For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In particular the relative rate of convergence is the square root of the sample size, which is known to be the best possible. The point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient, that is, an analog of the usual Fisher information, in this convergence. This establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform. It is seen that some existing methods attain the bound, others do not.

Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [LLY] that relates Gromov-Witten invariants and mirror symmetry computations. In this paper the A-twisted moduli stack for higher genus curves is defined and systematically studied. It is proved that such a moduli stack is an Artin stack. For genus 0, it has the A-twisted moduli space of [MP] as the coarse moduli space. The detailed proof of the regularity of the collapsing morphism by Jun Li in [LLY: I and II] can be viewed as a natural morphism from the moduli stack of genus 0 stable maps to the A-twisted moduli stack at genus 0.

We present a novel variation of the well-known infomax algorithm of blind source separation. Under natural gradient descent, the infomax algorithm converges to a stationary point of a limiting ordinary differential equation. However, due to the presence of saddle points or local minima of the corresponding likelihood function, the algorithm may be trapped around these bad stationary points for a long time, especially if the initial data are near them. To speed up convergence, we propose to add a sequence of random perturbations to the infomax algorithm to shake the iterating sequence so that it is captured by a path descending to a more stable stationary point. We analyze the convergence of the randomly perturbed algorithm, and illustrate its fast convergence through numerical examples on blind demixing of stochastic signals. The examples have analytical structures so that saddle points or local minima of the likelihood functions are explicit. The results may have implications for online learning algorithms in dissimilar problems.

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the l2 operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical Bakry–Émery curvature criterion implies subgaussian concentration for “matrix Lipschitz” functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron–Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.

The classification and construction of symmetry protected topological (SPT) phases have been intensively studied in interacting systems recently. To our surprise, in interacting fermion systems, there exists a new class of the so-called anomalous SPT (ASPT) states which are only well defined on the boundary of a trivial fermionic bulk system. We first demonstrate the essential idea by considering an anomalous topological superconductor with time-reversal symmetry T^2 = 1 in 2D. The physical reason for this is that the fermion parity might be changed locally by certain symmetry action, but it is conserved if we introduce a bulk. Then we discuss the layer structure and systematical construction of ASPT states in interacting fermion systems in 2D with a total symmetry G_f = G_b × \mathbb{Z}^f_2. Finally, potential experimental realizations of ASPT states are also addressed.