arXiv subjects

Ping Xu

Publications and source records attributed to Ping Xu.

At least 55 records · Page 3Linked to original sources

arXiv2023

Quantization of (-1)-Shifted Derived Poisson Manifolds

We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms of $\BV_\infty$-operators on the space of Berezinian half-densities. We prove that quantizing such a $(-1)$-shifted derived Poisson manifold is equivalent to the lifting of a consecutive sequences of Maurer-Cartan elements of short exact sequences of differential graded Lie algebras, where the obstruction is a certain class in the second Poisson cohomology. Consequently, a $(-1)$-shifted derived Poisson manifold is quantizable if the second Poisson cohomology group vanishes. We also prove that for any $Ł$-algebroid $\Cc{\aV}$, its corresponding linear $(-1)$-shifted derived Poisson manifold $\Cc{\aV}^\vee[-1]$ admits a canonical quantization. Finally, given a Lie algebroid $A$ and a one-cocycle $s\in \sections{A^\vee}$, the $(-1)$-shifted derived Poisson manifold corresponding to the derived intersection of coisotropic submanifolds determined by the graph of $s$ and the zero section of the Lie Poisson $A^\vee$ is shown to admit a canonical quantization in terms of Evens-Lu-Weinstein module.

math.SG↗

arXiv2023

A real-time hole depth diagnostic based on coherent imaging with plasma amendment during femtosecondlaser hole-drilling

An in-process coherent imaging diagnostic has been developed to real-time measure the hole depth during air-film hole drilling by a femtosecond laser. A super-luminescent diode with a wavelength of 830~13 nm is chosen as the coherent light source which determines a depth resolution of 12 μm. The drilled hole is coupled as a part of the sample arm and the depth variation can be extracted from the length variation of the optical path. Interference is realized in the detection part and a code has been written to discriminate the interference fringes. Density of plasma in the hole is diagnosed to evaluate its amendment to the optical path length and the depth measurement error induced by plasma is non-ignorable when drilling deep holes.

physics.ins-det↗

arXiv2023

Proteus: Simulating the Performance of Distributed DNN Training

DNN models are becoming increasingly larger to achieve unprecedented accuracy, and the accompanying increased computation and memory requirements necessitate the employment of massive clusters and elaborate parallelization strategies to accelerate DNN training. In order to better optimize the performance and analyze the cost, it is indispensable to model the training throughput of distributed DNN training. However, complex parallelization strategies and the resulting complex runtime behaviors make it challenging to construct an accurate performance model. In this paper, we present Proteus, the first standalone simulator to model the performance of complex parallelization strategies through simulation execution. Proteus first models complex parallelization strategies with a unified representation named Strategy Tree. Then, it compiles the strategy tree into a distributed execution graph and simulates the complex runtime behaviors, comp-comm overlap and bandwidth sharing, with a Hierarchical Topo-Aware Executor (HTAE). We finally evaluate Proteus across a wide variety of DNNs on three hardware configurations. Experimental results show that Proteus achieves $3.0\%$ average prediction error and preserves order for training throughput of various parallelization strategies. Compared to state-of-the-art approaches, Proteus reduces prediction error by up to $133.8\%$.

cs.DC↗

arXiv2022

Robust Distributed Learning Against Both Distributional Shifts and Byzantine Attacks

In distributed learning systems, robustness issues may arise from two sources. On one hand, due to distributional shifts between training data and test data, the trained model could exhibit poor out-of-sample performance. On the other hand, a portion of working nodes might be subject to byzantine attacks which could invalidate the learning result. Existing works mostly deal with these two issues separately. In this paper, we propose a new algorithm that equips distributed learning with robustness measures against both distributional shifts and byzantine attacks. Our algorithm is built on recent advances in distributionally robust optimization as well as norm-based screening (NBS), a robust aggregation scheme against byzantine attacks. We provide convergence proofs in three cases of the learning model being nonconvex, convex, and strongly convex for the proposed algorithm, shedding light on its convergence behaviors and endurability against byzantine attacks. In particular, we deduce that any algorithm employing NBS (including ours) cannot converge when the percentage of byzantine nodes is 1/3 or higher, instead of 1/2, which is the common belief in current literature. The experimental results demonstrate the effectiveness of our algorithm against both robustness issues. To the best of our knowledge, this is the first work to address distributional shifts and byzantine attacks simultaneously.

cs.LG↗

arXiv2022

Large-scale full-programmable quantum walk and its applications

With photonics, the quantum computational advantage has been demonstrated on the task of boson sampling. Next, developing quantum-enhanced approaches for practical problems becomes one of the top priorities for photonic systems. Quantum walks are powerful kernels for developing new and useful quantum algorithms. Here we realize large-scale quantum walks using a fully programmable photonic quantum computing system. The system integrates a silicon quantum photonic chip, enabling the simulation of quantum walk dynamics on graphs with up to 400 vertices and possessing full programmability over quantum walk parameters, including the particle property, initial state, graph structure, and evolution time. In the 400-dimensional Hilbert space, the average fidelity of random entangled quantum states after the whole on-chip circuit evolution reaches as high as 94.29$\pm$1.28$\%$. With the system, we demonstrated exponentially faster hitting and quadratically faster mixing performance of quantum walks over classical random walks, achieving more than two orders of magnitude of enhancement in the experimental hitting efficiency and almost half of the reduction in the experimental evolution time for mixing. We utilize the system to implement a series of quantum applications, including measuring the centrality of scale-free networks, searching targets on Erdös-Rényi networks, distinguishing non-isomorphic graph pairs, and simulating the topological phase of higher-order topological insulators. Our work shows one feasible path for quantum photonics to address applications of practical interests in the near future.

