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Ziang Chen

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Artificial Intelligence Machine Learning math-ph math.CA math.CO math.MP math.NA math.PR Numerical Analysis physics.comp-ph

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preprint2026arXiv

Almost-Orthogonality in Lp Spaces: A Case Study with Grok

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have $\Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p},$ where $c=2$ , $1/p+1/p'=1$ , and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$ . In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$ . We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$ . Finally, at the critical exponent $c=p'$ , we establish the inequality for all integer values $p\ge 2$ . In the second part of the paper we obtain a sharp three-function bound $\Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p},$ where $p \geq 3$ , $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$ . The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

preprint2022arXiv

Dual reparametrized Variational Generative Model for Time-Series Forecasting

This paper propose DualVDT, a generative model for Time-series forecasting. Introduced dual reparametrized variational mechanisms on variational autoencoder (VAE) to tighter the evidence lower bound (ELBO) of the model, prove the advance performance analytically. This mechanism leverage the latent score based generative model (SGM), explicitly denoising the perturbation accumulated on latent vector through reverse time stochastic differential equation and variational ancestral sampling. The posterior of denoised latent distribution fused with dual reparametrized variational density. The KL divergence in ELBO will reduce to reach the better results of the model. This paper also proposed a latent attention mechanisms to extract multivariate dependency explicitly. Build the local temporal dependency simultaneously in factor wised through constructed local topology and temporal wised. The proven and experiment on multiple datasets illustrate, DualVDT, with a novel dual reparametrized structure, which denoise the latent perturbation through the reverse dynamics combining local-temporal inference, has the advanced performance both analytically and experimentally.

preprint2020arXiv

Tensor Ring Decomposition: Optimization Landscape and One-loop Convergence of Alternating Least Squares

In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spurious local minima for the optimization landscape even when the tensor ring format is much over-parameterized, i.e., with bond dimension much larger than that of the true target tensor. On the other hand, when the bond dimension is further increased, we establish one-loop convergence for alternating least square algorithm for tensor ring decomposition. The theoretical results are complemented by numerical experiments for both local minimum and one-loop convergence for the alternating least square algorithm.

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