Small Subspace Research Articles

A parallel algorithm for reliability assessment of multi-state flow networks based on simultaneous finding of all multi-state minimal paths and performing state space decomposition

This article presents a new efficient method for reliability evaluation of two-terminal Multi-state Flow Networks (MFNs). Indirect State Space Decomposition (SSD) methods are usually more efficient than direct SSD methods. However, indirect SSD methods require searching for all d-MPs/d-MCs. The presented method requires only an MFN, a system demand requirement d, and the probability distributions of states of components as an input. The presented algorithm uses a partition technique to decompose a state space into smaller and distinct subspaces. The algorithm tries to determine a d-flow with a lower bound for each subspace. If a flow exists and it is a d-MP, the algorithm computes the probability that a capacity vector belongs to this subspace and is bounded below by the d-MP and decomposes this subspace into smaller subspaces. For all subspaces without d-MP, the presented algorithm uses parallel computations and the SSD method to compute the probability that a capacity vector belongs to this subspace and is bounded below by a d-MP. The numerical experiments show that the efficiency of the presented method improves in comparison with other indirect SSD methods if the number of available logical processes and the number of d-MPs increase.

Out-of-time ordered correlators in kicked coupled tops: Information scrambling in mixed phase space and the role of conserved quantities

We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify “information scrambling” due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system’s dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival’s conjecture to partition the eigenstates of the Floquet map into “regular” and “chaotic.” Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.

Synthetic unknown class learning for learning unknowns

This paper addresses the open set recognition (OSR) problem, where the goal is to correctly classify samples of known classes while detecting unknown samples to reject. In the OSR problem, “unknown” is assumed to have infinite possibilities because we have no knowledge about unknowns until they emerge. Intuitively, the more an OSR system explores the possibilities of unknowns, the more likely it is to detect unknowns. Even though several generative OSR models have been proposed to explore more by generating synthetic samples and learning them as unknowns, the generated samples are limited to a small subspace of the known classes. Thus, this paper proposes a novel synthetic unknown class learning method that constantly generates unknown-like samples while maintaining diversity between the generated samples. By learning the unknown-like samples and known samples in an alternating manner, the proposed method can not only experience diverse synthetic unknowns but also reduce overgeneralization with respect to known classes. Experiments on several benchmark datasets show that the proposed method significantly outperforms other state-of-the-art approaches by generating diverse realistic unknown samples.

Shape-Aware Synthon Search (SASS) for Virtual Screening of Synthon-Based Chemical Spaces.

Virtual screening of large-scale chemical libraries has become increasingly useful for identifying high-quality candidates for drug discovery. While it is possible to exhaustively screen chemical spaces that number on the order of billions, indirect combinatorial approaches are needed to efficiently navigate larger, synthon-based virtual spaces. We describe Shape-Aware Synthon Search (SASS), a synthon-based virtual screening method that carries out shape similarity searches in the synthon space instead of the enumerated product space. SASS can replicate results from exhaustive searches in ultralarge, combinatorial spaces with high recall on a variety of query molecules while only scoring a small subspace of possible enumerated products, thereby significantly accelerating large-scale, shape-based virtual screening.

Modeling personality language use with small semantic vector subspaces

Modeling personality language use with small semantic vector subspaces

A multiphase dynamic programming algorithm for the shortest path problem with resource constraints

The shortest path problem with resource constraints (SPPRC) finds a least cost path between two nodes in a network while respecting constraints on resource consumption. The problem is mainly used as a subproblem inside column generation for crew scheduling and vehicle routing problems. The standard approach for the subproblems is based on dynamic programming (DP). This class of methods is generally effective in practice when there are only a few resources, but it seems to be time-consuming for huge instances with many resources. To handle this issue, we propose a new exact primal algorithm called the multiphase dynamic programming algorithm (MPDPA) to solve the SPPRC in acyclic networks. The proposed approach splits the state-space into small disjoint subspaces. These subspaces are sequentially explored in several iterations, where each iteration builds on the previous ones, to reduce the dimension of the subspaces to explore and to quickly generate better paths. Computational experiments on vehicle and crew scheduling instances show the excellent performance of our approach compared to the standard DP method. On the one hand, MPDPA returns optimal solutions while achieving time reduction factors between 1.44 and 3.59 on average compared to DP. On the other hand, MPDPA is able to generate feasible paths with up to 90% of the optimal cost in less than 10% of the time required by standard DP. This feature is useful in column generation and may greatly reduce the computational effort, because we can stop the MPDPA solution process once columns with sufficiently negative reduced costs are obtained.

