arXiv subjects

Chen Lu

Publications and source records attributed to Chen Lu.

At least 55 records · Page 3Linked to original sources

arXiv2023

Query lower bounds for log-concave sampling

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension $d\ge 2$ requires $Ω(\log κ)$ queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension $d$ (hence also from general log-concave and log-smooth distributions in dimension $d$) requires $\widetilde Ω(\min(\sqrtκ\log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $κ$ denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in geometric measure theory, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature.

math.ST↗

arXiv2023

Theory of $d + id$ Second-Order Topological Superconductors

Topological superconductors are a class of unconventional superconducting materials featuring sub-gap zero-energy Majorana bound modes that hold promise as a building block for topological quantum computing. In this work, we study the realization of second-order topology that defines anomalous gapless boundary modes in a two-orbital superconductor with spin-orbital couplings. We reveal a time-reversal symmetry-breaking second-order topological superconducting phase with $d+id$-wave orbital-dependent paring without the need for the external magnetic field. Remarkably, this orbital-active $d$-wave paring gives rise to anomalous zero-energy Majorana corner modes, which is in contrast to conventional chiral $d$-wave pairing, accommodating one-dimensional Majorana edge modes. Our work not only reveals a unique mechanism of time-reversal symmetry breaking second-order topological superconductors but also bridges the gap between second-order topology and orbital-dependent pairings.

cond-mat.supr-con↗

arXiv2023

Effect of Rare-earth Element Substitution in Superconducting R$_3$Ni$_2$O$_7$ Under Pressure

Recently, high temperature ($T_c\approx 80$K) superconductivity (SC) has been discovered in La$_3$Ni$_2$O$_7$ (LNO) under pressure. Question arises whether the transition temperature $T_c$ could be further enhanced under suitable conditions. A possible route for realizing higher $T_c$ is element substitution. Similar SC could appear in rare-earth (RE) R$_3$Ni$_2$O$_7$ (RNO, R=RE element) material series under pressure. The electronic properties in the RNO materials are dominated by the Ni $3d$ orbitals in the bilayer NiO$_2$ plane. In the strong coupling limit, the SC could be fully characterized by a bilayer single $3d_{x^2-y^2}$-orbital $t$-$J_{\parallel}$-$J_{\perp}$ model. Under RE element substitution from La to RE element, the lattice constant decreases and the electronic hopping increases, leading to stronger superexchanges between the $3d_{x^2-y^2}$ orbitals. Based on the slave-boson mean-field theory, we explore the pairing nature and the evolution of $T_c$ in RNO materials. Consequently, it is found that the element substitution does not alter the pairing nature, i.e. the inter-layer $s$-wave pairing is always favored in RNO. However, the $T_c$ increases from La to Sm and a nearly doubled $T_c$ is achieved for SmNO. This work provides evidence for possible higher $T_c$ R$_3$Ni$_2$O$_7$ materials, which may be realized in further experiments.

cond-mat.supr-con↗

arXiv2022

Fisher information lower bounds for sampling

We prove two lower bounds for the complexity of non-log-concave sampling within the framework of Balasubramanian et al. (2022), who introduced the use of Fisher information (FI) bounds as a notion of approximate first-order stationarity in sampling. Our first lower bound shows that averaged LMC is optimal for the regime of large FI by reducing the problem of finding stationary points in non-convex optimization to sampling. Our second lower bound shows that in the regime of small FI, obtaining a FI of at most $\varepsilon^2$ from the target distribution requires $\text{poly}(1/\varepsilon)$ queries, which is surprising as it rules out the existence of high-accuracy algorithms (e.g., algorithms using Metropolis-Hastings filters) in this context.

stat.ML↗

arXiv2022

Chiral SO(4) spin-valley density wave and degenerate topological superconductivity in magic-angle-twisted bilayer-graphene

