In DM, kernels play an important role for capturing the nonlinear relationship of data.

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

This paper introduces a comprehensive unified framework for constructing multi-view diffusion geometries through intertwined multi-view diffusion trajectories (MDTs), a class of inhomogeneous diffusion processes that iteratively combine the random walk operators of multiple data views. Each MDT defines a trajectory-dependent diffusion operator with a clear probabilistic and geometric interpretation, capturing over time the interplay between data views. Our formulation encompasses existing multi-view diffusion models, while providing new degrees of freedom for view interaction and fusion. We establish theoretical properties under mild assumptions, including ergodicity of both the point-wise operator and the process in itself. We also derive MDT-based diffusion distances, and associated embeddings via singular value decompositions. Finally, we propose various strategies for learning MDT operators within the defined operator space, guided by internal quality measures. Beyond enabling flexible model design, MDTs also offer a neutral baseline for evaluating diffusion-based approaches through comparison with randomly selected MDTs. Experiments show the practical impact of the MDT operators in a manifold learning and data clustering context.

Textual network embedding leverages rich text information associated with the network to learn low-dimensional vectorial representations of vertices.

Markov state trajectories, where the transition kernel has a small intrinsic rank. In the spirit of diffusion map, we propose an efficient method for learning a low-dimensional state embedding and capturing the process's dynamics.

"I would like to see Theorem 4 reworded. It assumes that the underlying process has a correct clustering of states?" Thm 4 assumes there is an underlying partition that attains the smallest value of distortion (eq(4)). We will reword Thm 4 to make it easier to interpret. "How to find state pairs in DQN analysis" Then we screened the top 100 closest pairs and pick those with large raw-data distances.

We demonstrate that our visualization provides intuitive, detailed summaries of the learning dynamics beyond simple global measures (i.e., validation loss and accuracy), without the need to access validation data. Furthermore, M-PHA TE better captures both the dynamics and community structure of the hidden units as compared to visualization based on standard dimensionality reduction methods (e.g., ISOMAP, t-SNE). We demonstrate M-PHA TE with two vignettes: continual learning and generalization.

Fighting with the curse of dimensionality by leveraging low-dimensional intrinsic structures has become an important guiding principle in modern data science.

Textual network embedding leverages rich text information associated with the network to learn low-dimensional vectorial representations of vertices.

On the theoretical front, we analyze our approach with a toy example based on the stochastic block model.