Organizational meeting of the representation theory seminar

Regression analysis with probability measures as input predictors and output response has recently drawn great attention. However, it is challenging to handle multiple input probability measures due to the non-flat Riemannian geometry of the Wasserstein space, hindering the definition of arithmetic operations, hence additive linear structure is not well-defined. In this talk, a distribution-in-distribution-out regression model is proposed by introducing parallel transport to achieve provable commutativity and additivity of newly defined arithmetic operations in Wasserstein space. The appealing properties of the DIDO regression model can serve as a foundation for model estimation, prediction, and inference. Specifically, the Fréchet least squares estimator is employed to obtain the best linear unbiased estimate, supported by the newly established Fréchet Gauss-Markov Theorem. Furthermore, we investigate a special case when predictors and response are all univariate Gaussian measures, leading to a simple close-form solution of linear model coefficients and R^{2}metric. A simulation study and real case study in intra-operative cardiac output prediction are performed to evaluate the performance of the proposed method. Broader opportunities will be discussed.

During neural network training, the sharpness of the Hessian matrix of the training loss rises until training is on the edge of stability. As a result, even non-stochastic gradient descent does not accurately model the underlying dynamical system defined by the gradient flow of the training loss. We treat neural network training as a system of stiff ordinary differential equations and use an exponential Euler solver to train the network without entering the edge of stability. We demonstrate experimentally that the increase in the sharpness of the Hessian matrix is caused by the layerwise Jacobian matrices of the network becoming aligned, so that a small change in the preactivations at the front of the network can cause a large change in the outputs at the back of the network. We further demonstrate that the degree of layerwise Jacobian alignment scales with the size of the dataset by a power law with a coefficient of determination between 0.74 and 0.97.

Representation Theory seminar, Xinchun Ma.

We will define Dehn twists in all dimensions, discuss their properties, and explain how to calculate their orders in topological and smooth mapping class groups.

Plasma instabilities are a major concern in plasma science, for applications ranging from particle accelerators to nuclear fusion reactors. In this work, we consider the possibility of controlling such instabilities by adding an external electric field to the Vlasov–Poisson equations. Our approach to determining the external electric field is based on conducting a linear analysis of the resulting equations. We show that it is possible to select external electric fields that completely suppress the plasma instabilities present in the system when the equilibrium distribution and the perturbation are known. In fact, the proposed strategy returns the plasma to its equilibrium at a rate that is faster than exponential in time if the Fourier transform of the initial data decays super-exponentially with respect to the Fourier variable corresponding to velocity. We further perform numerical simulations of the nonlinear two-stream and bump-on-tail instabilities to verify our theory and to compare the different strategies that we propose in this work.

Representation theory seminar by Chengze Duan