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Let $\mathrm{Mp}(2n)$ be the metaplectic group of rank $n$ over a local field $F$ of characteristic zero. In this note, we determine the behavior of endoscopic transfer for $\mathrm{Mp}(2n)$ under variation of additive characters of $F$. The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan-Savin, which describes how the local Langlands correspondence for $\mathrm{Mp}(2n)$ depends on the additive characters., Comment: 20 pages

We analyse the entanglement structure of states generated by random constant-depth two-dimensional quantum circuits, followed by projective measurements of a subset of sites. By deriving a rigorous lower bound on the average entanglement entropy of such post-measurement states, we prove that macroscopic long-ranged entanglement is generated above some constant critical depth in several natural classes of circuit architectures, which include brickwork circuits and random holographic tensor networks. This behaviour had been conjectured based on previous works, which utilize non-rigorous methods such as replica theory calculations, or work in regimes where the local Hilbert space dimension grows with system size. To establish our lower bound, we develop new replica-free theoretical techniques that leverage tools from multi-user quantum information theory, which are of independent interest, allowing us to map the problem onto a statistical mechanics model of self-avoiding walks without requiring large local Hilbert space dimension. Our findings have consequences for the complexity of classically simulating sampling from random shallow circuits, and of contracting tensor networks: First, we show that standard algorithms based on matrix product states which are used for both these tasks will fail above some constant depth and bond dimension, respectively. In addition, we also prove that these random constant-depth quantum circuits cannot be simulated by any classical circuit of sublogarithmic depth., Comment: 22 + 12 pages, 11 figures

For metaplectic groups over a local field of characteristic zero, we define the Arthur packet attached to any Arthur parameter $\psi$ as a multi-set of unitary genuine irreducible representations, characterized by endoscopic character relations. Over number fields, we obtain a multiplicity formula for the genuine discrete $L^2$-automorphic spectrum in terms of global Arthur parameters and $\epsilon$-factors, by leveraging the trace formula for metaplectic groups. This confirms a conjecture of Gan, and extends earlier results of Gan-Ichino on the Shimura-Waldspurger correspondences, whereas their works play a critical role in our proof. Furthermore, all these are shown to be compatible with existing results in rank one (Waldspurger) and two (Gan-Ichino)., Comment: 90 pages

For handling intercurrent events in clinical trials, one of the strategies outlined in the ICH E9(R1) addendum targets the hypothetical scenario of non-occurrence of the intercurrent event. While this strategy is often implemented by setting data after the intercurrent event to missing even if they have been collected, g-estimation allows for a more efficient estimation by using the information contained in post-IE data. As the g-estimation methods have largely developed outside of randomised clinical trials, optimisations for the application in clinical trials are possible. In this work, we describe and investigate the performance of modifications to the established g-estimation methods, leveraging the assumption that some intercurrent events are expected to have the same impact on the outcome regardless of the timing of their occurrence. In a simulation study in Alzheimer disease, the modifications show a substantial efficiency advantage for the estimation of an estimand that applies the hypothetical strategy to the use of symptomatic treatment while retaining unbiasedness and adequate type I error control.

Scaling up deep learning models has been proven effective to improve intelligence of machine learning (ML) models, especially for industry recommendation models and large language models. The co-design of large distributed ML systems and algorithms (to maximize training performance) plays a pivotal role for its success. As it scales, the number of co-design hyper-parameters grows rapidly which brings challenges to feasibly find the optimal setup for system performance maximization. In this paper, we propose CubicML which uses ML to automatically optimize training performance of large distributed ML systems. In CubicML, we use an ML model as a proxy to predict the training performance for search efficiency and performance modeling flexibility. We proved that CubicML can effectively optimize training speed of in-house ads recommendation models with 73 billion parameters and large language models up to 405 billion parameters at Meta.

``Deep thermalization'' describes the emergence of universal wavefunction distributions in quantum many-body dynamics, appearing on a local subsystem upon measurement of its environment. In this work, we study in detail the effect of continuous internal symmetries and associated conservation laws on deep thermalization. Concretely, we consider quantum spin systems with a $U(1)$ symmetry associated with the conservation of magnetization (or `charge'), and analyze how the choice of initial states (specifically, their degree of charge fluctuations) and the choice of measurement basis (specifically, whether or not it can reveal information about the local charge density) determine the ensuing universal wavefunction distributions. We find a rich set of possibilities. First we focus on the case of a random state of well-defined charge subjected to charge-revealing masurements, and rigorously prove that the projected ensemble approaches a direct sum of Haar ensembles in the charge sectors of the subsystem of interest. We then analytically derive the limiting wavefunction distributions for more general initial states and measurement bases, finding results that include the Haar ensemble, the ``Scrooge ensemble'' (a distortion of the Haar ensemble by a density matrix), and the ``generalized Scrooge ensemble'' (a stochastic mixture of multiple Scrooge ensembles). These represent nontrivial higher-moment generalizations of the Gibbs state, and notably can depend on the entire charge distribution of the initial state, not just its average. Our findings demonstrate a rich interplay between symmetries and the information extracted by measurements, which allows deep thermalization to exhibit a range of universal behaviors far beyond regular thermalization., Comment: 15+15 pages, 8+3 figures

