arXiv research

New method for differentially private optimization with general Lipschitz conditions.

Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability…

A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.

New Lipschitz de Rham theorem for $L_p$-cohomology.

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

Privacy affects fairness in classification models, but not drastically.

We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous m…

Study improves Poincaré-Sobolev inequalities for differential forms.

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the c…

Optimizes privacy-preserving optimization for heavy-tailed data.

We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertib…

Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exteri…

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and…

Improved DP SO with large Lipschitz parameters, handling outliers and heavy-tailed data.

Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.

We study controlled systems which are uniformly observable and differentially observable with an order larger than the system state dimension. We establish that they may be transformed into a (partial) triangular canonical form but with possibly non locally Lipschitz functions. We characterize the points where this Lip…

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

The paper proves Rademacher's theorem for Heisenberg groups.

New approach achieves optimal rates for differentially private stochastic convex optimization with heavy-tailed gradients.

Paper proves convergence of Markovian iteration for FBSDEs with fully coupled drift and Z process.

In this paper, we provide conditions which ensure that stochastic Lipschitz BSDEs admit Malliavin differentiable solutions. We investigate the problem of existence of densities for the first components of solutions to general path-dependent stochastic Lipschitz BSDEs and obtain results for the second components in part…

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

Deep neural networks are notorious for being sensitive to small well-chosen perturbations, and estimating the regularity of such architectures is of utmost importance for safe and robust practical applications. In this paper, we investigate one of the key characteristics to assess the regularity of such methods: the Li…

We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise up…

We model how Lipschitz continuity changes during neural network training.

Generalization bounds derived for neural ODEs and deep residual networks.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

New method for efficient proximal mapping of 1-path-norm in shallow networks.

We establish that over a C^{2,1} manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomophism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelof approximation to prove the strong differentiability of the exponenti…

Let $Σ$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:Σ\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differen…

The paper sharpens a theorem about surfaces with zero Gaussian curvature.

Paper tackles robust control of SDEs with ambiguity, proving value function existence and applying to investment problems.

New privacy bounds for DP-SGD's last iterate, even with cyclic sampling.

We present a theoretical and experimental investigation of the quantization problem for artificial neural networks. We provide a mathematical definition of quantized neural networks and analyze their approximation capabilities, showing in particular that any Lipschitz-continuous map defined on a hypercube can be unifor…

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo…

Proves continuum limits of Lipschitz learning using Γ-convergence.

Let $ρ: G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $σ= (σ_1,\ldots,σ_n) : V \to \mathbb R^n$, where $σ_1,\ldots,σ_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} …

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_…

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of di…

We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^…

Paper proves DN map determination for simple surfaces with low regularity metrics.

Flow Matching improves Wasserstein 1 distance convergence in high dimensions.