quant-ph↗

arXiv2022

QC-ODKLA: Quantized and Communication-Censored Online Decentralized Kernel Learning via Linearized ADMM

This paper focuses on online kernel learning over a decentralized network. Each agent in the network receives continuous streaming data locally and works collaboratively to learn a nonlinear prediction function that is globally optimal in the reproducing kernel Hilbert space with respect to the total instantaneous costs of all agents. In order to circumvent the curse of dimensionality issue in traditional online kernel learning, we utilize random feature (RF) mapping to convert the non-parametric kernel learning problem into a fixed-length parametric one in the RF space. We then propose a novel learning framework named Online Decentralized Kernel learning via Linearized ADMM (ODKLA) to efficiently solve the online decentralized kernel learning problem. To further improve the communication efficiency, we add the quantization and censoring strategies in the communication stage and develop the Quantized and Communication-censored ODKLA (QC-ODKLA) algorithm. We theoretically prove that both ODKLA and QC-ODKLA can achieve the optimal sublinear regret $\mathcal{O}(\sqrt{T})$ over $T$ time slots. Through numerical experiments, we evaluate the learning effectiveness, communication, and computation efficiencies of the proposed methods.

cs.LG↗

arXiv2022

Hochschild cohomology of dg manifolds associated to integrable distributions

For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of $F$. In particular, for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ associated with a complex manifold $X$, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomology $HH^{\bullet}(X)$ of the complex manifold $X$. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ implies the Duflo-Kontsevich theorem for complex manifolds.

math.DG↗

arXiv2022

Arbitrary coherent distributions in a programmable quantum walk

The coherent superposition of position states in a quantum walk (QW) can be precisely engineered towards the desired distributions to meet the need of quantum information applications. The coherent distribution can make full use of quantum parallel in computation and simulation. Particularly, the uniform superposition provides the robust non-locality, which has wide applications such as the generation of genuine multi-bit random numbers without post-processing. We experimentally demonstrate that the rich dynamics featured with arbitrary coherent distributions can be obtained by introducing different sets of the time- and position-dependent operations. Such a QW is realized by a resource-constant and flexible optical circuit, in which the variable operation is executed based on a Sagnac interferometer in an intrinsically stable and precisely controlled way. Our results contribute to the practical realization of quantum-walk-based quantum computation, quantum simulations and quantum information protocols.

quant-ph↗

arXiv2022

Maximal coin-walker entanglement in a ballistic quantum walk

We report the position-inhomogeneous quantum walk (IQW) can be utilized to produce the maximal high dimensional entanglement while maintaining the quadratic speedup spread of the wave-function. Our calculations show that the maximal coin-walker entanglement can be generated in any odd steps or asymptotically in even steps, and the nearly maximal entanglement can be obtained in even steps after $2$. We implement the IQW by a stable resource-saving time-bin optical network, in which a polarization Sagnac loop is employed to realize the precisely tunable phase shift. Our approach opens up an efficient way for high-dimensional entanglement engineering as well as promotes investigations on the role of coin-walker interactions in QW based applications.

quant-ph↗

arXiv2021

Dg manifolds, formal exponential maps and homotopy Lie algebras

This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_\infty[1]$ algebras associated with dg manifolds in the $C^\infty$ context. Given a dg manifold, we prove that a `formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ arising from a complex manifold $X$, we prove that this $L_\infty[1]$ algebra structure is quasi-isomorphic to the standard $L_\infty[1]$ algebra structure on the Dolbeault complex $Ω^{0,\bullet}(T^{1,0}_X)$.

math.DG↗

arXiv2021

General Quantum Bernoulli Factory: Framework Analysis and Experiments

The unremitting pursuit for quantum advantages gives rise to the discovery of a quantum-enhanced randomness processing named quantum Bernoulli factory (QBF). This quantum enhanced process can show its priority over the corresponding classical process through readily available experimental resources, thus in the near term it may be capable of accelerating the applications of classical Bernoulli factories, such as the widely used sampling algorithms. In this work, we provide the framework analysis of the QBF. We thoroughly analyze the quantum state evolution in this process, discovering the field structure of the constructible quantum states. Our framework analysis shows that naturally, the previous works can be described as specific instances of this framework. Then, as a proof of principle, we experimentally demonstrate this framework via an entangled two-photon source along with a reconfigurable photonic logic, and show the advantages of the QBF over the classical model through a classically infeasible instance. These results may stimulate the discovery of advantages of the quantum randomness processing in a wider range of tasks, as well as its potential applications.