A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a specified canonical domain that preserves the local volume. This paper formulates the computation of ball-shaped volume-preserving parameterizations as an isovolumetric energy minimization (IEM) problem with the boundary points constrained on a unit sphere. In addition, we develop a new preconditioned nonlinear conjugate gradient algorithm for solving the IEM problem with guaranteed theoretical convergence and significantly improved accuracy and computational efficiency compared to other state-of-the-art algorithms. Applications to solid shape registration and deformation are presented to highlight the usefulness of the proposed algorithm., Comment: 23 pages, 10 figures

Point cloud segmentation is crucial for robotic visual perception and environmental understanding, enabling applications such as robotic navigation and 3D reconstruction. However, handling the sparse and unordered nature of point cloud data presents challenges for efficient and accurate segmentation. Inspired by the Mamba model's success in natural language processing, we propose the Serialized Point Cloud Mamba Segmentation Model (Serialized Point Mamba), which leverages a state-space model to dynamically compress sequences, reduce memory usage, and enhance computational efficiency. Serialized Point Mamba integrates local-global modeling capabilities with linear complexity, achieving state-of-the-art performance on both indoor and outdoor datasets. This approach includes novel techniques such as staged point cloud sequence learning, grid pooling, and Conditional Positional Encoding, facilitating effective segmentation across diverse point cloud tasks. Our method achieved 76.8 mIoU on Scannet and 70.3 mIoU on S3DIS. In Scannetv2 instance segmentation, it recorded 40.0 mAP. It also had the lowest latency and reasonable memory use, making it the SOTA among point semantic segmentation models based on mamba.

We uncover emergent universality arising in the equilibration dynamics of multimode continuous-variable systems. Specifically, we study the ensemble of pure states supported on a small subsystem of a few modes, generated by Gaussian measurements on the remaining modes of a globally pure bosonic Gaussian state. We find that beginning from sufficiently complex global states, such as random Gaussian states and product squeezed states coupled via a deep array of linear optical elements, the induced ensemble attains a universal form, independent of the choice of measurement basis: it is composed of unsqueezed coherent states whose displacements are distributed normally and isotropically, with variance depending on only the particle-number density of the system. We further show that the emergence of such a universal form is consistent with a generalized maximum entropy principle, which endows the limiting ensemble, which we call the "Gaussian Scrooge distribution", with a special quantum information-theoretic property of having minimal accessible information. Our results represent a conceptual generalization of the recently introduced notion of "deep thermalization" in discrete-variable quantum many-body systems -- a novel form of equilibration going beyond thermalization of local observables -- to the realm of continuous-variable quantum systems. Moreover, it demonstrates how quantum information-theoretic perspectives can unveil new physical phenomena and principles in quantum dynamics and statistical mechanics., Comment: 6+24 pages

Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their long-time dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical pseudo-randomness -- indistinguishability up to some large but finite moment, can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information., Comment: 22 pages, 4 figures

In this paper, we develop an $n$ dimensional volumetric stretch energy ($n$-VSE) functional for the volume-/mass-preserving parameterization of the $n$-manifolds topologically equivalent to $n$-ball. The $n$-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the $n$-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of $n$-VSE.

We are concerned with efficient numerical methods for stochastic continuous-time algebraic Riccati equations (SCARE). Such equations frequently arise from the state-dependent Riccati equation approach which is perhaps the only systematic way today to study nonlinear control problems. Often involved Riccati-type equations are of small scale, but have to be solved repeatedly in real time. Important applications include the 3D missile/target engagement, the F16 aircraft flight control, and the quadrotor optimal control, to name a few. A new inner-outer iterative method that combines the fixed-point strategy and the structure-preserving doubling algorithm (SDA) is proposed. It is proved that the method is monotonically convergent, and in particular, taking the zero matrix as initial, the method converges to the desired stabilizing solution. Previously, Newton's method has been called to solve SCARE, but it was mostly investigated from its theoretic aspect than numerical aspect in terms of robust and efficient numerical implementation. For that reason, we revisit Newton's method for SCARE, focusing on how to calculate each Newton iterative step efficiently so that Newton's method for SCARE can become practical. It is proposed to use our new inner-outer iterative method, which is provably convergent, to provide critical initial starting points for Newton's method to ensure its convergence. Finally several numerical experiments are conducted to validate the new method and robust implementation of Newton's method.