Density Ratio Estimation

We study density ratio estimation and importance-weighted regression under target shift with continuous outputs. Under target shift, the conditional distribution of the inputs given the outputs remains invariant across the training and test distributions, while the output marginal distribution may change. Although this problem has been extensively studied for discrete outputs, the continuous setting is substantially less understood: the importance weights are determined by an unknown density ratio function, for which existing estimation methods lack explicit finite-sample convergence rates. We propose a spectral regularization method in a reproducing kernel Hilbert space (RKHS) for estimating the continuous density ratio from labeled training samples and unlabeled test inputs. Under a source condition with regularity parameter $ι>0$, we establish high-probability finite-sample guarantees and show that the estimator achieves the capacity-independent minimax-optimal RKHS-norm rate $O(n_η^{-ι/(2ι+2)})$. We then incorporate the estimated density ratio into importance-weighted regression and characterize the propagation of density-ratio estimation error to the final predictor. When sufficiently many samples are available for density ratio estimation, the resulting regression estimator attains the minimax-optimal rates of standard kernel regression. These results establish a finite-sample theory for continuous density ratio estimation and importance-weighted learning under target shift.

Conformal prediction provides distribution-free uncertainty quantification under exchangeability. However, this assumption is violated by label shift, where the marginal distribution of labels changes while the conditional distribution of inputs given labels remains stable. Under such shifts, standard conformal procedures no longer maintain their intended coverage behavior. Existing approaches address this via importance weighting. They pair the reweighting with residual-based nonconformity scores that ignore predictive uncertainty. The resulting intervals have uniform width. Bayesian conformal methods produce adaptive intervals by leveraging predictive distributions. They evaluate conformity under the source predictive, which is misaligned with the target domain under label shift. We propose the \emph{Label-Shift-Adjusted Bayesian Score} (LSA score), a nonconformity score derived from a posterior predictive tilting identity. This identity shows that the target predictive is an importance-weighted transformation of the source predictive. We use it to derive a direct correction to the Bayesian score. We evaluate the method on molecular property prediction under controlled label shift. The LSA score consistently yields shorter intervals than residual-based and source-based Bayesian scores. Coverage in the target domain remains comparable. Under stronger shift, all methods incur some coverage loss due to pseudo-label-based density-ratio estimation. The LSA score is defined for any source predictive with a tractable log-density. We instantiate it with Bayesian Ridge Regression, where the correction admits a closed form.

Sparse-view X-ray 3D reconstruction is essential for reducing radiation exposure, but recovering a density field from a handful of X-ray projections is severely ill-posed. Recently, 3D Gaussian Splatting has achieved state-of-the-art performance in sparse-view reconstruction by representing the volume using explicit, optimized primitives, but it requires dozens of projected views. With fewer views, reconstruction quality degrades severely since the explicit primitives are optimized freely without any anatomical information. Anatomical structures, in contrast, share similar geometry and density across a population. Their variations are bounded within a limited range that statistical shape models can capture. This paper proposes a shape-guided Gaussian splatting framework for sparse-view X-ray 3D reconstructions. Our contribution lies in driving Gaussian positions toward anatomically valid configurations, alongside atlas-based density regularization. Our method ensures anatomically consistent reconstruction and improves PSNR by 2.83 dB over a state-of-the-art Gaussian splatting baseline with as few as 5 views. Code Available: https://github.com/polyshape-lab/ShapeGuidedGaussian

At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales $σ$ and $λσ$. For a finite union of $C^{2,α}$ half-branches in $\mathbb{R}^D$, the normalized score has the expansion $F_σ=F_0+σG+O(σ^{1+α})$. Matched subtraction cancels the tangent contribution and exposes $G$, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, $G$ uniquely identifies all $sD$ branch parameters, and $sD$ scalar component observations are necessary. An $O(σ^2)$ center error introduces $D$ translation modes, leading to $(s+1)D$ observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and $N^{-1/5}$ trends and remain full rank up to $D=20$ with 16 supplied branches. In end-to-end tests for $D=3$--$5$, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.

Pretraining corpus composition shapes LLM capabilities, but it often remains hidden even when model weights are released. Prior work has inferred corpus mixtures or traced specific token groups from released tokenizer vocabularies; in contrast, we estimate corpus ratios for arbitrary target tokens. We first show that BPE tokenizers trained on different corpora share stable token ID--ratio distributions, motivating distribution transfer from known corpora to a target tokenizer trained on hidden corpora. We then propose Quantile-Guided Density Estimation (QGDE), which approximates this distribution with multiple quantile trends and uses local density weighting to produce token-level estimates. In controlled settings and a realistic setting using the released SmolLM tokenizer, QGDE achieves mean relative errors as low as 3.00% for token-level estimation and 3.08% after aggregation into category-level mixtures. These results suggest that released tokenizer vocabularies provide a useful signal for fine-grained corpus estimation beyond coarse composition inference.

Density estimation underlies many unsupervised tasks on tabular data such as anomaly detection, out-of-distribution detection, and data augmentation. Although all these problems reduce to questions about where probability mass lies, they are typically solved individually by fitting a separate model to each dataset, with its own hyperparameters and tuning budget. We introduce ICED, an in-context, energy-based density estimator that removes this per-dataset cost. ICED is a transformer-based model pretrained once on a synthetic prior built specifically for density estimation under an objective that fits log-density where it is informative and preserves its ordering elsewhere. In the inference, it reads a dataset as context and returns an unnormalized log-density for any query point in a single forward pass, with no fitting, sampling, or hyperparameter selection. A single frozen ICED model then drives four tasks usually handled by four specialized pipelines: density estimation, out-of-distribution detection, unsupervised anomaly detection, and generative augmentation. Across all four, it is competitive with the strongest task-specific method, while being the only approach that needs no retraining, no tuning, and no labels to move between them. The code is available at https://github.com/gmum/iced.

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

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent $p$ using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.

Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source--target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a \emph{distributional distance} that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.

