Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time $\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right)$ where $\epsilon$ is the amount of error we are willing to tolerate. Here, $\kappa$ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever $\kappa$ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time $\widetilde{O}(m^{3/2} \log (1/\epsilon))$. In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Classical and quantum algorithms for scaling problems

This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

Enhanced Harris's Hawk algorithm for continuous multi-objective optimization problems

Multi-objective swarm intelligence-based (MOSI-based) metaheuristics were proposed to solve multi-objective optimization problems (MOPs) with conflicting objectives. Harris’s hawk multi-objective optimizer (HHMO) algorithm is a MOSIbased algorithm that was developed based on the reference point approach. The reference point is determined by the decision maker to guide the search process to a particular region in the true Pareto front. However, HHMO algorithm produces a poor approximation to the Pareto front because lack of information sharing in its population update strategy, equal division of convergence parameter and randomly generated initial population. A two-step enhanced non-dominated sorting HHMO (2SENDSHHMO) algorithm has been proposed to solve this problem. The algorithm includes (i) a population update strategy which improves the movement of hawks in the search space, (ii) a parameter adjusting strategy to control the transition between exploration and exploitation, and (iii) a population generating method in producing the initial candidate solutions. The population update strategy calculates a new position of hawks based on the flush-and-ambush technique of Harris’s hawks, and selects the best hawks based on the non-dominated sorting approach. The adjustment strategy enables the parameter to adaptively changed based on the state of the search space. The initial population is produced by generating quasi-random numbers using Rsequence followed by adapting the partial opposition-based learning concept to improve the diversity of the worst half in the population of hawks. The performance of the 2S-ENDSHHMO has been evaluated using 12 MOPs and three engineering MOPs. The obtained results were compared with the results of eight state-of-the-art multi-objective optimization algorithms. The 2S-ENDSHHMO algorithm was able to generate non-dominated solutions with greater convergence and diversity in solving most MOPs and showed a great ability in jumping out of local optima. This indicates the capability of the algorithm in exploring the search space. The 2S-ENDSHHMO algorithm can be used to improve the search process of other MOSI-based algorithms and can be applied to solve MOPs in applications such as structural design and signal processing

AI: Limits and Prospects of Artificial Intelligence

The emergence of artificial intelligence has triggered enthusiasm and promise of boundless opportunities as much as uncertainty about its limits. The contributions to this volume explore the limits of AI, describe the necessary conditions for its functionality, reveal its attendant technical and social problems, and present some existing and potential solutions. At the same time, the contributors highlight the societal and attending economic hopes and fears, utopias and dystopias that are associated with the current and future development of artificial intelligence