Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold—cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Diffusion models transform random noise into images of remarkable fidelity, yet the structure of this noise-to-image map remains largely unexplored. We investigate this relationship using patch-wise Principal Component Analysis (PCA) and empirically demonstrate that low-frequency components of the initial noise predominantly influence the compositional structure of generated images. Our analyses reveal that noise seeds inherently contain universal compositional cues, evident when identical seeds produce images with similar structural attributes across different datasets and model architectures. Leveraging these insights, we develop and theoretically justify a simple yet effective Patch PCA denoiser that extracts underlying structure from noise using only generic natural image statistics. The robustness of these structural cues is observed to persist across both pixel-space models and latent diffusion models, highlighting their fundamental nature. Finally, we introduce a zero-shot editing method that enables injecting compositional control over generated images, providing an intuitive approach to guided generation without requiring model fine-tuning or additional training.

(Directed) graphs with node attributes are a common type of data in various applications and there is a vast literature on developing metrics and efficient algorithms for comparing them. Recently, in the graph learning and optimization communities, a range of new approaches have been developed for comparing graphs with node attributes, leveraging ideas such as the Optimal Transport (OT) and the Weisfeiler-Lehman (WL) graph isomorphism test. Two state-of-the-art representatives are the OTC distance proposed in (O’Connor et al., 2022) and the WL distance in (Chen et al., 2022). Interestingly, while these two distances are developed based on different ideas, we observe that they both view graphs as Markov chains, and are deeply connected. Indeed, in this paper, we propose a unified framework to generate distances for Markov chains (thus including (directed) graphs with node attributes), which we call the Optimal Transport Markov (OTM) distances, that encompass both the OTC and the WL distances. We further introduce a special one-parameter family of distances within our OTM framework, called the discounted WL distance. We show that the discounted WL distance has nice theoretical properties and can address several limitations of the existing OTC and WL distances. Furthermore, contrary to the OTC and the WL distances, our new discounted WL distance can be differentiated after a entropy-regularization similar to the Sinkhorn distance, making it suitable to use in learning frameworks, e.g., as the reconstruction loss in a graph generative model.

The distinguishing power of graph transformers is tied to the choice of positional encoding: features used to augment the base transformer with information about the graph. There are two primary types of positional encoding: absolute positional encodings (APEs) and relative positional encodings (RPEs). APEs assign features to each node and are given as input to the transformer. RPEs instead assign a feature to each pair of nodes, e.g., shortest-path distance, and are used to augment the attention block. A priori, it is unclear which method is better for maximizing the power of the resulting graph transformer. In this paper, we aim to understand the relationship between these different types of positional encodings. Interestingly, we show that graph transformers using APEs and RPEs are equivalent in their ability to distinguish non-isomorphic graphs. In particular, we demonstrate how to interchange APEs and RPEs while maintaining their distinguishing power in terms of graph transformers. However, in the case of graphs with node features, we show that RPEs may have an advantage over APEs. Based on our theoretical results, we provide a study of different APEs and RPEs—including the shortest-path and resistance distance and the recently introduced stable and expressive positional encoding (SPE)—and compare their distinguishing power in terms of transformers. We believe our work will help navigate the vast number of positional encoding choices and provide guidance on the future design of positional encodings for graph transformers.

Laplace learning algorithms for graph-based semi-supervised learning have been shown to produce degenerate predictions at low label rates and in imbalanced class regimes, particularly near class boundaries. We propose CutSSL: a framework for graph-based semi-supervised learning based on continuous nonconvex quadratic programming, which provably obtains \emph{integer} solutions. Our framework is naturally motivated by an \emph{exact} quadratic relaxation of a cardinality-constrained minimum-cut graph partitioning problem. Furthermore, we show our formulation is related to an optimization problem whose approximate solution is the mean-shifted Laplace learning heuristic, thus providing new insight into the performance of this heuristic. We demonstrate that CutSSL significantly surpasses the current state-of-the-art on k-nearest neighbor graphs and large real-world graph benchmarks across a variety of label rates, class imbalance, and label imbalance regimes. Our implementation is available on Colab\footnote{\url{https://colab.research.google.com/drive/1tGU5rxE1N5d0KGcNzlvZ0BgRc7_vob7b?usp=sharing}}.

