 release introduces several adjustments to the published series that we compare with prior releases starting in 2017. On a common 1959-2016 sample, the mean of the GCB budget imbalance jumps from within +/-0.17 GtC/yr of zero for every vintage 2017-2024 to 0.61 GtC/yr in 2025, the only vintage whose 95% confidence interval for the imbalance mean excludes zero. The size of the imbalance changes much less: its mean absolute value rises from 0.61 to 0.76 GtC/yr. It is the mean, the quantity the budget identity constrains, that moves. We document and explore this shift in two ways. First, we conduct a model-free analysis, where we attribute the shift to a new adjustment that places the published land sink 0.40 GtC/yr below its ensemble mean (the average of the underlying models), a smaller adjustment in the ocean sink in the opposite direction, and a change in the composition of the bookkeeping ensemble. Second, we consider a dynamic statistical GCB model augmented with climate covariates. Its parameters are estimated for every GCB vintage 2017-2025. The coefficients of atmospheric concentrations in the sink equations shift in opposite directions on the 2025 issue, mirroring the model-free findings. A constant in the budget equation, statistically unnecessary in every vintage from 2017-2024, is required in 2025 and is estimated at -0.59 (0.09) GtC/yr. There is a persistent drifting imbalance across the entire sample in the budget equation. Each of the three adjustments is documented in the 2025 release and rests on evidence about the component it corrects. Their joint effect is a budget that closes over the last ten years and carries a mean imbalance of 0.61 GtC/yr over the full record. We argue that this cost to the full sample outweighs the gain on the last ten years.

Outcomes are regressed on a calibrated probability vector for unobserved class membership. Under a structural conditional mean excluding the score and conditional calibration, the observed-data model reduces to a partially linear regression. The probability vector is a Berkson-type surrogate for membership, so the effect vector $\tau$ is identified without attenuation. In practice the vector is often coarsened to a hard label - an argmax, a confidence threshold - and that label need not retain the Berkson property. For any coarsening the plug-in estimator converges to $\mathcal{A}\tau$, where $\mathcal{A}-I$ is determined by the regression of the discarded part of the score on the retained part. Coarsening therefore leaves $\tau$ undistorted exactly when that regression vanishes, and otherwise distorts some contrasts far more than others. The same operator determines the bias that drives coverage loss. Where that bias is of the order of the standard error, the Wald interval has limiting coverage $\Phi(z-\nu)-\Phi(-z-\nu)$, with $\nu$ their ratio. A fixed bias sends coverage to zero. Operator, standard error, and - through the uncoarsened estimator - the bias are estimable from observed data, so the implied coverage can be approximated before the interval is reported. Simulations show severe coverage loss after argmax coarsening. Three real-data audits exhibit the direction-specific distortion.

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral–nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of $\mathcal{O}(T^{-1/2})$ in the deterministic setting and $\mathcal{O}(T^{-1/4})$ under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Continual learning must absorb new tasks without erasing old ones, and replay—mixing a small buffer of past examples into current training—is among the most effective remedies for catastrophic forgetting. Yet its generalization behavior is shaped by two coupled effects that existing analyses fold into a single hypothesis-level quantity: finite memory replaces each past distribution with an empirical proxy, and repeated reuse couples the buffer, the current data, and the final hypothesis through a shared optimization trajectory. We develop a layer-wise information-theoretic framework that separates these effects at every depth. Our main result decomposes the expected generalization gap into a replay-induced representation drift and an optimization-dependence term, the latter further resolved into stability, plasticity, interaction, and residual-coupling components. Two refinements make the framework operational. A Wasserstein relaxation of the drift term, valid under support mismatch, yields a depth-dependent drift–sensitivity trade-off whose minimizer identifies which interior layer to stabilize. An SGLD instantiation of the optimization term reduces it to a trajectory-level log-determinant budget, exposing a curvature-aware gradient-alignment statistic that serves as an online diagnostic of task-wise forgetting. Controlled and benchmark experiments confirm the predicted memory scaling, the interior funnel, and the alignment signal's link to forgetting.

