Papers

We use geometric invariant theory (GIT) to study the deep linear network (DLN). The Kempf-Ness theorem is used to establish that the $L^2$ regularizer is minimized on the balanced manifold. We introduce related balancing flows using the Riemannian geometry of fibers. The balancing flow defined by the $L^2$ regularizer is shown to converge to the balanced manifold at a uniform exponential rate. The balancing flow defined by the squared moment map is computed explicitly and shown to converge globally. This framework allows us to decompose the training dynamics into two distinct gradient flows: a regularizing flow on fibers and a learning flow on the balanced manifold. It also provides a common mathematical framework for balancedness in deep learning and linear systems theory. We use this framework to interpret balancedness in terms of fast-slow systems, model reduction and Bayesian principles.

Online bilevel optimization (OBO) is a powerful framework for machine learning problems where both outer and inner objectives evolve over time, requiring dynamic updates. Current OBO approaches rely on deterministic window-smoothed regret minimization, which may not accurately reflect system performance when functions change rapidly. In this work, we introduce a novel search direction and show that both first- and zeroth-order (ZO) stochastic OBO algorithms leveraging this direction achieve sublinear {stochastic bilevel regret without window smoothing}. Beyond these guarantees, our framework enhances efficiency by: (i) reducing oracle dependence in hypergradient estimation, (ii) updating inner and outer variables alongside the linear system solution, and (iii) employing ZO-based estimation of Hessians, Jacobians, and gradients. Experiments on online parametric loss tuning and black-box adversarial attacks validate our approach.

The 2008 financial crisis exposed fundamental vulnerabilities in interconnected banking systems, yet existing frameworks fail to integrate spatial propagation with network contagion mechanisms. This paper develops a unified spatial-network framework to analyze systemic risk dynamics, revealing three critical findings that challenge conventional wisdom. First, banking consolidation paradoxically increased systemic fragility: while bank numbers declined 47.3% from 2007 to 2023, network fragility measured by algebraic connectivity rose 315.8%, demonstrating that interconnectedness intensity dominates institutional count. Second, financial contagion propagates globally with negligible spatial decay (boundary d* = 47,474 km), contrasting sharply with localized technology diffusion (d* = 69 km)–a scale difference of 688 times. Third, traditional difference-in-differences methods overestimate crisis impacts by 73.2% when ignoring network structure, producing severely biased policy assessments. Using bilateral exposure data from 156 institutions across 28 countries (2007-2023) and employing spectral analysis of network Laplacian operators combined with spatial difference-in-differences identification, we document that crisis effects amplified over time rather than dissipating, increasing fragility 68.4% above pre-crisis levels with persistent effects through 2023. The consolidation paradox exhibits near-perfect correlation (R = 0.97) between coupling strength and systemic vulnerability, validating theoretical predictions from continuous spatial dynamics. Policy simulations demonstrate network-targeted capital requirements achieve 11.3x amplification effects versus uniform regulations. These findings establish that accurate systemic risk assessment and macroprudential policy design require explicit incorporation of both spatial propagation and network topology.

Cryo-electron microscopy (cryo-EM) is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments (MoDM), which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.

Current neuroimaging studies on neurodegenerative diseases and psychological risk factors have been developed predominantly in non Hispanic White cohorts, with other populations markedly underrepresented. In this work, we construct directed hyper connectomes among large scale functional brain systems based on causal influences between brain regions, and examine their links to Alzheimer Disease progression and worry levels across racial groups. By using Health and Aging Brain Study Health Disparities (HABS HD) dataset, our experimental results suggest that neglecting racial variation in brain network architecture may reduce predictive performance in both cognitive and affective phenotypes. Important shared and population-specific hyper-connectome patterns related to both AD progression and worry levels were identified. We further observed distinct closed loop directed circuits across groups, suggesting that different populations may rely on distinct feedback based network regulation strategies when supporting cognition or managing emotional states. Together, these results indicate a common backbone of network vulnerability with population-dependent variations in regulatory coordination, underscoring the importance of population-aware neuroimaging models.