Task B of the 1st DAFx Parameter Estimation Challenge requires estimating the frequencies, decay rates, gains, and number of modes in a dense plate-reverb impulse response. Weak and overlapping modes make sparse peak detection prone to severe undercounting. We train an ExtraTrees regressor on simulator-generated data to predict mode counts in four frequency bands. These counts define dense frequency grids, after which a differentiable all-pole resonator model refines decay and gain while keeping frequency fixed. On two separate synthetic validation sets, the system reduces a local challenge-style error by about 66% relative to the official default peak-picking baseline. The improvement is mainly associated with lower mode-count mismatch, while decay and gain remain the largest error sources. These findings support separating modal-density estimation from continuous parameter fitting.

Density modes provide a localized and interpretable summary of multimodal distributions, but their estimation under rigorous differential privacy constraints remains largely unexplored. We study differentially private recovery of density modes for multivariate distributions under local smoothness, curvature, and separation conditions. We propose DP-GRAMS, a mean-shift inspired method that performs noisy ascent on a differentially private score estimator. Assuming the density belongs locally to a Hölder class with smoothness parameter $β> 2$, our score estimator uses bias-reducing higher-order kernels, and then enforces privacy in the gradient ascent steps via gradient clipping and calibrated Gaussian noise. A private initialization scheme combines a density-aware utility with a suppression rule and, with $k\asymp M\log n$ draws over a public $h_{\mathrm{DAP}}$-grid and suppression radius $ρ_{\mathrm{init}}\asymp (\log n)^{-1/d}$, achieves high-probability coverage of the modal basins by successively suppressing selected local neighborhoods in competitive regions, while correlated noise across multiple starts enables joint release under a single $(\varepsilon,δ)$-differential privacy guarantee. We prove that all population modes are recovered with high probability and establish asymptotic error rates of the form $O\!\left((\tfrac{\log n}{n})^{\frac{2(β-1)}{d+2β}}\right) + O\!\left((\tfrac{\mathrm{polylog}(n,δ)}{n^2\varepsilon^2})^{\frac{β-1}{d+β}}\right)$. We also provide minimax lower bounds for private mode estimation, and show that our estimators are nearly optimal, up to a logarithmic factor in the MSE. We present two natural extensions: DP-PMS, a private modal-regression method, and DP-GRAMS-C, a clustering pipeline. Extensive experiments on synthetic and real data demonstrate favorable privacy-utility trade-offs relative to common baselines.

We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts. Unlike the squared Maximum Mean Discrepancy (MMD), which discards directional information by squaring the RKHS distance, DKMD integrates the difference of kernel mean embeddings against a fixed odd weighting function. This construction yields three structural properties: antisymmetry, immunity to symmetric distributional differences, and directional monotonicity under stochastic dominance. We derive a data-driven Riemann estimator that ensures asymptotic consistency with the continuous formulation, strictly preserving the theoretical guarantees of the signed statistic in empirical evaluations. To overcome the quadratic computational cost of kernel methods, we develop an $O(N \log N)$ prefix--suffix scanning algorithm that exploits the total order of the real line while requiring only $O(N)$ memory. Experiments on synthetic benchmarks demonstrate that DKMD correctly isolates directional shifts from symmetric perturbations, remains robust to heavy-tailed outliers that can flip the sign of the mean difference, and scales to millions of samples in seconds.

Activation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. While many existing methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions, they do so heuristically rather than deriving it from a principled optimization problem. Moreover, these methods produce query-independent steering directions that can degrade performance on both in-distribution and out-of-distribution (OOD) inputs. We introduce \textsc{Cobras} (Conditional Optimal Bridge for Riemannian Activation Steering), which addresses both limitations by casting activation steering as a Schrödinger Bridge on the residual-stream hypersphere. This formulation yields, to our knowledge, the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Solving the bridge via entropic optimal transport and extracting the probability flow ODE recovers the widely used density-ratio gradient as a special case when the Sinkhorn potentials are uniform. Crucially, the Schrödinger potentials are evaluated at the current activation, making the resulting steering direction inherently query-adaptive. Empirically, across four models and three alignment axes (helpfulness, truthfulness, and detoxification), \textsc{Cobras} consistently outperforms prior activation steering baselines while avoiding the OOD degradation commonly observed in existing methods. The code can be found at https://github.com/arshandalili/cobras.

Implicit neural representations (INRs) offer compact encoding of volumes, but as lossy approximators, inevitably have prediction errors. We consider INRs that can simultaneously encode relative error scales by predicting distributions using tools from uncertainty estimation. Typically, uncertainty estimation relies on computationally expensive approaches or on predefined parametric assumptions about the predictive distribution (e.g., Gaussian). In this study, we propose a lightweight method that reformulates regression-based INR training as a classification task by discretizing continuous targets into bins, enabling flexible distribution modeling to capture complex multimodal behaviors. We analyze the trade-off between regression and classification for INR training and demonstrate that the classification setting tends to achieve high reconstruction quality and competitive error awareness through uncertainty estimation, compared to regression-based approaches.

Missing values undermine statistical inference and machine learning pipelines, yet most imputation methods rely on heuristics or restrictive parametric assumptions that ignore the joint data distribution. We recast imputation under missing completely at random (MCAR) as density estimation from masked observations: estimate a distribution whose observed marginals exactly match those in the data. Leveraging positive semi definite (PSD) kernel densities we obtain a convex empirical risk problem with closed form marginals, solvable by a Newton interior point method. The resulting PSD Impute model yields both single and multiple imputations from the same fitted density, enjoys statistical consistency with fast adaptive excess risk beating the curse of dimensionality for very regular probabilities. Preliminary experiments on one synthetic and eleven real world datasets already indicate competitive distributional accuracy compared with popular imputation baselines, suggesting strong practical promise.

The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.

