Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo

To sample from a general target distribution $p_*\propto e^{-f_*}$ beyond the isoperimetric condition, Huang et al. (2023) proposed to perform sampling through reverse diffusion, giving rise to Diffusion-based Monte Carlo (DMC). Specifically, DMC follows the reverse SDE of a diffusion process that transforms the target distribution to the standard Gaussian, utilizing a non-parametric score estimation. However, the original DMC algorithm encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $\epsilon$ of the obtained samples. In this paper, we demonstrate that the high complexity of DMC originates from its redundant design of score estimation, and proposed a more efficient algorithm, called RS-DMC, based on a novel recursive score estimation method. In particular, we first divide the entire diffusion process into multiple segments and then formulate the score estimation step (at any time step) as a series of interconnected mean estimation and sampling subproblems accordingly, which are correlated in a recursive manner. Importantly, we show that with a proper design of the segment decomposition, all sampling subproblems will only need to tackle a strongly log-concave distribution, which can be very efficient to solve using the Langevin-based samplers with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC only has a quasi-polynomial dependency on $\epsilon$, which significantly improves exponential gradient complexity in Huang et al. (2023). Furthermore, under commonly used dissipative conditions, our algorithm is provably much faster than the popular Langevin-based algorithms. Our algorithm design and theoretical framework illuminate a novel direction for addressing sampling problems, which could be of broader applicability in the community.Comment: 54 page

Highly tunable Terahertz filter with magneto-optical Bragg grating formed in semiconductor-insulator-semiconductor waveguides

A highly tunable terahertz (THz) filter with magneto-optical Bragg grating formed in semiconductor-insulator-semiconductor waveguides is proposed and demonstrated numerically by means of the Finite Element Method. The results reveal that a sharp peak with high Q-value presents in the band gap of Bragg grating waveguide with a defect, and the position of the sharp peak can be modified greatly by changing the intensity of the transverse magnetic field applied to the device. Compared to the situation without magnetic field applied, the shift of the filtered frequency (wavelength) reaches up to 36.1 GHz (11.4 μm) when 1 T magnetic field is applied. In addition, a simple model to predict the filtered frequency and an effective way to improve the Q-value of the filter are proposed by this paper