Qianyu Julie Zhu

Ayoub Belhadji, Qianyu Julie Zhu, Youssef Marzouk

Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a shift-invariant kernel with a sum using a quadrature rule. The design of the latter is meant to reduce the number of features required for high-precision approximation. Specifically, for the squared exponential kernel, one must design a quadrature rule that approximates the Gaussian measure on $\mathbb{R}^d$. Previous efforts in this line of research have faced difficulties in higher dimensions. We introduce a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions by exploiting its isotropy. These rules are constructed as a tensor product of a radial quadrature rule and a spherical quadrature rule. Compared to previous work, our approach leverages a thorough analysis of the approximation error, which suggests natural choices for both the radial and spherical components. We demonstrate that this family of Fourier features yields improved approximation bounds.

Qidong Yang, Qianyu Julie Zhu, Jonathan Giezendanner et al.

Conditional generative models map input variables to complex, high-dimensional distributions, enabling realistic sample generation in a diverse set of domains. A critical challenge with these models is the absence of calibrated uncertainty, which undermines trust in individual outputs for high-stakes applications. To address this issue, we propose a systematic conformal prediction approach tailored to conditional generative models, leveraging density estimation on model-generated samples. We introduce a novel method called CP4Gen, which utilizes clustering-based density estimation to construct prediction sets that are less sensitive to outliers, more interpretable, and of lower structural complexity than existing methods. Extensive experiments on synthetic datasets and real-world applications, including climate emulation tasks, demonstrate that CP4Gen consistently achieves superior performance in terms of prediction set volume and structural simplicity. Our approach offers practitioners a powerful tool for uncertainty estimation associated with conditional generative models, particularly in scenarios demanding rigorous and interpretable prediction sets.

Qianyu Julie Zhu, Youssef Marzouk, Konstantinos Spiliopoulos et al.

Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.