HOMER: Huber-of-Means for Efficient and Robust Estimation in Hilbert Spaces

Heavy tails weaken high-confidence control for the empirical mean. Geometric median-of-means (MOM) also lacks a threshold that moves toward mean efficiency. We propose \emph{HOMER}, or Huber-of-Means for Efficient and Robust Estimation. HOMER aggregates block means through a radial Huber center. Its canonical and pseudo-Huber forms bound each block score and interpolate between median-like robustness and the empirical mean. We establish a Hilbert-space majority theorem and a MOM-order deviation bound under a finite second moment. Canonical HOMER recovers the sample mean inside its quadratic region. Pseudo-HOMER approaches the sample mean as the threshold grows. It also admits asymptotic linearity and consistent sandwich covariance estimation around the population block-Huber target. Under a finite third moment, fixed finite-dimensional projections support mean inference at the usual parametric rate. This result requires growing block sizes and counts, with block sizes increasing faster. Heavy-tailed simulations show that HOMER remains stable when a minority of block summaries is displaced. On clean Gaussian data, both versions closely approach the empirical mean's efficiency. Finite-block sandwich intervals undercovered, especially for skewed functional data. Further studies show failure when contamination affects most blocks or compromises ordinary within-block means.

Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds

Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two Rényi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicate particles while matching ordinary effective sample size for well-separated particles. We prove monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension for compact manifolds without boundary. On spheres, the unlogged profile decomposes into spherical-harmonic energies that recover mean-direction, von Mises-Fisher-type, and Bingham-type summaries. Sphere-based experiments show that the profile reveals antipodal, girdle, multimodal, and duplicate-particle structure missed by weight-only and first-moment spherical summaries.

Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss. The corresponding minimum value is an integrated conditional Fréchet variance, which vanishes exactly for map-induced couplings and therefore defines a conditional-variance Monge defect. We also study a tangential log-exp projection, prove its Euclidean exactness, its compatibility with Brenier-McCann maps in the Monge case, and its interpretation as the first unit Riemannian gradient update for the intrinsic objective. For discrete couplings, both constructions decompose row-wise into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices support the proposed division of roles: the intrinsic projection is the variational representative, while the tangential projection is a useful local displacement surrogate.

Scale-Calibrated Median-of-Means for Robust Distributed Principal Component Analysis

Distributed principal component analysis (PCA) produces node-level estimates of both a mean vector and a principal subspace. Robustly aggregating these heterogeneous objects requires a relative scale between mean error and subspace error. We study a scale-calibrated median-of-means estimator for this problem using the product geometry of Euclidean space and the Grassmann manifold. A node-level PCA expansion shows that the mean component has the usual linear influence, whereas the subspace component is an eigengap-weighted covariance perturbation. We prove a local reduction showing that the proposed product-manifold median-of-means estimator is asymptotically equivalent to a scaled spatial median of node influence errors. This yields fixed-node non-Gaussian limits, growing-node Gaussian limits with finite-block bias, and an explicit scale-dependent covariance formula. We propose robust block-scale and inference-optimal calibration rules, establish high-probability median-of-means bounds, characterize factorwise bad-node influence, and prove node-bootstrap validity. Simulations and large-scale single-cell RNA-seq data show that scale calibration adapts to eigengap-driven subspace uncertainty and provides a robust distributed PCA summary.

Intrinsic effective sample size for manifold-valued Markov chain Monte Carlo via kernel discrepancy

Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise effective sample sizes can change under rotations, chart changes, or alternative embeddings of the same underlying path. We propose an intrinsic effective sample size based on kernel discrepancy. The proposed quantity is the number of independent draws that would yield the same expected squared kernel discrepancy between the empirical distribution and the target distribution. This gives an exact finite-sample risk interpretation, an asymptotic integrated-autocorrelation representation, and a coordinate-free diagnostic whenever the kernel respects the geometry of the state space. We establish invariance under transported kernels, operator and principal-direction interpretations, and consistency of a lag-window estimator under boundedness and absolute-regularity conditions. We also discuss valid kernel constructions on manifolds, emphasizing that geodesic Gaussian kernels are not generally positive definite on curved spaces. Sphere experiments illustrate rotation invariance and calibration of the proposed diagnostic against empirical distributional error.