Accurately estimating the unknown target label distribution is the critical first step for adapting to label shift. This task, widely known as quantification or class prevalence estimation, has recently seen significant advances through continuous KDE-based methods which model the density of multiclass classifier posteriors. Posterior vectors might be regarded as compositional data, since they lie on the probability simplex. However, existing KDE-based quantifiers typically rely on Euclidean Gaussian kernels, which ignore simplex geometry and incorrectly assign probability mass outside its boundaries. We introduce a geometry-aware KDE model for multiclass quantification based on log-ratio representations and Aitchison geometry, together with a shrinkage regularization that improves robustness near the simplex boundary. Combined with a maximum-likelihood interpretation of KDE-based quantification, we derive both point-estimation and Bayesian inference procedures for class prevalences. Experiments on 42 datasets across tabular, text, and image domains show that the proposed method is competitive with state-of-the-art quantifiers, often improving over standard KDE-based baselines, while also yielding strong results among Bayesian quantification methods.

The silhouette is one of the most widely used measures to assess the quality of a $k$-clustering of a dataset of $n$ elements. Its evaluation requires no information beyond the clustering assignment. In addition, the silhouette is extremely easy to interpret, providing a score to measure the quality of a clustering as a whole or for each element. The exact computation of the: (i) silhouette of each element of a dataset; and (ii) the global silhouette of the clustering; require $Θ(n^2)$ distance calculations, under general metrics. The quadratic complexity $Θ(n^2)$ is extremely prohibitive, especially on massive modern datasets. Surprisingly, existing approximate methods using $O(n^2)$ distance calculations are heuristics not offering provable and controllable guarantees on the quality of their results. We introduce the first rigorous and efficient algorithms to estimate: (i) the (local) silhouette of each element of a dataset; and (ii) the (global) silhouette; of any metric $k$-clustering. Our methods, based on sampling, perform $O(nk\varepsilon^{-2}\ln (nk/δ))$ distance computations, and provide estimates with additive error $O(\varepsilon)$ with probability at least $1-δ$. That is, parameters $\varepsilon$ and $δ$ in $(0,1)$ control the trade-off between accuracy and efficiency. We also introduce a scalable and distributed design of our methods for the MapReduce and Massively Parallel Computing (MPC) frameworks. Our distributed algorithms use a constant number of rounds and sublinear local memory. Finally, we perform extensive experiments against state-of-the-art approaches. The results show that our new techniques yield the best trade-off between accuracy and efficiency for both local and global silhouette estimation. In addition, our methods scale efficiently to massive datasets for which an exact computation of the silhouette is not practical.

We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($Δ$, I), with linear features, where $Δ$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.

Few-shot segmentation asks a model to delineate a target class in a query image from only a handful of annotated examples, a setting most acute in remote sensing, where labels are scarce and the imagery departs sharply from the natural images on which vision backbones are pretrained. Prevailing approaches either train a segmenter on labelled episodes, which raises accuracy within the training distribution but binds the model to it, or reduce each class to a lossy summary of frozen features, a single prototype, a few cluster prototypes, or a discrete clustering, none of which preserves the internal structure of a multimodal class. We argue that a class is better described by a distribution than by a point, and that frozen self-supervised features already carry enough structure to estimate that distribution directly. We introduce FROST, a training-free few-shot segmenter that treats the reference foreground and background as two point clouds on the unit sphere of frozen DINOv3 features and labels each query token by a nonparametric density ratio, with a threshold the Bayes rule fixes at zero under equal priors. Because the variance of a density estimate shrinks as its sample grows, the decision sharpens as references accumulate, and every remaining quantity from the kernel bandwidth to the spatial gate is read from the support set rather than tuned. We develop FROST for overhead imagery, where a class is typically a scatter of many small and dissimilar instances that a density tracks but a lossy summary blurs. Across seventeen remote-sensing benchmarks FROST surpasses both training-free and learning-based methods, leading by 5.6 mIoU from a single annotated example and widening its lead as the support set grows, all while remaining among the smallest models compared. Code is available at https://github.com/jhpark-ai/FROST.

The continuous integral R2 indicator is a Pareto-compliant refinement of the classical finite-weight-vector R2 indicator, used in performance assessment, bounded archiving for a-posteriori multi-objective optimization, and skyline selection in databases. This work introduces a bidirectional perspective mapping between continuous integral R2 computation and integration over unions of anchored axis-aligned boxes. After translating the ideal point of a minimization problem to the origin, approximation points become strictly positive loss vectors, and the subgraph of the lower weighted Tchebycheff envelope over the weight simplex maps to the complement of an anchored-box union in reciprocal objective space. The Jacobian gives an absolute R2 formula as a weighted complement volume with density $(x_1+\cdots+x_N)^{-(N+1)}$, while differences of R2 values become finite weighted hypervolume differences. Hence, hypervolume algorithms that emit box decompositions can be reused by replacing ordinary box volumes with closed-form weighted box integrals. For $N$ objectives, this gives an output-sensitive overhead $O(2^N M)$ for an $M$-box decomposition, or $O(M)$ for fixed $N$. Using existing box-decomposition approaches, the integral R2 can be computed in $O(n \log n)$ for $N=2,3$, in $O(n^2)$ for $N=4$, and in $O\left(n^{\lfloor (N-1)/2\rfloor+1}\right)$ for $N\geq4$, with $n$ denoting the size of the approximation set. On the lower-bound side, exact value computation has an $Ω(n\log n)$ lower bound in the algebraic decision-tree model already in two objectives, this bound lifts to every fixed $N\geq2$, and exact computation is $\#P$-hard when $N$ is part of the input. Together, the proposed perspective mapping provides a powerful tool for transferring algorithmic and structural results between anchored-box union and hypervolume theory and integral R2 computation.

