Study shows AD for neural nets with machine-representable numbers can be incorrect.
problem Correctness of AD for neural nets with machine-representable numbers.
method Analyzed two sets of parameters: incorrect and non-differentiable. Proved bounds and conditions for AD correctness.
result AD can be incorrect for machine-representable numbers, but provides a Clarke subderivative on non-differentiable set.
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensi…
Differential K-theory gets a λ-ring structure.
problem Establishing a λ-ring structure in differential K-theory. method Splitting principle for differential K-theory, Adams operations construction.
result Differential K0-ring admits a λ-ring structure. Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential inva…
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
Improved set prediction model using multiset-equivariant operations and approximate implicit differentiation.
problem Existing set prediction models struggle with multisets and cannot represent certain functions.
method Introduced multiset-equivariance, improved DSPN with approximate implicit differentiation, and applied to CLEVR object property prediction.
result Significantly improved object property prediction on CLEVR dataset.
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiabili…
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) th…
The curvature of almost Fuchsian immersions is concave in their Hopf differentials.
problem Understanding the geometry of almost Fuchsian immersions.
method Analyzing the extrinsic curvature and using properties of Hopf differentials.
result The set of Hopf differentials forms a convex subset of holomorphic quadratic differentials.
Differentiable clustering method using perturbed spanning forests.
problem Efficient clustering in trainable pipelines with noisy data.
method Stochastic perturbations of minimum-weight spanning forests.
result Method performs well even in challenging settings.
Preserving differential privacy has been well studied under centralized setting. However, it's very challenging to preserve differential privacy under multiparty setting, especially for the vertically partitioned case. In this work, we propose a new framework for differential privacy preserving multiparty learning in t…
We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized coho…
Proposes a method to generate private synthetic data in a decentralized setting using correlated noise.
problem Challenges of generating private synthetic data in a decentralized setting with limited client data.
method Integrates CAPE protocol into federated DP-CDA framework to generate anti-correlated noise.
result Improves privacy-utility trade-off in federated setting compared to centralized approach.
Differentially private conformal prediction improves statistical efficiency.
problem Quantifying uncertainty in private data analysis.
method Introducing differential conformal prediction and developing Differentially Private Conformal Prediction (DPCP).
result DPCP produces tighter prediction sets than existing private split conformal approaches.
Efficiently private regression for unbounded data.
problem Privacy constraints in regression settings with unbounded covariates.
method Differential privacy techniques on mean and covariance estimation extended to sub-gaussian regime.
result Unbiased estimate of true regression vector learned up to a scaling factor.
The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for eve…
Any discrete differential manifold M (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron P(M). This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes…
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
Paper links set derivatives to its orthogonal projections.
problem Understanding the relationship between set derivatives and projections.
method Derives equations from topological link between Minkowski functional partial derivatives and set boundary.
result System of equations for orthogonal projections derived.
Paper improves differential privacy analysis for machine learning.
problem Quantifying privacy leakage in noisy gradient descent.
method Shifted interpolation in f-differential privacy. result First exact privacy analysis for strongly convex optimization.
Revises Gauss's Lemma using metrical distortion and differential slip.
problem Revising Gauss's Lemma in Riemannian geometry.
method Defining metrical distortion and differential slip, showing their geometric implications.
result Geodesically radial volume and length preservation properties.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
We describe an elementary combinatorial move on the set of quadratic differentials with a horizontal one cylinder decom-position. Computer experiment suggests that the corresponding equivalent classes are in one-to-one correspondence with the con-nected component of the strata.
Extends unique continuation theorem to manifolds with boundary, proving zero set codimension.
problem Proving unique continuation for exterior differential forms on manifolds with boundary.
method Extends Aronszajn-Krzywicki-Szarski theorem to manifolds with boundary, assuming suitable boundary conditions.
result Hausdorff dimension of zero sets of harmonic forms and eigenfields has codimension at least 2.
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
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.
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independ…
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.
FuDGE estimates differences between functional graphs in high-dimensional settings.
problem Estimating differences between two undirected functional graphical models with shared structures.
method FuDGE: A method that directly estimates the functional differential graph without first estimating individual graphs.
result FuDGE consistently estimates the functional differential graph in high-dimensional settings.
Distributed machine learning is an approach allowing different parties to learn a model over all data sets without disclosing their own data. In this paper, we propose a weighted distributed differential privacy (WD-DP) empirical risk minimization (ERM) method to train a model in distributed setting, considering differ…
Study real logarithms of semi-simple matrices, focusing on differential structure.
problem Understanding the differential structure of real logarithms of semi-simple matrices.
method Examines the differential structure of real logarithms of semi-simple matrices under specific matrix types.
result Characterizes the differential structure of real logarithms of semi-simple matrices.
Paper improves privacy in SGD with low noise, achieving optimal risk rates.
problem Privacy-preserving machine learning with good performance.
method Differentially private SGD with low-noise analysis.
result Achieves optimal excess risk rates for non-smooth losses.
Research on refined algebraic domains respecting differential geometry.
problem Understanding shapes and regions of real algebraic curves.
method Investigates points in two curves, singular points, inflection points, and points of double tangent lines, considering differential geometry.
result Proves fundamental properties and investigates examples of refined algebraic domains.
Develops a high-dimensional differentially-private EM algorithm with near-optimal statistical guarantees.
problem Designing differentially-private EM algorithms for high-dimensional latent variable models.
method Noisy iterative hard-thresholding, statistical guarantees, near-optimal convergence rates.
result Near-optimal statistical guarantees and minimax rate optimality in high-dimensional settings.
In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certa…
New algorithms for differentially private optimization in convex and non-convex settings with near-optimal rates.
problem Differentially private optimization in convex and non-convex settings.
method Developed algorithms for convex and non-convex settings with near-optimal excess population risk.
result Achieved near-optimal rates in near-linear time for convex settings and nearly dimension independent rates for non-convex settings.
Privacy-preserving inference for clinical trials using differential privacy.
problem Balancing knowledge sharing and privacy in healthcare data.
method Differential privacy (DP) applied to log-linear belief updates in distributed settings.
result Differentially private, distributed inference methods outperform existing techniques.
Classifies connected components of meromorphic differentials with residue conditions.
problem Understanding the structure of meromorphic differentials with residue constraints.
method Analyzes the multi-scale compactification and residue conditions.
result Classified connected components of generalized strata of meromorphic differentials.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
In this paper we develop the first algorithms for online submodular minimization that preserve differential privacy under full information feedback and bandit feedback. A sequence of T submodular functions over a collection of n elements arrive online, and at each timestep the algorithm must choose a subset of $[n]…
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
Develops a computationally tractable high-dimensional differential privacy estimator.
problem Differential privacy in high dimensions is computationally intractable.
method Combines high-dimensional robust statistics with differential privacy techniques.
result A computationally tractable algorithm with dimension-independent privacy loss.