Starting from a realistic extended Hubbard model for a $p_{x,y}$-orbital tight-binding model on the Honeycomb lattice, we perform a thorough investigation on the possible electron instabilities in the MA-TBG near the van Hove (VH) dopings. Here we focus on the interplay between the approximate SU(2)$\times$SU(2) symmetry and the $D_3$ symmetry, which leads to intriguing quantum states relevant to recent experiments, as revealed by our systematic RPA based calculations followed by a succeeding mean-field energy minimization for the ground state energy. At the SU(2)$\times$SU(2) symmetric point, the degenerate inter-valley SDW and VDW are mixed into a new state of matter dubbed as the chiral SO(4) spin-valley DW. This state simultaneously hosts three 4-component vectorial spin-valley DW orders with each adopting one wave vector, and the polarization directions of the three DW orders are mutually perpendicular to one another. %in the $\mathbb{R}^4$ space. In the presence of a tiny inter-valley exchange interaction with coefficient $J_H\to 0^{-}$ which breaks the SU(2)$\times$SU(2) symmetry, a pure chiral SDW state is obtained. In the case of $J_H\to 0^{+}$, a nematic VDW+SDW state emerges which possesses a stripy distribution of the charge density, consistent with the recent STM observations. On the aspect of SC, while the triplet $p+ip$ and singlet $d+id$ topological SCs are degenerate at $J_H=0$ near the VH dopings, the former (latter) is favored for $J_H\to 0^{-}$ ($J_H\to 0^{+}$). In addition, the two asymmetric doping-dependent behaviors of the obtained pairing phase diagram are well consistent with experiments.

cond-mat.str-el↗

arXiv2021

Two-orbital model for possible superconductivity pairing mechanism in nickelates

The newly synthesized strontium doped RNiO$_2$ (R=Nd, La) superconductors have stimulated extensive interests in understanding their pairing mechanism and pairing nature. Here we study the pairing mechanism in this family from a two-orbital model comprising the Ni- $3d_{x^2-y^2}$- and $3d_{xy}$- orbitals, equipped with extended Hubbard interactions and induced low-energy effective superexchange interactions. We then study the pairing symmetry in this system by using large scale variational Monte Carlo approach. Our results yield the intraorbital $d_{x^2-y^2}$-wave singlet pairing as the leading pairing symmetry in the nickelates, which is analogous to the cuprates. However, there exist two important differences between the physical properties of the two families due to the fact that at the low Sr-doping regime, while the Ni-$3d_{x^2-y^2}$ orbitals remain half-filled and singly-occupied to form a Mott-insulating background, the Ni-$3d_{xy}$ orbitals accommodate nearly all the extra doped holes, which move freely on this background. The first difference lies in the single-particle aspect: while the $3d_{x^2-y^2}$ degree of freedom remains Mott insulating with spectra weight pinned down at zero at low dopings, the $3d_{xy}$ one behaves as Fermi liquid with spectra weight near 1. The second difference lies in the pairing aspect: while the huge intra-$3d_{x^2-y^2}$-orbital pairing gap is actually a pseudo gap which has nothing to do with the SC, the small intra-$3d_{xy}$-orbital pairing gap serves as the true superconducting pairing gap, which is related to the $T_c$ via the BCS relation. Both differences can be verified by the angle-resolved photo-emission spectrum.

cond-mat.str-el↗

arXiv2021

Nonparametric Heterogeneous Treatment Effect Estimation in Repeated Cross Sectional Designs

Identifying heterogeneity in a population's response to a health or policy intervention is crucial for evaluating and informing policy decisions. We propose a novel heterogeneous treatment effect estimator in the difference-in-differences design with repeated cross sectional data, where we observe different samples of a population at two time periods separated by the onset of a policy intervention, as well as samples of a population that serves as the control. Our estimator has orthogonality properties that enable fast rates on learning the treatment effect while allowing slower rates for estimating nuisance components. Our proposal shows promising empirical performance across a variety of simulation setups.

stat.ME↗

arXiv2021

The query complexity of sampling from strongly log-concave distributions in one dimension

We establish the first tight lower bound of $Ω(\log\logκ)$ on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number $κ$ in one dimension. Whereas existing guarantees for MCMC-based algorithms scale polynomially in $κ$, we introduce a novel algorithm based on rejection sampling that closes this doubly exponential gap.