Quaternionic metamonogenic functions in the unit disk

We construct a set of quaternionic metamonogenic functions (that is, in Ker(D+λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Ker}\\,}}(D+\\lambda )$$\\end{document} for diverse real λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the disk. The set is orthogonal except for the small subspace of elements of orders zero and one. These functions are used to express time-dependent solutions of the imaginary-time wave equation in the polar coordinate system.

Optimisation via encodings: a renormalisation group perspective

Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.

Reduced-Dimension Surrogate Modeling to Characterize the Damage Tolerance of Composite/Metal Structures

Complex engineering models are typically computationally demanding and defined by a high-dimensional parameter space challenging the comprehensive exploration of parameter effects and design optimization. To overcome this curse of dimensionality and to minimize computational resource requirements, this research demonstrates a user-friendly approach to formulating a reduced-dimension surrogate model that represents a high-dimensional, high-fidelity source model. This approach was developed specifically for a non-expert using commercially available tools. In this approach, the complex physical behavior of the high-fidelity source model is separated into individual, interacting physical behaviors. A separate reduced-dimension surrogate model is created for each behavior and then all are summed to formulate the reduced-dimension surrogate model representing the source model. In addition to a substantial reduction in computational resources and comparable accuracy, this method also provides a characterization of each individual behavior providing additional insight into the source model behavior. The approach encompasses experimental testing, finite element analysis, surrogate modeling, and sensitivity analysis and is demonstrated by formulating a reduced-dimension surrogate model for the damage tolerance of an aluminum plate reinforced with a co-cured bonded E-glass/epoxy composite laminate under four-point bending. It is concluded that this problem is difficult to characterize and breaking the problem into interacting mechanisms leads to improved information on influential parameters and efficient reduced-dimension surrogate modeling. The disbond damage at the interface between the resin and metal proved the most difficult mechanism for reduced-dimension surrogate modeling as it is only engaged in a small subspace of the full parameter space. A binary function was successful in engaging this damage mechanism when applicable based on the values of the most influential parameters.

The Enculturation of Liminality in Zine-Making: Acts of Associative Commonality That Enable Participation in a Media-Making Communitas

Participation in zine-making is enacted through a series of inter-linked making practices and intentions, constructed within both personal and interpersonal frames, and shared with others in different ways. Zines are catalysts for network formation within a cultural or sub-cultural community, with zine-makers sharing experiences, resources, and creativity to inform and associate with the community. Drawing on the constructivist grounded theory analysis of thirty-four semi-structured email interviews with zine-makers, this study sought to theorize a transdisciplinary understanding of the potential role zine-making communities play in the decision by an individual to participate in zine-making and how that participation was defined, enacted, and enculturated through zine-making. Using a constructivist grounded analysis, a model of participation was developed that defined and explored the bounded boundaryless of the zine-making communitas. In these spaces, members of the communitas deliberately transitioned into or through liminality and found affiliation and communion through engaging in and sharing the processes involved in zine-making (such as construction, distribution, and writing). Within this semi-fragile zine-making community, smaller sub-spaces were formed through engagement or affiliation with different interpretations and enactions of these processes, manifest in the production of different zine typologies and the creation and engagement with readership or circulation networks.

Adaptive Spectral Inversion for inverse medium problems

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting Adaptive Spectral Inversion (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using L 2-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

A consistent group-theoretic transformation for the block-diagonalization of structural matrices

Symmetry properties of physical systems may be studied through symmetry groups. In recent times, group theory has found application in the study of various problems in structural mechanics, specifically bifurcation, buckling, kinematics and vibration. Computational simplifications are achieved by decomposing the vector space of the problem into smaller subspaces that are independent of each other. When the basis vectors of a subspace are used as the symmetry-adapted variables of that subspace, a smaller problem (associated with a matrix of smaller dimensions) automatically results. However, the same decomposition may be achieved by first obtaining the structural matrix of the system, and then transforming this into a non-overlapping block-diagonal matrix, each independent block being associated with a subspace of the problem. The advantage of this approach is its greater amenability to computer programming, but it does not always give the correct results unless a very specific procedure is followed. The purpose of this contribution is to present a consistent group-theoretic approach for the block diagonalization of structural matrices.

