Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
Extends tangent functor to microformal morphisms, creating non-linear pullbacks for forms and cohomology.
problem Generalizing smooth maps to microformal morphisms for new types of mappings.
method Introduces microformal morphisms and shows how they act on functions and forms via non-linear pullbacks.
result Non-linear pullbacks of forms respect de Rham differentials and induce transformations of cohomology.
Defines simplicity of non-linear mappings using information geometry.
problem Finding a simple yet effective non-linear mapping from latent to observation space.
method Formalizes simplicity through information geometry, independent of empirical data.
result Proves basic properties of the defined simplicity measure.
New complex-valued maps found on complex geometries.
problem Finding new maps on complex geometries.
method Solving non-linear PDEs based on manifold geometry.
result Constructed new proper biharmonic and (2,1)-harmonic maps.
The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.
problem Equivariant neural networks on homogeneous spaces.
method Deriving generalized steerability constraints for non-linear equivariant layers.
result The universality of the derived construction for non-linear equivariant layers.
New harmonic self-maps constructed for specific Lie groups.
problem Harmonic self-maps of compact cohomogeneity one manifolds.
method Developed theory of equivariant harmonic self-maps; constructed new maps by solving non-linear boundary value problems.
result Harmonic self-maps of specific Lie groups with novel degrees.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Diffusion Maps improves on Functional PCA for non-linear functional data.
problem Functional PCA's linear manifold assumption fails for non-linear functional data.
method Extends Diffusion Maps to functional data and compares it to Functional PCA.
result Diffusion Maps outperforms Functional PCA in non-linear functional data analysis.
The performance of the Self-Organizing Map (SOM) algorithm is dependent on the initial weights of the map. The different initialization methods can broadly be classified into random and data analysis based initialization approach. In this paper, the performance of random initialization (RI) approach is compared to that…
New algorithm solves complex medical radiation therapy problems.
problem Optimizing radiation therapy treatment plans.
method Majorization-minimization principle applied to non-linear split feasibility problems.
result Euclidean norm in proximity function replaced by Bregman divergences.
Introduces MFVDM for high-dimensional data analysis.
problem Non-linear dimensionality reduction of high-dimensional datasets.
method Combines multiple unitary irreducible representations for nonlinear embeddings.
result Achieves better nearest neighbor search and alignment estimation on noisy data.
Modeling dynamical systems is important in many disciplines, e.g., control, robotics, or neurotechnology. Commonly the state of these systems is not directly observed, but only available through noisy and potentially high-dimensional observations. In these cases, system identification, i.e., finding the measurement map…
Develops an efficient method for real-time data analysis and visualization.
problem Challenges of analyzing high-dimensional data.
method Incremental non-linear manifold approximation using GMRA framework.
result Accurately represents non-linear manifolds with small initial samples.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal ℏ-differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms. result Developed constructions and examples of differential operators over maps.
KPCA improves OoD detection by separating InD and OoD data.
problem Insufficiency of PCA in detecting OoD data from InD data.
method Kernel PCA (KPCA) with task-specific kernels.
result KPCA achieves superior OoD detection performance.
Improved bounds for non-linear SA with fast convergence.
problem Stochastic approximation with non-linear mappings and multiple time scales.
method Mean squared error bounds with O(1/k) rate for contractive mappings. result First O(1/k) rate for non-linear two-time-scale SA without additional smoothness assumptions. We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from…
Enhances Cox model for survival analysis with symbolic non-linear log-risk functions.
problem Limited interpretability and non-linearity in traditional Cox models.
method Introduces GCPH model using Kolmogorov-Arnold Networks for symbolic non-linear log-risk functions.
result GCPH achieves competitive performance and superior interpretability.
Paper introduces non-linearity signature to measure deep neural network performance.
problem Difficulty in explaining performance differences among similar DNN architectures.
method Affine Optimal Transport mappings to measure non-linearity.
result Signature provides better understanding of DNN inner workings.
Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci fl…
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions …
In the two parts of this paper we solve a problem of De Rham, proving that Reidemeister torsion invariants determine topological equivalence of linear G-representations, for G a finite cyclic group. Methods in controlled K-theory and surgery theory are developed to establish, and effectively calculate, a necessary and …
This paper improves cross-modal learning to rank by using self-paced learning with non-linear mapping functions.
problem Challenges in learning cross-modal similarity, especially with linear mapping functions and equal importance assumption.
method Incorporates self-paced learning theory with diversity into cross-modal learning to rank, using non-linear mapping functions.
result Significant improvements over state-of-the-art methods in cross-modal retrieval tasks.
