Xinyu (Rain) Wei 魏昕雨

TpopT: Efficient Trainable Template Optimization on Low-Dimensional Manifolds

J. Yan, S. Wang, X. R. Wei, J. Wang, Z. Márka, S. Márka, and J. Wright

Apr 2023 – Jun 2024 · arXiv preprint, 2023

TpopT replaces exhaustive template-bank search with trainable optimization on a low-dimensional signal manifold. I led the core theoretical analysis, establishing curvature-dependent convergence guarantees for Riemannian gradient descent under Gaussian noise and estimation bounds governed primarily by intrinsic dimension. The analysis identifies the geometric conditions behind its computational advantage over matched filtering and the limits of that advantage.

Neural Network Theory: Training Dynamics and the Neural Tangent Kernel

T. Wang, X. R. Wei, and J. Wright

Apr 2022 – Apr 2023 · Collaborative theoretical research

How can we explain neural-network training as the network’s internal features change? We studied finite-width ReLU networks learning to classify data on low-dimensional manifolds. The neural tangent kernel (NTK) describes how parameter updates change predictions; approximating it by its value at initialization simplifies analysis but can impose restrictive conditions on network width and parameter movement.

Our approach investigates a time-dependent approximate NTK evaluated along a tractable proxy for the weight trajectory. The objective is to compare this approximation with the trained network and understand when useful kernel structure persists as features evolve.

I developed components of this framework, deriving forward-feature perturbation bounds and analyzing Gaussian small-ball estimates and changes in ReLU activation patterns. These calculations address how parameter movement changes intermediate representations and which neurons switch between active and inactive states—key steps in comparing the evolving kernels. In a related study of learning from noisy manifold observations, I contributed proofs involving local manifold geometry, smooth-function approximation, and small-norm kernel certificates, which connect the learning target to the kernel’s action.

This was my most sustained undergraduate research effort. The work produced intermediate theoretical analyses toward a broader convergence result; the full convergence theorem remains open.

Adaptive Statistical Estimation and Multi-Task Learning

Research with Kaizheng Wang

Aug 2022 – Jun 2024 · Research project

I studied Gaussian multi-task estimation to characterize when pooling related tasks improves on separate estimation, deriving the bias–variance tradeoff and the role of task heterogeneity. In related work, I derived K-fold cross-validation estimators for regularized Gaussian mean estimation and evaluated their risk against maximum-likelihood, Bayesian, and oracle benchmarks using analytical calculations and Monte Carlo experiments.

End-to-End Variational Inference for Robust Portfolio Construction

X. R. Wei, G. Costa, and G. N. Iyengar

Apr 2023 – Jun 2024 · Research project; poster presentation, 2023

This work integrates a variational inference neural network with a Monte Carlo-based portfolio-optimization layer, training the predictive distribution using both statistical and downstream task losses. I derived gradient expressions using KKT-based implicit differentiation and contributed to the PyTorch implementation and synthetic-data experiments evaluating decision quality.