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Quantum entanglement distribution between remote nodes is key to many promising quantum applications. Existing mechanisms have mainly focused on improving throughput and fidelity via entanglement routing or single-node scheduling. This paper considers entanglement scheduling and distribution among many source-destination pairs with different requests over an entire quantum network topology. Two practical scenarios are considered. When requests do not have deadlines, we seek to minimize the average completion time of the communication requests. If deadlines are specified, we seek to maximize the number of requests whose deadlines are met. Inspired by optimal scheduling disciplines in conventional single-queue scenarios, we design a general optimization framework for entanglement scheduling and distribution called ESDI, and develop a probabilistic protocol to implement the optimized solutions in a general buffered quantum network. We develop a discrete-time quantum network simulator for evaluation. Results show the superior performance of ESDI compared to existing solutions.

A quantum network distributes quantum entanglements between remote nodes, which is key to many quantum applications. However, unavoidable noise in quantum operations could lead to both low throughput and low quality of entanglement distribution. This paper aims to address the simultaneous exponential degradation in throughput and quality in a buffered multi-hop quantum network. Based on an end-to-end fidelity model with worst-case (isotropic) noise, we formulate the high-fidelity remote entanglement distribution problem for a single source-destination pair, and prove its NP-hardness. To address the problem, we develop a fully polynomial-time approximation scheme for the control plane of the quantum network, and a distributed data plane protocol that achieves the desired long-term throughput and worst-case fidelity based on control plane outputs. To evaluate our algorithm and protocol, we develop a discrete-time quantum network simulator. Simulation results show the superior performance of our approach compared to existing fidelity-agnostic and fidelity-aware solutions.

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is subject to error or natural changes over time. This motivated a new perspective on GSP, where the signal processing framework is developed for an entire class of graphs with similar structures. This approach can be formalized via the theory of graph limits, where graphs are considered as random samples from a distribution represented by a graphon. When the network under consideration has underlying symmetries, they may be modeled as samples from Cayley graphons. In Cayley graphons, vertices are sampled from a group, and the link probability between two vertices is determined by a function of the two corresponding group elements. Infinite groups such as the 1-dimensional torus can be used to model networks with an underlying spatial reality. Cayley graphons on finite groups give rise to a Stochastic Block Model, where the link probabilities between blocks form a (edge-weighted) Cayley graph. This manuscript summarizes some work on graph signal processing on large networks, in particular samples of Cayley graphons.

Quantum machine learning has the potential for a transformative impact across industry sectors and in particular in finance. In our work we look at the problem of hedging where deep reinforcement learning offers a powerful framework for real markets. We develop quantum reinforcement learning methods based on policy-search and distributional actor-critic algorithms that use quantum neural network architectures with orthogonal and compound layers for the policy and value functions. We prove that the quantum neural networks we use are trainable, and we perform extensive simulations that show that quantum models can reduce the number of trainable parameters while achieving comparable performance and that the distributional approach obtains better performance than other standard approaches, both classical and quantum. We successfully implement the proposed models on a trapped-ion quantum processor, utilizing circuits with up to $16$ qubits, and observe performance that agrees well with noiseless simulation. Our quantum techniques are general and can be applied to other reinforcement learning problems beyond hedging.

Factor models have been widely used in economics and finance. However, the heavy-tailed nature of macroeconomic and financial data is often neglected in the existing literature. To address this issue and achieve robustness, we propose an approach to estimate factor loadings and scores by minimizing the Huber loss function, which is motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms that use different penalty forms. The first algorithm, which we refer to as Huber PCA, minimizes the $\ell_2$-norm-type Huber loss and performs PCA on the weighted sample covariance matrix. The second algorithm involves an element-wise type Huber loss minimization, which can be solved by an iterative Huber regression algorithm. Our study examines the theoretical minimizer of the element-wise Huber loss function and demonstrates that it has the same convergence rate as conventional PCA when the idiosyncratic errors have bounded second moments. We also derive their asymptotic distributions under mild conditions. Moreover, we suggest a consistent model selection criterion that relies on rank minimization to estimate the number of factors robustly. We showcase the benefits of Huber PCA through extensive numerical experiments and a real financial portfolio selection example. An R package named ``HDRFA" has been developed to implement the proposed robust factor analysis.

Neural operators are gaining attention in computational science and engineering. PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate an underlying operator. The present work develops approximation theory for this approach, improving and significantly extending previous work in this direction. In terms of qualitative bounds, this paper derives a novel universal approximation result, under minimal assumptions on the underlying operator and the data-generating distribution. In terms of quantitative bounds, two potential obstacles to efficient operator learning with PCA-Net are identified, and made rigorous through the derivation of lower complexity bounds; the first relates to the complexity of the output distribution, measured by a slow decay of the PCA eigenvalues. The other obstacle relates the inherent complexity of the space of operators between infinite-dimensional input and output spaces, resulting in a rigorous and quantifiable statement of the curse of dimensionality. In addition to these lower bounds, upper complexity bounds are derived; first, a suitable smoothness criterion is shown to ensure a algebraic decay of the PCA eigenvalues. Then, it is shown that PCA-Net can overcome the general curse of dimensionality for specific operators of interest, arising from the Darcy flow and Navier-Stokes equations.

Invertible Neural Networks (INN) have become established tools for the simulation and generation of highly complex data. We propose a quantum-gate algorithm for a Quantum Invertible Neural Network (QINN) and apply it to the LHC data of jet-associated production of a Z-boson that decays into leptons, a standard candle process for particle collider precision measurements. We compare the QINN's performance for different loss functions and training scenarios. For this task, we find that a hybrid QINN matches the performance of a significantly larger purely classical INN in learning and generating complex data.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Dynamic neural network is an emerging research topic in deep learning. Compared to static models which have fixed computational graphs and parameters at the inference stage, dynamic networks can adapt their structures or parameters to different inputs, leading to notable advantages in terms of accuracy, computational efficiency, adaptiveness, etc. In this survey, we comprehensively review this rapidly developing area by dividing dynamic networks into three main categories: 1) instance-wise dynamic models that process each instance with data-dependent architectures or parameters; 2) spatial-wise dynamic networks that conduct adaptive computation with respect to different spatial locations of image data and 3) temporal-wise dynamic models that perform adaptive inference along the temporal dimension for sequential data such as videos and texts. The important research problems of dynamic networks, e.g., architecture design, decision making scheme, optimization technique and applications, are reviewed systematically. Finally, we discuss the open problems in this field together with interesting future research directions.

Deep convolutional neural networks (CNNs) have recently achieved great success in many visual recognition tasks. However, existing deep neural network models are computationally expensive and memory intensive, hindering their deployment in devices with low memory resources or in applications with strict latency requirements. Therefore, a natural thought is to perform model compression and acceleration in deep networks without significantly decreasing the model performance. During the past few years, tremendous progress has been made in this area. In this paper, we survey the recent advanced techniques for compacting and accelerating CNNs model developed. These techniques are roughly categorized into four schemes: parameter pruning and sharing, low-rank factorization, transferred/compact convolutional filters, and knowledge distillation. Methods of parameter pruning and sharing will be described at the beginning, after that the other techniques will be introduced. For each scheme, we provide insightful analysis regarding the performance, related applications, advantages, and drawbacks etc. Then we will go through a few very recent additional successful methods, for example, dynamic capacity networks and stochastic depths networks. After that, we survey the evaluation matrix, the main datasets used for evaluating the model performance and recent benchmarking efforts. Finally, we conclude this paper, discuss remaining challenges and possible directions on this topic.