quant-ph↗

arXiv2021

COKE: Communication-Censored Decentralized Kernel Learning

This paper studies the decentralized optimization and learning problem where multiple interconnected agents aim to learn an optimal decision function defined over a reproducing kernel Hilbert space by jointly minimizing a global objective function, with access to their own locally observed dataset. As a non-parametric approach, kernel learning faces a major challenge in distributed implementation: the decision variables of local objective functions are data-dependent and thus cannot be optimized under the decentralized consensus framework without any raw data exchange among agents. To circumvent this major challenge, we leverage the random feature (RF) approximation approach to enable consensus on the function modeled in the RF space by data-independent parameters across different agents. We then design an iterative algorithm, termed DKLA, for fast-convergent implementation via ADMM. Based on DKLA, we further develop a communication-censored kernel learning (COKE) algorithm that reduces the communication load of DKLA by preventing an agent from transmitting at every iteration unless its local updates are deemed informative. Theoretical results in terms of linear convergence guarantee and generalization performance analysis of DKLA and COKE are provided. Comprehensive tests on both synthetic and real datasets are conducted to verify the communication efficiency and learning effectiveness of COKE.

cs.LG↗

arXiv2021

Weyl quantization of degree 2 symplectic graded manifolds

Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle $(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra $\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the associated graded algebra $\mathrm{gr}\mathcal{D}(M,S)$ is isomorphic to the algebra $\mathcal{O}(\mathcal{M})$ of functions on $\mathcal{M}$, where $\mathcal{M}$ is the symplectic graded manifold of degree $2$ canonically associated to $(E,\mathrm{g})$. Accordingly, we define the Weyl quantization on $\mathcal{M}$ as a map $\mathcal{WQ}_\hbar:\mathcal{O}(\mathcal{M})\to\mathcal{D}(M,S)$, and prove that $\mathcal{WQ}_\hbar$ satisfies all desired usual properties. As an application, we obtain a bijection between Courant algebroid structures $(E,\mathrm{g},ρ,[\cdot,\cdot])$, that are encoded by Hamiltonian generating functions on $\mathcal{M}$, and skew-symmetric Dirac generating operators $D\in\mathcal{D}(M,S)$. The operator $D^2$ gives a new invariant of $(E,\mathrm{g},ρ,[\cdot,\cdot])$, which generalizes the square norm of the Cartan $3$-form of a quadratic Lie algebra. We study in detail the particular case of $E$ being the double of a Lie bialgebroid $(A,A^*)$.

math.DG↗

arXiv2021

Derived Differentiable Manifolds

We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore, we can make sense of "homotopy fibered product" and "derived intersection" of submaifolds in a smooth manifold in the homotopy category of derived manifolds. We construct a factorization of the diagonal using path spaces. First we construct an infinite-dimensional factorization using actual path spaces motivated by the AKSZ construction, then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. We also prove the inverse function theorem for derived manifolds, and investigate the relationship between weak equivalence and quasi-isomorphism for derived manifolds.

math.DG↗

arXiv2021

Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds

We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over the same base manifold $M$ corresponds a Kapranov dg-manifold structure on $A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence, $Γ(Λ^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$ algebra structure whose unary bracket is the Chevalley--Eilenberg differential corresponding to the Bott representation of $A$ on $L/A$ and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair $(L,A)$. To this end, we construct explicit isomorphisms of $C^\infty(M)$-coalgebras $Γ\big(S(L/A)\big)\xrightarrow{\sim}\frac{\mathcal{U}(L)}{\mathcal{U}(L)Γ(A)}$, which we elect to call Poincaré--Birkhoff--Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map $S(\mathfrak{g})\to\mathcal{U}(\mathfrak{g})$ of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold $A[1]\oplus L/A$ is linearizable if and only if the Atiyah class of the Lie pair $(L,A)$ vanishes.

math.DG↗

arXiv2021

Polyvector fields and polydifferential operators associated with Lie pairs

We prove that the spaces $\operatorname{tot}\big(Γ(Λ^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and $\operatorname{tot}\big(Γ(Λ^\bullet A^\vee)\otimes_R\mathcal{D}_{\operatorname{poly}}^{\bullet}\big)$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to an $L_\infty$ isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair $(L,A)$. Consequently, both $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{T}_{\operatorname{poly}}^{\bullet})$ and $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{D}_{\operatorname{poly}}^{\bullet})$ admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold).

math.QA↗

arXiv2021

Tracking quantum state evolution by the Berry curvature with a two-level system

We investigate two kinds of topological structures (sphere and torus) spanned by the controlled parameters of a driven two-level system's Hamiltonian, and consider the connection between the structures and the system's dynamics. We discuss the Berry curvature obtained through the dynamical response method, show the certain physical and observable manifolds including the gapped region probed by integrating the Berry curvature, and demonstrate the system's state evolution can be tracked and manipulated by extracting the Berry curvature.

quant-ph↗

arXiv2021

Symplectic realizations of holomorphic Poisson manifolds

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, π)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subseteq T^*X$.

math.DG↗