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.
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.
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.
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…
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…
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.
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.
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…
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …
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 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.
This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate r…
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.
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 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.
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. Proposes RSP model for efficient big data analysis.
problem Efficiently partitioning big data sets for analysis.
method Random sample partition (RSP) data model and block-level sampling.
result RSP data blocks can estimate statistics and build models equivalent to whole data set.
Data preprocessing improves data quality for robust data mining.
problem Noisy and incomplete data hinders data mining models.
method Overview of data cleaning, transformation, and preprocessing methods.
result Preprocessing significantly affects data mining model performance.
A new method for handling imbalanced big data using ensembles and smart data.
problem Imbalanced data distribution in big data scenarios.
method Smart Data driven Decision Trees Ensemble (SD_DeTE) methodology.
result SD_DeTE outperforms Random Forest in handling imbalanced binary classification problems in big data.
Prevents sensitive data generation in diffusion models using labeled and unlabeled data.
problem Generating sensitive data in diffusion models using unlabeled data.
method Positive-Unlabeled Diffusion Models, approximating ELBO with labeled and unlabeled data.
result Prevents the generation of sensitive data without compromising image quality.
Study reveals Data Shapley's inconsistent performance in data selection tasks.
problem Inconsistency of Data Shapley's performance in data selection across different settings.
method Hypothesis testing framework and identification of utility functions.
result Data Shapley's performance is no better than random selection without specific constraints.
Survey on data collection challenges in machine learning.
problem Data scarcity and need for labeled data in machine learning.
method Comprehensive study of data acquisition, labeling, and improvement techniques.
result Identification of research challenges in data collection.
PRRO generates synthetic tabular data that improves SL performance and class distribution.
problem Low SL utility of synthetic data due to class imbalance and overlooked data relationships.
method Data pruning and column reordering to optimize SL utility.
result Synthetic data generated with PRRO enhances predictive performance and class distribution.
Defines data science as a natural ecosystem with challenges and missions.
problem Challenges and missions in data science due to 5D complexities and data life cycle phases.
method Systemic and data-centric view of data science as a fusion of data universe and its challenges, formalizing a general-purpose architecture.
result Essential data science as a natural ecosystem integrating specific disciplines and high-impact applications.
Synthetic data enhances analytics but requires careful volume management.
problem Accuracy of statistical methods on synthetic data vs. raw data.
method Synthetic Data Generation for Analytics framework using tabular diffusion models.
result Error rate decreases with more synthetic data but may stabilize or increase.
Data science redefines causal inference from observational data, classifying tasks into description, prediction, and counterfactual prediction.
problem Widespread misunderstandings about data science's role in causal inference from observational data.
method Organizing data science tasks into three classes: Description, prediction, and counterfactual prediction (including causal inference).
result The necessity of subject-matter expert knowledge for causal analyses in data science.
This paper evaluates how dirty data affects data mining and machine learning results.
problem Negative impacts of dirty data on data mining and machine learning results.
method Experimental comparison of missing, inconsistent, and conflicting data on classification and clustering algorithms.
result Guidelines for algorithm selection and data cleaning based on experimental findings.
DPASF stream preprocesses Big Data streams efficiently.
problem Efficient preprocessing of streaming Big Data.
method Implemented six preprocessing algorithms in Apache Flink.
result Preprocessing improves data accuracy in streaming Big Data.
This paper introduces C-DSL to improve data mining outcomes by considering context.
problem Data collection ambiguities, data imbalance, hidden biases, lack of domain info, and data incompleteness.
method Developed Context-Driven Data Science Lifecycle (C-DSL) to address data quality issues.
result Tangible improvements to data mining outcomes were achieved through C-DSL.
Proposes using probabilistic models for privacy-preserving synthetic data.
problem Designing high-quality synthetic data for privacy preservation.
method Formulate the problem through probabilistic modelling, choosing a model for the data.
result Statistical discoveries can be reliably reproduced from synthetic data.
Unlabeled data helps stop active learning better than labeled data.
problem Reducing the need for manual annotation in text classification.
method Compared stopping methods based on labeled, unlabeled, and training data.
result Stopping methods using unlabeled data are more effective.
New test ensures quality of shared data in machine learning.
problem Ensuring quality of external data in machine learning tasks.
method Distribution-free two-sample testing procedures grounded in conformal outlier detection.
result Identifies valuable external data agents for model personalization.
Paper creates fair synthetic data ensuring equal predictions across sensitive attributes.
problem Ensuring fair predictions across sensitive attributes in synthetic data.
method Equalizing target probability distributions across sensitive attributes in synthetic data generation.
result Synthetic data provides strong fair predictions, equal across all thresholds.
A new method classifies multiple correlated data streams simultaneously.
problem Classifying multiple correlated data streams in practical scenarios.
method Double-Coupling Support Vector Machines (DC-SVM) considers both internal and external correlations.
result The proposed method outperforms traditional methods on artificial and real-world data streams.
This paper improves neural machine translation training by selecting and denoising data.
problem Reduces negative impact of noisy data on neural machine translation training.
method Measures and selects domain data, applies denoising curriculum using online data selection.
result Significant effectiveness for training on noisy data.
DPA preserves data distribution in reduced dimensions.
problem Loss of data distribution in dimension reduction.
method DPA combines encoder and decoder to match data distribution.
result DPA successfully reconstructs data distribution.
Framework captures missing data in sparse data sets.
problem Capturing missing data in extremely sparse data sets.
method Coupled compound Poisson factorization with stochastic variational inference.
result Explicitly modeling missing data improves results in clustering, prediction, and matrix factorization.