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Ji Li

Publications and source records attributed to Ji Li.

At least 145 records · Page 8Linked to original sources
arXiv2022

Singular integral operators, $T1$ theorem, Littlewood-Paley theory and Hardy spaces in Dunkl Setting

The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles. New tools developed in this paper include the weak-type discrete Calderón reproducing formulae, new test functions, and distributions, the Littlewood-Paley, the wavelet-type decomposition, and molecule characterizations of the Dunkl-Hardy space, Coifman's approximation to the identity and the decomposition of the identity operator on $L^2$, Meyer's commutation Lemma, and new almost orthogonal estimates in the Dunkl setting.

math.CA↗
arXiv2022

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

math.FA↗
arXiv2022

Schatten classes and commutator in the two weight setting, I. Hilbert transform

We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

math.FA↗
arXiv2022

A Secure and Efficient Federated Learning Framework for NLP

In this work, we consider the problem of designing secure and efficient federated learning (FL) frameworks. Existing solutions either involve a trusted aggregator or require heavyweight cryptographic primitives, which degrades performance significantly. Moreover, many existing secure FL designs work only under the restrictive assumption that none of the clients can be dropped out from the training protocol. To tackle these problems, we propose SEFL, a secure and efficient FL framework that (1) eliminates the need for the trusted entities; (2) achieves similar and even better model accuracy compared with existing FL designs; (3) is resilient to client dropouts. Through extensive experimental studies on natural language processing (NLP) tasks, we demonstrate that the SEFL achieves comparable accuracy compared to existing FL solutions, and the proposed pruning technique can improve runtime performance up to 13.7x.

cs.CR↗
arXiv2021

Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications

We investigate the Cauchy--Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy--Szegő kernel and prove that the Cauchy--Szegő kernel is non-zero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy--Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space $ H^p$ on quaternionic Siegel upper half space for $2/3<p\leq1$. Moreover, we establish the characterisation of singular values of the commutator of Cauchy--Szegő projection based on the kernel estimates and on the recent new approach by Fan--Lacey--Li. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

math.CV↗
arXiv2021

Phonon-laser ultrasensitive force sensor

Developing nano-mechanical oscillators for ultrasensitive force detection is of great importance in exploring science. We report our achievement of ultrasensitive detection of the external force regarding the radio-frequency electric field by a nano-sensor made of a single trapped $^{40}$Ca$^{+}$ ion under injection-locking, where squeezing is additionally applied to detection of the smallest force in the ion trap. The employed ion is confined stably in a surface electrode trap and works as a phonon laser that is very sensitive to the external disturbance. The injection-locking drove the ion's oscillation with phase synchronization, yielding the force detection with sensitivity of 347 $\pm$ 50 yN/$\sqrt{Hz}$. Further with 3 dB squeezing applied on the oscillation phase variance, we achieved a successful detection of the smallest force to be 86.5 $\pm$ 70.1 yN.

quant-ph↗
arXiv2021

TAG: Gradient Attack on Transformer-based Language Models

Although federated learning has increasingly gained attention in terms of effectively utilizing local devices for data privacy enhancement, recent studies show that publicly shared gradients in the training process can reveal the private training images (gradient leakage) to a third-party in computer vision. We have, however, no systematic understanding of the gradient leakage mechanism on the Transformer based language models. In this paper, as the first attempt, we formulate the gradient attack problem on the Transformer-based language models and propose a gradient attack algorithm, TAG, to reconstruct the local training data. We develop a set of metrics to evaluate the effectiveness of the proposed attack algorithm quantitatively. Experimental results on Transformer, TinyBERT$_{4}$, TinyBERT$_{6}$, BERT$_{BASE}$, and BERT$_{LARGE}$ using GLUE benchmark show that TAG works well on more weight distributions in reconstructing training data and achieves 1.5$\times$ recover rate and 2.5$\times$ ROUGE-2 over prior methods without the need of ground truth label. TAG can obtain up to 90$\%$ data by attacking gradients in CoLA dataset. In addition, TAG has a stronger adversary on large models, small dictionary size, and small input length. We hope the proposed TAG will shed some light on the privacy leakage problem in Transformer-based NLP models.

cs.CR↗
arXiv2021

Schatten classes and commutators of Riesz transform on Heisenberg group and applications

We study commutators with the Riesz transforms on the Heisenberg group. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson--Wolff and Rochberg--Semmes. The method of proof extends the earlier methods, allowing us to address not just the Riesz transforms, but also the Cauchy--Szego projection and second order Riesz transforms on the Heisenberg group, among other settings.

math.FA↗
arXiv2021

A weak type estimate for regular fractional sparse operators

In this note the weak type estimates for fractional integrals are studied. More precisely, we adapt the arguments of Domingo-Salazar, Lacey, and Rey to obtain improvements for the endpoint weak type estimates for regular fractional sparse operators.

math.CA↗
arXiv2021

Orbital Stability of the sum of Smooth solitons in the Degasperis-Procesi Equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the $L^2\cap L^\infty$ orbital stability of a wave train containing $N$ smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce \textit{a priori } estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.

