arXiv subjects

Brian Choi

Publications and source records attributed to Brian Choi.

16 recordsLinked to original sources

arXiv2025

ChainMarks: Securing DNN Watermark with Cryptographic Chain

With the widespread deployment of deep neural network (DNN) models, dynamic watermarking techniques are being used to protect the intellectual property of model owners. However, recent studies have shown that existing watermarking schemes are vulnerable to watermark removal and ambiguity attacks. Besides, the vague criteria for determining watermark presence further increase the likelihood of such attacks. In this paper, we propose a secure DNN watermarking scheme named ChainMarks, which generates secure and robust watermarks by introducing a cryptographic chain into the trigger inputs and utilizes a two-phase Monte Carlo method for determining watermark presence. First, ChainMarks generates trigger inputs as a watermark dataset by repeatedly applying a hash function over a secret key, where the target labels associated with trigger inputs are generated from the digital signature of model owner. Then, the watermarked model is produced by training a DNN over both the original and watermark datasets. To verify watermarks, we compare the predicted labels of trigger inputs with the target labels and determine ownership with a more accurate decision threshold that considers the classification probability of specific models. Experimental results show that ChainMarks exhibits higher levels of robustness and security compared to state-of-the-art watermarking schemes. With a better marginal utility, ChainMarks provides a higher probability guarantee of watermark presence in DNN models with the same level of watermark accuracy.

cs.CR↗

arXiv2025

Nonexistence of traveling wave solutions in the fractional Rosenau-Hyman equation via homotopy perturbation method

We apply the homotopy perturbation method to construct series solutions for the fractional Rosenau-Hyman (fRH) equation and study their dynamics. Unlike the classical RH equation where compactons arise from truncated periodic solutions, we show that spatial nonlocality prevents the existence of compactons, and therefore periodic traveling waves are considered. By asymptotic analyses involving the Mittag-Leffler function, it is shown that the quadratic fRH equation exhibits bifurcation with respect to the order of the temporal fractional derivative, leading to the eventual pinning of wave propagation. Additionally, numerical results suggest potential finite time blow-up in the cubic fRH. While HPM proves effective in constructing analytic solutions, we identify cases of divergence, underscoring the need for further research into its convergence properties and broader applicability.

math.AP↗

arXiv2024

Nonlinear excitations in multi-dimensional nonlocal lattices

We study the formation of breathers in multi-dimensional lattices with long-range interactions. By variational methods, the exact relationship between various parameters (dimension, nonlinearity, nonlocal parameter $\alpha$) that defines positive excitation thresholds is characterized. We establish a sharp mass-threshold dichotomy: no positive threshold in the mass-subcritical regime, and a strictly positive threshold at and above the critical regime. In the anti-continuum regime, a family of unique ground states characterizes the excitation thresholds, enabling explicit computations. Analytic formulas of the excitation thresholds, determined by the ground states, are derived and corroborated with numerical simulations. We not only characterize the sharp spatial decay of ground states, which varies continuously in $\alpha$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $\alpha$.

nlin.PS↗

arXiv2024

Dynamics of periodic fractional discrete NLS in the continuum limit

The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that the discrete system converges to the fractional NLS as $h \rightarrow 0$ below the energy space by directly estimating the difference between the discrete and continuum solutions in $L^2(\mathbb{T})$ using the discrete periodic Strichartz estimates. The sharp convergence rate via the finite difference method (FDM) is shown to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. To further illustrate the convergent behavior of fDNLS, we survey various dynamical behaviors of the continuous wave (CW) solutions in the context of modulational instability, emphasizing the interplay between linear dispersion (or lattice diffraction), characterized by the nonlocal lattice coupling, and nonlinearity. In particular, the transition as $h \rightarrow 0$ from the linear dependence of maximum gain $\Omega_m$ on the amplitude $A$ of CW solutions to the quadratic dependence is shown analytically and numerically.

math.AP↗

arXiv2023

Uncertainty quantification and complex analyticity of the nonlinear Poisson-Boltzmann equation for the interface problem with random domains

