Efficiently integrates stiff ODEs with vectorized methods.
problem Stiff systems and sparse training data in ODEs.
method Implicit, vectorized time integration with adjoint method.
result Achieves speed ups of greater than 100x on modern GPUs.
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48 results for “Implicit integration”
New methods improve integration of external LMs with AED models.
problem Improving performance of AED models by integrating external LMs.
method Comparing and proposing novel methods to estimate implicit LM from AED models.
result Proposed methods outperform previous approaches.
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid may be described in terms of Lagrangian implicit difference equations …
New Hamiltonian Monte Carlo method for non-canonical dynamics.
problem Incompatibility of canonical symplectic structure with non-canonical dynamics.
method Developed a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures with implicit integration.
result Non-canonical Hamiltonian Monte Carlo provides sampling advantages.
Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensionsstat.ML
SGD outperforms GD in high dimensions via implicit conditioning, revealed by asymptotic analysis.
problem Understanding why SGD outperforms GD in high-dimensional convex problems.
method Asymptotic analysis of multi-pass SGD on high-dimensional convex quadratics, establishing an equivalence to HSGD.
result SGD's efficiency is explained by implicit conditioning, not regularization.
Implicit generative models are difficult to train as no explicit density functions are defined. Generative adversarial nets (GANs) present a minimax framework to train such models, which however can suffer from mode collapse due to the nature of the JS-divergence. This paper presents a learning by teaching (LBT) approa…
Implicit Generative Copulasstat.ML
Flexible copula model using implicit generative neural networks.
problem Limited flexibility of parametric copulas and curse of dimensionality in non-parametric methods.
method Implicit generative neural networks to model high-dimensional copula distributions with unspecified marginals.
result Demonstrated flexibility and performance on various datasets.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
We study the problem to provide a triangular form based on implicit differential equations for non-linear multi-input systems with respect to the flatness property. Furthermore, we suggest a constructive method for the transformation of a given system into that special triangular shape, if possible. The well known Brun…
A new framework for offline RL improves policy flexibility and regularity.
problem Lack of environmental interactions in offline RL leads to poor policy performance.
method Proposes a behavior-regularized implicit policy framework with modified policy-matching methods.
result The framework improves policy effectiveness and robustness beyond static datasets.
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…
Improved speech recognition with language model integration in sequence-to-sequence models.
problem Improving word error rate in speech recognition models.
method Log-linear combination of acoustic and language models with per-token renormalization.
result The proposed method shows good improvements over standard model combination on Librispeech system.
Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Here, we present a compact neural circuit for two common numerical integrators: the explicit …
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall…
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
Paper presents an ADMM-based approach to efficiently integrate quadratic programming layers into neural networks.
problem Integrating quadratic programs into neural networks for optimization.
method An ADMM-based network layer architecture for solving quadratic programs efficiently.
result The ADMM layer is approximately an order of magnitude faster than existing methods for medium scaled problems.
Exact formulas for volumes of specific knot cone-manifolds.
problem Finding exact volumes of cone-manifolds with two-bridge knots.
method Provided exact integral formulas using Chebyshev polynomials and algebraic equations.
result Exact formulas for hyperbolic and spherical volumes of cone-manifolds.
We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of sto…
Discretization of Dirac systems and port-Hamiltonian systems: the role of the constraint algorithmmath.NA
Study discretizes Dirac and port-Hamiltonian systems using manifolds.
problem Discretization of Dirac and port-Hamiltonian systems.
method Retraction and discretization maps on manifolds for Dirac structures, applied to port-Hamiltonian systems.
result Numerical integrators for port-Hamiltonian systems derived from discretization techniques.
Efficient numerical valuation of European options under the two-asset Kou jump-diffusion modelmath.NA
The paper efficiently solves a complex option valuation equation for two assets.
problem Valuation of European options under a two-asset Kou jump-diffusion model.
method Extends an efficient algorithm for a one-dimensional integral to a two-dimensional one, using operator splitting schemes for time discretization.
result The method achieves optimal computational cost and stable convergence for various operator splitting schemes.
For sampling from a log-concave density, we study implicit integrators resulting from -method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the resulting sampling methods for and a range of step sizes are established. Ou…
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
Efficiently simulates the Heston model with large time steps using a novel method.
problem Challenges in simulating the Heston model with large time steps.
method Implicit integrated variance scheme exploiting the near-linear nature between stochastic driver and conditional integrated variance process.
result Achieves near-exact accuracy with coarse discretizations, efficient for large time steps.
In the complex setting, let be an analytic or algebraic differential equation with -degree . We deal with the qualitative study of such equations through the geometry of the planar -web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discu…
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are t…
Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussianq-fin.MF
A new simulation method for Volterra processes improves convergence for rough kernels.
problem Simulating Volterra processes with singular kernels.
method iVi (integrated Volterra implicit) scheme based on Inverse Gaussian distribution.
result The iVi scheme achieves weak convergence with few time steps, especially for rough kernels.
The paper details local forms of morphisms in colored supermanifolds.
problem Understanding local forms of morphisms in colored supermanifolds.
method Detailed account of -differential calculus and local theorems.
result Detailed insights into local forms of morphisms in colored supermanifolds.