math.AP↗
arXiv2021

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\geq 2$. Let $H^1_L(X)$ be the Hardy space associated with $L.$ In this paper we show sharp endpoint estimate for the Schrödinger group $e^{itL}$ associated with $L$ such that \begin{eqnarray*} \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ L^1(X)} + \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ H^1_L(X)} \leq C(1+|t|)^{n/2}\|f\|_{H^1_L(X)}, \ \ \ t\in{\mathbb R} \end{eqnarray*} for some constant $C=C(n, m)>0$ independent of $t$. By a duality and interpolation argument, it gives a new proof of a recent result of \cite{CDLY} for { sharp} endpoint $L^p$-Sobolev bound for $e^{itL}$: $$ \left\| (I+L)^{-s }e^{itL} f\right\|_{ L^p(X)} \leq C (1+|t|)^{s} \|f\|_{ L^p(X)}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big| $$ for every $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound, which extends the classical results due to Miyachi ) for the Laplacian on the Euclidean space ${\mathbb R}^n$.

math.AP↗
arXiv2021

Mapping the Galactic Disk with the LAMOST and Gaia Red Clump Sample VII: the Stellar Disk Structure Revealed by the Mono-abundance Populations

Using a sample of 96,201 primary red clump (RC) stars selected from the LAMOST and Gaia surveys, we investigate the stellar structure of the Galactic disk. The sample stars show two separated sequences of high-[α/Fe] and low-[α/Fe] in the [α/Fe]-[Fe/H] plane. We divide the sample stars into five mono-abundance populations (MAPs) with different ranges of [α/Fe] and [Fe/H], named as the high-[α/Fe], high-[α/Fe] & high-[Fe/H], low-[Fe/H], solar, high-[Fe/H] MAPs respectively. We present the stellar number density distributions in the R R Z plane, and the scale heights and scale lengths of the individual MAPs by fitting their vertical and radial density profiles. The vertical profiles, the variation trend of scale height with the Galactocentric radius, indicate that there is a clear disk flare in the outer disk both for the low-[α/Fe] and the high-[α/Fe] MAPs. While the radial surface density profiles show a peak radius of 7 kpc and 8 kpc for the high-[α/Fe] and low-[α/Fe] MAPs, respectively. We also investigate the correlation between the mean rotation velocity and metallicity of the individual MAPs, and find that the mean rotation velocities are well separated and show different trends between the high-[α/Fe] and the low-[α/Fe] MAPs. At last, we discuss the character of the high-[α/Fe] & high-[Fe/H] MAP and find that it is more similar to the high-[α/Fe] MAP either in the radial and vertical density profiles or in the rotation velocity.

astro-ph.GA↗
arXiv2021

Quantitative weighted estimates for rough singular integrals on homogeneous groups

In this paper, we study weighted $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of singular integrals with homogeneous convolution kernel $K(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$, {under the assumption that $K_0$, the restriction of $K$ to the unit annulus, is mean zero and $L^{q}$ integrable for some $q_{0}<q\leq \infty$,} where $q_{0}$ is a fixed constant depending on $w$. We obtain a quantitative weighted bound, which is consistent with the one obtained by Hytönen--Roncal--Tapiola in the Euclidean setting, for this operator on $L^{p}(w)$. Comparing to the previous results in the Euclidean setting, our assumptions on the kernel and on the underlying space are weaker. Moreover, we investigate the quantitative weighted bound for the bi-parameter rough singular integrals on product homogeneous Lie groups.

math.AP↗
arXiv2021

Nonhomogeneous $T(1)$ Theorem on Product Quasimetric Spaces

In this paper, we provide a non-homogeneous $T(1)$ theorem on product spaces $(X_1 \times X_2, ρ_1 \times ρ_2, μ_1 \times μ_2)$ equipped with a quasimetric $ρ_1 \times ρ_2$ and a Borel measure $μ_1 \times μ_2$, which, need not be doubling but satisfies an upper control on the size of quasiballs.

math.CA↗
arXiv2021

Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces

We provide a boundedness criterion for the integral operator $S_φ$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_φ$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_φF(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w}) dλ(w) \end{eqnarray*} with $φ$ in the Fock space $F^2(\mathbb C^n)$ and $dλ(w): = π^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $\mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.

math.CV↗
arXiv2020

Marcinkiewicz multipliers associated with the Kohn Laplacian on the Shilov boundary of the product domain in $\mathbb C ^{2n}$

Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $Ω^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the product domain $Ω^{(1)}\times\cdots\times Ω^{(n)}$. We show that if a function $F(λ_1, \ldots ,λ_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{(n)})$ is a product Calderón--Zygmund operator of Journé type.

math.CV↗
arXiv2020

Efficient Transformer-based Large Scale Language Representations using Hardware-friendly Block Structured Pruning

Pre-trained large-scale language models have increasingly demonstrated high accuracy on many natural language processing (NLP) tasks. However, the limited weight storage and computational speed on hardware platforms have impeded the popularity of pre-trained models, especially in the era of edge computing. In this work, we propose an efficient transformer-based large-scale language representation using hardware-friendly block structure pruning. We incorporate the reweighted group Lasso into block-structured pruning for optimization. Besides the significantly reduced weight storage and computation, the proposed approach achieves high compression rates. Experimental results on different models (BERT, RoBERTa, and DistilBERT) on the General Language Understanding Evaluation (GLUE) benchmark tasks show that we achieve up to 5.0x with zero or minor accuracy degradation on certain task(s). Our proposed method is also orthogonal to existing compact pre-trained language models such as DistilBERT using knowledge distillation, since a further 1.79x average compression rate can be achieved on top of DistilBERT with zero or minor accuracy degradation. It is suitable to deploy the final compressed model on resource-constrained edge devices.

cs.CL↗