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In this paper, we formulate the deep residual network (ResNet) as a control problem of transport equation. In ResNet, the transport equation is solved along the characteristics. Based on this observation, deep neural network is closely related to the control problem of PDEs on manifold. We propose several models based on transport equation, Hamilton-Jacobi equation and Fokker-Planck equation. The discretization of these PDEs on point cloud is also discussed.

The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm, where $p$ is in $(0,1)$. In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the $l_p$ thresholding functions ($p$ is in $[0,1]$) are in closed form only for $p=0;1;1=2;2=3$, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds (TL1IT-s1 for short), consistently performs the best in sparse signal recovery with and without measurement noise.

For a random quantum state on $\cH=\C^d \otimes \C^d$ obtained by partial tracing a random pure state on $\cH \otimes \C^s$, we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold $s_0=s_0(d)$ of order roughly $d^3$. More precisely, for any $\e>0$ and for $d$ large enough, such a random state is entangled with very large probability when $s \leq (1-\e)s_0$, and separable with very large probability when $s \geq (1+\e) s_0$. One consequence of this result is as follows: for a system of $N$ identical particles in a random pure state, there is a threshold $k_0 = k_0(N)\sim N/5$ such that two subsystems of $k$ particles each typically share entanglement if $k>k_0$, and typically do not share entanglement if $k<k_0$. Our methods work also for multipartite systems and for ``unbalanced'' systems such as $\C^{d_1} \otimes \C^{d_2} $, ${d_1} \neq {d_2} $. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value.

We formulate the Riemannian calculus of the probability set embedded with L^2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability manifold as a submanifold of the positive measure space with a nonlinear metric tensor. Here the nonlinearity comes from the linear weighted Laplacian operator. By this viewpoint, we establish torsion-free Christoffel symbols, Levi-Civita connections, curvature tensors, and volume forms in the probability manifold by Euclidean coordinates. As a consequence, the Jacobi equation, Laplace-Beltrami and Hessian operators on the probability manifold are derived. These geometric computations are also provided in the infinite-dimensional density space (density manifold) supported on a finite-dimensional manifold. In particular, an identity is given connecting the Baker-{É}mery Γ2 operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.