The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
Paper shows certain algebra types are not differentially smooth.
problem Characterizing smoothness in double extension regular algebras.
method Analyzing algebra type (14641) for differential smoothness.
result Double extension regular algebras of type (14641) are not differentially smooth.
Introduces non-regular spacetime geometry without smooth calculus.
problem Defining gravity without smooth spacetime geometry.
method Discusses non-regular spacetime geometry and curvature without differential calculus.
result Curvature and gravity can be defined without smooth spacetime calculus.
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G-SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Paper develops a new probabilistic method for American options using entropy regularization.
problem Finding optimal stopping times for American options with entropy regularization.
method Entropy-regularized penalization scheme based on Doob-Meyer-Mertens decomposition and reflected backward stochastic differential equations.
result Explicit convergence rates and policy improvement algorithm for American options.
Paper introduces a differentiable regularizer for condition number to improve neural network stability.
problem Maintaining numerical stability in neural networks to ensure reliable and performant models.
method Introduces a novel differentiable regularizer for the condition number of weight matrices.
result Derives a differentiable formula for the gradient of the regularizer, promoting matrices with low condition numbers.
New regularizer for machine learning using private data.
problem Machine learning with private data.
method Distributionally-robust optimization with locally-differentially-private datasets.
result New regularizer for training linear regression models.
Generalized meshes for non-regular geometries, including fractures.
problem Discretization of partial differential equations in non-regular geometries.
method Introduces generalized meshes with overlapping elements and flexible adjacency relations.
result Discrete differential forms on virtually inflated meshes characterize the trace space of forms in surrounding volumes.
Differentiable PF via entropy-regularized OT for better inference.
problem Non-differentiability of traditional PF resampling methods.
method Entropy-regularized optimal transport for differentiable resampling.
result Convergent differentiable PF method with improved gradient estimates.
The paper improves ALO for ℓ1-regularized models.
problem Estimating out-of-sample error for ℓ1-regularized models. method Developed a novel theory for ℓ1-regularized problems, bounding ALO error. result For ℓ1-regularized problems, ALO error goes to zero as p goes to infinity. Study evolutes of curves with varying smoothness.
problem Understanding evolutes of curves with low smoothness.
method Analyzing the relationship between curve smoothness and evolute regularity.
result Evolutes have one less order of smoothness than the parent curve in generic cases.
In this paper we study the general affine differential geometry of surfaces in affine space A3. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…
The paper improves model robustness by regularizing posterior differences.
problem Improving model robustness in noisy input scenarios.
method Posterior differential regularization with f-divergence. result Regularizing with f-divergence improves model robustness. Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We …
Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they…
In seeking for sparse and efficient neural network models, many previous works investigated on enforcing L1 or L0 regularizers to encourage weight sparsity during training. The L0 regularizer measures the parameter sparsity directly and is invariant to the scaling of parameter values, but it cannot provide useful gradi…
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
NAPP-ERM improves ERM with differential privacy guarantees by iteratively achieving target regularization and delivering strong convexity.
problem Over-regularization in privacy-preserving ERM approaches.
method Noise-Augmented Privacy-Preserving Empirical Risk Minimization (NAPP-ERM) with a dual-purpose l2 regularizer and privacy budget retrieval strategy.
result Mitigates over-regularization and achieves strong convexity through a single regularizer.
Determinants remain constant along specific families of differential operators.
problem Local constancy of regularized determinants for differential operators.
method Analyzing families of operators Dτ=[δτ,d∇], showing flat-regularized determinant's constancy. result The flat-regularized determinant is constant in τ when restricted to im(δτ) under suitable assumptions. The paper proposes a gradient-based method for multi-penalty Ridge regression.
problem Optimizing multiple regularization hyperparameters for linear regression.
method Gradient-based optimization through matrix differential calculus.
result The method outperforms traditional regularization techniques like LASSO and Ridge.
Continuum Dropout improves neural differential equations by preventing overfitting.
problem Overfitting in Neural Differential Equations (NDEs).
method Introduces Continuum Dropout, a regularization technique based on alternating renewal processes.
result Continuum Dropout outperforms existing methods in various tasks, improving generalization and uncertainty quantification.