Paper studies solutions to a specific equation in conformal geometry with singular sets.
problem Singular solutions to a fully non-linear equation in conformal geometry.
method Uses a classical gluing method adapted to the fully non-linear setting.
result Shows the classical gluing method can be applied to the σ2--Yamabe equation. Diffusion maps sped up with Nyström method for big data.
problem High computational complexity of diffusion maps for long time-series data.
method Integrate Nyström method with diffusion maps to reduce computational demand.
result Achieved a speedup of roughly two to four times for dominant diffusion map components.
This work extends identifiability analysis to sequential latent variable models, focusing on Switching Dynamical Systems.
problem Identifying latent variables in sequential data models.
method Proved identifiability of Markov Switching Models and established conditions for Switching Dynamical Systems.
result Identifiability of latent variables and non-linear mappings in Switching Dynamical Systems up to affine transformations.
LCC algorithm maps instances to a central space for better classification.
problem Improving classification accuracy for various datasets.
method Formulated as a quadratic program, simplified to a linear program, uses kernel functions for non-linear cases.
result LCC outperforms other methods in accuracy on standard datasets.
Develops goodness-of-fit tests for noisy submanifold samples.
problem Testing non-linear models on noisy submanifold samples.
method Non-linear least-square problem solution and χ2 distribution application. result Residual follows χ2 distribution with parameters related to model order and dimension. Gaussian processes emulate complex non-linear models efficiently.
problem Efficiently simulate and analyze highly non-linear, time-evolving systems.
method Gaussian process emulators to approximate model output, considering input uncertainty and time series correlation.
result High predictive performance and uncertainty measures for Lorenz and Van der Pol equations.
We discuss in rather general terms quantum field theories dealing with spaces of maps between Riemannian manifolds. In particular we explore the well--known connection between the renormalization group flow for non--linear sigma models and the Ricci flow.
Method orders Pareto solutions using transformed objective scores.
problem Lack of orderability in multi-objective optimization solutions.
method Probability integral transform to map objectives to scores.
result Pareto efficient solutions can be ordered in score space.
This research evaluates learning models for bionic robots, focusing on transfer function identification.
problem Developers need guidance on selecting and constructing transfer functions for bionic robots.
method Comprehensive evaluation strategy including data collection, learning model selection, comparative analysis, and transfer function identification.
result A framework for effectively dealing with multi-input multi-output robotic data.
Gaussian processes improve system identification models.
problem Improving system identification models for non-linear dynamics.
method Using Gaussian processes to create time series prediction models.
result Gaussian processes enhance model accuracy in system identification.
Study of H1 geometry of symplectomorphisms on manifolds.
problem Understanding the geometry of symplectomorphism groups in Sobolev spaces.
method Analysis of H1 metric properties and geodesics. result Existence of globally defined geodesics and non-linear Fredholm map properties.
Unified method for CNNs to approximate equivariant maps across various groups.
problem Limited universal approximation theorems for CNNs with specific groups and settings.
method Unified approach to derive universal approximation theorems for equivariant maps by CNNs in diverse settings.
result Ability to handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.
In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of Dθ(M) in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic correspon…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line of work and present low distortion embeddings for dot product kernels into linear…
Adversarial robust models have more interpretable saliency maps.
problem Understanding the relationship between adversarial robustness and saliency map interpretability.
method Quantifying the alignment between input images and saliency maps, hypothesizing and testing the relationship with linear and neural network models.
result The alignment between input images and saliency maps increases as the distance to the decision boundary grows, especially in linear models.
Kernel principal component analysis (KPCA) provides a concise set of basis vectors which capture non-linear structures within large data sets, and is a central tool in data analysis and learning. To allow for non-linear relations, typically a full n×n kernel matrix is constructed over n data points, but this…
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
This paper examines how energy in feature maps decays in deep neural networks.
problem Understanding energy decay in deep convolutional neural networks.
method Analyzes energy conservation and decay rates in various deep neural network architectures.
result Energy in feature maps decays polynomially or exponentially across layers.
We introduce a toy probabilistic model to analyze job-matching processes in recent Japanese labor markets for university graduates by means of statistical physics. We show that the aggregation probability of each company is rewritten by means of non-linear map under several conditions. Mathematical treatment of the map…
Proposes a new model for non-linear regression of multivariate time series data.
problem Regression models for non-scalar variables, especially time series, have limitations.
method Develops a non-linear function-on-function model using neural networks.
result Demonstrates effectiveness through real-world applications.
We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of Sn whose inclusion is non harmonic and indexed-biha…
Paper investigates Lipschitz constants of self-attention modules in neural networks.
problem Lipschitz constants of self-attention modules in neural networks.
method Proved standard dot-product self-attention is not Lipschitz for unbounded input domain. Proposed L2 self-attention that is Lipschitz. Derived upper bound on L2 self-attention's Lipschitz constant.
result Proved standard self-attention is not Lipschitz for unbounded input domain and proposed an alternative L2 self-attention that is Lipschitz.
Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not utilize the capacity of the projected feature space effectively. To address this chal…