Daily Papers

Self-attention is a ubiquitous primitive in modern sequence models, yet its operator-level geometry is only partially understood. We view a token sequence as a vector field over the token-position graph and identify attention as a connection walk: messages are aggregated by a nonnegative walk matrix while being transported along each edge by a learned linear map. Within this framework, we prove that single-head attention (SHA) is exactly a connection propagation step with constant transport, and that multi-head attention (MHA) is exactly a single edge-dependent connection walk whose effective transport is an attention-gated mixture of headwise transports. We further clarify the conditions under which the corresponding generator reduces to a random-walk connection Laplacian, highlighting the roles of stochasticity, reversibility, and metric-compatible transports. Empirically, we find that trained Transformers across scales (from 124M to 8B) and structures (encoder/decoder) exhibit geometric structure consistent with our theory: effective attention graphs converge to stable geometric operators in deeper layers, learned transports self-organize into approximate scaled isometries, and both phenomena strengthen consistently with scale. Overall, the paper provides a precise connection-walk formalism that links self-attention to classical geometric operators, along with a set of operator-level tools for analyzing transformer models from a geometric perspective.

We develop the GKM theory for the torus-equivariant cohomology of the affine flag variety using the combinatorics of alcove walks. Dual to the usual GKM setup, which depicts the orbits of the small torus action on a graph, alcove walks take place in tessellations of Euclidean space. Walks in affine rank two occur on triangulations of the plane, providing a more direct connection to splines used for approximating surfaces. Alcove walks in GKM theory also need not be minimal length, and can instead be randomly generated, giving rise to more flexible implementation. This work reinterprets and recovers classical results in GKM theory on the affine flag variety, generalizing them to both non-minimal and folded alcove walks, all motivated by applications to splines.

Virtual Reality (VR) is quickly establishing itself in various industries, including training, education, medicine, and entertainment, in which users are frequently required to carry out multiple complex cognitive and physical activities. However, the relationship between cognitive activities, physical activities, and familiar feelings of cybersickness is not well understood and thus can be unpredictable for developers. Researchers have previously provided labeled datasets for predicting cybersickness while users are stationary, but there have been few labeled datasets on cybersickness while users are physically walking. Thus, from 39 participants, we collected head orientation, head position, eye tracking, images, physiological readings from external sensors, and the self-reported cybersickness severity, physical load, and mental load in VR. Throughout the data collection, participants navigated mazes via real walking and performed tasks challenging their attention and working memory. To demonstrate the dataset's utility, we conducted a case study of training classifiers in which we achieved 95% accuracy for cybersickness severity classification. The noteworthy performance of the straightforward classifiers makes this dataset ideal for future researchers to develop cybersickness detection and reduction models. To better understand the features that helped with classification, we performed SHAP(SHapley Additive exPlanations) analysis, highlighting the importance of eye tracking and physiological measures for cybersickness prediction while walking. This open dataset can allow future researchers to study the connection between cybersickness and cognitive loads and develop prediction models. This dataset will empower future VR developers to design efficient and effective Virtual Environments by improving cognitive load management and minimizing cybersickness.

We study the black-box attacks on graph neural networks (GNNs) under a novel and realistic constraint: attackers have access to only a subset of nodes in the network, and they can only attack a small number of them. A node selection step is essential under this setup. We demonstrate that the structural inductive biases of GNN models can be an effective source for this type of attacks. Specifically, by exploiting the connection between the backward propagation of GNNs and random walks, we show that the common gradient-based white-box attacks can be generalized to the black-box setting via the connection between the gradient and an importance score similar to PageRank. In practice, we find attacks based on this importance score indeed increase the classification loss by a large margin, but they fail to significantly increase the mis-classification rate. Our theoretical and empirical analyses suggest that there is a discrepancy between the loss and mis-classification rate, as the latter presents a diminishing-return pattern when the number of attacked nodes increases. Therefore, we propose a greedy procedure to correct the importance score that takes into account of the diminishing-return pattern. Experimental results show that the proposed procedure can significantly increase the mis-classification rate of common GNNs on real-world data without access to model parameters nor predictions.