Unsupervised tabular anomaly detection requires methods that are accurate, robust across heterogeneous datasets, and computationally efficient. Classical statistical detectors are often efficient, but they usually rely on a fixed data view and a single notion of abnormality. Deep anomaly detectors can learn more flexible scoring functions, but they are substantially slower and difficult to tune in unsupervised settings due to the lack of a reliable supervisory signal. We propose RGLD, a randomized global-local density estimator for efficient unsupervised tabular anomaly detection. RGLD combines a global random-feature density branch, which identifies samples in broadly low-density regions, with a local neighbor branch, which detects samples that are weakly supported by nearby observations. Both branches operate over feature-bagged randomized views, allowing RGLD to expose anomaly evidence that may be hidden in any single representation. We conduct experiments on 47 tabular datasets against 23 statistical and deep anomaly detection baselines under fully unsupervised setting. RGLD achieves the strongest dataset-level AUROC performance, ranking 1st in dataset wins, and ranks 2nd in AUPRC wins. RGLD is also faster than all evaluated deep detectors, achieving 50x-580x speedups, and remains competitive with statistical methods in runtime, yielding a favorable accuracy-efficiency tradeoff.

Detectors for GNSS radio-frequency impairments (jamming, spoofing, multipath) are usually reported with a single AUC measured on the distribution they were tuned on. That number falls once conditions move, and the size of the drop is rarely known in advance because labelled field data is scarce. We ask whether this optimism can be predicted before any out-of-distribution data is seen. On an open, parameter-grounded synthetic testbed with a tunable severity shift, we evaluate thirteen detectors (five physics baselines, full-feature logistic regression and multilayer perceptrons, and single-feature learned controls) across four impairment classes. The optimism gap, the difference between in-distribution and shifted AUC, grows monotonically as the shift deepens (mean Spearman correlation 0.50). It is driven by how many observables a detector uses rather than by whether it is learned, and it varies systematically by class. Centrally, a ridge model built only from in-distribution score statistics predicts the gap for a detector it has never seen (R^2 = 0.47) and for an impairment class it has never seen (R^2 = 0.46); both are significant against a 2000-fold permutation null (p < 0.001) and survive removing the feature that is, by construction, part of the target. The headline findings are synthetic. We then run the pre-registered protocol on three open field corpora: on Jammertest 2024 the cross-detector prediction holds (R^2 = 0.11, p = 0.009), and on SatGrid, whose spoofer power sweep gives a calibrated severity axis, in-distribution AUC overstates higher-severity AUC by up to 0.22 and to the point of sign inversion, with in-distribution AUC and realised gap perfectly rank-correlated (Spearman rho = 1.0). The mechanism survives contact with real data, at smaller magnitude than in simulation. We release the testbed, a software-receiver front end, the ingest adapters and the protocol.

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a popular and prominent evolutionary algorithm for solving multi-objective optimisation problems. Despite its popularity, theoretical analyses of SPEA2 have only appeared recently. Moreover, these analyses focus exclusively on how SPEA2 handles non-dominated solutions and disregard the algorithmic components responsible for handling dominated solutions. We conduct a first runtime analysis of SPEA2 for which these components are analysed. We prove that, unlike other prominent algorithms, including NSGA-II, NSGA-III and SMS-EMOA under the same setting of constant population size and duplicate elimination, SPEA2 is unable to cover the Pareto front of the OneTrapZeroTrap benchmark efficiently. Our results indicate that using k-th nearest-neighbour distance in the fitness assignment provides an insufficient signal to maintain diversity among dominated individuals. To address this issue, we propose an improved variant, SPEA2$^+$, that considers all pairwise distances. The new algorithm achieves the same performance guarantees as the other prominent algorithms on OneTrapZeroTrap, while matching the performance of the original SPEA2 on simpler problems. Experimental results complement our theoretical findings.

We study the task of density estimation, where we hope to accurately estimate a probability density from $n$ samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.

Humans rely on spatially dense, geometry and force-aware tactile feedback at high temporal resolution for dexterous manipulation. While vision-based tactile sensors enable dense force estimation, they are limited by camera frame rates, motion blur, and data bandwidth. Event-based optical tactile sensors offer an attractive alternative with microsecond temporal resolution and low motion blur, but existing methods are restricted to predicting only net forces. We introduce the first framework for dense 3D force field reconstruction using event-based optical tactile sensors. Our approach estimates 3D surface displacements from event data and maps them to forces via the inverse Finite Elements Method (iFEM). Shear displacements are recovered through the proposed event-based marker tracking algorithm, while normal displacements are predicted by a convolutional neural network trained on a collected dataset of synchronized force-displacement-event data. Experiments demonstrate accurate reconstruction of physically grounded forces, achieving a mean absolute error of (0.14 N, 0.10 N, 0.93 N) over force ranges up to (4 N, 4 N, 20 N), while operating at an average of 100 Hz. This work constitutes a first step toward enabling dense force feedback for high-frequency control in robotic grasping and dexterous manipulation.

Selecting a clustering algorithm and its hyperparameters without labels is a common difficulty in engineering machine learning pipelines that work with unsupervised analysis of sensor, image, or process data. Clustering validation indices (CVIs) provide internal scores for ranking candidate clusterings, but most popular CVIs are built from Euclidean compactness and separation terms and so tend to favour compact, convex partitions. Their performance is known to degrade on non convex, irregular, or variable density data, where kernel transformations or alternative distance measures are typically used at the cost of additional tuning and computation. This paper introduces the Central Description Length (CDL) clustering validation index. CDL uses the observed within cluster compactness, the estimated cluster centers, and the estimated cluster covariances to compute a probabilistic upper bound on the description length associated with the unobservable true cluster centers. The bound condenses intra cluster compactness and centroid displacement into a single computable quantity and is evaluated on the partition produced by any clustering algorithm. The implementation uses only observable quantities (the data, the partition, the estimated centers, and the estimated covariances) and does not use ground truth labels. On synthetic benchmarks with non convex and arbitrary shape clusters, CDL-CVI selected the reference number of clusters more often and reached higher Adjusted Rand Index (ARI) values than the conventional CVIs we tested, without an additional kernel preprocessing stage. On image benchmarks (MNIST, CIFAR-10, STL-10) clustered from frozen unsupervised embeddings, CDL-CVI returned cluster numbers close to the reference class counts across K-means, DBSCAN, and spectral clustering in the reported trials.