Large language models (LLMs) trained for step-by-step reasoning often become excessively verbose, raising inference cost. Standard Reinforcement Learning with Verifiable Rewards (RLVR) pipelines filter out “easy” problems for training efficiency, leaving the model to train primarily on harder problems that require longer reasoning chains. This skews the output length distribution upward, resulting in a model that conflates “thinking longer” with “thinking better”. In this work, we show that retaining and modestly up-weighting moderately easy problems acts as an implicit length regularizer. Exposing the model to solvable short-chain tasks constrains its output distribution and prevents runaway verbosity. The result is \textbf{emergent brevity for free}: the model learns to solve harder problems without inflating the output length, despite the absence of any explicit length penalization. RLVR experiments using this approach on Qwen3-4B-Thinking-2507 (with a 16k token limit) achieve baseline pass@1 AIME25 accuracy while generating solutions that are, on average, nearly twice as short. The code is available at \href{https://github.com/MBZUAI-Paris/Frugal-AI}{GitHub}, with datasets and models on \href{https://huggingface.co/collections/MBZUAI-Paris/k2-think-mini-68dcfa8b114686a4bd3dc2bc}{Hugging Face}.

The problem of optimal linear estimation of functionals depending on the unknown values of a random field $\zeta(t,x)$, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $\zeta(t,x)+\theta(t,x)$ at points $(t,x):t<0,x\in S_{n}$, where $\theta(t,x)$ is an uncorrelated with $\zeta(t,x)$ random field, which is mean-square continuous periodically correlated with respect to time argument $t\in\mathbb R$ and isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

This paper introduces a new stochastic diffusion process to model the electricity production from natural gas sources (as a percentage of total electricity production) in the United States. The method employs trend function analysis to generate fits and forecasts with both conditional and unconditional estimated trend functions. Parameters are estimated using the maximum likelihood (ML) method, based on discrete sampling paths of the variable "electricity production from natural gas sources in the United States" with annual data from 1990 to 2021. The results show that the proposed model effectively fits the data and provides dependable medium-term forecasts for 2022-2023.

In a widely cited paper, Xie and Liu (henceforth XL) proposed to use inverse probability of treatment weighting (IPTW) to account for possible confounding in observational studies with survival endpoints subject to right censoring. Their proposal includes an IPTW Kaplan-Meier (KM) estimator for the survival function of a treatment-specific potential failure time, which can be used to evaluate the causal effect of one treatment versus another. The IPTW KM estimator is remarkably simple and highly effective for confounding bias correction. The method has been implemented in SAS's popular procedure LIFETEST for analyzing survival data and has seen widespread use. This letter is concerned with variance estimation for the IPTW KM estimator. The variance estimator provided by XL does not account for the variability of the IPTW weight when the propensity score is estimated from data, as is usually the case in observational studies. In this letter, we provide a rigorous asymptotic analysis of the IPTW KM estimator based on an estimated propensity score. Our analysis indicates that estimating the propensity score does tend to result in a smaller asymptotic variance, which can be estimated consistently using a plug-in variance estimator. We also present a simulation study comparing the variance estimator we propose with the XL variance estimator. Our simulation results confirm that the proposed variance estimator is more accurate than the XL variance estimator, which tends to over-estimate the sampling variance of the IPTW KM estimator.

The limit distribution of the nonparametric maximum likelihood estimator for interval censored data with more than one observation time per unobservable observation, is still unknown in general. For the so-called separated case, where one has observation times which are at a distance larger than a fixed positive epsilon, the limit distribution was derived in [5]. For the non-separated case there is a conjectured limit distribution, given in [10], Section 5.2 of Part 2. Whether this conjecture holds is still unknown, but the present paper shows that for sample sizes 1000 and 10,000 this limit behavior is still not clearly seen. We prove consistency of a related nonparametric isotonic least squares estimator and sketch of the proof for its limit distribution. We also provide simulation results to show how the nonparametric MLE and least squares estimator behave in comparison. Moreover, we discuss a simpler least squares estimator that can be computed in one step, but is inferior to the other least squares estimator, since it does not use all information. For the simplest model of interval censoring, the current status model, the nonparametric maximum likelihood and least squares estimators are the same. This equivalence breaks down if there are more observation times per unobservable observation. The computations for the simulation of the more complicated interval censoring model were performed by using the iterative convex minorant algorithm. They are provided in the GitHub repository [7].