math.ST↗

arXiv2021

Rejection sampling from shape-constrained distributions in sublinear time

We consider the task of generating exact samples from a target distribution, known up to normalization, over a finite alphabet. The classical algorithm for this task is rejection sampling, and although it has been used in practice for decades, there is surprisingly little study of its fundamental limitations. In this work, we study the query complexity of rejection sampling in a minimax framework for various classes of discrete distributions. Our results provide new algorithms for sampling whose complexity scales sublinearly with the alphabet size. When applied to adversarial bandits, we show that a slight modification of the Exp3 algorithm reduces the per-iteration complexity from $\mathcal O(K)$ to $\mathcal O(\log^2 K)$, where $K$ is the number of arms.

cs.LG↗

arXiv2021

TouchRoller: A Rolling Optical Tactile Sensor for Rapid Assessment of Large Surfaces

Tactile sensing is important for robots to perceive the world as it captures the texture and hardness of the object in contact and is robust to illumination and colour variances. However, due to the limited sensing area and the resistance of the fixed surface, current tactile sensors have to tap the tactile sensor on target object many times when assessing a large surface, i.e., pressing, lifting up and shifting to another region. This process is ineffective and time consuming. It is also undesirable to drag such sensors as this often damages the sensitive membrane of the sensor or the object. To address these problems, we propose a cylindrical optical tactile sensor named TouchRoller that can roll around its center axis. It maintains being in contact with the assessed surface throughout the entire motion, which allows for measuring the object continuously and effectively. Extensive experiments show that the TouchRoller sensor can cover a textured surface of 8cm*11cm in a short time of 10s, much more effectively than a flat optical tactile sensor (in 196s). The reconstructed map of the texture from the collected tactile images has a high Structural Similarity Index (SSIM) of 0.31 on average, when compared with the visual texture. In addition, the contacts on the sensor can be localised with a low localisation error, 2.63mm in the center regions and 7.66mm on average. The proposed sensor will enable the fast assessment of large surfaces with high-resolution tactile sensing, and also the effective collection of tactile images.

cs.RO↗

arXiv2020

Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm

Conventional wisdom in the sampling literature, backed by a popular diffusion scaling limit, suggests that the mixing time of the Metropolis-Adjusted Langevin Algorithm (MALA) scales as $O(d^{1/3})$, where $d$ is the dimension. However, the diffusion scaling limit requires stringent assumptions on the target distribution and is asymptotic in nature. In contrast, the best known non-asymptotic mixing time bound for MALA on the class of log-smooth and strongly log-concave distributions is $O(d)$. In this work, we establish that the mixing time of MALA on this class of target distributions is $\widetildeΘ(d^{1/2})$ under a warm start. Our upper bound proof introduces a new technique based on a projection characterization of the Metropolis adjustment which reduces the study of MALA to the well-studied discretization analysis of the Langevin SDE and bypasses direct computation of the acceptance probability.

math.ST↗

arXiv2020

Contextual Stochastic Block Model: Sharp Thresholds and Contiguity

We study community detection in the contextual stochastic block model arXiv:1807.09596 [cs.SI], arXiv:1607.02675 [stat.ME]. In arXiv:1807.09596 [cs.SI], the second author studied this problem in the setting of sparse graphs with high-dimensional node-covariates. Using the non-rigorous cavity method from statistical physics, they conjectured the sharp limits for community detection in this setting. Further, the information theoretic threshold was verified, assuming that the average degree of the observed graph is large. It is expected that the conjecture holds as soon as the average degree exceeds one, so that the graph has a giant component. We establish this conjecture, and characterize the sharp threshold for detection and weak recovery.

cs.SI↗

arXiv2020

Intrinsic topological superconductivity with exactly flat surface bands in the quasi-one-dimensional A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors

