Papers

We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is generally not Markov, its stationary one-step conditional law defines a genuine Markov kernel. For every fixed strictly positive reversible kernel \(P\) on a finite state space, we present a Poincaré inequality for the induced count kernel $\tP_n$ of length $n$. In other words, we derive the lower bound of the spectral gap $\Gap(\tP_n)$ of $\tP_n$ as \[ \Gap(\tP_n)\ge \frac{c(P)}{n}, \] where \(c(P)>0\) depends only on \(P\). The proof combines a martingale oscillation inequality for the stationary path law with a direct comparison of coordinate oscillations to the Dirichlet form of the count kernel. A linear statistic of the count vector gives the matching \(O(1/n)\) upper bound, so for every fixed strictly positive reversible \(P\) one has \(\Gap(\tP_n)=\Theta_P(1/n)\). The resulting count-space Poincaré inequality yields a local-to-global variance bound for finite-window count statistics and, together with a general matrix-concentration principle, operator-norm concentration for matrix-valued empirical averages.

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.

Fraud operations must allocate events among automatic approval, analyst review, and automatic blocking even though the labels needed to evaluate these actions are selective and delayed. Predictive scores order cases, but they do not show whether the evidence is current and representative enough to delegate an action to the model. We develop freshness-constrained audit capacity (FCAC), a decision-support framework that treats automation as an authorization decision constrained by action risk, evidence freshness, and shared review capacity. It evaluates candidate action regions from mature randomized audits and a prespecified temporal allowance. Supported regions are automated; unsupported regions remain in review. The resulting decision record reports evidence age, audit demand, total review workload, value exposure, and compatible temporal change. We show that current action risk is unidentified without restricting unobserved label evolution. Under representative randomized audits, label-independent evidence windows, and a prespecified condition linking historical and current action risk, we derive simultaneous finite-sample control of unsafe authorization. Chronological evaluations with simulated audits on IEEE-CIS, ULB-Worldline, and Elliptic++ yield zero-drift automation rates of 84.4%, 67.4%, and 81.3%, with total review workloads of 24.1%, 46.0%, and 43.1%. The experiments reveal an audit-capacity trade-off: sparse auditing delays authorization, whereas intensive auditing eventually increases workload. A separately specified BAF stress test further indicates that fallback thresholds must reflect candidate-specific evidence rather than a common fraction of the risk limit. These findings identify audit freshness and analyst capacity as joint design considerations for fraud decision support.

Neural network classifiers trained by cross-entropy minimization are highly sensitive to label noise and adversarial contamination. While robust alternatives offer bounded influence and resistance to corruption, their statistical foundations in the deep learning setting are insufficient due to a fundamental difficulty: neural parameterizations are non-identifiable, so the population loss minimizer is an equivalence class of parameters, not a unique point. We develop a consistency theory for robust neural classifiers based on the S-divergence family that requires no identifiability assumption. Casting training as stochastic optimization over a non-identifiable parameter space, we prove that empirical S-divergence minimizers converge to the population-optimal equivalence class under mild regularity conditions, and verify these conditions for three architecture choices. We further establish that limit points of the robust training algorithm are stationary points of the empirical objective. Experiments on vision and language benchmark datasets confirm that S-divergence training maintains clean-data accuracy while exhibiting performance competitive with existing robust methods.

Conformal changepoint localization turns any score into a confidence set for the changepoint with finite-sample coverage. Coverage is universal; efficiency is not. The oracle score is a likelihood ratio, so practical scores estimate density ratios, and set length deteriorates under heavy tails, skewness, and distribution shift, where no length guarantee applies. We propose ARC (Augmented-Rank Conformalization), a family of scores depending on the data only through within-segment ranks: rank-CUSUM location and scale channels, their fixed combinations, and a lightweight neural score frozen after synthetic training. Every ARC score inherits finite-sample coverage for every frozen weight configuration, including random initialization and mistraining. The main result is an efficiency transfer theorem: the entire ARC confidence set is almost surely invariant under strictly increasing marginal transforms, so the set length distribution depends on the data pair only through its rank structure, and lengths certified once hold verbatim across its monotone orbit, whereas a plug-in score's length changes with every re-expression. Across different rank structures lengths do change, and are reported as such. Classical rank-test theory positions ARC as targeting the optimal invariant score at bounded cost. Simulations confirm nominal coverage for all scores, including sabotaged networks, identical sets under monotone transforms where plug-in scores inflate, and smooth degradation where plug-in sets become vacuous; on the well-log benchmark ARC localizes annotated shifts to three to five candidates and flags misfit by an empty set. Two boundaries are stated rather than hidden: serial dependence destroys exactness, and trend-type alternatives lie outside the piecewise-exchangeable model.

