arXiv subjects
Publications and source records attributed to Li Luo.
We introduce and study Lyndon bases of split $\imath$quantum groups $\mathbf{U}^\imath(\mathfrak{g})$. A relationship between the Lyndon bases and PBW-type bases was provided. As an application, we establish the existence of canonical bases for the type A split $\imath$quantum groups $\mathbf{U}^\imath(\mathfrak{sl}_n)$.
Unobserved effect modifiers can induce bias when generalizing causal effect estimates to target populations. In this work, we extend a sensitivity analysis framework assessing the robustness of study results to unobserved effect modification that adapts to various generalizability scenarios, including multiple (conditionally) randomized trials, observational studies, or combinations thereof. This framework is interpretable and does not rely on distributional or functional assumptions about unknown parameters. We demonstrate how to leverage the multi-study setting to detect violation of the generalizability assumption through hypothesis testing, showing with simulations that the proposed test achieves high power under real-world sample sizes. Finally, we apply our sensitivity analysis framework to analyze the generalized effect estimate of secondhand smoke exposure on birth weight using cohort sites from the Environmental influences on Child Health Outcomes (ECHO) study.
We study the affine quantum Schur algebras corresponding to the affine Hecke algebras of type C with three parameters. Multiplication formulas for semisimple generators are derived for these algebras. We prove that they admit a stabilization property in the sense of Beilinson-Lusztig-MacPherson, by which we construct quasi-split $\imath$quantum groups of affine type AIII with three parameters. Upon specialization of parameters, we recover some known multiplication formulas for affine quantum Schur algebras of types C and D with equal parameters. We take this opportunity to demonstrate in the appendix that different forms of multiplication formulas for affine quantum Schur algebras of types C with equal parameters across the literature are indeed equivalent.
The restricted Boltzmann machine (RBM) is a neural network based on the Ising model, well known for its ability to learn probability distributions and stochastically generate new content. However, the high computational cost of Gibbs sampling in content generation tasks imposes significant bottlenecks on electronic implementations. Here, we propose a photonic restricted Boltzmann machine (PRBM) that leverages photonic computing to accelerate Gibbs sampling, enabling efficient content generation. By introducing an efficient encoding method, the PRBM eliminates the need for computationally intensive matrix decomposition and reduces the computational complexity of Gibbs sampling from $O(N)$ to $O(1)$. Moreover, its non-Von Neumann photonic computing architecture circumvents the memory storage of interaction matrices, providing substantial advantages for large-scale RBMs. We experimentally validate the photonic-accelerated Gibbs sampling by simulating a two-dimensional Ising model, where the observed phase transition temperature closely matches the theoretical predictions. Beyond physics-inspired tasks, the PRBM demonstrates robust capabilities in generating and restoring diverse content, including images and temporal sequences, even in the presence of noise and aberrations. The scalability and reduced training cost of the PRBM framework underscore its potential as a promising pathway for advancing photonic computing in generative artificial intelligence.
Generating accurate and stable long rollouts is a notorious challenge for time-dependent PDEs (Partial Differential Equations). Recently, motivated by the importance of high-frequency accuracy, a refiner model called PDERefiner utilizes diffusion models to refine outputs for every time step, since the denoising process could increase the correctness of modeling high frequency part. For 1-D Kuramoto-Sivashinsky equation, refiner models can degrade the amplitude of high frequency part better than not doing refinement process. However, for some other cases, the spectrum might be more complicated. For example, for a harder PDE like Navior-Stokes equation, diffusion models could over-degrade the higher frequency part. This motivates us to release the constraint that each frequency weighs the same. We enhance our refiner model with doing adjustments on spectral space, which recovers Blurring diffusion models. We developed a new v-prediction technique for Blurring diffusion models, recovering the MSE training objective on the first refinement step. We show that in this case, for different model backbones, such as U-Net and neural operators, the outputs of PDE-SpectralRefiner are more accurate for both one-step MSE loss and rollout loss.
