dc decomposition

This study addresses the lack of a priori convergence assessment for the Difference-of-Convex Algorithm (DCA) in Gaussian radial basis function kernel support vector regression (RBF-SVR). By exploiting the analytical structure of the RBF kernel, the authors construct an explicit DC decomposition and, for the first time, derive closed-form expressions for the lower bound μ of the strong convexity parameter and the upper bound L of the gradient Lipschitz constant of the DC components. They further identify the scalar Cαρ—determined by the hyperparameters C and γ—as the key quantity governing DCA’s convergence rate and dependence on initialization. Numerical experiments on six benchmark functions confirm that Cαρ alone effectively predicts DCA’s convergence behavior both before and after training, establishing the first analytical link between RBF-SVR hyperparameters and DCA convergence.

Optimizing non-convex functionals such as the Maximum Mean Discrepancy (MMD) and energy distance in Wasserstein space is highly challenging due to the lack of geodesic convexity, which hinders the theoretical analysis of standard first-order methods. This work introduces, for the first time, a difference-of-convex (DC) optimization framework into Wasserstein space by constructing effective DC decompositions for such functionals and extending the classical Convex–Concave Procedure (CCCP) to this setting. Under mild assumptions, we establish the local convergence of the proposed algorithm. Both theoretical analysis and empirical experiments demonstrate that the method converges faster and more stably than Wasserstein gradient descent, offering a provably convergent new pathway for optimizing non-convex probabilistic functionals.

This work addresses the “curse of dimensionality” in computing convex conjugates (Legendre transforms) of high-dimensional convex functions. We propose a novel deep learning framework grounded in the implicit Fenchel duality formula, bypassing conventional numerical discretization and existing optimal transport–based approaches. Instead, it employs differentiable implicit modeling to directly approximate the conjugate function, enabling gradient-based optimization and posterior error estimation. To enhance interpretability, we integrate Kolmogorov–Arnold networks with symbolic regression, automatically recovering closed-form analytical expressions from numerical approximations. Experiments on multiple high-dimensional benchmarks demonstrate high-accuracy conjugate approximation; notably, the method is the first to recover exact analytical conjugates—such as quadratic, exponential, and entropy functions—directly from data. This establishes a new computational paradigm for convex analysis that is differentiable, interpretable, and scalable.

This work proposes an improved accelerated difference-of-convex (DC) algorithm, termed IBDCA, to address the issue that conventional accelerated methods in nonsmooth nonconvex DC optimization may generate ascent directions and fail to ensure monotonic descent. The method is tailored for problems where the objective function is expressed as the difference between a nonsmooth convex function and a smooth convex function. By constructing a valid descent direction and incorporating a monotone line search, IBDCA achieves, for the first time in nonsmooth DC optimization, an accelerated algorithm that guarantees both monotonicity and global convergence. The theoretical analysis leverages DC decomposition, extrapolation strategies, and the Kurdyka–Łojasiewicz property. Numerical experiments on image restoration tasks demonstrate that IBDCA significantly outperforms classical DCA and other state-of-the-art methods in terms of both iteration count and computational time.

Ill-posed inverse problems—such as sparse-angle or limited-angle CT reconstruction—pose a fundamental challenge in balancing reconstruction performance and interpretability of regularization methods under small-data, weakly supervised regimes. Method: We propose the first learnable Difference-of-Convex (DC) regularizer framework, introducing the DC structure into data-driven regularizer design for the first time. A deep neural network parameterizes the DC function, and optimization is efficiently performed via the Difference-of-Convex Algorithm (DCA) combined with proximal subgradient methods. Contribution/Results: Theoretically, we establish a sufficient condition for a regularizer to admit a DC representation and rigorously prove the strong convergence of the proposed algorithm. Experimentally, our framework achieves state-of-the-art reconstruction accuracy under weak supervision, demonstrating both rigorous theoretical guarantees and superior empirical performance across multiple benchmarks.

This work addresses the challenge of modeling convex-concave structures in minimax problems by proposing a structured separable decomposition approach. By imposing sign and concavity constraints on the output layer of neural networks, the method inherently preserves convexity in variable \(x\) and concavity in variable \(y\). The study establishes, for the first time, a one-dimensional universal approximation theorem under mixed Monge-type convexity conditions, which directly informs the design of a neural network architecture that strictly maintains the underlying convex-concave geometry. Combining convexity-preserving networks with a simple output transformation, the approach achieves high-accuracy approximation of smooth, nonsmooth, and high-rank convex-concave functions, demonstrating superior performance on both one-dimensional and five-dimensional benchmark problems.

This work addresses the challenge of sampling from non-log-concave distributions arising from nonsmooth difference-of-convex (DC) regularized models. It introduces, for the first time, a DC decomposition within the Langevin sampling framework: the convex component is smoothed via the Moreau envelope, while the concave part is incorporated into the data fidelity term, yielding a novel proximal Langevin algorithm. This approach substantially relaxes the assumptions on the target distribution required by existing theory, thereby extending the applicability of non-log-concave sampling methods. Through Wasserstein convergence analysis, the algorithm accurately recovers the target distribution on synthetic data and demonstrates reliable uncertainty quantification in real-world CT imaging tasks.

This work addresses the limitations of existing universal approximation theories for neural networks, which typically rely on uniform hypercube partitions and struggle to capture the local irregularities of target functions near singularities. To overcome this, the authors propose a task-oriented approximation strategy based on polyhedral decomposition, integrating kernel polynomial constructions with Totik–Ditzian-type moduli of continuity. Within each subdomain, ReLU networks are individually tailored to the local geometry and regularity of the function. This approach significantly enhances approximation efficiency and flexibility in regions containing singularities and achieves faster convergence rates for analytic functions compared to conventional uniform partitioning methods.