This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
problem Optimizing trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
method Relative entropy-regularized robust optimal control problem, modeled as a stochastic differential game.
result Analytical expressions for optimal strategy and trajectory are derived under specific assumptions.
SmoothDARTS stabilizes DARTS-based architecture search by smoothing loss landscapes.
problem DARTS-based NAS methods suffer from instability, leading to deteriorating architectures.
method SmoothDARTS (SDARTS) uses perturbation-based regularization to smooth the loss landscape.
result SmoothDARTS improves the generalizability and performance of DARTS-based methods.
Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this …
We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on Rn and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential fo…
New divergences help audit DP in high dimensions.
problem Challenges in auditing DP in high-dimensional data.
method Propose kernel Rényi divergence and its regularized version for auditing.
result Regularized kernel Rényi divergence can be estimated from samples in high dimensions.
Injectivity of geodesic X-ray transform on low-regularity manifolds.
problem Injectivity of geodesic X-ray transform on manifolds with low regularity.
method Calculus of differential and curvature operators on non-smooth structures.
result Injectivity of geodesic X-ray transform on simple Riemannian manifolds with C1,1-regularity. We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize and become real analytic in space and time. The regularity result is obtained by i…
Develops geometric integration for rough differential forms.
problem Integrating rough differential forms with low regularity.
method Uses rough path theory to construct geometric integration.
result Constructs geometric integration for rough differential forms.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level …
Method estimates parameters of complex nonlinear systems.
problem Parameter estimation for nonlinear systems with derivative states.
method Regularized linear regression using differentiation filtering and least squares.
result Finite-sample bound on mean absolute error of estimation.
We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…
This work is focused on the solvability of initial-boundary value problems for degenerate parabolic partial differential equations that arise in the pricing of Asian options, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability for such …
In this paper, we present a differential privacy version of convex and nonconvex sparse classification approach. Based on alternating direction method of multiplier (ADMM) algorithm, we transform the solving of sparse problem into the multistep iteration process. Then we add exponential noise to stable steps to achieve…
Study shows how neural networks generalize with minimal training data.
problem Understanding how neural networks generalize with limited data.
method Mean-field analysis of KL-regularized empirical risk minimization.
result Generalization error rate is O(1/n) for large n. Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Deep weight factorization improves neural network training through smooth optimization of sparse penalties.
problem Challenges in applying sparse regularization in neural networks due to non-differentiability of penalties.
method Introduces deep weight factorization, decomposing weights into multiple factors for smooth optimization of L1-penalized networks. result Deep weight factorization outperforms shallow factorization and pruning methods consistently across various architectures and datasets.
Efficient echo state network with explicit memory performs well on benchmark tasks.
problem Training differentiable neural computers is difficult and time-consuming.
method Echo state network with an explicit memory.
result Echo state network can recognize all regular languages, including those contractive networks cannot.
Extends elliptic operator regularity to maximally hypoelliptic operators.
problem Maximally hypoelliptic differential operators and their regularity.
method Define a principal symbol for arbitrary differential operators involving vector fields and their commutators.
result Proves the invertibility of the principal symbol is equivalent to maximally hypoellipticity, answering a conjecture.
New privacy framework tailored to specific data distributions.
problem Protecting individual data points in decision-making processes.
method Introducing tangent differential privacy, a new form of differential privacy.
result Entropic regularization guarantees tangent differential privacy under general conditions.
This paper tackles gauge fixing and regularity for perturbations around spherical backgrounds.
problem Understanding gauge freedom and regularity in perturbation theory for symmetric tensors.
method Analyzing Hodge-type decomposition for axially symmetric and axistationary tensors, showing existence and uniqueness of gauge tensors.
result Stationary and axially symmetric second order perturbations can be rendered in a canonical form with only one degree of differentiability loss near the origin.
Following the publication of an attack on genome-wide association studies (GWAS) data proposed by Homer et al., considerable attention has been given to developing methods for releasing GWAS data in a privacy-preserving way. Here, we develop an end-to-end differentially private method for solving regression problems wi…