The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification.

math.NA↗

arXiv2023

Analytic regularity of strong solutions for the complexified stochastic non-linear Poisson Boltzmann Equation

Semi-linear elliptic Partial Differential Equations (PDEs) such as the non-linear Poisson Boltzmann Equation (nPBE) is highly relevant for non-linear electrostatics in computational biology and chemistry. It is of particular importance for modeling potential fields from molecules in solvents or plasmas with stochastic fluctuations. The extensive applications include ones in condensed matter and solid state physics, chemical physics, electrochemistry, biochemistry, thermodynamics, statistical mechanics, and materials science, among others. In this paper we study the complex analytic properties of semi-linear elliptic Partial Differential Equations with respect to random fluctuations on the domain. We first prove the existence and uniqueness of the nPBE on a bounded domain in $\mathbb{R}^3$. This proof relies on the application of a contraction mapping reasoning, as the standard convex optimization argument for the deterministic nPBE no longer applies. Using the existence and uniqueness result we subsequently show that solution to the nPBE admits an analytic extension onto a well defined region in the complex hyperplane with respect to the number of stochastic variables. Due to the analytic extension, stochastic collocation theory for sparse grids predict algebraic to sub-exponential convergence rates with respect to the number of knots. A series of numerical experiments with sparse grids is consistent with this prediction and the analyticity result. Finally, this approach readily extends to a wide class of semi-linear elliptic PDEs.

math.NA↗

arXiv2023

Localization and global dynamics in the long-range discrete nonlinear Schr\"odinger equation

We study localization, pinning, and mobility in the fractional discrete nonlinear Schr\"odinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $\alpha$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $\alpha$.

math.CA↗

arXiv2023

Radiomap Inpainting for Restricted Areas based on Propagation Priority and Depth Map

Providing rich and useful information regarding spectrum activities and propagation channels, radiomaps characterize the detailed distribution of power spectral density (PSD) and are important tools for network planning in modern wireless systems. Generally, radiomaps are constructed from radio strength measurements by deployed sensors and user devices. However, not all areas are accessible for radio measurements due to physical constraints and security consideration, leading to non-uniformly spaced measurements and blanks on a radiomap. In this work, we explore distribution of radio spectrum strengths in view of surrounding environments, and propose two radiomap inpainting approaches for the reconstruction of radiomaps that cover missing areas. Specifically, we first define a propagation-based priority and integrate exemplar-based inpainting with radio propagation model for fine-resolution small-size missing area reconstruction on a radiomap. Then, we introduce a novel radio depth map and propose a two-step template-perturbation approach for large-size restricted region inpainting. Our experimental results demonstrate the power of the proposed propagation priority and radio depth map in capturing the PSD distribution, as well as the efficacy of the proposed methods for radiomap reconstruction.

eess.SP↗

arXiv2023

Speed of propagation of fractional dispersive waves

In this paper, we show that all non-trivial solutions of a broad class of nonlinear dispersive equations, whose linear evolution is governed by a dispersion relation under minimal regularity assumptions, cannot remain compactly supported for any non-trivial time interval. Our approach, based on complex-analytic arguments and the Paley-Wiener-Schwartz theorem, yields a stronger result: if linear solutions are compactly supported at two distinct times, then the dispersion relation must admit an analytic extension. This extends previous results beyond polynomial dispersion relations and applies to more general settings, including fractional-order systems. As an application, we examine the generalized space-time fractional Schr\"{o}dinger equation, illustrating the role of memory effects in wave propagation.

math.AP↗

arXiv2022

Exemplar-Based Radio Map Reconstruction of Missing Areas Using Propagation Priority

Radio map describes network coverage and is a practically important tool for network planning in modern wireless systems. Generally, radio strength measurements are collected to construct fine-resolution radio maps for analysis. However, certain protected areas are not accessible for measurement due to physical constraints and security considerations, leading to blanked spaces on a radio map. Non-uniformly spaced measurement and uneven observation resolution make it more difficult for radio map estimation and spectrum planning in protected areas. This work explores the distribution of radio spectrum strengths and proposes an exemplar-based approach to reconstruct missing areas on a radio map. Instead of taking generic image processing approaches, we leverage radio propagation models to determine directions of region filling and develop two different schemes to estimate the missing radio signal power. Our test results based on high-fidelity simulation demonstrate efficacy of the proposed methods for radio map reconstruction.