Ranking data arise in scientific and machine learning applications, including recommendation systems, information retrieval, voting, marketing, and AI preference ranking from human feedback. Existing statistical work has primarily focused on inference tasks such as preference estimation, rank aggregation, and ranking prediction. However, generating realistic synthetic rankings from an observed population is important for privacy-preserving data sharing, benchmark construction, simulation, and uncertainty quantification. This task is challenging because rankings are high-dimensional combinatorial objects with non-Euclidean dependence structures, while ranking populations often exhibit substantial preference heterogeneity. We propose a framework for population-level generative modeling through a latent preference simplex embedding. It estimates a low-dimensional latent preference simplex through a likelihood-based ranking model, leverages flow matching to learn the population distribution of latent preferences, and generates new rankings through the fitted probabilistic ranking model. We show that ranking generation admits an oracle reduction to latent distribution learning and derive finite-sample generative guarantees that clarify how the number of items, ranking length, and latent dimension affect accuracy. Experiments on synthetic and real datasets demonstrate improved population-level fidelity and provide a statistically interpretable representation of preference heterogeneity.

We study constrained statistical learning over infinite-dimensional hypothesis classes in the fully nonconvex setting, and establish universal PACC learnability of the solutions of dual algorithms: Probably Approximately Correct on Constraints, guaranteeing optimality and constraint satisfaction at once. This strengthens near-PACC results, whose feasibility residual no amount of data can remove. Optimality is caught between generalization, governed by Rademacher complexity and favoring small classes, and strong Lagrangian duality, which rests on Lyapunov convexity for vector measures and needs decomposability, a demand pulling the other way. We reconcile the two by posing the population problem over a universal RKHS $\mathcal{H}_K$, dense in a decomposable envelope, and learning over norm balls of growing radius. This yields the Tikhonov complexity $\mathfrak{T}^{\varepsilon}_{n}$, the least RKHS norm reaching an $\varepsilon$-optimal Lagrangian level set; we prove it finite, obtain exact learnability of the optimal value, and make the sample threshold explicit and polynomial in $1/\varepsilon$ under a source condition. Feasibility is harder: absent convexity the Lagrangian may not attain its infimum, and dual information pins down only an averaged constraint-risk vector, not the risks of any returned predictor. We introduce the closure-realization gap $\varepsilon^\star_\infty$, an index of how well $\mathcal{H}_K$ retrieves feasible solutions from dualization; it is a property of the problem, not of a modeling choice. Learnability is exact when $\varepsilon^\star_\infty=0$, in particular under dual differentiability, and near-PACC with residual exactly $\varepsilon^\star_\infty$ otherwise. Finally, no distribution-free threshold exists already in the unconstrained specialization, so universality is the canonical frame for dual algorithms over large hypothesis classes.

The instance-wise $F_1$ measure is a central performance measure for multi-label classification. For a problem with $s$ labels, it defines a $2^s\times 2^s$ loss matrix. Previous work exhibited $s^2+1$-coordinate affine and shifted low-rank representations and used them to construct quadratic-dimensional convex calibrated surrogates. We determine the exact rank. Under the convention $F_1(\varnothing,\varnothing)=1$, the $F_1$ score matrix, the shifted loss matrix, and the unshifted loss matrix all have rank $s^2-s+2$, while the column-affine dimension of the loss is $s^2-s+1$. The proof factors the nonempty score matrix through subset-incidence matrices and a positive-definite Cauchy matrix. Exact rank does not, by itself, lower-bound the dimension of an arbitrary convex calibrated surrogate. We therefore analyze the Bayes geometry of $F_1$ directly. We construct a distribution for which precisely all supersets of a fixed core label set are Bayes optimal, and show that the corresponding active loss columns, restricted to the witness support, have affine dimension $hn$, where $n=s-\lfloor s/3\rfloor$ and $h=\lceil(s\lfloor s/3\rfloor)^{1/2}\rceil-1$. Applying the feasible-subspace lower bound for convex calibration dimension gives \[ \operatorname{CCdim}(L^{F_1}) \ge \left(\frac{2}{3\sqrt{3}}-o(1)\right)s^2. \] Together with the quadratic upper bound, this establishes $\operatorname{CCdim}(L^{F_1})=\Theta(s^2)$.

