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Gaussian graphical models provide a powerful framework to reveal the conditional dependency structure between multivariate variables. The process of uncovering the conditional dependency network is known as structure learning. Bayesian methods can measure the uncertainty of conditional relationships and include prior information. However, frequentist methods are often preferred due to the computational burden of the Bayesian approach. Over the last decade, Bayesian methods have seen substantial improvements, with some now capable of generating accurate estimates of graphs up to a thousand variables in mere minutes. Despite these advancements, a comprehensive review or empirical comparison of all recent methods has not been conducted. This paper delves into a wide spectrum of Bayesian approaches used for structure learning and evaluates their efficacy through a simulation study. We also demonstrate how to apply Bayesian structure learning to a real-world data set and provide directions for future research. This study gives an exhaustive overview of this dynamic field for newcomers, practitioners, and experts.

Two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently with the help of the extended subresultant theorem for parametric cases, two ideas are given to conduct the further partition of parameter space for choosing and checking the separating element. Based on these, we give two algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the two algorithms have been implemented on the computer algebra system Singular. Experimental data show that the second algorithm has the better performance in contrast to the first one.

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

(Economic) nonlinear model predictive control ((e)NMPC) requires dynamic models that are sufficiently accurate and computationally tractable. Data-driven surrogate models for mechanistic models can reduce the computational burden of (e)NMPC; however, such models are typically trained by system identification for maximum prediction accuracy on simulation samples and perform suboptimally in (e)NMPC. We present a method for end-to-end reinforcement learning of Koopman surrogate models for optimal performance as part of (e)NMPC. We apply our method to two applications derived from an established nonlinear continuous stirred-tank reactor model. The controller performance is compared to that of (e)NMPCs utilizing models trained using system identification, and model-free neural network controllers trained using reinforcement learning. We show that the end-to-end trained models outperform those trained using system identification in (e)NMPC, and that, in contrast to the neural network controllers, the (e)NMPC controllers can react to changes in the control setting without retraining.

We study a general factor analysis framework where the $n$-by-$p$ data matrix is assumed to follow a general exponential family distribution entry-wise. While this model framework has been proposed before, we here further relax its distributional assumption by using a quasi-likelihood setup. By parameterizing the mean-variance relationship on data entries, we additionally introduce a dispersion parameter and entry-wise weights to model large variations and missing values. The resulting model is thus not only robust to distribution misspecification but also more flexible and able to capture non-Gaussian covariance structures of the data matrix. Our main focus is on efficient computational approaches to perform the factor analysis. Previous modeling frameworks rely on simulated maximum likelihood (SML) to find the factorization solution, but this method was shown to lead to asymptotic bias when the simulated sample size grows slower than the square root of the sample size $n$, eliminating its practical application for data matrices with large $n$. Borrowing from expectation-maximization (EM) and stochastic gradient descent (SGD), we investigate three estimation procedures based on iterative factorization updates. Our proposed solution does not show asymptotic biases, and scales even better for large matrix factorizations with error $O(1/p)$. To support our findings, we conduct simulation experiments and discuss its application in three case studies.

Sampling from the output distributions of quantum computations comprising only commuting gates, known as instantaneous quantum polynomial (IQP) computations, is believed to be intractable for classical computers, and hence this task has become a leading candidate for testing the capabilities of quantum devices. Here we demonstrate that for an arbitrary IQP circuit undergoing dephasing or depolarizing noise, whose depth is greater than a critical $O(1)$ threshold, the output distribution can be efficiently sampled by a classical computer. Unlike other simulation algorithms for quantum supremacy tasks, we do not require assumptions on the circuit's architecture, on anti-concentration properties, nor do we require $\Omega(\log(n))$ circuit depth. We take advantage of the fact that IQP circuits have deep sections of diagonal gates, which allows the noise to build up predictably and induce a large-scale breakdown of entanglement within the circuit. Our results suggest that quantum supremacy experiments based on IQP circuits may be more susceptible to classical simulation than previously thought.