We introduce causal density functions: Radon-Nikodym derivatives that compare interventional laws to observational laws and therefore act as local density ratios for causal effects. Whereas many causal-strength measures compare whole distributions after graph surgery, causal density functions provide a pointwise change-of-measure object that can be estimated, calibrated, and used to score directed influence. The basic identity $$\mathbb{E}_{\mathrm{do}}[f(Y)] = \mathbb{E}_{\mathrm{obs}}\!\left[f(Y)ρ(X,Y)\right]$$ makes causal density directly testable: if the estimated density ratio is correct, observational expectations reweighted by $ρ$ reproduce interventional expectations. We derive practical estimators for do-curves and directed edge scores, relate the construction to Radon-Nikodym/Kan semantics for conditioning and intervention, and evaluate the resulting estimators on synthetic and real perturbation benchmarks.

Counterfactual explanations (CEs) are essential for actionable recourse, yet their reliability is often compromised in low-density regions, where classifiers exhibit high variance. Unlike existing methods that rely on expensive ensemble intersections to define stability, we propose \textit{DensityFlow}, a generative framework that constructs robust CEs by adhering to the high-confidence data manifold. Specifically, we model the counterfactual generation as continuous-time dynamics parameterized by Neural ODE, guided by a differentiable density score to actively avoid uncertain, low-density areas. This density score is learned via Noise Contrastive Estimation, effectively leveraging a $(K{+}1)$-way discriminator to estimate density ratios. For black-box settings, we introduce a local proxy distillation mechanism that aligns a lightweight surrogate with the target model strictly within the trajectory of CE generation, enabling efficient gradient-based optimization with minimal queries. Experiments demonstrate that \textit{DensityFlow} achieves superior validity under model multiplicity while significantly reducing query costs compared to ensemble-based baselines. Our implementation is available at https://github.com/G-AILab/DensityFlow.

Discrete probability laws underpin statistical modeling, yet the catalog of interpretable distributions has expanded only gradually through centuries of case-by-case mathematical derivations. We introduce symbolic density estimation (SDE), an unsupervised framework that automatically recovers closed-form probability mass functions by composing elementary analytic operations within a structured search space. Our method integrates domain-specific structural priors with evolutionary search and a validity-aware inference stage, and it extends to richer distribution families such as zero inflation and finite mixtures. To support systematic evaluation and future research, we contribute a benchmark dataset spanning a broad collection of commonly used discrete distributions. The proposed algorithm recovers all benchmark families with accurate parameter estimates. A real data application shows that it identifies concise and interpretable mixture models that improve goodness-of-fit over standard models.

Uniform sampling on implicitly defined manifolds is a core primitive in motion planning, constrained simulation, and probabilistic machine learning. MASEM addresses this problem by entropy-maximizing resampling, but its resampling weights depend on a local k-nearest-neighbour density estimate whose errors can be amplified by aggressive resampling temperatures. We ask whether a polynomial-maximization moment estimator can replace the plug-in density rule without changing the surrounding MASEM architecture. The proposed PMM-MASEM module computes shell spacings from nested k-nearest-neighbour radii, estimates their standardized cumulants, and uses a gated PMM2/PMM3 estimator only when the spacing distribution departs from the flat Exp(1) regime; otherwise it falls back to the plug-in/MLE rule. This fallback is essential: on a flat homogeneous manifold the plug-in estimator is already the MLE, so PMM should not outperform it. A local Known-DGP Monte Carlo experiment confirms this gate: the selector returns MLE on flat Exp(1) spacings and reduces density MSE by 22--36% on asymmetric gamma and boundary-spacing regimes. The evidence is not uniformly positive: PMM3 worsens a platykurtic uniform spacing law, and a lightweight resampling-proxy experiment improves seven-lobes coverage but degrades the sine and swiss-roll proxies. The current evidence therefore supports an applicability-boundary result rather than a general MASEM improvement claim.

Estimating forest aboveground biomass (AGB) from Earth observation combines two structurally incompatible label sources: spaceborne lidar provides canopy structure at millions of locations but no biomass estimate, and ground-based plots provide biomass at thousands of biased locations but no metrics of structure. No single training sample carries labels for all target variables, plot labels are missing not at random (MNAR), and biomass is linked to the structural variables by known but biome-specific allometric laws. We formalise this as multi-task dense regression under heterogeneous disjoint partial supervision with MNAR labels and inter-task physical constraints, and propose StruMPL to address it jointly. A shared encoder feeds per-variable regression, imputation, and propensity heads for spatial MNAR correction, and a learnable physics module that evaluates the inter-task constraint on the model's own predictions at every pixel. The supervised loss uses an Augmented IPW (AIPW) pseudo-outcome with stop-gradients on the propensity and on the imputation baseline; we show analytically and empirically that both are necessary for joint optimisation to recover IPW-weighted stationary points while keeping the loss bounded. On two ecologically distinct biomes, StruMPL outperforms ablation variants and the closest published method on AGB RMSE and bias, with a stratified analysis showing AIPW reduces high-AGB bias by ~54%.

Occupancy estimation and crowd counting are critical tasks in designing smart and efficient public transport vehicles. Given that public transport loading can vary from sparse to crowded, classical models for occupancy estimation must be adapted to suit this purpose. Attention mechanisms have shown remarkable capability in enhancing the representational power of deep neural networks for crowd counting in congested scenes with occlusion, complex backgrounds, and perspective distortion. However, conventional approaches, often implemented as parameterized sub-networks within convolutional layers, inevitably increase model size and computational cost, limiting deployment on resource-constrained edge devices. This paper investigates the effectiveness of state-of-the-art parameter-free attention mechanisms for crowd counting and density map estimation in highly congested scenes. We evaluate channel-wise (PFCA), spatial-wise (SA), and 3-D (SimAM) modules and compare their performance with parameterized attention modules constrained to introduce no more than 1% additional parameters. Furthermore, we present a novel combination of attention mechanisms that combines the strengths of PFCA and SA (PFCASA) customized for analyzing video streams onboard public transport systems. Using CSRNet as the backbone, experiments on the ShanghaiTech dataset demonstrate that parameter-free attention mechanisms achieve comparable or superior accuracy without introducing additional model parameters. A detailed performance analysis further reveals that PFCASA outperforms other attention modules in scenes with fewer than 40 individuals, while PFCA shows greater effectiveness as crowd density increases, underscoring their potential applicability for integration into smart public transport modalities.

