The ability to perform fast and accurate atomistic simulations is crucial for advancing the chemical sciences. By learning from high-quality data, machine-learned interatomic potentials achieve accuracy on par with ab initio and first-principles methods at a fraction of their computational cost. The success of machine-learned interatomic potentials arises from integrating inductive biases such as equivariance to group actions on an atomic system, e.g., equivariance to rotations and reflections. In particular, the field has notably advanced with the emergence of equivariant message passing. Most of these models represent an atomic system using spherical tensors, tensor products of which require complicated numerical coefficients and can be computationally demanding. Cartesian tensors offer a promising alternative, though state-of-the-art methods lack flexibility in message-passing mechanisms, restricting their architectures and expressive power. This work explores higher-rank irreducible Cartesian tensors to address these limitations. We integrate irreducible Cartesian tensor products into message-passing neural networks and prove the equivariance and traceless property of the resulting layers. Through empirical evaluations on various benchmark data sets, we consistently observe on-par or better performance than that of state-of-the-art spherical and Cartesian models.

Graph coarsening aims to reduce the size of a large graph while preserving some of its key properties, which has been used in many applications to reduce computational load and memory footprint. For instance, in graph machine learning, training Graph Neural Networks (GNNs) on coarsened graphs leads to drastic savings in time and memory. However, GNNs rely on the Message-Passing (MP) paradigm, and classical spectral preservation guarantees for graph coarsening do not directly lead to theoretical guarantees when performing naive message-passing on the coarsened graph. In this work, we propose a new message-passing operation specific to coarsened graphs, which exhibit theoretical guarantees on the preservation of the propagated signal. Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is often oriented, even when the original graph is undirected. We conduct node classification tasks on synthetic and real data and observe improved results compared to performing naive message-passing on the coarsened graph.

The last decade has witnessed an explosion in the development of models, theory and computational algorithms for "big data" analysis. In particular, distributed computing has served as a natural and dominating paradigm for statistical inference. However, the existing literature on parallel inference almost exclusively focuses on Euclidean data and parameters. While this assumption is valid for many applications, it is increasingly more common to encounter problems where the data or the parameters lie on a non-Euclidean space, like a manifold for example. Our work aims to fill a critical gap in the literature by generalizing parallel inference algorithms to optimization on manifolds. We show that our proposed algorithm is both communication efficient and carries theoretical convergence guarantees. In addition, we demonstrate the performance of our algorithm to the estimation of Fréchet means on simulated spherical data and the low-rank matrix completion problem over Grassmann manifolds applied to the Netflix prize data set.

In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable AMP iterates.

Message passing plays a vital role in graph neural networks (GNNs) for effective feature learning. However, the over-reliance on input topology diminishes the efficacy of message passing and restricts the ability of GNNs. Despite efforts to mitigate the reliance, existing study encounters message-passing bottlenecks or high computational expense problems, which invokes the demands for flexible message passing with low complexity. In this paper, we propose a novel dynamic message-passing mechanism for GNNs. It projects graph nodes and learnable pseudo nodes into a common space with measurable spatial relations between them. With nodes moving in the space, their evolving relations facilitate flexible pathway construction for a dynamic message-passing process. Associating pseudo nodes to input graphs with their measured relations, graph nodes can communicate with each other intermediately through pseudo nodes under linear complexity.

With the increasing size of data and models, today's instances demand parallel and distributed computing for real-world machine

Message Passing Neural Networks (MPNNs) have emerged as the de facto standard in graph representation learning. However, when it comes to link prediction, they are not always superior to simple heuristics such as Common Neighbor (CN). This discrepancy stems from a fundamental limitation: while MPNNs excel in node-level representation, they stumble with encoding the joint structural features essential to link prediction, like CN. To bridge this gap, we posit that, by harnessing the orthogonality of input vectors, pure message-passing can indeed capture joint structural features. Specifically, we study the proficiency of MPNNs in approximating CN heuristics. Based on our findings, we introduce the Message Passing Link Predictor (MPLP), a novel link prediction model. MPLP taps into quasiorthogonal vectors to estimate link-level structural features, all while preserving the node-level complexities. We conduct experiments on benchmark datasets from various domains, where our method consistently outperforms the baseline methods, establishing new state-of-the-arts.

Figure 3: The rooted subtree of node v1 with 1-hop message passing and K-hop message passing. Here we assume that K = 2 and the number of layers is 2. In this section, we further discuss two different types of K-hop kernel and K-hop message passing. A.1 More about K-hop kernel First, recall the shortest path distance kernel and graph diffusion kernel defined in Definition 1 and 2. Given two definitions, the first thing we can conclude is that the K-hop neighbors of node v under two different kernels will be the same, namely N A.2 More about K-hop message passing Here, we use an example shown in Figure 3 to illustrate how K-hop message passing works and compare it with 1-hop message passing. The input graph is shown on the left top of the figure. Suppose we want to learn the representation of node v1 using 2 layer message passing GNNs. First, if we perform 1-hop message passing, it will encode a 2-height rooted subtree, which is shown on the right top of the figure. Note that each node is learned using the same set of parameters, which is indicated by filling each node with the same color (white in the figure). Now, we consider performing a 2-hop message passing GNN with the shortest path distance kernel. The rooted subtree of node v1 is shown in the middle of the figure. Furthermore, different sets of parameters are used for different hops, which is indicated by filling nodes in the different hops with different colors (blue for 1st hop and yellow for 2nd hop). Finally, at the bottom of the figure, we show the 2-hop message passing GNN with graph diffusion kernel. It is easy to see the rooted subtree is different from the one that uses the shortest path distance kernel, as nodes can appear in both the 1st hop and 2nd hop of neighbors.