JKO scheme adds deceleration in rapidly changing metric curvature directions.
problem Understanding the implicit bias of the JKO scheme in Wasserstein gradient flow.
method Characterized the implicit bias of the JKO scheme at second order in η, modifying the energy functional.
result JKO scheme adds deceleration in directions where metric curvature of J is rapidly changing.
Operator splitting schemes for American options under the two-asset Merton jump-diffusion modelmath.NA
This paper deals with the efficient numerical solution of the two-dimensional partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Merton jump-diffusion model. We consider the adaptation of various operator splitting schemes of both the impl…
This paper shows how to train only the implicit layer of overparameterized implicit neural networks.
problem Understanding how the implicit layer contributes to the training of overparameterized implicit neural networks.
method Restricting training to only the implicit layer and analyzing the generalization error for ReLU-activated networks.
result Global convergence is guaranteed even if only the implicit layer is trained, and gradient flow with proper random initialization can achieve small generalization errors.
Paper studies the theoretical equivalence between implicit and explicit neural networks in high dimensions.
problem Lack of theoretical analysis of implicit and explicit neural networks.
method Examined high-dimensional implicit neural networks and established their equivalence to explicit networks.
result Equivalence between implicit and explicit neural networks in high dimensions.
Gradient matching method estimates implicit regularization in complex deep learning systems.
problem Estimating implicit regularization in modern deep learning systems with complex modifications.
method Gradient matching methods to empirically estimate implicit regularization.
result Empirical estimation of implicit regularization in arbitrary networks, including dropout.
LiLaN uses linear latent networks to solve stiff ODEs efficiently.
problem Solving stiff ordinary differential equations (StODEs) requires expensive methods.
method LiLaN integrates latent dynamics analytically, avoiding explicit/implicit integration.
result LiLaN can approximate stiff nonlinear systems to any accuracy epsilon.
Continuous semi-implicit models enable faster training and better performance in generative modeling.
problem Slow convergence in hierarchical semi-implicit models during training.
method CoSIM, a continuous semi-implicit model that incorporates a continuous transition kernel for efficient training.
result CoSIM achieves superior performance on image generation tasks compared to existing methods.
Symplectic GP regression models Hamiltonian systems for particle tracing.
problem Efficiently modeling long-term Hamiltonian flow maps for charged particles.
method Multi-output Gaussian process regression with symplectic matrix-valued covariance function.
result Symplectic methods outperform existing approaches in learning Hamiltonian functions.
RINS-T solves time series inverse problems robustly without pretraining.
problem Recovering original signals from corrupted time series data.
method Implicit neural solvers with robust optimization techniques.
result RINS-T achieves high recovery performance without pretraining.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
Contact Tulczyjew Geometry for Continuous and Discrete Dissipative Dynamics on Skew Algebroidsmath-ph
Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.
problem Dissipative dynamics on skew algebroids
method Contact Tulczyjew formalism
result Intrinsic explanation of contact term and Euler-Lagrange-Herglotz equations
Study finds implicit government guarantee improves municipal investment bond ratings.
problem Questioning the objectivity of municipal investment bond ratings due to implicit government guarantee.
method Text mining of policy documents and PMC index model for implicit guarantee strength calculation.
result Implicit government guarantee boosts municipal investment bond ratings, especially in less developed regions.
Study shows SGD's generalization is not explained by implicit bias.
problem Explaining the generalization ability of overparameterized learning algorithms.
method Revisited Stochastic Convex Optimization with SGD, demonstrating limitations of implicit bias.
result No distribution-independent or distribution-dependent implicit regularizer can explain SGD's generalization.
This paper measures the intensity of implicit government guarantees using PMC index model.
problem Excessive local government debt due to implicit government guarantees.
method Text mining of policy documents related to municipal investment bonds, PMC index model.
result Recent policies have reduced the intensity of implicit government guarantees.
Implicit Regularization in Hierarchical Tensor Factorization and Deep Convolutional Neural Networkscs.LG
The paper explains implicit regularization in hierarchical tensor factorization and deep CNNs.
problem Understanding implicit regularization in complex neural network architectures.
method Theoretical analysis using dynamical systems to overcome challenges in hierarchy.
result Established implicit regularization towards low hierarchical tensor rank, equivalent to locality in CNNs.
In this paper, we describe the "implicit autoencoder" (IAE), a generative autoencoder in which both the generative path and the recognition path are parametrized by implicit distributions. We use two generative adversarial networks to define the reconstruction and the regularization cost functions of the implicit autoe…
A new method improves flow matching by dynamically weighting density estimates.
problem High-dimensional integration inefficiency in flow matching.
method Density-weighted Dynamic Stein operators.
result Significant improvement in vector field smoothness and sampling efficiency.
Implicit discourse relation classification is of great challenge due to the lack of connectives as strong linguistic cues, which motivates the use of annotated implicit connectives to improve the recognition. We propose a feature imitation framework in which an implicit relation network is driven to learn from another …
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Gradient descent on ReLU networks with square loss implicitly favors balanced weights.
problem Understanding implicit regularization in nonlinear neural networks with regression losses.
method Analyzing gradient descent dynamics on ReLU networks with square loss.
result It is impossible to characterize the implicit regularization of ReLU networks with square loss by any explicit function of model parameters.