Foundational marked temporal point process (MTPP) models, such as the Hawkes process, often use inexpressive model families in order to offer interpretable parameterizations of event data. On the other hand, neural MTPPs models forego this interpretability in favor of absolute predictive performance. In this work, we present a new family MTPP models: the hyper Hawkes process (HHP), which aims to be as flexible and performant as neural MTPPs, while retaining interpretable aspects. To achieve this, the HHP extends the classical Hawkes process to increase its expressivity by first expanding the dimension of the process into a latent space, and then introducing a hypernetwork to allow time- and data-dependent dynamics. These extensions define a highly performant MTPP family, achieving state-of-the-art performance across a range of benchmark tasks and metrics. Furthermore, by retaining the linearity of the recurrence, albeit now piecewise and conditionally linear, the HHP also retains much of the structure of the original Hawkes process, which we exploit to create direct probes into how the model creates predictions. HHP models therefore offer both state-of-the-art predictions, while also providing an opportunity to “open the box” and inspect how predictions were generated.

This paper presents a study of power series distributions (PSD) with prescribed covariance characteristics. Such distributions constitute a fundamental class in probability theory and mathematical statistics, as they generalize a wide range of well-known discrete distributions and enable the description of various stochastic phenomena with a predetermined variance structure. The aim of the research is to develop analytical methods for constructing power series distributions with given covariances and to establish the conditions under which a particular function can serve as the covariance of a certain PSD. The paper derives a first-order differential equation for the generating function of the distribution, which determines the relationship between its parameters and the form of the covariance function. It is shown that the choice of an analytical or polynomial covariance completely specifies the structure of the corresponding generating function. The analysis made it possible to construct new families of PSDs that generalize the classical Bernoulli, Poisson, geometric, and other distributions while preserving a given covariance structure. The proposed approach is based on the analytical relationship between the generating function and the covariance function, providing a framework for constructing stochastic models with predefined dispersion properties. The results obtained expand the theoretical framework for describing discrete distributions and open up opportunities for practical applications in statistical estimation, modeling of complex systems, financial processes, machine learning where it is crucial to control the dependence between the mean and the variation. Further research may focus on constructing continuous analogues of such distributions, studying their limiting properties, and applying them to problems of regression and Bayesian analysis.

This position paper challenges the "scaling fundamentalism" dominating AI research, where unbounded growth in model size and computation has led to unsustainable environmental impacts and widening resource inequality. We argue that LLM development should be fundamentally reoriented toward capability-per-resource rather than capability alone. We present a theoretical framework demonstrating that resource-allocation decisions guided by gradient influence patterns can dramatically improve efficiency throughout the AI lifecycle. Our analysis shows that in transformer-based models, where a small fraction of parameters exert outsized influence (following heavy-tailed distributions), three critical insights emerge: (1) updating only high-influence parameters strictly outperforms full-parameter tuning on a performance-per-resource basis; (2) simple gradient norms provide computationally efficient proxies for identifying these high-influence components; and (3) coordinated parameter and data selection yields multiplicative efficiency gains, potentially reducing resource requirements by orders of magnitude. Building on these theoretical foundations, we propose a two stage paradigm marginal-return pretraining for foundation developers and influence guided adaptation for downstream users bridged by gradient blueprints, metadata describing which parameters matter most for various tasks. This capability-per-resource perspective transforms what were once considered pragmatic hardware workarounds into theoretically optimal strategies, democratizing access to cutting-edge AI capabilities while significantly reducing environmental impact. By embedding resource consciousness into how we develop, adapt, and evaluate models, we can reshape AI progress toward a more sustainable and equitable future.

In this paper, we propose a novel fairness framework grounded in the concept of happiness, a measure of the utility each group gains fromdecisionoutcomes. Bycapturingfairness through this intuitive lens, we not only offer a more human-centered approach, but also one that is mathematically rigorous: In order to compute the optimal, fair post-processing strategy, only a linear program needs to be solved. This makes our method both efficient and scalable with existing optimization tools. Furthermore, it unifies and extends several well-known fairness definitions, and our empirical results highlight its practical strengths across diverse scenarios.