Throughout the last decade, Gaussian universality has been widely studied for high-dimensional estimation problems. Most of the literature focuses on i.i.d. sensing matrices or accounts for special forms of dependence, such as block dependence or other specific row/column dependencies. More general simultaneous row and column mixing has not yet been fully studied. In this paper, we focus on that setting. We prove a Gaussian universality theorem for the lasso in the sparse regime, where the non- Gaussian covariates have linearly dependent rows and columns. To the best of our knowledge, our setting permits a broader simultaneous row and column dependence structure than those treated in much of the prior universality literature. Numerical illustrations for various sparse profiles support the universality claims of this paper.

Augmented inverse-probability weighting (AIPW), targeted maximum likelihood estimation (TMLE), and double/debiased machine learning (DML) are three routes to the same efficient influence function for the average treatment effect — settled theory we treat as background. This tutorial's contribution is its worked, shared-nuisance reconciliation on real data: what a practitioner must actually match for the routes, and the software packages, to agree. Working the effect of smoking cessation on weight change in the open NHEFS data (n=1566) with one shared Super Learner library and identical cross-fitting folds, we build all three estimators by hand from one influence function, in full-sample, cross-fit, and double-cross-fit variants; the six resulting doubly-robust estimates span only 3.32–3.42 kg, consistent with the established benchmark. We then reconcile the same estimand across our engine and the tmle, AIPW, DoubleML, and tmle3 packages. At their defaults the estimates span 3.32–3.49 kg. Once the library and folds are matched and single-split noise is averaged out, the three library-sharing implementations agree to within 0.01 kg — so the residual spread traces to the nuisance library, folds, and repetitions, not to the estimator label. Because the estimators share one influence function, they agree under good overlap; under a positivity violation the pooled ATE is not identified without additional extrapolation assumptions, and their finite-sample estimates can then diverge sharply. We therefore place a positivity diagnosis ahead of estimator choice, illustrate the failure on a no-overlap example, and close with a reporting checklist. Open-source R code reproduces every number.

Estimating counterfactual outcomes over time from longitudinal observational data is central to clinical decision support. Existing methods rely on domain confusion – adversarial training that renders representations invariant to treatment assignment – yet this invariance creates a mutual information conflict: it suppresses treatment-correlated covariate signals necessary for accurate outcome prediction. We formalise this tension via a Jensen-Shannon divergence bound on counterfactual prediction error and develop two complementary models. CSSD (Causal State-Space model with Direct decoder) adapts selective State Space Models with a parallel multi-step decoder that eliminates accumulated rollout error by producing all prediction horizons simultaneously in a single forward pass. CSSPD (Causal State-Space model with Predictive regularisation and Direct decoder) augments CSSD with Contrastive Predictive Coding and Local Information Maximisation to reinforce temporal predictability in the balancing representation and recover local covariate information destroyed by domain confusion. On MIMIC-III, CSSPD achieves lower counterfactual RMSE than the Causal Transformer at every horizon tau >= 2 at O(T) encoder cost, with gains from 0.02 (2-step) to 0.07 (6-step). On Cancer Simulation across confounding strengths gamma in {0,1,2,3,4}, CSSPD outperforms CT at gamma <= 3 (margins 25.9%–37.0%), and CSSD achieves the lowest overall average RMSE (12.7% reduction over CT), confirming the MI conflict analysis. To our knowledge, this is the first work to formalise the balancing-prediction MI conflict and propose a structured resolution through complementary predictive and information-theoretic training objectives.