Tabular data generation considers a large table with multiple columns -- each column comprised of numerical, categorical, or sometimes ordinal values. The goal is to produce new rows for the table that replicate the distribution of rows from the original data -- without just copying those initial rows. The last 4 years have seen enormous progress on this problem, mostly using computational expensive methods that employ one-hot encoding, VAEs, and diffusion. This paper describes a new approach to the problem of tabular data generation. By employing copula transformations and modeling the distribution as a kernel density estimate we can nearly match the accuracy and leakage-avoidance achievements of the previous methods, but with almost no training time. Our method is very scalable, and can be run on data sets orders of magnitude larger than prior state-of-the-art on a simple laptop. Moreover, because we employ kernel density estimates, we can store the model as a coreset of the original data -- we believe the first for generative modeling -- and as a result, require significantly less space as well. Our code is available here: \url{https://github.com/tabkde/tabkde-main}

Estimating the number of distinct elements in a data stream is well understood when repeated elements are identical. In modern settings, however, observations are high-dimensional and noisy, so repeated instances of the same object are only approximately similar -- for example, different images of the same individual may vary significantly at the pixel level. Classical sketches such as HyperLogLog rely on consistent hash values for identical elements and break down in this regime. Recent work on robust distinct counting in general metric spaces achieves $\widetildeΘ(\sqrt{n})$ memory, which is tight in the worst case. We show that substantially improved memory guarantees are possible under geometric structure common in learned representations. We introduce MaxSketch, a simple max-linear sketch built from random Gaussian projections, and prove that it succeeds in estimating the number of distinct latent objects. Concretely, we show that under this assumption $m = \widetilde{O} (\log n / \varepsilon^2)$ random projections (and hence $\widetilde{O} (\log n/\varepsilon^2)$ memory) suffice to recover the true distinct count within a $(1+\varepsilon)$ factor. Experiments on image streams confirm that MaxSketch accurately estimates distinct counts and generalizes beyond the training regime. Our results bridge classical streaming algorithms and modern representation learning, showing how geometric structure can fundamentally reduce the complexity of distinct counting.

Density estimation in high-dimensional settings is an important and challenging statistical problem.Traditional methods based on kernel smoothing are inefficient in high dimensions due to the difficulties in specifying appropriate location-adaptive kernels. In this work, we introduce pre-training, a key idea behind many cutting-edge AI technologies, to the context of non-parametric density estimation. By establishing a pre-trained neural network that can recommend an appropriate location-adaptive kernel for each sample point, efficient density estimation with adaptive kernels is achieved in high dimensions. A wide range of numerical experiments show that this strategy is highly effective for improving density-estimation accuracy, when the target distribution is close to the distribution family for pre-training. When the target distribution is substantially different from the pre-training distribution family, the benefit from the proposed pre-training strategy may be diluted, but can be reactivated by an additional fine-tuning procedure.

We propose a closed-form spectral framework for relative log-density estimation in linearly parameterized probabilistic models, including unnormalized and conditional models. This is achieved by representing the Kullback-Leibler (KL) divergence as an integral of weighted chi-squared divergences, converting KL estimation into a family of least-squares problems. We derive an explicit spectral formula based only on first- and second-order feature moments, yielding closed-form estimators of both divergences and log-density potentials for fixed features. The framework extends to a broad class of f-divergences and can be combined with kernelization or feature learning with neural networks. We prove convergence guarantees for the resulting estimators and empirically compare them on synthetic data with optimization-based variational formulations, including logistic and softmax regression for normalized conditional models.

Distributional causal inference requires estimating not only average treatment effects but also interventional outcome distributions, including quantiles, tail risks, and policy-dependent uncertainty. As a method for distributional causal inference, generative adversarial network (GAN)-based counterfactual methods are flexible tools for this task. However, these methods have several limitations. First, the objectives of certain techniques do not coincide with the statistical risk of the identifiable causal target, and therefore provide limited theoretical guarantees regarding estimable counterfactual distributions or optimality. Second, they tend to rely on unstable density-based methods, such as density ratio estimation. In this paper, we propose GANICE (GAN for Interventional Conditional Estimation) with several advantages: it (i) clarifies the conditional interventional distribution for each treatment--covariate state as the causal estimation target; (ii) estimates the conditional distribution such that its averaged Wasserstein risk is minimized; (iii) establishes minimax optimality. GANICE achieves these advantages through the introduction of the extended Wasserstein distance, the incorporation of a cellwise critic in its dual, and an optimality proof based on Besov space theory. Our experiments demonstrate that GANICE consistently outperforms existing methods.

Density estimation is a central primitive in probabilistic modeling, yet continuous, discrete, and mixed-variable domains are often treated by separate objectives, limiting the ability to exploit a common statistical structure across data types. Continuous score-based methods rely on log-density gradients, while discrete extensions typically use concrete score whose unbounded targets become unstable near low-probability states. We introduce Constant-Target Energy Matching (CTEM), a unified energy-based framework for density estimation on general state spaces. CTEM replaces ordinary density-ratio regression with a bounded energy-difference transform and derives from it a sample-only training objective with the constant target 1. The learned scalar potential recovers log p without partition-function estimation or explicit unbounded ratio regression. Across continuous, discrete, and mixed-variable benchmarks, CTEM substantially improves density estimation over competitive baselines and yields higher-quality samples under standard sampling procedures.