eess.SP↗

arXiv2022

Continuum Limit of 2D Fractional Nonlinear Schr\"odinger Equation

We prove that the solutions to the discrete Nonlinear Schr\"odinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schr\"odinger Equation (FNLSE), as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $\alpha \in (1,2)$ approaches the boundaries.

math.AP↗

arXiv2021

Well-posedness of the mixed-Fractional Nonlinear Schr\"odinger Equation on $\mathbb{R}^2$

We investigate the well-posedness theory of the 2-D fractional nonlinear Schr\"odinger equation (NLSE) with a mixed degree of derivatives. Motivated by models in optics and photonics where the light propagation is governed by non-quadratic, fractional, and anisotropic dispersion profile, this paper presents first results in this direction. Dispersive estimates are developed in the context of anisotropic Sobolev spaces defined by inhomogeneous symbols. The main model is shown to exhibit scattering for small data in the scaling-critical space. Furthermore the continuity of solution with respect to the dispersion parameter is shown on a compact time interval.

math.AP↗

arXiv2021

Existence of Strong Solution for the Complexified Non-linear Poisson Boltzmann Equation

We prove the existence and uniqueness of the complexified Nonlinear Poisson-Boltzmann Equation (nPBE) in a bounded domain in $\mathbb{R}^3$. The nPBE is a model equation in nonlinear electrostatics. The standard convex optimization argument to the complexified nPBE no longer applies, but instead, a contraction mapping argument is developed. Furthermore, we show that uniqueness can be lost if the hypotheses given are not satisfied. The complixified nPBE is highly relevant to regularity analysis of the solution of the real nPBE with respect to the dielectric (diffusion) and Debye-H\"uckel coefficients. This approach is also well-suited to investigate the existence and uniqueness problem for a wide class of semi-linear elliptic Partial Differential Equations (PDEs).

math.AP↗

arXiv2020

Observations on Porting In-memory KV stores to Persistent Memory

Systems that require high-throughput and fault tolerance, such as key-value stores and databases, are looking to persistent memory to combine the performance of in-memory systems with the data-consistent fault-tolerance of nonvolatile stores. Persistent memory devices provide fast bytea-ddressable access to non-volatile memory. We analyze the design space when integrating persistent memory into in-memory key value stores and quantify performance tradeoffs between throughput, latency, and and recovery time. Previous works have explored many design choices, but did not quantify the tradeoffs. We implement persistent memory support in Redis and Memcached, adapting the data structures of each to work in two modes: (1) with all data in persistent memory and (2) a hybrid mode that uses persistent memory for key/value data and non-volatile memory for indexing and metadata. Our experience reveals three actionable design principles that hold in Redis and Memcached, despite their very different implementations. We conclude that the hybrid design increases throughput and decreases latency at a minor cost in recovery time and code complexity

cs.DB↗

arXiv2019

Small Time Behavior and Summability for the Schr\"odinger Equation

We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schr\"odinger equation, both linear and nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the Schr\"odinger flow, that the (sharp) result proved by Dahlberg and Kenig for initial data in Sobolev spaces still holds when one considers the full Schr\"odinger equation with a certain class of potentials. As for $s<\frac{1}{4}$, the failure of $L^p$-boundedness of the (localized) maximal operator is investigated.

math.AP↗

arXiv2019

Remark on the Adiabatic Limit of Quantum Zakharov System

This paper is concerned with the low regularity well-posedness of the adiabatic limit of the quantum Zakharov system, which is a modified nonlinear Schr\"odinger equation (NLSE). As the quantum parameter tends to zero, the modified NLSE formally converges to the standard NLSE, and we show this rigorously.

math.AP↗