Many functional data analyses reduce random functions to scalar summaries or conditional mean curves. This is limiting when we wish to understand how covariates affect the distribution of entire functional responses, including their shape, timing, or variability. We study the problem of estimating conditional laws of functional outcomes and show that these objects can be estimated and evaluated in a practical nonparametric framework. To do this, we introduce functional distributional random forests, which estimate each conditional law as a covariate-dependent distribution over sampled functions by training a random forest to minimize a kernel-based maximum mean discrepancy within the leaf nodes of the decision tree. This supports inference on arbitrary functionals of the conditional distribution while keeping predictive samples tied to realistic curves. We consider a variety of kernels defined on function spaces, including Sobolev and operator-induced kernels. We also provide conditions for consistency of our estimator and develop scoring rules for comparing it to baseline estimators. In simulations, our method recovers distributional changes that are missed by baseline methods. In an application to NHANES accelerometer data, it identifies interesting covariate-associated changes in both median activity profiles and predictive dispersion.

This work proposes deep nonparametric Instrumental variable quantile regression (IVQR), a two-stage estimator that combines conditional diffusion modeling with a kernel-smoothed conditional moment formulation. In the first stage, we estimate the joint conditional distribution of the outcome and endogenous covariates given the instrument using a variance-preserving conditional diffusion model. In the second stage, we approximate the conditional moment operator through Monte Carlo sampling and a kernel-smoothed surrogate for the indicator function, and then estimate the structural quantile function by empirical risk minimization over deep neural networks. We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model under unbounded support, explicitly accounting for score estimation, early stopping, and discretization errors. Our theory is developed under a polynomial-tail envelope on the data distribution and degenerates continuously to the exponential setting: as the tail index grows, the obtained excess-risk rate converges to the minimax-optimal rate of nonparametric regression, thus our heavy-tailed theory covers the classical light-tailed nonparametric guarantees as a limiting case. Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches, with gains that become increasingly pronounced as the dimensionality of the covariates and instruments increases.

As machine learned models increase in complexity and expressive power, features of simpler models, such as intrinsic coefficient transparency and control over the shape of the modeled function are lost. On the one edge of the spectrum we have simple linear models that possess coefficient transparency, but have a limited expressive power. On the other edge we have neural networks, that have expressive power that improves with scaling, but are mostly opaque. In this work we develop the spectral neuron concept: a scalar model given by $f(x)=\lambda_k (A_0 + A_1 x + ... + A_n x_n)$, with learned real symmetric matrices $A_0, ..., A_n$. The input enters the model through an affine matrix function, but the prediction is obtained by reading one of its eigenvalues. Thus, the model is nonlinear, but the source of nonlinearity is still mathematically explicit. This gives us a useful middle ground: the model can become more expressive as the matrix dimension grows, while retaining coefficient transparency through the learned matrices. For example, extremal eigenvalues yield convex or concave functions, semidefinite constraints on the coefficient matrices impose monotonicity, and the associated eigenspaces characterize local feature influence. We study coefficient transparency, feature-influence bounds, and shape-control properties of this model family, and then test whether it can be learned and scaled in practice. We develop a systematic study of this model family, bringing together spectral results from several mathematical literatures to characterize its expressivity, coefficient transparency, feature influence, and shape-control properties. Code available at https://github.com/alexshtf/spectral_neuron_paper.

Lipschitz bandits admit near-optimal regret $\widetilde O_d(T^{(d+1)/(d+2)})$ with little memory under full adaptivity, or with few batches under unrestricted memory. We characterize the minimax expected pseudo-regret over $T$ rounds with $W$ bits of memory and at most $B$ batches in dimension $d$. For every memory budget $W$, we prove a lower bound of order $T^{\frac{d+2}{d+3}} (1+(B-1)W)^{-\frac{1}{d(d+3)}}$. When $W\gtrsim_d\log(eT)$, algorithms with fixed batch boundaries attain, up to logarithmic factors, the larger of this bound and the optimal $B$-batch regret with unrestricted memory. The analysis separates fine-scale comparisons from the regional information needed to allocate their samples at low regret. Attaining $\widetilde O_d(T^{(d+1)/(d+2)})$ regret requires both $B=\Omega_d(\log\log T)$ and $(B-1)W=\widetilde{\Omega}_d(T^{d/(d+2)})$. With one bit of memory, minimax regret is $\widetilde\Theta_d(TB^{-1/(d+2)})$ under adaptive batch boundaries while $\Theta_d(T)$ under fixed boundaries.

