We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
Improved 2-bit covariance estimator with reduced operator norm error and no tuning needed.
problem Improving 2-bit covariance estimation with reduced operator norm error and no tuning needed.
method Proposed a new 2-bit covariance matrix estimator using triangular dithering scales.
result Improved operator norm error rate that depends on effective rank of covariance matrix, closing theoretical gap.
We establish a theoretical link between adversarial training and operator norm regularization for deep neural networks. Specifically, we prove that ℓp-norm constrained projected gradient ascent based adversarial training with an ℓq-norm loss on the logits of clean and perturbed inputs is equivalent to data-…
Optimal scaling found to depend on operator norm across large models and datasets.
problem Lack of unifying principle for optimal hyperparameter scaling across models and datasets.
method Discovered that optimal scaling is conditioned on the operator norm of the output layer.
result The optimal learning rate/batch size pair (η∗,B∗) consistently has the same operator norm value. New optimizers control network width scaling, improving stability and transfer across different model sizes.
problem Designing stable optimizers for networks of varying widths.
method Interpreting optimizers as steepest descent under mean-normalized operator norms, enabling layerwise composability and width-independent bounds.
result New optimizers like row normalization and column normalization provide stable learning-rate transfer across different model widths.
The Jacobian Conjecture is proven for all Jacobian maps.
problem Proving the Jacobian Conjecture for all Jacobian maps.
method Using the Weyl algebra and holonomic modules, the paper shows that the Jacobian module is 1-generated and has finite length.
result The Jacobian Conjecture is true for all Jacobian maps.
This study connects Jacobian regularization to adversarial robustness and improves generalization.
problem Adversarial attacks make deep neural networks vulnerable.
method Developed a connection between Jacobian regularization and adversarial training, and established robust generalization gaps.
result Jacobian norms are related to both standard and robust generalization.
Study shows connections between Jacobian torsors and Fermat curves.
problem Understanding torsors of Jacobian of universal Fermat curves.
method Analyzes torsors of Jacobian of universal family of degree-m Fermat curves. result Every torsor is a connected component of the Picard scheme.
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Efficiently regularizes deep learning models using Jacobian nuclear norm.
problem Regularizing deep learning models to prevent overfitting and improve generalization.
method Proposes a denoising-style approximation to penalize the Jacobian nuclear norm without computing the Jacobian matrix.
result Demonstrates that penalizing the average squared Frobenius norm of Jg and Jh is equivalent to penalizing the Jacobian nuclear norm for function compositions. Abstract: Unknown status of Jacobian Conjecture, proof has a gap.
problem Status of Jacobian Conjecture
method Analysis of proof of theorem 2.1
result Proof of theorem 2.1 contains a gap
A well-conditioned Jacobian spectrum has a vital role in preventing exploding or vanishing gradients and speeding up learning of deep neural networks. Free probability theory helps us to understand and handle the Jacobian spectrum. We rigorously show almost sure asymptotic freeness of layer-wise Jacobians of deep neura…
The Jacobian conjecture is simplified using polynomial mappings.
problem Simplifying the Jacobian conjecture over the real field.
method Using polynomial mappings to restrict transitions on manifolds.
result An equivalent statement of the Jacobian conjecture.
Paper tackles Hessian/Jacobian-free stochastic bilevel optimization with O(ε−1.5) complexity.
problem Nonconvex-strongly-convex bilevel optimization problem.
method FdeHBO optimizer with finite-difference Hessian/Jacobian-vector approximation and momentum.
result FdeHBO achieves O(ε−1.5) iterations for ε-accurate stationary point. Following Hartigan, a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two step…
New algorithm adds Hessian regularization to improve neural network robustness.
problem Improving neural network robustness against adversarial attacks.
method Proposes an efficient algorithm to train neural networks with Hessian operator-norm regularization.
result Hessian operator-norm regularization increases neural network robustness over input gradient regularization.
Recent work (Pennington et al, 2017) suggests that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning. Motivated by this, we study the distribution of singular values of the Jacobian of the generator in Generative Adversarial Networks (GANs). We find th…
The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.
problem Understanding Shimura subvarieties in Jacobian loci for curves of positive genus.
method Analyzing Galois covers of curves and their Shimura subvarieties under specific numerical conditions.
result The Jacobian locus contains infinitely many Shimura subvarieties of positive dimension for g≤4. The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…
New algorithms estimate Jacobian matrices for large-scale machine learning.
problem Efficiently computing search directions for large nonlinear least squares.
method Exploit low-rank structure in Hessian to estimate Jacobian matrices.
result Two algorithms perform well compared to state-of-the-art methods.
The paper extends infinite-width analysis to neural network Jacobians, revealing convergence to Gaussian processes and linear ODEs.
problem Understanding the training dynamics of neural networks in the infinite-width limit.
method Extending infinite-width analysis to Jacobians, characterizing convergence to Gaussian processes and linear ODEs.
result The evolution of MLPs under robust training in the infinite-width limit is described by a linear ODE.
