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.
Study on fractional mass for codimension-two currents, proving equi-coercivity and Γ-convergence.
problem Defining and studying fractional mass for codimension-two currents on manifolds.
method Energy minimization with Jacobian constraint, equi-coercivity, Γ-convergence, weak linking.
result Equivalence of two formulations of fractional mass, improved regularity for s-harmonic maps. To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the ent…
Derives derivatives and geometric framework for functions with non-independent variables.
problem Characterizing functions with non-independent variables in probabilistic models.
method Derives actual and dependent partial derivatives, dependent Jacobian matrix, and tensor metric.
result Derives gradient, Hessian, and Taylor expansion for functions with non-independent variables.
We prove two-sided inequalities for the Lp-norm of a pushforward or pullback (with respect to an orientation-preserving diffeomorphism) on oriented volume and Riemannian manifolds. For a function or density on a volume manifold, these bounds depend only on the Jacobian determinant, which arises through the change of…
Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network's input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spe…
A new method for faster bandwidth selection in Gaussian kernel ridge regression.
problem Efficiently selecting the bandwidth in Gaussian kernel ridge regression.
method Formulated an approximate Jacobian expression for bandwidth selection, proposing a closed-form heuristic.
result Our method is as accurate as cross-validation and marginal likelihood maximization but up to six orders of magnitude faster.
Generalization in nonlinear least squares can be studied via algorithmic stability and effective dimension.
problem Generalization in nonlinear least squares models
method Deriving error bounds for local minimizers using algorithmic stability and effective dimension
result Bounds depend on learned geometry rather than parameter count
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.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
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.
Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We…
Maxout networks study gradients and propose initialization strategies.
problem Complexity in input-output Jacobian distribution complicates stable parameter initialization.
method Obtained bounds on moments of gradients and formulated initialization strategies.
result Parameter initialization strategies improve training of deep maxout networks.
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.
Paper improves robustness of GNNs against adversarial attacks.
problem Understanding robust generalization of GNNs in adversarial settings.
method Develops a sensitivity-aware PAC-Bayesian framework for MPGNNs.
result Derives tighter robust generalization bounds for MPGNNs.
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…
It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condi…
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. 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 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.
Contrastive embeddings improve neural architecture search performance.
problem Improving performance of neural architecture search algorithms.
method Contrastive learning to identify networks based on data Jacobians and produce embeddings.
result Traditional black-box optimization algorithms can reach state-of-the-art performance with contrastive embeddings.
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
Study on sphere-valued maps, proving energy convergence and current limits.
problem Understanding the behavior of sphere-valued Sobolev maps as their energy grows.
method Proving Gamma-convergence of p-energies to the mass of an integral current. result Jacobian convergence to an area-minimizing current in a cobordism class.
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.
New Hessian-free method improves bilevel optimization for meta-learning.
problem Efficiently solving bilevel optimization problems with limited second-order information.
method Proposes a new Hessian-free method that approximates the response Jacobian matrix via optimization path differences.
result Demonstrates superior performance on meta-learning tasks compared to baseline methods.
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.
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.
Modern neural network architectures often generalize well despite containing many more parameters than the size of the training dataset. This paper explores the generalization capabilities of neural networks trained via gradient descent. We develop a data-dependent optimization and generalization theory which leverages…
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…
We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structur…
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.
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.