Algorithm learns principal curves from data streams in a sequential manner.
problem Summarizing large data streams using PCA is challenging due to theoretical and algorithmic issues.
method Proposes a novel sequential algorithm for learning principal curves from data streams.
result Supports regret bounds with optimal sublinear remainder terms.
SPCA extracts nonlinear features for feature extraction.
problem Nonlinear feature extraction in data.
method Unsupervised, nonlinear, invertible feature extraction technique.
result Identifies curvilinear features interpretable as nonlinear sensors.
New method explains color vision's nonlinearities and adaptability.
problem Understanding human color vision's nonlinear and adaptive aspects.
method Sequential Principal Curves Analysis (SPCA) with local metric.
result Color discrimination thresholds and illuminant discount emerge from realistic data.
New approach for principal curves on spherical data.
problem Dimension reduction of spherical data.
method Projection of data onto a continuous curve on a sphere.
result Stationary principal curves on a sphere.
A new method improves uncertainty quantification in Bayesian inference.
problem Poor uncertainty quantification in traditional Gibbs posteriors.
method Sequential Gibbs posteriors with a Bernstein-von Mises theorem.
result Sequential Gibbs posteriors provide better frequentist coverage.
This paper analyzes how errors accumulate in PCA's deflation method.
problem Error accumulation in PCA's deflation method.
method Mathematical analysis of inexact Hotelling's deflation method in two scenarios.
result Characterization of error propagation in PCA's deflation method.
Novel prior for orthogonal functions improves functional component estimation.
problem Improving orthogonality in functional principal component analysis.
method Sequential adaptive priors for orthogonal functions using hierarchical conditionally normal distributions.
result Proposed prior leads to nearly orthogonal posterior estimates.
The paper introduces a method for interpretable principal component analysis of high-dimensional time series.
problem Inconsistent and difficult-to-interpret principal component estimates in high-dimensional regimes.
method Localized sparse principal component analysis of spectral density matrices in frequency domain.
result Efficient algorithm for sparse-localized estimates of principal subspaces.
The paper introduces a method to find multiple one-dimensional structures in noisy data.
problem Finding one-dimensional structures in noisy data.
method Developed a new functional allowing for multiple curves and an efficient algorithm to find near minimizers.
result The method can find one-dimensional structures even in noisy data.
Principal Component Analysis (PCA) is the most common nonparametric method for estimating the volatility structure of Gaussian interest rate models. One major difficulty in the estimation of these models is the fact that forward rate curves are not directly observable from the market so that non-trivial observational e…
The paper extends square root velocity framework to curves in homogeneous spaces.
problem Computing metrics and analyzing curves in homogeneous spaces.
method Generalized square root velocity framework to homogeneous spaces, identifying curves with horizontal lifts in G, computing geodesics, and performing quotient operations. result Geodesics and Karcher means can be computed in quotient spaces of curves in homogeneous spaces.
Sequential or online dimensional reduction is of interests due to the explosion of streaming data based applications and the requirement of adaptive statistical modeling, in many emerging fields, such as the modeling of energy end-use profile. Principal Component Analysis (PCA), is the classical way of dimensional redu…
Study explores K-means clustering of variables and its relation to PCA.
problem Exploring the relationship between K-means clustering of variables and PCA.
method Apply PCA to original data and K-means to transposed data, quantify variable contributions to principal components.
result Identifies how variable clusters contribute to principal components identified by PCA.
New algorithm clusters hyperplanes with improved accuracy.
problem Clustering data from a union of hyperplanes.
method Dual Principal Component Pursuit (DPCP) with geometric analysis.
result DPCP can uniquely identify the dominant hyperplane under certain conditions.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
Analyzes dependencies in sequential datasets to improve deep neural architectures.
problem Improving deep recurrent neural architectures by understanding long distance dependencies.
method Detailed analysis of dependency decay curves in various datasets, testing factors affecting decay, generating synthesized datasets.
result Factors influencing dependency decay curves (number of unique symbols, dataset size, interacting symbols, distance between symbols) can inform optimal hyper-parameters.
A new method uses Gram matrix for efficient multivariate functional principal components.
problem Efficiently estimating eigencomponents of multidimensional functional datasets.
method Proposes using inner-product matrix to estimate eigenelements of multivariate and multidimensional functional datasets.
result Established relationship between eigenelements of covariance operator and inner-product matrix.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
A new method for analyzing shapes using FDA techniques.
problem Statistical shape analysis of deformed contours.
method Functional Data Analysis (FDA) with basis expansion and principal component analysis.
result Successfully identifies deformation parameters and captures contour distributions.
PPA models data directions as curves for manifold learning.
problem Nonlinear data structure in manifold learning.
method Principal Polynomial Analysis (PPA) models data directions as curves.
result PPA reduces to simple univariate regressions, making it computationally feasible and robust.
A new method selects PCA components based on residual memory, outperforming existing techniques.
problem Selecting the optimal number of components in PCA for data with long memory effects.
method Sequentially removes components, stopping when maximum memory accounted for.
result Our method outperforms existing techniques in computational efficiency and accuracy.