Density estimation and reliable prediction regions for outputs are crucial in supervised and unsupervised learning. While conformal prediction effectively generates coverage-guaranteed regions, it struggles with multi-dimensional outputs due to reliance on one-dimensional nonconformity scores. To address this, we introduce CONTRA: CONformal prediction region via normalizing flow TRAnsformation. CONTRA utilizes the latent spaces of normalizing flows to define nonconformity scores based on distances from the center. This allows for the mapping of high-density regions in latent space to sharp prediction regions in the output space, surpassing traditional hyperrectangular or elliptical conformal regions. Further, for scenarios where other predictive models are favored over flow-based models, we extend CONTRA to enhance any such model with a reliable prediction region by training a simple normalizing flow on the residuals. We demonstrate that both CONTRA and its extension maintain guaranteed coverage probability and outperform existing methods in generating accurate prediction regions across various datasets. We conclude that CONTRA is an effective tool for (conditional) density estimation, addressing the under-explored challenge of delivering multi-dimensional prediction regions.

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$, and farthest-point-sampling positions $2$-approximating the optimal $k$-center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ($91.3 \pm 1.0\%$, matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ($25\%$ on 4-class data).

Empirical fixation densities, spatial distributions estimated from human eye-tracking data, are foundational to saliency benchmarking. They directly shape benchmark conclusions, leaderboard rankings, failure case analyses, and scientific claims about human visual behavior. Yet the standard estimation method, fixed-bandwidth isotropic Gaussian KDE, has gone essentially unchanged for decades. This matters now more than ever: as the field shifts toward sample-level evaluation (failure case analysis, inverse benchmarking, per-image model comparison), reliable per-image density estimates become critical. We propose a principled mixture model that combines an adaptive-bandwidth KDE based on Abramson's method, center bias and uniform components, and a state-of-the-art saliency model, to capture different spatial and semantic types of interobserver consistency, and optimize all parameters per image via leave-one-subject-out cross-validation. Our method yields substantially higher interobserver consistency estimates across multiple benchmarks, with median per-image gains of 5-15% in log-likelihood and up to 2 percentage points in AUC. For the most affected images -- precisely those most relevant to failure case analysis -- improvements exceed 25%. We leverage these improved estimates to identify and analyze remaining failure cases of state-of-the-art saliency models, demonstrating that significant headroom for model improvement remains. More broadly, our findings highlight that empirical fixation densities should not be treated as fixed ground truths but as evolving estimates that improve with better methodology.

Conformal prediction (CP) provides powerful, distribution-free prediction sets, but its guarantees rely on the exchangeability of training and test data, which is often violated in practice due to covariate shifts. While weighted conformal prediction (WCP) is designed to handle such shifts, it can suffer from significant undercoverage when the density ratio between the distributions is unbounded and/or must be learned. This is because of both overfitting in learning the density ratio, and high variance in estimating the nonconformity score threshold. To address this, we introduce clipped least-squares importance fitting (CLISF) as a reduced-variance method for density ratio estimation. Specifically, we show that density ratios learned using CLISF, when plugged into WCP, have bounded expected undercoverage. Furthermore, we show that the undercoverage can be corrected by running WCP with a slightly inflated coverage target; crucially, we are able to estimate the required level of inflation from the data. We provide the first theoretical guarantees for weight clipping in conformal inference, achieving dataset-conditional coverage with a sample complexity that does not blow up with the higher moments of the true density ratio -- a key limitation of prior work. We verify our results on real-world benchmarks and synthetic data.

Dimensionality reduction methods such as t-SNE are designed to preserve local neighborhood structure but do not explicitly account for how probability mass is distributed, often leading to distortions of data density. We reformulate dimensionality reduction as the joint alignment of two components: (i) conditional structure, capturing local relationships, and (ii) relative density structure, captured via local density statistics. Based on this perspective, we introduce Density-Regularized SNE (DR-SNE), which augments the stochastic neighbor embedding objective with a density regularization term derived from normalized log-density estimates. Unlike prior approaches such as DensMAP and DenSNE, which rely on local scale consistency, DR-SNE directly aligns normalized density estimates, providing a simple and scale-invariant mechanism for preserving relative density variations. Empirically, DR-SNE improves density preservation while maintaining competitive neighborhood fidelity, and yields gains on density-sensitive tasks such as anomaly detection across multiple datasets. These results suggest that incorporating density information complements geometry-focused objectives in dimensionality reduction.

While diffusion models enable new approaches for estimating Local Intrinsic Dimension (LID), existing methods fail in high-dimensional spaces where noise from vast normal directions overwhelms the tangent signal. We propose Local Hessian Spectral Dimension (LHSD), which resolves this by applying spectral filtering to the log-density Hessian, explicitly cutting off large eigenvalues associated with normal directions to count zero-curvature tangent directions. Implemented using Stochastic Lanczos Quadrature (SLQ), LHSD avoids full Hessian construction, achieving linear scalability with dimension $D$. Experiments on synthetic and real data confirm LHSD's superior robustness and its utility in detecting memorization in large-scale diffusion models. The code is available at github.com/geosada/LHSD

SFT-then-RLVR is widely used for post-training reasoning models, but why this specific ordering, and why RLVR-only stalls at cold start, have lacked a unifying theoretical account. We provide that account under a unified loss family $J_Q$ using the Tsallis $q$-logarithm. $J_Q$ is a single-parameter family that interpolates between RLVR (at $q{=}0$, the \textit{exploitation pole}) and the log-marginal-likelihood over latent trajectories (at $q{=}1$, the \textit{density-estimation pole}), under which the standard pipeline corresponds to a stepwise $q{=}1 \to 0$ schedule. All members share the same per-example gradient direction, differing only by a per-instance amplification $P_θ^{-q}$ that reweights each instance independently of the learning rate. Under gradient flow analysis, we show that the exploitation pole requires $Ω(\frac{1}{p_0})$ time to escape cold start but is robust to label noise, while the density-estimation pole escapes in $Θ\big(\log(\frac{1}{p_0})\big)$ but memorizes label noise. This separation explains how SFT ($q{=}1$) first moves the model out of the cold-start regime, followed by the more robust RLVR ($q{=}0$), under the SFT-then-RLVR paradigm. We further derive two Monte Carlo estimators that directly optimize fixed-$q$ on the $J_Q$ continuum, without annotated rationales: Gradient-Amplified RL (GARL) and Posterior-Attenuated Fine-Tuning (PAFT), with shared bias $O\big(\frac{q}{M P_θ^q}\big)$ but different variance and stability properties. On FinQA, HotPotQA, and MuSiQue, GARL at sufficiently high $q$ substantially mitigates cold-start stalling, escaping cold start where GRPO fails entirely. In warm start, GARL at low $q$ dominates FinQA where training is stable; on HotPotQA and MuSiQue, GARL destabilizes and PAFT at $q{=}0.75$ remains stable, reaching $47.9$ \texttt{m@16} on HotPotQA ($+13.9$ over GRPO).