Preferential Bayesian optimization (PBO) optimizes objectives accessible only through pairwise user comparisons. The standard approach fits a Gaussian process surrogate for observed pairwise comparisons (PairwiseGP) using the Laplace approximation and selects queries with the Expected Utility of Best Option (EUBO) acquisition function. EUBO queries new candidates at each step, producing pairs that share no candidates with previous queries. Each such pair forms an isolated component in the comparison graph, removing one degree of freedom from the likelihood Hessian and making it rank-deficient. This deficiency is structural and cannot be resolved by changing the surrogate modeling approach. Existing approaches to remedy this issue either waste query budget by forcing comparisons to stay connected, or apply uniform regularization that also perturbs directions already well-constrained by the observed comparisons. We propose KappaSharp that enables a diagonal correction to the Hessian to reduce its condition number, with larger corrections where the prior uncertainty is higher. The correction is only applied in the model fitting step, not query selection. An adaptive variant of KappaSharp is also presented that activates the correction only when the surrogate is confident about recent comparisons, avoiding unnecessary corrections when the problem is well-conditioned. On 11 benchmarks (5–20 dimensions), including a 16-dimensional controller tuning problem in plasma medicine, Adaptive KappaSharp outperforms the standard PBO baseline, with up to +10.9% ($p{=}0.003$).

Prediction intervals for multi-modal regression with tabular variables, text, images, or other input sources are difficult to calibrate when those sources disagree or one is missing. A single global quantile averages these regimes together instead of calibrating to the modality pattern observed at test time. We address this through a modality-aware conformal calibration layer. The layer trains or reuses one predictor per modality, computes a disagreement score from their predictions, and uses that score in split conformal calibration under a strict split protocol. We use the score in two complementary ways. First, a continuous disagreement-scaled method reallocates interval width across examples while preserving the usual marginal split-conformal guarantee. Second, a Mondrian (stratified) method calibrates within groups defined by disagreement or modality availability fixed before calibration, giving group guarantees under joint exchangeability of the calibration and test examples. Across four multi-modal datasets, the disagreement-scaled layer matches or improves the marginal conformal baseline in 59 of 60 paired runs for interval continuous ranked probability score (CRPS) and in 52 of 60 for interval width, while keeping empirical coverage near the 95% target. In stress tests with missing modalities, mask-matched recalibration recovers up to 19.5 percentage points of coverage in the hardest fixed-mask regime. The result is a simple, model-agnostic reliability layer for multi-modal regression systems. A project page is available at https://unco3892.github.io/modality-aware-conformal.

Sampling high-dimensional probability distributions is a central task in scientific computing, with applications ranging from Bayesian inference to statistical physics and molecular simulation. Despite decades of methodological developments, two major challenges remain: scaling to high dimensions and efficiently exploring multimodal distributions characterized by metastable states. Classical approaches such as Markov chain Monte Carlo, tempering methods, or enhanced sampling based on collective variables have achieved major successes, but they also face intrinsic limitations. This tutorial review explores a new paradigm that has recently emerged at the interface of machine learning and computational statistical physics: the use of generative models as tools for sampling. In this context, models such as normalizing flows and diffusion models are not used in their traditional data-driven setting, but rather as flexible probabilistic models that can assist the sampling of distributions known only up to a normalization constant. This manuscript reviews the early development of this rapidly evolving field and discusses several methodological directions, including exact samplers based on generative models and strategies to train such models in the absence of data. While an exhaustive survey of the literature is not attempted, we present a selection of key ideas and methods, along with a discussion of their strengths and limitations. The review is intended to be an accessible tutorial for both physics and machine learning audiences, and it aims to provide a starting point for researchers interested in exploring this exciting area of research.

We adapt two classical statistical estimators for quantifying uncertainty to modern deep learning, in order to provide clearer insights into uncertainty attributable to two sources : aleatoric uncertainty, or locally scarce data. Our approach leverages recent advances in approximate Fisher Information Matrices, to enable scaling to actual architectures. Experimental results demonstrate how each test points is differentially impacted by both sources, highlighting the practical utility of our estimators in improving the robustness of real-world applications.

Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.

