PDE-SSM: A Spectral State Space Approach to Spatial Mixing in Diffusion Transformers

The success of vision transformers-especially for generative modeling-is limited by the quadratic cost and weak spatial inductive bias of self-attention. We propose PDE-SSM, a spatial state-space block that replaces attention with a learnable convection-diffusion-reaction partial differential equation. This operator encodes a strong spatial prior by modeling information flow via physically grounded dynamics rather than all-to-all token interactions. Solving the PDE in the Fourier domain yields global coupling with near-linear complexity of $O(N \log N)$, delivering a principled and scalable alternative to attention. We integrate PDE-SSM into a flow-matching generative model to obtain the PDE-based Diffusion Transformer PDE-SSM-DiT. Empirically, PDE-SSM-DiT matches or exceeds the performance of state-of-the-art Diffusion Transformers while substantially reducing compute. Our results show that, analogous to 1D settings where SSMs supplant attention, multi-dimensional PDE operators provide an efficient, inductive-bias-rich foundation for next-generation vision models.

Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization

We introduce Probabilistic Gaussian Homotopy (PGH), a probability-space continuation framework for nonconvex optimization. Unlike classical Gaussian homotopy, which smooths the objective and uniformly averages gradients, PGH deforms the associated Boltzmann distribution and induces Boltzmann-weighted aggregation of perturbed gradients, which exponentially biases descent directions toward low-energy regions. We show that PGH corresponds to a log-sum-exp (soft-min) homotopy that smooths a nonconvex objective at scale $λ>0$ and recovers the original objective as $λ\to 0$, yielding a posterior-mean generalization of the Moreau envelope, and we derive a dynamical system governing minimizer evolution along an annealed homotopy path. This establishes a principled connection between Gaussian continuation, Bayesian denoising, and diffusion-style smoothing. We further propose Probabilistic Gaussian Homotopy Optimization (PGHO), a practical stochastic algorithm based on Monte Carlo gradient estimation, and demonstrate strong performance on high-dimensional nonconvex benchmarks and sparse recovery problems where classical gradient methods and objective-space smoothing frequently fail.

Preconditioned Score and Flow Matching

Flow matching and score-based diffusion train vector fields under intermediate distributions $p_t$, whose geometry can strongly affect their optimization. We show that the covariance $Σ_t$ of $p_t$ governs optimization bias: when $Σ_t$ is ill-conditioned, and gradient-based training rapidly fits high-variance directions while systematically under-optimizing low-variance modes, leading to learning that plateaus at suboptimal weights. We formalize this effect in analytically tractable settings and propose reversible, label-conditional \emph{preconditioning} maps that reshape the geometry of $p_t$ by improving the conditioning of $Σ_t$ without altering the underlying generative model. Rather than accelerating early convergence, preconditioning primarily mitigates optimization stagnation by enabling continued progress along previously suppressed directions. Across MNIST latent flow matching, and additional high-resolution datasets, we empirically track conditioning diagnostics and distributional metrics and show that preconditioning consistently yields better-trained models by avoiding suboptimal plateaus.

Towards Efficient Training of Graph Neural Networks: A Multiscale Approach

Graph Neural Networks (GNNs) have emerged as a powerful tool for learning and inferring from graph-structured data, and are widely used in a variety of applications, often considering large amounts of data and large graphs. However, training on such data requires large memory and extensive computations. In this paper, we introduce a novel framework for efficient multiscale training of GNNs, designed to integrate information across multiscale representations of a graph. Our approach leverages a hierarchical graph representation, taking advantage of coarse graph scales in the training process, where each coarse scale graph has fewer nodes and edges. Based on this approach, we propose a suite of GNN training methods: such as coarse-to-fine, sub-to-full, and multiscale gradient computation. We demonstrate the effectiveness of our methods on various datasets and learning tasks.