Quotient-Based Posterior Analysis for Euclidean Latent Space Models

Latent space models are widely used in statistical network analysis and are often fit by Markov chain Monte Carlo. However, posterior summaries of latent coordinates are not canonical because the likelihood depends only on pairwise distances and is invariant under rigid motions of the latent space. Standard post hoc alignment can aid visualization, but the resulting summaries depend on an arbitrary reference configuration. We propose a quotient-based posterior analysis for Euclidean latent space models using the centered Gram map, which represents identifiable latent structure while removing nonidentifiability. This yields intrinsic posterior summaries of mean structure and uncertainty that can be computed directly from posterior samples, together with basic theoretical guarantees including canonicality, existence, and stability. Through simulations and analyses of the Florentine marriage network and a statisticians' coauthorship network, the proposed framework clarifies when alignment-based summaries are stable, when they become reference-sensitive, and which nodes or relationships are weakly identified. These results show how coherent posterior analysis can reveal latent relational structure beyond a single embedding.

Constant Metric Scaling in Riemannian Computation

Constant rescaling of a Riemannian metric appears in many computational settings, often through a global scale parameter that is introduced either explicitly or implicitly. Although this operation is elementary, its consequences are not always made clear in practice and may be confused with changes in curvature, manifold structure, or coordinate representation. In this note we provide a short, self-contained account of constant metric scaling on arbitrary Riemannian manifolds. We distinguish between quantities that change under such a scaling, including norms, distances, volume elements, and gradient magnitudes, and geometric objects that remain invariant, such as the Levi--Civita connection, geodesics, exponential and logarithmic maps, and parallel transport. We also discuss implications for Riemannian optimization, where constant metric scaling can often be interpreted as a global rescaling of step sizes rather than a modification of the underlying geometry. The goal of this note is purely expository and is intended to clarify how a global metric scale parameter can be introduced in Riemannian computation without altering the geometric structures on which these methods rely.

Semantics at an Angle: When Cosine Similarity Works Until It Doesn't

Cosine similarity has become a standard metric for comparing embeddings in modern machine learning. Its scale-invariance and alignment with model training objectives have contributed to its widespread adoption. However, recent studies have revealed important limitations, particularly when embedding norms carry meaningful semantic information. This informal article offers a reflective and selective examination of the evolution, strengths, and limitations of cosine similarity. We highlight why it performs well in many settings, where it tends to break down, and how emerging alternatives are beginning to address its blind spots. We hope to offer a mix of conceptual clarity and practical perspective, especially for quantitative scientists who think about embeddings not just as vectors, but as geometric and philosophical objects.

Learning over von Mises-Fisher Distributions via a Wasserstein-like Geometry

We introduce a novel, geometry-aware distance metric for the family of von Mises-Fisher (vMF) distributions, which are fundamental models for directional data on the unit hypersphere. Although the vMF distribution is widely employed in a variety of probabilistic learning tasks involving spherical data, principled tools for comparing vMF distributions remain limited, primarily due to the intractability of normalization constants and the absence of suitable geometric metrics. Motivated by the theory of optimal transport, we propose a Wasserstein-like distance that decomposes the discrepancy between two vMF distributions into two interpretable components: a geodesic term capturing the angular separation between mean directions, and a variance-like term quantifying differences in concentration parameters. The derivation leverages a Gaussian approximation in the high-concentration regime to yield a tractable, closed-form expression that respects the intrinsic spherical geometry. We show that the proposed distance exhibits desirable theoretical properties and induces a latent geometric structure on the space of non-degenerate vMF distributions. As a primary application, we develop the efficient algorithms for vMF mixture reduction, enabling structure-preserving compression of mixture models in high-dimensional settings. Empirical results on synthetic datasets and real-world high-dimensional embeddings, including biomedical sentence representations and deep visual features, demonstrate the effectiveness of the proposed geometry in distinguishing distributions and supporting interpretable inference. This work expands the statistical toolbox for directional data analysis by introducing a tractable, transport-inspired distance tailored to the geometry of the hypersphere.

Scalable Geometric Learning with Correlation-Based Functional Brain Networks

The correlation matrix is a central representation of functional brain networks in neuroimaging. Traditional analyses often treat pairwise interactions independently in a Euclidean setting, overlooking the intrinsic geometry of correlation matrices. While earlier attempts have embraced the quotient geometry of the correlation manifold, they remain limited by computational inefficiency and numerical instability, particularly in high-dimensional contexts. This paper presents a novel geometric framework that employs diffeomorphic transformations to embed correlation matrices into a Euclidean space, preserving salient manifold properties and enabling large-scale analyses. The proposed method integrates with established learning algorithms - regression, dimensionality reduction, and clustering - and extends naturally to population-level inference of brain networks. Simulation studies demonstrate both improved computational speed and enhanced accuracy compared to conventional manifold-based approaches. Moreover, applications in real neuroimaging scenarios illustrate the framework's utility, enhancing behavior score prediction, subject fingerprinting in resting-state fMRI, and hypothesis testing in electroencephalogram data. An open-source MATLAB toolbox is provided to facilitate broader adoption and advance the application of correlation geometry in functional brain network research.