6 citations · 18 across the 13 of their papers we have counts for
13 papers
Pierce-Birkhoff conjecture is true for splines
Zehua Lai, Lek-Heng Lim
We prove the Pierce--Birkhoff conjecture for splines, i.e., continuous piecewise polynomials of degree in variables on a hyperplane partition of , can be writ…
Stiefel optimization is NP-hard
Zehua Lai, Lek-Heng Lim, Tianyun Tang
We show that linearly constrained linear optimization over a Stiefel or Grassmann manifold is NP-hard in general. We show that the same is true for unconstrained quadratic optimiza…
Euclidean distance degree in manifold optimization
Zehua Lai, Lek-Heng Lim, Ke Ye
We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we wi…
Attention is a smoothed cubic spline
Zehua Lai, Lek-Heng Lim, Yucong Liu
We highlight a perhaps important but hitherto unobserved insight: The attention module in a transformer is a smoothed cubic spline. Viewed in this manner, this mysterious but criti…
Grassmannian optimization is NP-hard
Zehua Lai, Lek-Heng Lim, Ke Ye
We show that unconstrained quadratic optimization over a Grassmannian is NP-hard. Our results cover all scenarios: (i) when and are both allowed to…
Simple matrix expressions for the curvatures of Grassmannian
Zehua Lai, Lek-Heng Lim, Ke Ye
We show that modeling a Grassmannian as symmetric orthogonal matrices $\operatorname{Gr}(k,\mathbb{R}^n) \cong\{Q \in \mathbb{R}^{n \times n} : Q^{\scriptscriptstyle\mathsf{T}} Q =…
Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent
Xiangyu Chang, Xi Chen, Zehua Lai +3
With the fast development of big data, learning the optimal decision rule by recursively updating it and making online decisions has been easier than before. We study the online st…
Simpler flag optimization
Zehua Lai, Lek-Heng Lim, Ke Ye
We study the geometry of flag manifolds under different embeddings into a product of Grassmannians. We show that differential geometric objects and operations -- tangent vector, me…
Haagerup bound for quaternionic Grothendieck inequality
Shmuel Friedland, Zehua Lai, Lek-Heng Lim
We present here several versions of the Grothendieck inequality over the skew field of quaternions: The first one is the standard Grothendieck inequality for rectangular matrices,…
Stochastic Steffensen method
Minda Zhao, Zehua Lai, Lek-Heng Lim
Is it possible for a first-order method, i.e., only first derivatives allowed, to be quadratically convergent? For univariate loss functions, the answer is yes -- the Steffensen me…
Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods
Xi Chen, Zehua Lai, He Li +1
This paper investigates the problem of online statistical inference of model parameters in stochastic optimization problems via the Kiefer-Wolfowitz algorithm with random search di…
Simpler Grassmannian optimization
Zehua Lai, Lek-Heng Lim, Ke Ye
There are two widely used models for the Grassmannian , as the set of equivalence classes of orthogonal matrices $\operatorname{O}(n)/(\operatorname{O}(k) \…
Recht-Ré Noncommutative Arithmetic-Geometric Mean Conjecture is False
Zehua Lai, Lek-Heng Lim
Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement…