Kernel Density Estimation

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.

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.

As deep learning models are increasingly deployed in high-stakes applications, providing well-calibrated uncertainty estimates has become as critical as achieving high predictive accuracy. While Kernel Density Estimation (KDE) has emerged as a smooth and continuous alternative to traditional binning for quantifying miscalibration, its reliability is heavily dependent on the choice of the kernel bandwidth. Standard selection techniques, such as Maximum Likelihood Estimation (MLE), often fail to produce optimal bandwidths for calibration tasks. In this work, we introduce Risk Alignment (RA), a novel optimization framework that determines the optimal bandwidth by aligning KDE-reconstructed risk with empirical risk. We theoretically demonstrate that this alignment minimizes calibration estimation bias across the data distribution, establishing a principled bandwidth selection criterion applicable to various metrics, including the challenging case of canonical calibration error. Extensive experiments across multiple architectures and datasets show that RA consistently outperforms standard bandwidth selection methods, yielding more reliable calibration assessments.

Real-world time-series data in industrial sensing, healthcare, and energy systems is often corrupted by a mixture of Gaussian noise and occasional large-magnitude impulse outliers. For tasks that depend on local shape, such as ECG morphology analysis and battery degradation monitoring, the main requirement is not only low reconstruction error but also preservation of derivative peaks and task-critical features. We propose Cascade-KDE, a training-free restoration framework for corrupted time series. The method first estimates a two-dimensional temporal-amplitude density, then applies a Density-Truncated Robust Expectation to limit the influence of distant abnormal points, and finally refines the sequence through an exponential cascade with adaptive stopping. This design aims to improve robustness under out-of-distribution impulse corruptions while keeping the restored trajectory close to the original local structure. Across several benchmark datasets, the proposed method shows consistent gains over classical filters and representative learning-based baselines on curve fidelity, derivative preservation, downstream classification, and runtime efficiency. These results suggest that bounded density-based restoration is a practical option for feature-preserving preprocessing in noisy time-series pipelines.

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.

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.

Spatial computation in geographic systems increasingly requires query-conditioned, local, interpretable aggregation under metric constraints. Many classical approaches rely on global summation and treat approximation as an implementation concern, limiting interpretability and scalability at large scales. We propose the Adaptive Density Field (ADF), a geometric attention framework that formulates spatial aggregation as a query-conditioned, metric-induced attention operator in continuous space. Given a set of labelled spatial points with associated scalar scores, ADF defines a continuous intensity field over space. For a given query location, the field value is obtained via a local adaptive Gaussian kernel mixture centered on the query's nearest neighbors, where kernel bandwidths are modulated by point-specific scores to evaluate local aggregated influence. Additionally, approximate nearest-neighbor search is introduced, enabling scalable execution while preserving locality. The proposed ADF bridges concepts from adaptive kernel methods, classical GIS methods, and attention mechanisms by reinterpreting spatial influence as geometry-embedded attention, grounded in physical distance rather than learned latent projections. The proposed framework is formulation-level rather than algorithm-specific, allowing flexible kernel choices, score-to-bandwidth mappings, and approximation parameters. This approach provides a unifying perspective on spatial influence modeling that emphasizes structure, scalability, and geometric interpretability, with relevance to geographic information systems and spatial machine learning.

Approximate Nearest Neighbor (ANN) search and Approximate Kernel Density Estimation (A-KDE) are fundamental problems at the core of modern machine learning, with broad applications in data analysis, information systems, and large-scale decision making. In massive and dynamic data streams, a central challenge is to design compact sketches that preserve essential structural properties of the data while enabling efficient queries. In this work, we develop new sketching algorithms that achieve sublinear space and query time guarantees for both ANN and A-KDE for a dynamic stream of data. For ANN in the streaming model, under natural assumptions, we design a sublinear sketch that requires only $\mathcal{O}(n^{(1-η)(1+ρ)})$ memory by storing only a sublinear ($n^{-η}$) fraction of the total inputs, where $ρ$ is a parameter of the LSH family, and $0<η<1$. Our method supports sublinear query time, batch queries, and extends to the more general Turnstile model. While earlier works have focused on Exact NN, this is the first result on ANN that achieves near-optimal trade-offs between memory size and approximation error. Next, for A-KDE in the Sliding-Window model, we propose a sketch of size $\mathcal{O}\left(LW \cdot \frac{1}{\sqrt{1+ε} - 1} \log^2 N\right)$, where $L$ is the number of sketch rows, $W$ is the LSH range, $N$ is the window size, and $ε$ is the approximation error. This, to the best of our knowledge, is the first theoretical sublinear sketch guarantee for A-KDE in the Sliding-Window model. We complement our theoretical results with experiments on various real-world datasets, which show that the proposed sketches are lightweight and achieve consistently low error in practice.