Graph-based regularization is widely used to stabilize noisy RNA-velocity estimates by encouraging transcriptionally similar cells to share similar velocity vectors. Near developmental bifurcations, however, proximity-based graphs may connect cells from distinct daughter lineages, reducing estimation variance at the cost of attenuating biologically meaningful lineage-specific dynamics. We formulate graph-regularized RNA velocity as a statistical estimation problem on a potentially misspecified cell-state graph and develop a theoretical framework for analyzing this trade-off. We derive an exact graph-spectral bias–variance decomposition that characterizes how Laplacian regularization suppresses estimation noise while introducing systematic smoothing bias. To quantify lineage preservation, we introduce a branch-sensitive risk that separates within-lineage denoising from cross-lineage information leakage. We further show that persistent cross-branch connectivity can induce nonvanishing branch bias, implying that standard proximity-graph regularization may remain asymptotically inconsistent near developmental bifurcations. These results provide a mathematical foundation for understanding both the statistical benefits and the geometric limitations of graph regularization for RNA-velocity estimation.

Consider two PAC learning algorithms, both having access to quantum computation, but differing in the types of examples they obtain: one is provided with classical samples, while the other is given quantum samples. Are there any learning tasks that can be efficiently performed by the latter, but not by the former? This question, the focus of our work, is surprisingly still open. Our main result is to show that relative to an oracle, there are distributions that can be efficiently generated by a quantum learner with access to quantum samples, but not by a quantum learner with access to only classical samples, making progress to answering this question in the affirmative.

Digital light processing (DLP) enables rapid fabrication of polymer structures, but fracture performance depends on multiple interacting processing variables, making exhaustive experimental characterization impractical. This work presents a data-efficient framework for process-fracture mapping of a DLP-printed photopolymer using Bayesian active learning and digital image correlation (DIC)-assisted Mode I fracture experiments. Four processing parameters were considered: layer angle, UV exposure time, layer height, and print temperature. Fracture resistance was quantified by the critical J-integral, $J_c$, obtained from three-point-bending tests with DIC-based evaluation of crack-mouth opening displacement and hinge-point kinematics. Beginning with two randomly selected conditions, Gaussian process regression (GPR) and a modified upper confidence bound (UCB)-style acquisition function selected 26 additional experiments, yielding 28 processing conditions with three replicates each. The final GPR surrogate reproduced the training data with $R^2=0.99$ and achieved leave-one-out cross-validation performance of $R^2=0.63$ and Pearson $r=0.81$. Surrogate-based sensitivity analysis quantified parameter effects and global contributions. One-at-a-time response curves revealed nonlinear conditional trends, while global Sobol analysis identified UV exposure time as the dominant processing variable, with first-order and total-order indices of 0.6780 and 0.7581, respectively. Based on total-order influence, the parameters ranked as UV exposure time, layer angle, print temperature, and layer height. The first-order Sobol indices summed to 0.8058, indicating non-negligible interaction and higher-order effects. These results demonstrate that Bayesian-active-learning-guided experimentation can efficiently recover process-fracture relationships and parameter interactions from a sparse experimental campaign.

Practitioners enrich customer-return models with ever more signals (lifetime value, category, recency/frequency, calendar, geography), and the temporal-point-process (TPP) literature follows suit with covariate- and external-covariate-conditioned intensities. But does any of it improve the timing, and how would you know? A null ("feature X doesn't help") is only meaningful if the model could have found a signal. We make two contributions–a method and a measurement–to answer this credibly. (i) A screen-and-confirm protocol that certifies whether a candidate signal improves a TPP's event-timing likelihood: a positive control plants a coupling of known strength and confirms the model recovers it, so a real-data null can be read as "no signal" rather than "weak method." The control is validated for categorical and continuous encodings, and on a real clock-driven dataset (NYC taxi hour-of-day). (ii) A model-free ceiling quantifying how little of customer-return timing is point-predictable at all (a single-digit percentage of gap variance from any covariate; returns are near-memoryless). With these we certify a clean result on three public benchmarks (Amazon, Taobao, RetailRocket) and a real marketplace (Thumbtack): the inter-event clock–continuous-time decay, long known to beat frozen-intensity models–is nearly sufficient, and the conditioning the field keeps adding is redundant or harmful on top of it (statistically null on the public benchmarks, at most 0.06 NLL; null to mildly harmful on the marketplace). We do not claim to discover that decay helps; our contribution is the tools that turn "conditioning doesn't help" into a checkable, certified statement–plus an honest-evaluation account of the read-out/leakage pitfalls we hit and retracted.