Learning disentangled representations is a key requirement for developing versatile, general-purpose, and sustainable models in multi-modal wearable computing. However, existing approaches do not operate as full-stack wearable processors, i.e., they do not simultaneously address task-specific classification performance, disentangled and interpretable representation learning, fusion, and generative modeling of highly heterogeneous multi-modal time series. To address this gap, we introduce Omni-modal Variational Decomposition Autoencoders (OmniDecVAEs), a framework that efficiently learns multi-purpose representations in a unified and scalable manner from arbitrarily many modalities. OmniDecVAEs extend DecVAEs by learning modality-conditioned time-frequency latent subspaces through a multi-view self-supervised decomposition loss and a shared asymmetric autoencoder (AE) architecture. Results on a challenging omni-modal human activity recognition (HAR) setting with up to thirty modalities, demonstrate the ability of OmniDecVAEs to learn full-stack wearable representations. When compared to transformer-based and VAE-based methods, OmniDecVAEs full-stack disentangled representation properties lead to accuracy improvements of 1.01% and 6.75% in activity and identity recognition, respectively. Furthermore, OmniDecVAEs synthesize realistic omni-modal time-frequency data that manifest with enhanced reconstructions (mean absolute error improves by 76.84%) and distributional similarity between real and synthetic data (maximum mean discrepancy improves by 13.85%). Our results highlight OmniDecVAEs potential as a lightweight model suitable for intelligent edge wearables and clinical healthcare, unifying processing requirements and abilities in a single model, through its enhanced representational capacity, modality-invariant spatial complexity (4.1M parameters), and real-time latency.

This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension $p$ and the sample size $n$ grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the number of spiked eigenvalues may remain finite or increase with $n$, and the spiked eigenvalues may be bounded or diverge at arbitrary rates. Beyond characterizing the impact of covariance spectra, we reveal a new mechanism underlying benign overfitting: the prediction behavior of ridgeless interpolation is fundamentally governed by the alignment between the regression coefficient $\boldsymbol\beta$ and the spiked eigenspaces of the population covariance matrix. In particular, we show that the signal energy distributed along latent spike directions determines whether interpolation leads to benign, tempered, or catastrophic overfitting. Our theoretical framework establishes sharp prediction risk limits under minimal moment conditions, requiring only finite fourth moments rather than Gaussianity. We characterize how the number, strength, and geometric structure of the spikes jointly influence the double-descent phenomenon. These results provide a unified understanding of when latent covariance structures facilitate or hinder generalization in overparameterized regression.

Many economic and financial relationships may change gradually rather than abruptly. We study panel data models in which the coefficient vector is continuous and piecewise linear in calendar time, with a finite number of unknown kink dates at which its slope changes. We propose a penalised least squares estimator that applies adaptive weighted group penalties to the second differences of the coefficient path, and develop asymptotic theory showing that it recovers both the number and the locations of the kinks with probability approaching one. To our knowledge, this is the first panel framework to estimate an unknown number of common kink dates in a time-varying coefficient path under fixed effects. We establish that endpoint slopes converge at the usual cubic regime-length rate and interior slopes at rates determined by their own and adjacent regime lengths. We also develop a coefficient-by-coefficient extension allowing individual regressors to kink at different dates. Monte Carlo evidence supports the good finite sample properties, and we illustrate the method through an application in macro-finance, specifically the relationship between debt and growth.

We consider the optimization of the Optimized Certainty Equivalent (OCE) risk, with applications including portfolio optimization in finance, and uncertainty quantification, classification, and regression in machine learning. Our contributions cover popular special cases of OCE, such as entropic risk, mean-variance risk, and smooth variants of Conditional Value-at-Risk. Our treatment sets out the conditions that facilitate the extension of OCE to unbounded r.v.s.. We provide a useful characterization of OCE that links OCE to utility-based shortfall risk (UBSR). Our characterization enables us to form an OCE estimator from the classic sample-average approximation (SAA) of UBSR. We derive mean-squared error (MSE) bounds for our proposed OCE estimator. For OCE optimization, we first derive an expression for the OCE gradient using the characterization linking OCE to UBSR. This expression serves as the basis for a gradient estimator for the OCE. We derive non-asymptotic bounds on the MSE for the proposed OCE gradient estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm to optimize OCE and quantify its convergence rate using non-asymptotic bounds that we derive. Finally, we present three experiments that use our OCE optimization algorithm to solve portfolio optimization and uncertainty quantification problems.