In this work, maximal $\alpha$-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed version via a quantum privacy mechanism. We first show that an adversary's maximal expected $\alpha$-gain using optimal measurement is characterized by measured conditional R\'enyi entropy. This can be viewed as a parametric generalization of K\"onig et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the $\alpha$-leakage and maximal $\alpha$-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured R\'enyi capacity, respectively. Various properties of maximal $\alpha$-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized $\alpha$-leakage and regularized maximal $\alpha$-leakage for identical and independent quantum privacy mechanisms coincide with $\alpha$-tilted sandwiched R\'enyi information and sandwiched R\'enyi capacity, respectively.

In many applications, a combinatorial problem must be repeatedly solved with similar, but distinct parameters. Yet, the parameters $w$ are not directly observed; only contextual data $d$ that correlates with $w$ is available. It is tempting to use a neural network to predict $w$ given $d$. However, training such a model requires reconciling the discrete nature of combinatorial optimization with the gradient-based frameworks used to train neural networks. When the problem in question is an Integer Linear Program (ILP), one approach to overcome this training issue is to consider a continuous relaxation of the combinatorial problem. While existing methods utilizing this approach have shown to be highly effective on small problems, they do not always scale well to large problems. In this work, we draw on ideas from modern convex optimization to design a network and training scheme which scales effortlessly to problems with thousands of variables. Our experiments verify the computational advantage our proposed method enjoys on two representative problems, namely the shortest path problem and the knapsack problem.

Limit theorems for the magnetization in the $p$-spin Curie-Weiss model, for $p \geq 3$, has been derived recently by Mukherjee et al. (2021). In this paper, we strengthen these results by proving Cram\'er-type moderate deviation theorems and Berry-Esseen bounds for the magnetization (suitably centered and scaled). In particular, we show that the rate of convergence is $O(N^{-\frac{1}{2}})$ when the magnetization has asymptotically Gaussian fluctuations, and it is $O(N^{-\frac{1}{4}})$ when the fluctuations are non-Gaussian. As an application, we derive a Berry-Esseen bound for the maximum pseudolikelihood estimate of the inverse temperature in $p$-spin Curie-Weiss model with no external field, for all points in the parameter space where consistent estimation is possible.

The maximum coverage problem is to select $k$ sets from a collection of sets such that the cardinality of the union of the selected sets is maximized. We consider $(1-1/e-\epsilon)$-approximation algorithms for this NP-hard problem in three standard data stream models. 1. {\em Dynamic Model.} The stream consists of a sequence of sets being inserted and deleted. Our multi-pass algorithm uses $\epsilon^{-2} k \cdot \text{polylog}(n,m)$ space. The best previous result (Assadi and Khanna, SODA 2018) used $(n +\epsilon^{-4} k) \text{polylog}(n,m)$ space. While both algorithms use $O(\epsilon^{-1} \log n)$ passes, our analysis shows that when $\epsilon$ is a constant, it is possible to reduce the number of passes by a $1/\log \log n$ factor without incurring additional space. 2. {\em Random Order Model.} In this model, there are no deletions and the sets forming the instance are uniformly randomly permuted to form the input stream. We show that a single pass and $k \text{polylog}(n,m)$ space suffices for arbitrary small constant $\epsilon$. The best previous result, by Warneke et al.~(ESA 2023), used $k^2 \text{polylog}(n,m)$ space. 3. {\em Insert-Only Model.} Lastly, our results, along with numerous previous results, use a sub-sampling technique introduced by McGregor and Vu (ICDT 2017) to sparsify the input instance. We explain how this technique and others used in the paper can be implemented such that the amortized update time of our algorithm is polylogarithmic. This also implies an improvement of the state-of-the-art insert only algorithms in terms of the update time: $\text{polylog}(m,n)$ update time suffices whereas the best previous result by Jaud et al.~(SEA 2023) required update time that was linear in $k$.