Variational inference (VI) approximates a target density $p$ by the best match $q$ in a family of tractable distributions. The best variational approximation is found by minimizing a divergence between distributions, $D(p||q)$, and several divergences have been proposed as objective functions for VI, with different choices leading to different approximations. We show that even when these divergences have different minimizers, the resulting approximations all abide by certain symmetry-matching principles. Specifically, our results hold for all $f$-divergences, a broad class which includes the reverse and forward Kullback-Leibler divergences and the $\alpha$-divergences. We show that in the presence of even symmetry, any stationary point of an $f$-divergence is guaranteed to recover the mean of $p$ and likewise, in the presence of elliptical symmetry, any stationary point is guaranteed to recover its correlation matrix. To obtain these guarantees we assume that $p$ and $q$ are unimodal, but notably we do not require them to be log-concave, light-tailed, or even everywhere-smooth. These guarantees generalize a previous result obtained for the reverse Kullback-Leibler divergence when $p$ is log-concave. They also extend to cases where the target density $p$ only exhibits symmetry along some but not all of its coordinates. These partial symmetries arise naturally in Bayesian hierarchical models, where the prior induces a challenging geometry but still possesses axes of symmetry.

Structural equation modeling (SEM) and path analysis have long been central tools for studying complex causal relationships in the social and behavioral sciences, yet their reliance on parametric assumptions can lead to biased inference under model misspecification. To bridge traditional SEM with modern causal machine learning, this paper introduces targeted maximum likelihood estimation (TMLE), a doubly robust framework built on nonparametric structural equation modeling. We formally connect TMLE to classical path analysis, showing that standard SEM estimators arise as special cases of TMLE under restrictive parametric specifications and that both approaches can estimate common causal quantities such as direct, indirect, and total effects. Through simulation studies under both correctly specified and misspecified models, we demonstrate that while the two methods perform similarly when models are correctly specified, TMLE consistently achieves lower bias, reduced mean squared error, and improved confidence interval coverage when parametric assumptions are violated. We further illustrate these differences using an applied mediation analysis examining the role of poverty in access to high school education, where path analysis suggests a significant direct effect, whereas TMLE does not, highlighting the practical consequences of robustness in causal inference. Overall, this tutorial offers SEM researchers a conceptual and practical introduction to targeted learning, providing guidance on leveraging TMLE to enhance causal analysis beyond traditional parametric frameworks.

Recent studies suggest that asymmetric binary perceptron (ABP) likely exhibits the so-called statistical-computational gap characterized with the appearance of two phase transitioning constraint density thresholds: \textbf{(i)} the satisfiability threshold $\alpha_c$, below/above which ABP succeeds/fails to operate as a storage memory; and \textbf{(ii)} algorithmic threshold $\alpha_a$, below/above which one can/cannot efficiently determine ABP's weight so that it operates as a storage memory. We consider a particular parametric utilization of fully lifted random duality theory (fl RDT) [85] and study its potential ABP's algorithmic implications. A remarkable structural parametric change is uncovered as one progresses through fl RDT lifting levels. On the first two levels, the so-called $\c$ sequence – a key parametric fl RDT component – is of the (natural) decreasing type. A change of such phenomenology on higher levels is then connected to the $\alpha_c$ – $\alpha_a$ threshold change. Namely, on the second level concrete numerical values give for the critical constraint density $\alpha=\alpha_c\approx 0.8331$. While progressing through higher levels decreases this estimate, already on the fifth level we observe a satisfactory level of convergence and obtain $\alpha\approx 0.7764$. This allows to draw two striking parallels: \textbf{(i)} the obtained constraint density estimate is in a remarkable agrement with range $\alpha\in (0.77,0.78)$ of clustering defragmentation (believed to be responsible for failure of locally improving algorithms) [17,88]; and \textbf{(ii)} the observed change of $\c$ sequence phenomenology closely matches the one of the negative Hopfield model for which the existence of efficient algorithms that closely approach similar type of threshold has been demonstrated recently [87].

We investigate a generalized quantum perceptron architecture characterized by an oscillating activation function with a tunable frequency ranging from zero to infinity. Employing analytical techniques from statistical mechanics, we derive the optimal storage capacity and demonstrate that the classical result is recovered in the limit of vanishing frequency. As the frequency increases, however, the architecture exhibits enhanced quantum storage capabilities. Notably, this improvement stems solely from the specific form of the activation function and, in principle, could be emulated within a classical framework. Accordingly, we refer to this enhancement as a pseudo quantum advantage.

