arXiv subjects

Nanjing Huang

Publications and source records attributed to Nanjing Huang.

4 recordsLinked to original sources

arXiv2026

Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness

Despite Neural Ordinary Differential Equations (Neural ODEs) exhibiting intrinsic robustness, existing methods often impose Lyapunov stability for formal guarantees. However, these methods still face a fundamental accuracy-robustness trade-off, which stems from a core limitation: their applied stability conditions are rigid and inappropriate, creating a mismatch between the model's regions of attraction (RoAs) and its decision boundaries. To resolve this, we propose Zubov-Net, a novel framework that unifies dynamics and decision-making. We first employ learnable Lyapunov functions directly as the multi-class classifier, ensuring the prescribed RoAs (PRoAs, defined by the Lyapunov functions) inherently align with a classification objective. Then, for aligning prescribed and true regions of attraction (PRoAs-RoAs), we establish a Zubov-driven stability region matching mechanism by reformulating Zubov's equation into a differentiable consistency loss. Building on this alignment, we introduce a new paradigm for actively controlling the geometry of RoAs by directly optimizing PRoAs to reconcile accuracy and robustness. Theoretically, we prove that minimizing the tripartite loss guarantees consistency alignment of PRoAs-RoAs, non-overlapping PRoAs, trajectory stability, and a certified robustness margin. Moreover, we establish stochastic convex separability with tighter probability bounds and lower dimensionality requirements to justify the convex design in Lyapunov functions.

cs.LG↗

arXiv2026

Difference-of-Convex Elastic Net for Compressed Sensing

This work proposes a novel and unified sparse recovery framework, termed the difference of convex Elastic Net (DCEN). This framework effectively balances strong sparsity promotion with solution stability, and is particularly suitable for high-dimensional variable selection involving highly correlated features. Built upon a difference-of-convex (DC) structure, DCEN employs two continuously tunable parameters to unify classical and state-of-the-art models--including LASSO, Elastic Net, Ridge, and $\ell_1-α\ell_2$--as special cases. Theoretically, sufficient conditions for exact and stable recovery are established under the restricted isometry property (RIP), an oracle inequality and recovery bound are derived for the global solution, and a closed-form expression of the DCEN regularization proximal operator is obtained. Moreover, two efficient optimization algorithms are developed based on the DC algorithm (DCA) and the alternating direction method of multipliers (ADMM). Within the Kurdyka-Łojasiewicz (KŁ) framework, the global convergence of DCA and its linear convergence rate are rigorously established. Furthermore, DCEN is extended to image reconstruction by incorporating total variation (TV) regularization, yielding the DCEN-TV model, which is efficiently solved via the Split Bregman method. Numerical experiments demonstrate that DCEN consistently outperforms state-of-the-art methods in sparse signal recovery, high-dimensional variable selection under strong collinearity, and Magnetic Resonance Imaging (MRI) image reconstruction, achieving superior recovery accuracy and robustness.

math.OC↗

arXiv2025

An Efficient ADMM Method for Ratio-Type Nonconvex and Nonsmooth Minimization in Sparse Recovery

Sparse signal recovery based on nonconvex and nonsmooth optimization problems has significant applications and demonstrates superior performance in signal processing and machine learning. This work deals with a scale-invariant $\ell_{1/2}/\ell_{2}$ sparse minimization with nonconvex, nonseparable, ratio-type regularization to enhance the accuracy and stability of sparse recovery. Within the framework of the null space property, we analyze the conditions for exact and stable recovery in constrained minimization problem. For the unconstrained regularized minimization problem, we develop an alternating direction method of multipliers (ADMM) based on a splitting strategy and rigorously analyze its global convergence and linear convergence rate under reasonable assumptions. Numerical experiments demonstrate that the proposed method consistently outperforms existing approaches across diverse noise levels and measurement settings. Furthermore, experiments on neural network sparsity and generalization performance demonstrate that the method effectively improves prediction accuracy.

math.OC↗

arXiv2024

FxTS-Net: Fixed-Time Stable Learning Framework for Neural ODEs

Neural Ordinary Differential Equations (Neural ODEs), as a novel category of modeling big data methods, cleverly link traditional neural networks and dynamical systems. However, it is challenging to ensure the dynamics system reaches a correctly predicted state within a user-defined fixed time. To address this problem, we propose a new method for training Neural ODEs using fixed-time stability (FxTS) Lyapunov conditions. Our framework, called FxTS-Net, is based on the novel FxTS loss (FxTS-Loss) designed on Lyapunov functions, which aims to encourage convergence to accurate predictions in a user-defined fixed time. We also provide an innovative approach for constructing Lyapunov functions to meet various tasks and network architecture requirements, achieved by leveraging supervised information during training. By developing a more precise time upper bound estimation for bounded non-vanishingly perturbed systems, we demonstrate that minimizing FxTS-Loss not only guarantees FxTS behavior of the dynamics but also input perturbation robustness. For optimising FxTS-Loss, we also propose a learning algorithm, in which the simulated perturbation sampling method can capture sample points in critical regions to approximate FxTS-Loss. Experimentally, we find that FxTS-Net provides better prediction performance and better robustness under input perturbation.

math.OC↗