When task-specific labels are not available, it becomes difficult to select an embedding model for a specific target corpus. Existing labelless measures based on kernel estimators or Gaussian mixes fail in high-dimensional space, resulting in unstable rankings. We propose a flow-based labelless representation embedding evaluation (FLARE), which utilizes normalized streams to estimate information sufficiency directly from log-likelihood and avoid distance-based density estimation. We give a finite sample boundary, indicating that the estimation error depends on the intrinsic dimension of the data manifold rather than the original embedding dimension. On 11 datasets and 8 embedders, FLARE reached Spearman's $ρ$ of 0.90 under the supervised benchmark and remained stable in high-dimensional embeddings ($d \geq 3{,}584$) as the existing labelless baseline collapsed.

We study mean estimation for a Gaussian distribution with identity covariance in $\mathbb{R}^d$ under a missing data scheme termed realizable $ε$-contamination model. In this model an adversary can choose a function $r(x)$ between 0 and $ε$ and each sample $x$ goes missing with probability $r(x)$. Recent work Ma et al., 2024 proposed this model as an intermediate-strength setting between Missing Completely At Random (MCAR) -- where missingness is independent of the data -- and Missing Not At Random (MNAR) -- where missingness may depend arbitrarily on the sample values and can lead to non-identifiability issues. That work established information-theoretic upper and lower bounds for mean estimation in the realizable contamination model. Their proposed estimators incur runtime exponential in the dimension, leaving open the possibility of computationally efficient algorithms in high dimensions. In this work, we establish an information-computation gap in the Statistical Query model (and, as a corollary, for Low-Degree Polynomials and PTF tests), showing that algorithms must either use substantially more samples than information-theoretically necessary or incur exponential runtime. We complement our SQ lower bound with an algorithm whose sample-time tradeoff nearly matches our lower bound. Together, these results qualitatively characterize the complexity of Gaussian mean estimation under $ε$-realizable contamination.

Evidential Deep Learning (EDL) is a popular framework for uncertainty-aware classification that models predictive uncertainty via Dirichlet distributions parameterized by neural networks. Despite its popularity, its theoretical foundations and behavior under distributional shift remain poorly understood. In this work, we provide a principled statistical interpretation by proving that EDL training corresponds to amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood. This perspective reveals a major drawback: standard EDL conflates epistemic and aleatoric uncertainty, leading to systematic overconfidence on out-of-distribution (OOD) inputs. To address this, we introduce Density-Informed Pseudo-count EDL (DIP-EDL), a new parametrization that decouples class prediction from the magnitude of uncertainty by separately estimating the conditional label distribution and the marginal covariate density. This separation preserves evidence in high-density regions while shrinking predictions toward a uniform prior for OOD data. Theoretically, we prove that DIP-EDL achieves asymptotic concentration. Empirically, we show that our method enhances interpretability and improves robustness and uncertainty calibration under distributional shift.

Clustering is a cornerstone of modern data analysis. Detecting clusters in exploratory data analyses (EDA) requires algorithms that make few assumptions about the data. Density-based clustering algorithms are particularly well-suited for EDA because they describe high-density regions, assuming only that a density exists. Applying density-based clustering algorithms in practice, however, requires selecting appropriate hyperparameters, which is difficult without prior knowledge of the data distribution. For example, DBSCAN requires selecting a density threshold, and HDBSCAN* relies on a minimum cluster size parameter. In this work, we propose Persistent Leaves Spatial Clustering for Applications with Noise (PLSCAN), a multiscale density-based clustering algorithm that replaces HDBSCAN*'s fixed minimum cluster size pruning of a mutual-reachability linkage hierarchy with a persistence-based cluster selection procedure. Effectively, PLSCAN identifies all minimum cluster sizes for which HDBSCAN* produces stable (leaf) clusters. In concept, PLSCAN applies scale-space clustering principles and is equivalent to persistent homology on a novel metric space. We compare its performance to HDBSCAN* on several real-world datasets, demonstrating that it achieves a higher median ARI, is less sensitive to changes in the number of mutual reachability neighbours, and has higher stability under resampling. Additionally, we compare PLSCAN's computational costs to $k$-Means++, demonstrating competitive run-times on low-dimensional datasets. At higher dimensions, run times scale more similarly to HDBSCAN*.

We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints. We obtain three sets of results. First, for measures with densities bounded away from zero on a bounded Lipschitz domain $Ω$, we prove that any approximation scheme for functions in $\mathrm{L}_p(Ω)$ transfers, with linear rate, to a corresponding approximation scheme for measures in $\mathrm{W}_p(Ω)$. The argument applies a theorem of Bogovskii on regularity of solutions to the continuity equation in the Benamou-Brenier formulation of optimal transport. We exhibit concrete approximation schemes (polynomials, shift-invariant spaces, cardinal interpolation with radial basis functions, kernel density estimators, and piecewise approximations on nonuniform Voronoi partitions) that fit the framework. As a matter of independent interest, we prove a negative Sobolev lower bound that generalizes existing bounds from $p=2$ to all $p\in(1,\infty)$. We also consider deterministic bounds for discrete approximations to arbitrary measures in terms of the mesh norm of a quasi-uniform set of points. We specialize these bounds to show that compactly supported measures admit a deterministic $N$-term approximation $μ_N$ such that $\mathrm{W}_p(μ,μ_N) = O(N^{-\frac{1}{d}})$ for all $d\geq 1$, which matches the asymptotic optimal quantizer rate. We also extend these results to non-compactly supported measures with appropriate tail decay.