Training deep neural networks (DNNs) is an important and challenging optimization problem in machine learning due to its non-convexity non-separable structure. The alternating minimization (AM) approaches split the composition structure of DNNs have drawn great interest communities. In this paper, we propose a unified framework for analyzing convergence rate AM-type network training methods. Ou...

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG ×BunH yields the theta-lifting functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories. Assu...

We study model transition for representations occurring in the local theta correspondence between split even special orthogonal groups and symplectic groups, over a non-archimedean local field of characteristic zero.

In Brownian last-passage percolation (BLPP), the Busemann functions $\mathcal B^{\theta}(\mathbf x,\mathbf y)$ are indexed by two points $\mathbf y \in \mathbb Z \times R$, and a direction parameter $\theta > 0$. We derive joint distribution of across all directions. The set directions where process is discontinuous, denoted $\Theta$, provides detailed information about uniqueness coalescence s...

In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, b...

The free vibration characteristics of a dual split torque transmission are studied in this work. A transversetorsional coupling dynamic model is developed using lumped parameter method. The natural frequencies and mode shapes are calculated by solving the eigenvalue problem of the differential equations of motion. The vibration modes of the system are classified into translational mode and rota...

We analyze a numerical instability that occurs in the well-known split-step Fourier method on the background of a soliton. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finite-difference schemes. Moreover, the principle of “frozen coefficients”, in which variable coefficients are treat...

We analyze a numerical instability that occurs in the well-known split-step Fourier method on the background of a soliton. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finite-difference schemes. Moreover, the principle of “frozen coefficients”, in which variable coefficients are treat...