This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA's effectiveness for both 2D contour and 3D shape analysis.

We discuss Bayesian inference on a low-dimensional targeted parameter in the presence of possibly highly complex nuisance components within the semi-parametric inference framework using an estimating function approach. We obtain a posterior distribution using non-parametric Bayesian methods through the Dirichlet process and the Bayesian bootstrap. We relax the commonly deployed notion of stochastic equicontinuity and develop a framework leading to posterior inference with good frequentist properties, specifically we demonstrate that the posterior distribution is asymptotically Normal and concentrates at the true value of the parameter. We emphasize the specific assumptions that are required to obtain these results, and how relaxing any of them alters the conclusions. We verify the analytical results in simulation.

High-throughput screening (HTS) assays are central to early-stage drug discovery but are often limited by extreme data sparsity, as primary screens typically use only a single replicate per test substance. This sparsity makes conventional machine-learning performance metrics, such as sensitivity, specificity, and area under the receiver operating characteristic curve (AUROC), difficult to estimate empirically because they require adequately sized labeled samples. Here, we introduce a model-based framework that derives these classification metrics from the strictly standardized mean difference (SSMD), a well-established HTS effect-size parameter. Under a Gaussian equal-variance assumption, we derive closed-form relationships linking SSMD to Youden-optimal sensitivity and specificity, and sensitivity at a preset specificity, yielding explicit estimators and exact confidence intervals from the noncentral t-distribution, even under single-replicate designs. Unlike classical statistical power, which approaches 1 as sample size grows regardless of how small the true non-zero difference between group means is, the SSMD-derived sensitivity converges to a finite population value that reflects the true degree of separation between two groups, making it a more meaningful and stable performance measure for hit selection. We demonstrate the utility of this framework in a hepatitis C virus primary siRNA screen comprising approximately 22,000 single-replicate measurements, showing that SSMD, AUROC, and sensitivity-based thresholds yield equivalent and interpretable hit sets. This work bridges classical HTS statistics and machine-learning evaluation theory, providing a statistically principled, reproducible way to estimate classification performance in ultra-low-replication screening workflows.

The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.

Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an $\varepsilon$-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over $(s,a)$-rectangular total-variation uncertainty sets of radius at most $\sigma$. Let $H_0$ and $H_\sigma$ denote the nominal and robust optimal bias spans, respectively. We identify $\sigma H_0$ as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is $$ NSA \asymp \frac{SA}{\varepsilon^2}\begin{cases} \min\{H_0,H_\sigma\}, & \varepsilon\gtrsim\sigma H_0,\\ \min\{H_0,H_\sigma\}+\sigma H_\sigma^2, & \varepsilon\lesssim\sigma H_0. \end{cases} $$ Here $S$ and $A$ are the numbers of states and actions, and $N$ is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction—nominal or robust—and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.

Vector autoregressive moving-average (VARMA) models have long been considered impractical beyond moderate dimensions: the likelihood is non-convex, the parametrization is identified only up to equivalence, and every evaluation costs a pass over the entire series. Yet their moving-average term captures with a few parameters what a pure autoregression matches only with many lags. We introduce an estimation framework that removes this computational barrier: each optimization iteration is independent of the series length $T$. The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on the reparametrized coefficients with separate scales for diagonal and off-diagonal entries, and losses that depend on the data only through fixed-size sufficient statistics, evaluated by a Parseval (Fourier) identity at near-linear cost in the truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. We prove that both recover the infinite-autoregressive representation of the true process at a near-parametric rate in fixed dimension, so the truncation introduces no asymptotic bias. The same machinery extends, at the same leading cost, to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits. Empirically, the estimators stay close to the oracle forecast error from $d=10$ to $d=40$ (where classical conditional MLE returns non-invertible fits whose forecasts diverge) and match or beat VAR, Bayesian-VAR, component-wise ARMA, and sparse-VARMA baselines on retail-demand, meteorological, and air-quality data. This brings likelihood-based VARMA estimation, at a per-iteration cost independent of the series length, to the problem sizes where practitioners have so far relied on VAR models.