MC-MCL improves MCL for nonlinear clustering.
problem Nonlinear clustering in data science.
method MC-MCL combines MCL with Minimum Curvilinearity for nonlinear distances.
result MC-MCL outperforms classical MCL and baseline clustering algorithms in nonlinear datasets.
Method constructs orthogonal curvilinear coordinates in constant curvature spaces.
problem Creating orthogonal coordinates in spaces of constant curvature.
method Modification of Krichever's method for Euclidean space, applied to constant curvature spaces.
result Examples of orthogonal coordinate systems on the sphere and hyperbolic plane constructed.
We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.
Formula for Laplacian determinants on polygonal domains with slits.
problem Determining the ζ-regularized determinant of the Laplacian on polygonal domains with slits. method Patchwork method for heat trace asymptotics, comparison formula for smooth conformal metrics.
result Polyakov-Alvarez type formula for Laplacian determinants on polygonal domains with slits.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
The monster tower's spaces are stratified naturally.
problem Describing the monster tower's spaces.
method Natural stratification of parameter spaces.
result A natural stratification of the monster tower's spaces.
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 unsupervised method for dimensionality reduction via regression in hyperspectral imagery.
problem Collinearity and ill-determination problems in high-dimensional spectral data.
method DRR (Dimensionality Reduction via Regression) using multivariate regression.
result DRR outperforms linear PCA and other nonlinear methods in reducing dimensionality and improving classification accuracy.
We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
This note is the updated outline of the article "Interpolational properties of planar spiral curves", Fund. and Applied Math., 2001, Vol.7, N.2, 441-463, published in Russian. The main result establishes boundary regions for spiral and piecewise spiral splines, matching given data. The width of such region can serve as…
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface Π is considered. A problem of determination of such surfaces when both principal curvatures are given over a line in Π is formulated and studied. The geometric problem is reduced to th…
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
This paper extends Mirror Descent to Riemannian manifolds for optimization.
problem Optimization on Riemannian manifolds.
method Developed a Riemannian Mirror Descent (RMD) framework and a stochastic variant.
result Established non-asymptotic convergence guarantees for RMD and stochastic RMD.
Study on 3-metrics and their evolutions in conformally flat spaces.
problem Understanding the space of conformally flat 3-metrics with the Guichard condition.
method Evolution of orthogonal 2-metrics and determination of conformally flat 3-metrics.
result One-parameter family of conformally flat 3-metrics derived from 2-metrics.
Study differential operators and their solutions on manifolds, proving upper bounds and curvature.
problem Understanding the dimension of solution spaces for differential equations on manifolds.
method Analyzing ordinary and calibrated differential operators, constructing vector bundles and connections.
result Upper bounds and curvature obstructions for solution spaces, proving concentration theorems.
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
Paper defines untangling number to measure entanglement complexity in 3-periodic networks.
problem Measuring the complexity of entanglement in 3-periodic networks.
method Defining ground states through knot-theoretic crossing diagrams and measuring untangling number.
result Introduced untangling number as a measure of entanglement complexity.
While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…
Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control prob…
Study on conical singularities in 2D surfaces, deriving Polyakov formulas.
problem Analyzing zeta-regularized determinants in surfaces with conical singularities.
method Demonstrated variational and integrated Polyakov formulas for conical singularities, circular sectors, and cones.
result Explicit formulas for the determinant of conical sectors and cones derived.
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between the continuous differential for…
Unified theory for curved shell deformations with elastic and inelastic components.
problem Coupled nonlinear elastic and inelastic deformations of curved thin shells.
method Multiplicative decomposition of surface deformation gradient, detailed kinematics analysis, surface balance laws, constitutive relations derived from thermodynamics.
result Unified constitutive relations for growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity of shells.
The paper develops algorithms for solving complex optimization problems over Riemannian manifolds.
problem Nonconvex and nonsmooth multi-block optimization over Riemannian manifolds with coupled constraints.
method Develops an ADMM-like primal-dual approach with decoupled solvable subroutines.
result The algorithms achieve an iteration complexity of O(1/ε^2) to reach an ε-stationary solution.
Total variation and mean curvature flows on a Lie group quotient enhance and denoise crossing structures.
problem Preserving crossing curvilinear structures in image enhancement and denoising.
method Lifting images to the homogeneous space M=RdtimesSd−1, applying PDEs for TVF and MCF, and using locally optimal differential frames. result Better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings compared to data-driven diffusions.
The paper studies webs formed by rational curves on moduli spaces and their abelian relations.
problem Analyzing the structure and abelian relations of webs formed by rational curves on moduli spaces.
method Recalling classical results, focusing on the 6-web, using abelian 2-forms, and applying Damiano's approach.
result The (n+3)-web W0,n+3 has maximal rank with rational abelian relations for any n≥2. Identifies features most relevant to concept drift in data.
problem Identifying features most relevant to concept drift.
method Distinguishing between drift inducing and faithfully drifting features; deriving minimal subsets of features to characterize drift.
result Derives a detection algorithm for concept drift.