The problem of optimal linear estimation of a linear functional depending on the unknown values of periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.

We introduce the Neutrality Boundary Framework (NBF), a set of geometric metrics for quantifying statistical robustness and fragility as the normalized distance from the neutrality boundary, the manifold where the effect equals zero. The neutrality boundary value nb in [0,1) provides a threshold-free, sample-size invariant measure of stability that complements traditional effect sizes and p-values. We derive the general form nb = |Delta - Delta_0| / (|Delta - Delta_0| + S), where S>0 is a scale parameter for normalization; we prove boundedness and monotonicity, and provide domain-specific implementations: Risk Quotient (binary outcomes), partial eta^2 (ANOVA), and Fisher z-based measures (correlation). Unlike threshold-dependent fragility indices, NBF quantifies robustness geometrically across arbitrary significance levels and statistical contexts.

Normalization layers are critical components of modern AI systems, such as ChatGPT, Gemini, DeepSeek, etc. Empirically, they are known to stabilize training dynamics and improve generalization ability. However, the underlying theoretical mechanism by which normalization layers contribute to both optimization and generalization remains largely unexplained, especially when using many normalization layers in a deep neural network (DNN). In this work, we develop a theoretical framework that elucidates the role of normalization through the lens of capacity control. We prove that an unnormalized DNN can exhibit exponentially large Lipschitz constants with respect to either its parameters or inputs, implying excessive functional capacity and potential overfitting. Such bad DNNs are uncountably many. In contrast, the insertion of normalization layers provably can reduce the Lipschitz constant at an exponential rate in the number of normalization layers. This exponential reduction yields two fundamental consequences: (1) it smooths the loss landscape at an exponential rate, facilitating faster and more stable optimization; and (2) it constrains the effective capacity of the network, thereby enhancing generalization guarantees on unseen data. Our results thus offer a principled explanation for the empirical success of normalization methods in deep learning.

We develop jump-robust estimators of the leverage effect and volatility of volatility using high-frequency data. Our construction begins with a spot volatility estimator based on the empirical characteristic function of high-frequency increments. This method can mitigate the contamination from jumps, which can be of infinite variation. We then construct estimators of the leverage effect and volatility of volatility and correct for the bias induced by spot volatility estimation. We establish consistency and central limit theorems under conditions that allow greater jump activity than existing methods. We also develop consistent estimators of the asymptotic variances, making the limiting results feasible for statistical inference. Simulation studies demonstrate the improved finite-sample performance of the proposed estimators, particularly in the presence of infinite variation jumps. An empirical application provides evidence of nonzero leverage effect and volatility of volatility, when the jump activity is intensive.

We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for classical persistence statistics-total and maximum persistence-under general distributions, with explicit variance-scaling behavior derived for Gaussian mixture models. Building on these results, we introduce a bounded and normalized statistic based on persistence landscapes combined with the Jensen-Shannon divergence, and we prove its Holder continuity with respect to perturbations of the input point clouds. The resulting measure is stable, scale- and shift-invariant, and well suited for finite-sample nonparametric inference via permutation testing. An illustrative numerical study using dynamic attribute vectors from decentralized governance data demonstrates the practical applicability of the proposed method. Overall, this work provides a statistically rigorous and computationally stable approach to change-point detection in complex, high-dimensional data.

In this paper, the classification task for a family of sets representing the realisation of some random set models is solved. Both unsupervised and supervised classification methods are utilised using the similarity measure between two realisations derived as empirical estimates of $\mathcal N$-distances quantified based on geometric characteristics of the realisations, namely the boundary curvature and the perimeter over area ratios of obtained samples of connected components from the realisations. To justify the proposed methodology, a simulation study is performed using random set models. The methods are used further for classifying histological images of mastopathy and mammary cancer tissue.

Spiking neural networks (SNNs) are a promising paradigm for energy-efficient computation, yet their theoretical foundations-especially regarding stability and robustness-remain limited compared to artificial neural networks. In this work, we study discrete-time leaky integrate-and-fire (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classification tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum on low-frequency components. Motivated by this, we introduce the notion of spectral simplicity, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analysis to the simplicity bias observed in deep networks. Within this framework, we show that random LIF-SNNs are biased toward simple functions. Experiments on trained networks confirm that these stability properties persist in practice. Together, these results provide new insights into the stability and robustness properties of SNNs.

