Tharrshann Jayan Logarajah

arXiv2026

On the Optimisation Landscape of Overparameterised Weighted MMD

We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.

stat.ML↗

arXiv2026

Parameterized Complexity of $L_p$-Lipschitz Constants for Input Convex Neural Networks and $L_p$-Norm Maximization over Zonotopes

Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.

cs.CC↗