New method evaluates feature interactions using orthogonal variance decomposition.
problem Feature selection fails to account for interactions between features.
method Orthogonal variance decomposition to evaluate feature subsets considering interactions.
result Our method accurately identifies relevant features and improves model accuracy.
Paper predicts EEG features from acoustic features using RNN and GAN.
problem Predicting EEG features from acoustic features.
method Recurrent Neural Network (RNN) and Generative Adversarial Network (GAN).
result Lower RMSE and normalized RMSE values compared to generating acoustic features from EEG features.
Introduces RFI for assessing feature importance relative to any subset of features.
problem Lack of nuanced feature importance computation.
method Generalizes PFI and CFI to assess relative feature importance.
result Derives general interpretation rules for RFI.
A single pre-trained agent guides feature selection using knockoffs.
problem Feature selection challenges in AI-readiness of data.
method Generates knockoff features and uses reinforcement learning.
result Optimal feature subset identified with reduced dependency on target variable.
Feature networks link ML features via graph structure for enhanced learning.
problem Enhancing feature expressiveness and learning efficiency in machine learning.
method Graph representation of feature vectors, leveraging Fourier and functional analysis.
result Feature networks enable novel, complex feature dependencies.
Approach for selecting features by discarding nuisance and correlated ones.
problem Large datasets with correlated and nuisance features.
method Laplacian score criterion, autoencoder architecture, concrete layer.
result Outperforms similar approaches in clustering performance.
Improved text classification using human-understandable features.
problem Text classification accuracy with traditional methods.
method Solicited human-comprehensible features from a teacher.
result Models with human-comprehensible features are competitive with traditional methods.
New stability measures for similar features improve feature selection accuracy.
problem Existing stability measures fail to distinguish similar features in highly correlated datasets.
method Introduce new adjusted stability measures that consider feature similarities.
result One new stability measure considers highly similar features as interchangeable.
Counterexamples show HSIC feature selection misses critical features.
problem Feature selection using HSIC misses important features.
method Feature selection via HSIC maximization.
result HSIC feature selection can miss critical features.
This paper shows feature importance remains valid even in low-performing models.
problem Feature importance validity in low-performing machine learning models for biomedical data.
method Experiments with synthetic and real biomedical datasets to compare feature rank stability under different data reductions.
result Feature importance can be maintained even at low performance levels if data size is adequate.
Proposes finding missing features in Lasso solutions.
problem Lasso overlooks features not selected in its optimal solution.
method Computes alternate features efficiently without redundant computations.
result Reasonable alternate features found in 20 newsgroup data.
A novel online feature selection method using DPP for diversity.
problem Online feature selection for diverse feature sets.
method DPP-based framework with three stages: sampling, local criteria, and global criteria.
result Demonstrated better compactness and comparable/outsuperior performance.
New algorithms select and rank features from MTS without feature extraction.
problem Feature extraction step for MTS classification.
method Directly computes similarity between time series and assesses cluster structure matching labels.
result Techniques match labels well without feature extraction.
Pipeline learns topological features for protein stability prediction.
problem Predicting protein stability using topological features.
method Data-driven method to learn topological features, comparing with expert features.
result Topological features achieve 92%-99% of SME-based models' performance.
A new learning method for evolving features in streaming data.
problem Learning with data streams where features can change over time.
method Develops a learning paradigm for feature evolvable streaming data, combining predictions from old and new features.
result Improves performance on new features by leveraging recovered old features.
FeAT improves OOD generalization by learning richer features.
problem Improving feature learning for out-of-distribution (OOD) generalization.
method Feature Augmented Training (FeAT) iteratively augments and retains features from different subsets of training data.
result FeAT effectively learns richer features, boosting OOD performance.
The paper defines and analyzes feature complexity in DNNs, proposing metrics for feature disentanglement and evaluation.
problem Understanding and quantifying the complexity of features learned by deep neural networks.
method Proposes a definition and disentanglement of feature complexity orders, introduces metrics for reliability and over-fitting evaluation.
result Establishes a relationship between feature complexity and DNN performance, and proposes a generic mathematical tool for network compression and knowledge distillation.
Conventional mutual information (MI) based feature selection (FS) methods are unable to handle heterogeneous feature subset selection properly because of data format differences or estimation methods of MI between feature subset and class label. A way to solve this problem is feature transformation (FT). In this study,…
A new feature selection method using structural correlation between samples.
problem Feature selection in high-dimensional data overlooks structural correlation information.
method Converts features into graph representations, uses fused lasso for feature selection.
result Demonstrates effectiveness of the proposed approach through experiments.
A new method for measuring conditional feature importance using generative models.
problem Challenges in evaluating feature importance given other feature values.
method Adversarial Random Forest (ARF) for generating on-manifold data points.
result cARFi method yields robust importance scores adaptable for various feature importance notions.