Kisung You

Kisung You

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.

Kisung You

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.

Kisung You, Yelim Lee, Hae-Jeong Park

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.

Kisung You

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.

Kisung You

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.

Colleen Chan, Kisung You, Sunny Chung et al.

Applications of large language models (LLMs) like ChatGPT have potential to enhance clinical decision support through conversational interfaces. However, challenges of human-algorithmic interaction and clinician trust are poorly understood. GutGPT, a LLM for gastrointestinal (GI) bleeding risk prediction and management guidance, was deployed in clinical simulation scenarios alongside the electronic health record (EHR) with emergency medicine physicians, internal medicine physicians, and medical students to evaluate its effect on physician acceptance and trust in AI clinical decision support systems (AI-CDSS). GutGPT provides risk predictions from a validated machine learning model and evidence-based answers by querying extracted clinical guidelines. Participants were randomized to GutGPT and an interactive dashboard, or the interactive dashboard and a search engine. Surveys and educational assessments taken before and after measured technology acceptance and content mastery. Preliminary results showed mixed effects on acceptance after using GutGPT compared to the dashboard or search engine but appeared to improve content mastery based on simulation performance. Overall, this study demonstrates LLMs like GutGPT could enhance effective AI-CDSS if implemented optimally and paired with interactive interfaces.

Kisung You, Dennis Shung, Mauro Giuffrè

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.

Kisung You

The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport plans used in place of Monge maps when the latter do not exist. This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable. We establish theoretical guarantees, including consistency of barycentric projections, monotone descent and convergence to stationary points, stability with respect to perturbations of the inputs, and resolution consistency as the number of atoms increases. Empirical studies on averaging probability distributions, Bayesian posterior aggregation, image prototypes and classification, and large-scale clustering demonstrate accuracy and scalability of the proposed particle-flow approach, positioning it as a principled alternative to both linear programming and regularized solvers.

Byeongsu Yu, Kisung You

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection $L$ which preserves the persistent diagram of a point cloud $\mathbb{X}$ via simulated annealing. The projection $L$ induces a set of canonical simplicial maps from the Rips (or Čech) filtration of $\mathbb{X}$ to that of $L\mathbb{X}$. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism $μ_{\operatorname{quasi-iso}}$ or strong homotopy equivalence $μ_{\operatorname{equiv}}$. These $μ_{\operatorname{quasi-iso}}$ and $μ_{\operatorname{equiv}}$ measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.

Kisung You

Discovering patterns of the complex high-dimensional data is a long-standing problem. Dimension Reduction (DR) and Intrinsic Dimension Estimation (IDE) are two fundamental thematic programs that facilitate geometric understanding of the data. We present Rdimtools - an R package that supports 133 DR and 17 IDE algorithms whose extent makes multifaceted scrutiny of the data in one place easier. Rdimtools is distributed under the MIT license and is accessible from CRAN, GitHub, and its package website, all of which deliver instruction for installation, self-contained examples, and API documentation.

Dianbin Bao, Kisung You, Lizhen Lin

Distance plays a fundamental role in measuring similarity between objects. Various visualization techniques and learning tasks in statistics and machine learning such as shape matching, classification, dimension reduction and clustering often rely on some distance or similarity measure. It is of tremendous importance to have a distance that can incorporate the underlying structure of the object. In this paper, we focus on proposing such a distance between network objects. Our key insight is to define a distance based on the long term diffusion behavior of the whole network. We first introduce a dynamic system on graphs called Laplacian flow. Based on this Laplacian flow, a new version of diffusion distance between networks is proposed. We will demonstrate the utility of the distance and its advantage over various existing distances through explicit examples. The distance is also applied to subsequent learning tasks such as clustering network objects.