Algorithm learns optimal contracts for unaware principals.
problem Learning optimal contracts when principal is unaware of agent's utility and action space.
method Sequential contract offers with observed outcomes, using algorithm to approximate optimal contract.
result Algorithm learns optimal contract within epsilon of optimal net profit with bounded samples.
Proposes a deep spectral Q-learning for mobile health data.
problem Personalized treatment assignment for patients with time-varying covariates.
method Integrates PCA with deep Q-learning for mixed frequency data.
result Mean return converges to optimal under estimated optimal policy.
Study identifies and estimates treatment effect heterogeneity within principal stratification subpopulations.
problem Causal inference with intermediate outcomes and treatment effect heterogeneity.
method Proposes a novel doubly cross-fit doubly robust machine learner to efficiently learn conditional principal causal effects under principal ignorability.
result Demonstrates informative patterns of treatment effect heterogeneity within the always-survivor subpopulation in an acute lung injury trial.
We propose to describe the variety of galaxies from SDSS by using only one affine parameter. To this aim, we build the Principal Curve (P-curve) passing through the spine of the data point cloud, considering the eigenspace derived from Principal Component Analysis of morphological, physical and photometric galaxy prope…
In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …
Enhances Random Forest for imbalanced functional data classification.
problem Challenges in classifying imbalanced functional data.
method Functional Random Forest with Adaptive Cost-Sensitive Splitting (FRF-ACS).
result Significantly improves minority class recall and predictive performance.
Unified method visualizes curvature on curves and surfaces.
problem Visualizing curvature on planar curves and surfaces of revolution.
method Tangential angle parameterization.
result Clear, consistent visualizations without arbitrary parameter tuning.
In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…
Two new algorithms unwrap manifolds using density ridges.
problem Unwrapping manifolds using density ridges.
method Kernel density estimation and gradient flow.
result Clear and intuitive unwrapping results comparable to state-of-the-art algorithms.
Missing data reduces signal-to-noise ratio, not sample size, for PCA.
problem Effect of missing data on PCA signal structure learning.
method Analytic and simulation studies of probabilistic PCA with missing data.
result Missing data effectively reduces signal-to-noise ratio, not sample size.
New numerical methods for analyzing shapes of curves.
problem Discretization of Sobolev metrics on curve spaces.
method Developed algorithms for solving geodesic problems.
result Computed Karcher means and performed shape analysis.
Study the geometry of a surface formed by extending a Whitney umbrella.
problem Investigate the geometric properties of a specific surface formed by extending a Whitney umbrella.
method Analyze the intersection with the normal plane, geodesic and normal curvatures, Gaussian and mean curvatures.
result Determine the zeros of curvature functions and deduce geometric relationships.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …
Paper describes principal boundaries of moduli spaces for abelian and quadratic differentials.
problem Understanding the structure of moduli spaces of abelian and quadratic differentials.
method Flat geometric degeneration and smoothing techniques.
result Described the principal boundary for each configuration in terms of twisted differentials.
A principal Higgs bundle (P,φ) over a singular curve X is a pair consisting of a principal bundle P and a morphism φ:X→AdP⊗ΩX1. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve X using the theory of decorated vector bundles. More precisely, given…
Paper extends RPD for better handling multiple modalities and non-convexity.
problem Handling multiple modalities and non-convexity in data clouds.
method Computes RPD in a reproducing kernel Hilbert space using kernel principal component analysis.
result The method outperforms RPD and is comparable to other models on benchmark datasets.
The paper defines and classifies special curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Defined and characterized anti-torqued slant helices and torqued curves through differential equations.
result Characterized and classified anti-torqued slant helices and torqued curves.
Proposes estimators for complex dose-response curves using kernel methods.
problem Estimating complex dose-response curves with continuous treatments, mediators, and covariates.
method Kernel ridge regression with sequential kernel embedding technique.
result Simple estimators for mediated and time-varying dose response curves with nonasymptotic uniform rates.
Lower bound derived for spectral threshold in curved quantum layers.
problem Finding a lower bound for the spectral threshold in curved quantum layers.
method Deriving a lower bound using the lowest eigenvalue of a one-dimensional operator.
result The derived lower bound is optimal for non-negatively curved surfaces.
Paper develops a framework for learning interpretable representations of sequential decision behavior.
problem Obtaining a transparent description of existing behavior.
method Inverse decision modeling framework, formalizing both forward and inverse problems.
result Learning interpretable representations of behavior, including suboptimal actions, biased beliefs, and imperfect knowledge.
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
Study on autoencoder denoising in high dimensions.
problem Denoising data from Gaussian mixtures.
method Two-layer non-linear autoencoder with skip connection in high-dimensional limit.
result Closed-form expressions for denoising mean-squared test error.
This paper introduces a novel monotone curve estimation framework based on convex duality.
problem Estimating smooth, continuous, and monotonic curves in data.
method Convex duality and optimal transport theories.
result Established statistical guarantees for monotone curve estimates.
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this problem under the optimal transport (Wasserstein) geometry, using curves that are re…