SE(3)-MeanFlow: Few-Step Protein Backbone Generation on Lie Groups

Generative modeling of protein backbones promises the de novo design of proteins with prescribed structural and functional properties. Existing diffusion and flow-matching models produce high-quality backbones on SE(3)^N, but inference requires numerically integrating an ODE over hundreds of network evaluations, each involving a Lie group exponential map - a bottleneck for high-throughput design campaigns. We introduce SE(3)-MeanFlow, a few-step generative framework that extends MeanFlow from Euclidean space to the Lie group geometry of protein frames. Working natively in the Lie algebra so(3) and in R^3, we derive closed-form average-velocity identities for rotations and translations, giving simulation-free training targets. We further introduce an SE(3) alpha-Flow objective that removes the Jacobian-vector product from the rotation branch and serves as a warm-up stage, after which training switches to a small-t stabilized MeanFlow loss that is used for the remainder of pretraining and for rectification-based post-training. In protein backbone generation, SE(3)-MeanFlow matches or exceeds flow-matching baselines that use several times more sampling steps, and its advantage widens in the few-step regime, where rectification lets it lead at every matched budget - at a modest cost in diversity.

The Distribution of Dependency Distance and Hierarchical Distance in Contemporary Written Japanese and Its Influencing Factors

To explore the relationship between dependency distance (DD) and hierarchical distance (HD) in Japanese, we compared the probability distributions of DD and HD with and without sentence length fixed, and analyzed the changes in mean dependency distance (MDD) and mean hierarchical distance (MHD) as sentence length increases, along with their correlation coefficient based on the Balanced Corpus of Contemporary Written Japanese. It was found that the valency of the predicates is the underlying factor behind the trade-off relation between MDD and MHD in Japanese. Native speakers of Japanese regulate the linear complexity and hierarchical complexity through the valency of the predicates, and the relative sizes of MDD and MHD depend on whether the threshold of valency has been reached. Apart from the cognitive load, the valency of the predicates also affects the probability distributions of DD and HD. The effect of the valency of the predicates on the distribution of HD is greater than on that of DD, which leads to differences in their probability distributions and causes the mean of MDD to be lower than that of MHD.