In many high-stakes domains, extreme events carry substantial consequences, yet learning the heavy-tailed distributions that govern them from finite samples remains challenging: the quantities of interest are driven by a few extreme observations, so the tail is under-sampled relative to its importance. Even generative models tailored for heavy tails capture the tail region inadequately in practice. We propose the Conditional Value-at-Risk (CVaR)-penalized Generative Particle Algorithm (CVaR-GPA), a tail-agnostic algorithm for fine-tuning generative models toward heavy-tailed targets, built as a time discretization of the Wasserstein gradient flow of the Lipschitz-regularized KL divergence penalized by a CVaR discrepancy term. Such a flow can be initialized from the output samples of the pre-trained model, which are then transported along the gradient descent of the loss functional, without requiring access to the pre-trained model's internal architecture. The Lipschitz-regularized KL divergence requires minimal assumptions on the target, while the CVaR penalty focuses the flow on the tail. The CVaR penalty depends on the target only through a scalar tail statistic, inducing a velocity field that remains active in the under-sampled tail region at a dimension-free estimation cost. The resulting velocity field, and hence the depth of the transport map, implicitly adapts to the target, without target-specific modifications. Across four targets, including two real-world, high-dimensional datasets (daily streamflow in the Ohio River basin ($d=64$) and the Fama-French portfolios ($d=25$) with tail indices ranging from $1.05$ to $3.34$), fine-tuning with CVaR-GPA reduces global and tail errors by geometric-mean factors of $14.0 \times$ and $9.8 \times$, respectively, across seven pre-trained models spanning GANs, diffusion models, and other generative flows, with a single set of hyperparameters.

The continuous calibration of pilot items embedded in operational assessments is challenging when pilot samples are small and adaptively routed. We formalize a Bayesian modularization framework for local item calibration that blocks feedback from pilot responses to the operational latent scale: Plausible Values are drawn from the operational posterior, and each pilot item is calibrated by its own local logistic regression. This construction is computationally scalable, protects operational trait estimates from malfunctioning pilot items, and admits Firth's penalized likelihood for sparse routed samples. Because treating imputed traits as fixed predictors induces attenuation, we derive closed-form disattenuation mappings that recover the generating item parameters under normal-ogive, posterior-normality, homoscedasticity, and joint-normality approximations. The resulting Modular Local Calibration (MLC) estimator reaches the target of marginal-likelihood calibration without per-item numerical integration. A multivariate delta-method covariance combined with Rubin's-rules pooling propagates operational item-parameter uncertainty into the focal item standard errors. Monte Carlo simulations for the unidimensional 2PL show that corrected MLC substantially reduces attenuation bias and yields nominal-to-conservative interval coverage in the studied conditions, including under restricted-range MAR routing.

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer's method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been proposed in the literature, that seem to achieve satisfactory practical efficiency for difficult instances, the Stochastic Gradient Method being the most rudimentary, while still outperforming more recent algorithms at a number of learning tasks. A major open question about the current methods used in deep learning is to understand their convergence properties. Following a line of previous works about the long time behavior of gradient-type algorithms, %and in particular the recent contributions from Azizian et al., we present a new approach for studying the asymptotic properties of a wide family of methods from an ergodic theoretical viewpoint. Our main results include a study of the expected time for a stochastic optimisation algorithm to reach a certain small neighborhood of a minimizer and show that this reaching time distributes exponentially around its average, which is given by the inverse of the stationary measure of the target. The assumptions on the Stochastic Gradient noise include the Gaussian and the Sub-Exponential assumptions.