The (combinatorial) graph Laplacian is a fundamental object in the analysis of, and optimization on, graphs. Via a topological view, this operator can be extended to a simplicial complex K and therefore offers a way to perform "signal processing" on p-(co)chains of K. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes K ↪ L connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we significantly expand the scope of the persistent Laplacian by generalizing it to a pair of weighted simplicial complexes connected by a weight preserving simplicial map f: K → L. Such a simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets, i.e. cases in which it does not hold that K ⊂ L. However, the simplicial map setting is much more challenging than the inclusion setting since the underlying algebraic structure is much more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint positive semi-definite operator to a given subspace. As a consequence of this relation, we prove that the qth persistent Betti number of the simplicial map f: K → L equals the nullity of the qth persistent Laplacian Δ_q^{K,L}. We then propose an algorithm for finding the matrix representation of Δ_q^{K,L} which in turn yields a fundamentally different algorithm for computing the qth persistent Betti number of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under weight-preserving simplicial maps and establish monotonicity results for their eigenvalues.

Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes. Understanding and mitigating oversquashing has recently received significant attention from the research community. In this paper, we continue this line of work by analyzing oversquashing through the lens of the *effective resistance* between nodes in the input graph. Effective resistance intuitively captures the ``strength'' of connection between two nodes by paths in the graph, and has a rich literature spanning many areas of graph theory. We propose to use *total effective resistance* as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use. We further develop an algorithm to identify edges to be added to an input graph to minimize the total effective resistance, thereby alleviating oversquashing. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies for improving the performance of GNNs.

Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, the higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encode both the homology of a data set, and some additional geometric information not captured by the homology. Here, we provide the first investigation into the efficacy of the persistence Laplacian as an embedding of data for downstream classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images, then evaluate its performance as an embedding method on the MNIST and MoleculeNet datasets, demonstrating that it consistently outperforms persistent homology across tasks.

The hyperbolic space is widely used for representing hierarchical datasets due to its ability to embed trees with small distortion. However, this property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we analyze the limitations of two popular models for the hyperbolic space, namely, the Poincaré ball and the Lorentz model. We find that, under the 64-bit arithmetic system, the Poincaré ball has a relatively larger capacity than the Lorentz model for correctly representing points. However, the Lorentz model is superior to the Poincaré ball from the perspective of optimization, which we theoretically validate. To address these limitations, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these issues. We further extend this Euclidean parametrization to hyperbolic hyperplanes and demonstrate its effectiveness in improving the performance of hyperbolic SVM.

The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and compares subtle structures of the underlying LMMCs and, as a consequence of this, it is more discriminating than the distance between graphs used for defining the state-of-the-art Wasserstein Weisfeiler-Lehman graph kernel. Inspired by the structure of the WL distance we identify a neural network architecture on LMMCs which turns out to be universal w.r.t. continuous functions defined on the space of all LMMCs (which includes all graphs) endowed with the WL distance. Finally, the WL distance turns out to be stable w.r.t. a natural variant of the Gromov-Wasserstein (GW) distance for comparing metric Markov chains that we identify. Hence, the WL distance can also be construed as a polynomial time lower bound for the GW distance which is in general NP-hard to compute.

We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces. The construction draws inspiration from Optimal Transportation ideas. We establish the stability of our method under data perturbation and, when the dataset is assumed to be Euclidean, we also exhibit a precise connection between the Wasserstein transform and the mean shift family of algorithms. We then use this connection to prove that mean shift also inherits stability under perturbations. We study the performance of the Wasserstein transform method on different datasets as a preprocessing step prior to clustering and classification tasks.