Study robust estimation of principal components under adversarial perturbations.
problem Estimating principal components in high-dimensional data under adversarial perturbations.
method Design of a computationally efficient algorithm for recovering the top-r principal subspace.
result The algorithm recovers an estimate of the top-r principal subspace with error depending on the robustness parameter κ.
Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.
problem Learning linear operators between infinite-dimensional Hilbert spaces in an online setting.
method Online learning approach for linear operators with bounded p-Schatten norm, proving impossibility for operator norm. result Separation between online learnability and uniform convergence for bounded linear operators.
GrokAlign aligns Jacobians to accelerate grokking in deep networks.
problem Accelerating the training dynamics of deep networks to avoid delayed generalisation and robustness.
method Aligning the Jacobians of a deep network with the training data to ensure grokking under a low-rank assumption.
result GrokAlign regularizes Jacobians to induce grokking sooner than conventional methods.
We provide a characterization for complex analytic curves among two-dimensional minimal graphs in R4 via the Jacobian
We extend the well-known result that any f∈W1,n(Ω,Rn), Ω⊂Rn with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces Ws,sn(Ω) for any s≥n+1n, where the sign condition on the Jacobian is understood in a distr…
New method reduces deep learning training costs by approximating vector-jacobian products.
problem Efficiently training deep neural networks with reduced computational and memory costs.
method Randomized, unbiased approximations of vector-jacobian products during backpropagation.
result Validated potential for reducing deep learning training costs through unbiased estimates.
Unified analysis of MPLE for Ising models with bounded operator norm or infinity norm.
problem Estimating Ising models in Total Variation distance with limited samples.
method Maximum Pseudo-Likelihood Estimator (MPLE) for two general classes of Ising models.
result Unified framework for polynomial-time estimation in TV distance for two general classes of Ising models.
Normalizing flows optimize Jacobian determinant for unique likelihood objective.
problem Optimizing normalizing flows for unique likelihood.
method Showed Jacobian determinant is unique for given distributions, leading to a unique global optimum. Used eigenvalues of auto-correlation matrix for explicit likelihood expression.
result Explicit expression of likelihood for flows, independent of neural network parameterization, with theoretical optimal value.
The Jacobian of Douady-Earle extension equals 1 only for isometries.
problem Investigating the Jacobian of Douady-Earle extension maps.
method Analyzing the Jacobian of the Douady-Earle extension map and constructing sequences of hyperbolic surfaces.
result The Jacobian of the Douady-Earle extension map is 1 only when the map is an isometry, and it can grow arbitrarily large for certain sequences of surfaces.
We study the adaptive estimation of copula correlation matrix Σ for the semi-parametric elliptical copula model. In this context, the correlations are connected to Kendall's tau through a sine function transformation. Hence, a natural estimate for Σ is the plug-in estimator Σ^ with Kendall's tau statistic. We …
The aim here is to continue the investigation in \cite{AB} of Jacobians of a Klein surface and also to correct an error in \cite{AB}.
To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the …
We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian.
Design of reliable systems must guarantee stability against input perturbations. In machine learning, such guarantee entails preventing overfitting and ensuring robustness of models against corruption of input data. In order to maximize stability, we analyze and develop a computationally efficient implementation of Jac…
Generative adversarial networks (GANs) are notoriously difficult to train and the reasons underlying their (non-)convergence behaviors are still not completely understood. By first considering a simple yet representative GAN example, we mathematically analyze its local convergence behavior in a non-asymptotic way. Furt…
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…
Recovering hidden influence networks from cascade data using Jacobian-based machine learning.
problem Recovering influence networks behind dynamic cascades.
method CascadeNet, a Jacobian-based machine learning framework.
result CascadeNet achieves high accuracy in network recovery.
We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples y1,y2,…,yn into an appropriate basis (dictionary) A∗ and sparse vectors x1∗,…,xn∗. Our algorithm …
Geometrically represents path integral reduction Jacobian for interacting systems.
problem Quantizing a model mechanical system with dependent coordinates.
method Geometric representation using scalar curvature and Christoffel symbols in a nonholonomic basis.
result Found a geometric representation for the path integral reduction Jacobian.
This work relaxes energy constraints in self-attention layers for a more general analysis.
problem Understanding inherent biases and dynamics in self-attention layers without energy functions.
method Dynamical systems analysis and Jacobian matrix examination.
result Normalized dynamics are close to a critical state, indicating high inference performance.
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.
problem Enforcing structure on derivatives of neural network mappings.
method Proposes using a neural network to directly learn the Jacobian of the input-output function, allowing control over derivative structure.
result Demonstrates learning invertible approximations to simple and 1-Lipschitz functions.
The paper studies global invertibility of maps on Finsler manifolds.
problem Global invertibility of locally Lipschitz maps on Finsler manifolds.
method Introduces pseudo-Jacobian and studies its relations with local metric properties of the map.
result Conditions for a map to be globally invertible and covering.
The Jacobian matrix (or the gradient for single-output networks) is directly related to many important properties of neural networks, such as the function landscape, stationary points, (local) Lipschitz constants and robustness to adversarial attacks. In this paper, we propose a recursive algorithm, RecurJac, to comput…
We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.