A spin-U(1)-symmetry protected momentum-dependent integer-$Z$-valued topological invariant is proposed to time-reversal-invariant (TRI) superconductivity (SC) whose nonzero value will lead to exactly flat surface band(s). The theory is applied to the weakly spin-orbit coupled quasi-1D A$_2$Cr$_3$As$_3$ (A=Na, K, Rb, Cs) superconductors family with highest $T_c$ up to 8.6 K with $p_z$-wave pairing in the $S_z=0$ channel. It's found that up to the leading atomic spin-orbit-coupling (SOC), the whole (001) surface Brillouin zone is covered with exactly-flat surface bands, with some regime hosting three flat bands and the remaining part hosting two. Such exactly-flat surface bands will lead to very sharp zero-bias conductance peak in the scanning tunneling microscopic spectrum. When a tiny subleading spin-flipping SOC is considered, the surface bands will only be slightly split. The application of this theory can be generalized to other unconventional superconductors with weak SOC, particularly to those with mirror-reflection symmetry.

cond-mat.supr-con↗

arXiv2020

SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence

Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport. We introduce a new perspective on SVGD that instead views SVGD as the (kernelized) gradient flow of the chi-squared divergence which, we show, exhibits a strong form of uniform exponential ergodicity under conditions as weak as a Poincaré inequality. This perspective leads us to propose an alternative to SVGD, called Laplacian Adjusted Wasserstein Gradient Descent (LAWGD), that can be implemented from the spectral decomposition of the Laplacian operator associated with the target density. We show that LAWGD exhibits strong convergence guarantees and good practical performance.

math.ST↗

arXiv2020

Exponential ergodicity of mirror-Langevin diffusions

Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization and highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

math.ST↗

arXiv2020

Spectroscopic evidence for a spin and valley polarized metallic state in a non-magic-angle twisted bilayer graphene

In the magic-angle twisted bilayer graphene (MA-TBG), strong electron-electron (e-e) correlations caused by the band-flattening lead to many exotic quantum phases such as superconductivity, correlated insulator, ferromagnetism, and quantum anomalous Hall effects, when its low-energy van Hove singularities (VHSs) are partially filled. Here our high-resolution scanning tunneling microscope and spectroscopy measurements demonstrate that the e-e correlation in a non-magic-angle TBG with a twist angle θ = 1.49 still plays an important role in determining its electronic properties. Our most interesting observation on that sample is that when one of its VHS is partially filled, the one associated peak in the spectrum splits into four peaks. Our analysis based on the continuum model suggests that such a one-to-four split of the VHS originates from the formation of an interaction-driven spin-valley-polarized metallic state near the VHS, lifting both the spin and valley degeneracies. Our results for this non-magic-angle TBG reveal a new symmetry-breaking phase, which has not been identified in the MA-TBG or in other systems.

cond-mat.mtrl-sci↗

arXiv2019

Surface Following using Deep Reinforcement Learning and a GelSightTactile Sensor

Tactile sensors can provide detailed contact in-formation that can facilitate robots to perform dexterous, in-hand manipulation tasks. One of the primitive but important tasks is surface following that is a key feature for robots while exploring unknown environments or workspace of inaccurate modeling. In this paper, we propose a novel end-to-end learning strategy, by directly mapping the raw tactile data acquired from a GelSight tactile sensor to the motion of the robot end-effector.Experiments on a KUKA youBot platform equipped with theGelSight sensor show that 80% of the actions generated by a fully trained SFDQN model are proper surface following actions; the autonomous surface following test also indicates that the proposed solution works well on a test surface.

cs.RO↗

arXiv2019

Single-orbital realization of high-temperature $s^\pm$ superconductivity in the square-octagon lattice

We propose possible high-temperature superconductivity (SC) with singlet $s^\pm$-wave pairing symmetry in the single-orbital Hubbard model on the square-octagon lattice with only nearest-neighbor hopping terms. Three different approaches are engaged to treat with the interacting model for different coupling strengths, which yield consistent result for the $s^\pm$ pairing symmetry. We propose octagraphene, i.e., a monolayer of carbon atoms arranged into this lattice, as a possible material realization of this model. Our variational Monte Carlo study for the material with realistic coupling strength yields a pairing strength comparable with the cuprates, implying a similar superconducting critical temperature between the two families. This study also applies to other materials with similar lattice structure.

cond-mat.supr-con↗