arXiv subjects

Ziang Chen

Publications and source records attributed to Ziang Chen.

48 records · Page 3Linked to original sources

arXiv2024

One-dimensional Tensor Network Recovery

We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.

math.NA↗

arXiv2023

Power-balanced Memristive Cryptographic Implementation Against Side Channel Attacks

Memristors, as emerging nano-devices, offer promising performance and exhibit rich electrical dynamic behavior. Having already found success in applications such as neuromorphic and in-memory computing, researchers are now exploring their potential for cryptographic implementations. In this study, we present a novel power-balanced hiding strategy utilizing memristor groups to conceal power consumption in cryptographic logic circuits. Our approach ensures consistent power costs of all 16 logic gates in Complementary-Resistive-Switching-with-Reading (CRS-R) logic family during writing and reading cycles regardless of Logic Input Variable (LIV) values. By constructing hiding groups, we enable an effective power balance in each gate hiding group. Furthermore, experimental validation of our strategy includes the implementation of a cryptographic construction, xor4SBox, using NOR gates. The circuit construction without the hiding strategy and with the hiding strategy undergo T-test analysis, confirming the significant improvement achieved with our approach. Our work presents a substantial advancement in power-balanced hiding methods, offering enhanced security and efficiency in logic circuits.

cs.CR↗

arXiv2023

On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross-Pitaevskii energy functional with respect to the $H^1_0$-metric and two other equivalent metrics on $H_0^1$, including the iterate-independent $a_0$-metric and the iterate-dependent $a_u$-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross-Pitaevskii energy for the discrete-time $H^1$ and $a_0$-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.

math.NA↗

arXiv2023

On Representing Linear Programs by Graph Neural Networks

Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML). In particular, the graph neural network (GNN) is considered a suitable ML model for optimization problems whose variables and constraints are permutation--invariant, for example, the linear program (LP). While the literature has reported encouraging numerical results, this paper establishes the theoretical foundation of applying GNNs to solving LPs. Given any size limit of LPs, we construct a GNN that maps different LPs to different outputs. We show that properly built GNNs can reliably predict feasibility, boundedness, and an optimal solution for each LP in a broad class. Our proofs are based upon the recently--discovered connections between the Weisfeiler--Lehman isomorphism test and the GNN. To validate our results, we train a simple GNN and present its accuracy in mapping LPs to their feasibilities and solutions.

cs.LG↗

arXiv2023

On Representing Mixed-Integer Linear Programs by Graph Neural Networks

While Mixed-integer linear programming (MILP) is NP-hard in general, practical MILP has received roughly 100--fold speedup in the past twenty years. Still, many classes of MILPs quickly become unsolvable as their sizes increase, motivating researchers to seek new acceleration techniques for MILPs. With deep learning, they have obtained strong empirical results, and many results were obtained by applying graph neural networks (GNNs) to making decisions in various stages of MILP solution processes. This work discovers a fundamental limitation: there exist feasible and infeasible MILPs that all GNNs will, however, treat equally, indicating GNN's lacking power to express general MILPs. Then, we show that, by restricting the MILPs to unfoldable ones or by adding random features, there exist GNNs that can reliably predict MILP feasibility, optimal objective values, and optimal solutions up to prescribed precision. We conducted small-scale numerical experiments to validate our theoretical findings.

cs.LG↗

arXiv2022

On the global convergence of randomized coordinate gradient descent for non-convex optimization

In this work, we analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes for non-convex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely escape strict saddle points of the objective function. As a result, the algorithm is guaranteed to converge to local minima if all saddle points are strict. Our proof is based on viewing coordinate descent algorithm as a nonlinear random dynamical system and a quantitative finite block analysis of its linearization around saddle points.

math.OC↗

arXiv2022

A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces

Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper we study the regularity of solutions to the whole-space static Schrödinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space $\mathcal{B}^s(\mathbb{R}^d)$ and the potential function admitting a non-negative lower bound decomposes as a positive constant plus a function in $\mathcal{B}^s(\mathbb{R}^d)$, then the solution lies in the spectral Barron space $\mathcal{B}^{s+2}(\mathbb{R}^d)$.

math.AP↗

arXiv2022

Physics inspired compact modelling of BiFeO$_3$ based memristors for hardware security applications

With the advent of the Internet of Things, nanoelectronic devices or memristors have been the subject of significant interest for use as new hardware security primitives. Among the several available memristors, BiFe$\rm O_{3}$ (BFO)-based electroforming-free memristors have attracted considerable attention due to their excellent properties, such as long retention time, self-rectification, intrinsic stochasticity, and fast switching. They have been actively investigated for use in physical unclonable function (PUF) key storage modules, artificial synapses in neural networks, nonvolatile resistive switches, and reconfigurable logic applications. In this work, we present a physics-inspired 1D compact model of a BFO memristor to understand its implementation for such applications (mainly PUFs) and perform circuit simulations. The resistive switching based on electric field-driven vacancy migration and intrinsic stochastic behaviour of the BFO memristor are modelled using the cloud-in-a-cell scheme. The experimental current-voltage characteristics of the BFO memristor are successfully reproduced. The response of the BFO memristor to changes in electrical properties, environmental properties (such as temperature) and stress are analyzed and consistent with experimental results.

cs.ET↗

arXiv2022

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.

cs.LG↗

arXiv2021

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is $ε$-close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension $d$ of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the $H^1$ norm with a dimension-explicit convergence rate.

math.NA↗

arXiv2021

A Trust-Region Method For Nonsmooth Nonconvex Optimization

We propose a trust-region type method for a class of nonsmooth nonconvex optimization problems where the objective function is a summation of a (probably nonconvex) smooth function and a (probably nonsmooth) convex function. The model function of our trust-region subproblem is always quadratic and the linear term of the model is generated using abstract descent directions. Therefore, the trust-region subproblems can be easily constructed as well as efficiently solved by cheap and standard methods. When the accuracy of the model function at the solution of the subproblem is not sufficient, we add a safeguard on the stepsizes for improving the accuracy. For a class of functions that can be "truncated", an additional truncation step is defined and a stepsize modification strategy is designed. The overall scheme converges globally and we establish fast local convergence under suitable assumptions. In particular, using a connection with a smooth Riemannian trust-region method, we prove local quadratic convergence for partly smooth functions under a strict complementary condition. Preliminary numerical results on a family of $\ell_1$-optimization problems are reported and demonstrate the efficiency of our approach.

math.OC↗

arXiv2020

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.

math.NA↗