On the Optimisation Landscape of Overparameterised Weighted MMD

Tharrshann Jayan Logarajah; Jeremias Knoblauch; Louis Sharrock

Home/Archive/arXiv

arXiv · 2610.19861

On the Optimisation Landscape of Overparameterised Weighted MMD

Tharrshann Jayan Logarajah·Jeremias Knoblauch·Louis Sharrock

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

stat.ML cs.LG math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tharrshann Jayan Logarajah, Jeremias Knoblauch, Louis Sharrock. 2026-10-14. On the Optimisation Landscape of Overparameterised Weighted MMD. https://arxiv.org/abs/2610.19861

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Noise-Aware Differentially Private Variational Inference

Differential privacy (DP) provides robust privacy guarantees for statistical inference, but this can lead to unreliable results and biases in downstream applications. While several noise-aware approaches have been proposed which integrate DP perturbation into the inference, they are limited to specific types of simple probabilistic models. In this work, we propose a novel method for noise-aware approximate Bayesian inference based on stochastic gradient variational inference which can also be applied to high-dimensional and non-conjugate models. We also propose a more accurate evaluation method for noise-aware posteriors. Empirically, our inference method has similar performance to existing methods in the domain where they are applicable. Outside this domain, we obtain accurate coverages on high-dimensional Bayesian linear regression and well-calibrated predictive probabilities on Bayesian logistic regression with the UCI Adult dataset.

stat.ML↗

arXiv2026

Is Flow Matching Just Trajectory Replay for Sequential Data?

Flow matching (FM) is increasingly used in scientific domains for time series generation and forecasting, where data often arise from underlying dynamical systems. However, it is not well-understood whether it learns transferable dynamical structure or simply performs an effective ''trajectory replay''. We study this question by deriving the optimal velocity field targeted by the empirical FM objective on sequential data. For the Gaussian conditional paths commonly used in practice, we show that the resulting sampler is an ODE whose dynamics describe a nonparametric, memory-augmented continuous-time dynamical system. The driving velocity field admits a closed-form expression as a weighted mixture of instantaneous velocities determined by observed transitions. This characterization provides an interpretable view of neural FM models as parametric estimation of an underlying nonparametric target and naturally leads to FreeFM: a closed-form, training-free realization of this target. FreeFM provides a transparent reference for studying sequential FM while yielding effective probabilistic forecasts directly from historical transitions.

stat.ML↗

arXiv2026

Sharp Statistical Thresholds for Attributed Gaussian Graph Matching

We investigate the problem of matching networks where both edge weights and node features are correlated following a specific node correspondence. While prior work has established sharp thresholds for matching either graphs or features individually, the joint setting has been significantly less explored. In this study, we derive precise information-theoretic thresholds for exact recovery, almost-exact recovery, and partial recovery of the underlying alignment. These are expressed in terms of graph and feature correlation strengths, the number of nodes, and the dimension of the features. Interestingly, whereas an all-or-nothing phase transition is observed in the standard graph-matching scenario, the inclusion of contextual information at the node level introduces a richer structure: the exact-recovery threshold separates from the shared threshold for almost-exact and partial recovery. Closing gaps left by prior works, we provide a sharp characterization of the thresholds governing how structural and contextual information interact in graph matching, and thereby establish a benchmark for designing efficient algorithms in the high-dimensional setting.

stat.ML↗