Markov Chain Monte Carlo (MCMC) algorithms are standard approaches to solve imaging inverse problems and quantify estimation uncertainties, a key requirement in absence of ground-truth data. To improve estimation quality, Plug-and-Play MCMC algorithms, such as PnP-ULA, have been recently developed to accommodate priors encoded by a denoising neural network. Designing scalable samplers for high-dimensional imaging inverse problems remains a challenge: drawing and storing high-dimensional samples can be prohibitive, especially for high-resolution images. To address this issue, this work proposes a distributed sampler based on approximate data augmentation and PnP-ULA to solve very large problems. The proposed sampler uses lightweight denoising convolutional neural network, to efficiently exploit multiple GPUs on a Single Program Multiple Data architecture. Reconstruction performance and scalability are evaluated on several imaging problems. Communication and computation overheads due to the denoiser are carefully discussed. The proposed distributed approach noticeably combines three very precious qualities: it is scalable, enables uncertainty quantification, for a reconstruction performance comparable to other PnP methods.

This study presents a spatiotemporal dual Bayesian model that examines both the occurrence and number of conflict fatalities using event-level data from Ethiopia (1997-2024), sourced from the Armed Conflict Location and Event Data (ACLED) project. Fatalities are treated as two linked outcomes: the binary occurrence of deaths and the count of deaths when they occur. The model combines additive fixed effects for covariates with random effects capturing spatiotemporal influences, allowing for outcome-specific effects. Covariates include event type and season as categorical variables, proximity to cities and borders as nonlinear effects, and population as an offset term in the count model. A latent spatiotemporal process accounts for shared spatial and temporal dependence, with the spatial structure modeled using a Matérn field prior and inference via Integrated Nested Laplace Approximation (INLA). Results show strong spatial clustering and temporal variation in fatality risk, emphasizing the importance of modeling both dimensions for better understanding and prediction. Airstrikes, shelling, and attacks show the highest fatality likelihood and counts, while communal and rebel actors cause the most deaths. Multiple fatalities are more likely in summer, and proximity to borders drives intense violence, whereas remoteness from urban centers is linked to lower-intensity events. These results provide insight for planning, policy, and resource allocation to protect vulnerable communities.

Out-of-distribution (OOD) detection is essential for deploying deep learning models in open-world environments. Existing approaches, such as energy-based scoring and gradient-projection methods, typically rely on high-dimensional representations to separate in-distribution (ID) and OOD samples. We introduce P-OCS (Perturbations in the Orthogonal Complement Subspace), a lightweight and theoretically grounded method that operates in the orthogonal complement of the principal subspace defined by ID features. P-OCS applies a single projected perturbation restricted to this complementary subspace, enhancing subtle ID-OOD distinctions while preserving the geometry of ID representations. We show that a one-step update is sufficient in the small-perturbation regime and provide convergence guarantees for the resulting detection score. Experiments across multiple architectures and datasets demonstrate that P-OCS achieves state-of-the-art OOD detection with negligible computational cost and without requiring model retraining, access to OOD data, or changes to model architecture.

We develop a collection of methods for adjusting the predictions of quantile regression to ensure coverage. Our methods are model agnostic and can be used to correct for high-dimensional overfitting bias with only minimal assumptions. Theoretical results show that the estimates we develop are consistent and facilitate accurate calibration in the proportional asymptotic regime where the ratio of the dimension of the data and the sample size converges to a constant. This is further confirmed by experiments on both simulated and real data. A key component of our work is a new connection between the leave-one-out coverage and the fitted values of variables appearing in a dual formulation of the quantile regression problem. This facilitates the use of cross-validation in a variety of settings at significantly reduced computational costs.

In this paper, we study stochastic ordering results between two finite mixtures with single and multiple outliers, assuming subpopulations follow general exponentiated location-scale distributions. For single-outlier mixtures, several sufficient conditions are derived under which the mixture variables are ordered in the usual stochastic, reversed hazard rate, and likelihood ratio orders, using majorization concepts. For multiple-outlier mixtures, results are obtained for the reversed hazard rate, likelihood ratio, and ageing faster orders in reversed hazard rate. Numerical examples and counterexamples are presented to illustrate and support the established theoretical findings.