This paper presents a hybrid model combining Transformer and CNN for predicting the current waveform in signal lines. Unlike traditional approaches such as current source models, driver linear representations, waveform functional fitting, or equivalent load capacitance methods, our model does not rely on fixed simplified models of standard-cell drivers or RC loads. Instead, it replaces the complex Newton iteration process used in traditional SPICE simulations, leveraging the powerful sequence modeling capabilities of the Transformer framework to directly predict current responses without iterative solving steps. The hybrid architecture effectively integrates the global feature-capturing ability of Transformers with the local feature extraction advantages of CNNs, significantly improving the accuracy of current waveform predictions. Experimental results demonstrate that, compared to traditional SPICE simulations, the proposed algorithm achieves an error of only 0.0098. These results highlight the algorithm's superior capabilities in predicting signal line current waveforms, timing analysis, and power evaluation, making it suitable for a wide range of technology nodes, from 40nm to 3nm.
The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $\mathfrak{sl}$ and $\mathfrak{gl}$ types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak{sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak{gl}$ and its canonical basis.
Momentum-based optimizers are widely adopted for training neural networks. However, the optimal selection of momentum coefficients remains elusive. This uncertainty impedes a clear understanding of the role of momentum in stochastic gradient methods. In this paper, we present a frequency domain analysis framework that interprets the momentum method as a time-variant filter for gradients, where adjustments to momentum coefficients modify the filter characteristics. Our experiments support this perspective and provide a deeper understanding of the mechanism involved. Moreover, our analysis reveals the following significant findings: high-frequency gradient components are undesired in the late stages of training; preserving the original gradient in the early stages, and gradually amplifying low-frequency gradient components during training both enhance performance. Based on these insights, we propose Frequency Stochastic Gradient Descent with Momentum (FSGDM), a heuristic optimizer that dynamically adjusts the momentum filtering characteristic with an empirically effective dynamic magnitude response. Experimental results demonstrate the superiority of FSGDM over conventional momentum optimizers.
Simulation of high-order nonlinear system requires extensive computational resources, especially in modern VLSI backend design where bifurcation-induced instability and chaos-like transient behaviors pose challenges. We present S-Crescendo - a nested transformer weaving framework that synergizes S-domain with neural operators for scalable time-domain prediction in high-order nonlinear networks, alleviating the computational bottlenecks of conventional solvers via Newton-Raphson method. By leveraging the partial-fraction decomposition of an n-th order transfer function into first-order modal terms with repeated poles and residues, our method bypasses the conventional Jacobian matrix-based iterations and efficiently reduces computational complexity from cubic $O(n^3)$ to linear $O(n)$.The proposed architecture seamlessly integrates an S-domain encoder with an attention-based correction operator to simultaneously isolate dominant response and adaptively capture higher-order non-linearities. Validated on order-1 to order-10 networks, our method achieves up to 0.99 test-set ($R^2$) accuracy against HSPICE golden waveforms and accelerates simulation by up to 18(X), providing a scalable, physics-aware framework for high-dimensional nonlinear modeling.
Assessing artistic creativity has long challenged researchers, with traditional methods proving time-consuming. Recent studies have applied machine learning to evaluate creativity in drawings, but not paintings. Our research addresses this gap by developing a CNN model to automatically assess the creativity of human paintings. Using a dataset of six hundred paintings by professionals and children, our model achieved 90% accuracy and faster evaluation times than human raters. This approach demonstrates the potential of machine learning in advancing artistic creativity assessment, offering a more efficient alternative to traditional methods.
We provide the geometric construction of a series of generalized Schur algebras of any type via Borel-Moore homologies and equivariant K-groups of generalized Steinberg varieties. As applications, we obtain a Schur algebra analogue of the local geometric Langlands correspondence of any type, provide an equivariant K-theoretic realization of quasi-split $\imath$quantum groups of affine type AIII, and establish a geometric Howe duality for affine ($\imath$-)quantum groups.
In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the "zero-energy-contribution" property while maintaining conservative time discretization for the "non-zero-energy-contribution" terms. Our algorithm simplifies to solving three independent linear elliptic systems per time step, two of them with constant coefficients. The update of all scalar auxiliary variables is explicit and decoupled from solving the phase field variable and velocity field. We rigorously prove unconditional energy stability of the scheme and perform extensive benchmark simulations to demonstrate accuracy and efficiency of the method.
We develop an invariant theory of quasi-split $\imath$quantum groups $\mathbf{U}_n^\imath$ of type AIII on a tensor space associated to $\imath$Howe dualities. The first and second fundamental theorems for $\mathbf{U}_n^\imath$-invariants are derived.
Although thick disk is a structure prevalent in local disk galaxies and also present in our home Galaxy, its formation and evolution is still unclear. Whether the thick disk is born thick and/or gradually heated to be thick after formation is under debate. To disentangle these two scenarios, one effective approach is to inspect the thickness of young disk galaxies in the high redshift Universe. In this work we study the vertical structure of 191 edge-on galaxies spanning redshift from 0.2 to 5 using JWST NIRCAM imaging observations. For each galaxy, we retrieve the vertical surface brightness profile at 1 ${R_e}$ and fit a sech$^2$ function that has been convolved with the line spread function. The obtained scale height of galaxies at $z>1.5$ show no clear dependence on redshift, with a median value in remarkable agreement with that of the Milky Way's thick disk. This suggests that local thick disks are already thick when they were formed in the early times and secular heating is unlikely the main driver of thick disk formation. For galaxies at $z<1.5$, however, the disk scale height decreases systematically towards lower redshift, with low-redshift galaxies having comparable scale height with that of the Milky Way's thin disk. This cosmic evolution of disk thickness favors an upside-down formation scenario of galaxy disks.
In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource constraints, which has been the backbone of several solutions for several classical problems like vehicle routing problems. However, approaches that perform well on these problems cannot be used since most constraints in the nurse rostering problem are soft. Based on ideas borrowed from global constraints in constraint programming to model rostering problems, an efficient dynamic programming algorithm with novel label definitions and dominating rules specific to soft time-related constraints is proposed. In addition, several acceleration strategies are employed to improve the branch and price algorithm. Computational results on instances of different sizes indicate that the proposed algorithm is a promising solution for the nurse rostering problem with multiple units.
We prove that the q-Schur algebras of finite type introduced in [LW22] are cellular in the sense of Graham and Lehrer, which is a generalization of Geck's theorem on the cellularity of Hecke algebras of finite type. Moreover, we study special modules of the associated asymptotic Schur algebras and left cell representations of Schur algebras, which generalize Lusztig's work about special modules of asymptotic Hecke algebras and left cell representations of Weyl groups, respectively.
Recently, spatial photonic Ising machines (SPIMs) have demonstrated the abilities to compute the Ising Hamiltonian of large-scale spin systems, with the advantages of ultrafast speed and high power efficiency. However, such optical computations have been limited to specific Ising models with fully connected couplings. Here we develop a wavelength-division multiplexing SPIM to enable programmable spin couplings and external magnetic fields as well for general Ising models. We experimentally demonstrate such a wavelength-division multiplexing SPIM with a single spatial light modulator, where the gauge transformation is implemented to eliminate the impact of pixel alignment. To show the programmable capability of general spin coupling interactions, we explore three spin systems: $\pm J$ models, Sherrington-Kirkpatrick models, and only locally connected ${{J}_{1}}\texttt{-}{{J}_{2}}$ models and observe the phase transitions among the spin-glass, the ferromagnetic, the paramagnetic and the stripe-antiferromagnetic phases. These results show that the wavelength-division multiplexing approach has great programmable flexibility of spin couplings and external magnetic fields, which provides the opportunities to solve general combinatorial optimization problems with large-scale and on-demand SPIM.
We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that the quantum Howe duality derived from partial flag varieties coincides with the one constructed by quantum coordinate (co)algebras. Moreover, the explicit multiplicity-free decompositions for these Howe dualities are obtained.