Machine learning aided dimensionality reduction toward a resource efficient projective quantum eigensolver: Formal development and pilot applications.

In recent times, a variety of hybrid quantum-classical algorithms have been developed that aim to calculate the ground state energies of molecular systems on Noisy Intermediate-Scale Quantum (NISQ) devices. Albeit the utilization of shallow depth circuits in these algorithms, the optimization of ansatz parameters necessitates a substantial number of quantum measurements, leading to prolonged runtimes on the scantly available quantum hardware. Through our work, we lay the general foundation for an interdisciplinary approach that can be used to drastically reduce the dependency of these algorithms on quantum infrastructure. We showcase these pertinent concepts within the framework of the recently developed Projective Quantum Eigensolver (PQE) that involves iterative optimization of the nonlinearly coupled parameters through repeated residue measurements on quantum hardware. We demonstrate that one may perceive such a nonlinear optimization problem as a collective dynamic interplay of fast and slow relaxing modes. As such, the synergy among the parameters is exploited using an on-the-fly supervised machine learning protocol that dynamically casts the PQE optimization into a smaller subspace by reducing the effective degrees of freedom. We demonstrate analytically and numerically that our proposed methodology ensures a drastic reduction in the number of quantum measurements necessary for the parameter updates without compromising the characteristic accuracy. Furthermore, the machine learning model may be tuned to capture the noisy data of NISQ devices, and thus the predicted energy is shown to be resilient under a given noise model.

Rare-event sampling analysis uncovers the fitness landscape of the genetic code.

The genetic code refers to a rule that maps 64 codons to 20 amino acids. Nearly all organisms, with few exceptions, share the same genetic code, the standard genetic code (SGC). While it remains unclear why this universal code has arisen and been maintained during evolution, it may have been preserved under selection pressure. Theoretical studies comparing the SGC and numerically created hypothetical random genetic codes have suggested that the SGC has been subject to strong selection pressure for being robust against translation errors. However, these prior studies have searched for random genetic codes in only a small subspace of the possible code space due to limitations in computation time. Thus, how the genetic code has evolved, and the characteristics of the genetic code fitness landscape, remain unclear. By applying multicanonical Monte Carlo, an efficient rare-event sampling method, we efficiently sampled random codes from a much broader random ensemble of genetic codes than in previous studies, estimating that only one out of every 1020 random codes is more robust than the SGC. This estimate is significantly smaller than the previous estimate, one in a million. We also characterized the fitness landscape of the genetic code that has four major fitness peaks, one of which includes the SGC. Furthermore, genetic algorithm analysis revealed that evolution under such a multi-peaked fitness landscape could be strongly biased toward a narrow peak, in an evolutionary path-dependent manner.

Machine learning-enabled globally guaranteed evolutionary computation

Evolutionary computation, for example, particle swarm optimization, has impressive achievements in solving complex problems in science and industry; however, an important open problem in evolutionary computation is that there is no theoretical guarantee of reaching the global optimum and general reliability; this is due to the lack of a unified representation of diverse problem structures and a generic mechanism by which to avoid local optima. This unresolved challenge impairs trust in the applicability of evolutionary computation to a variety of problems. Here we report an evolutionary computation framework aided by machine learning, named EVOLER, which enables the theoretically guaranteed global optimization of a range of complex non-convex problems. This is achieved by: (1) learning a low-rank representation of a problem with limited samples, which helps to identify an attention subspace; and (2) exploring this small attention subspace via the evolutionary computation method, which helps to reliably avoid local optima. As validated on 20 challenging benchmarks, this method finds the global optimum with a probability approaching 1. We use EVOLER to tackle two important problems: power grid dispatch and the inverse design of nanophotonics devices. The method consistently reached optimal results that were challenging to achieve with previous state-of-the-art methods. EVOLER takes a leap forwards in globally guaranteed evolutionary computation, overcoming the uncertainty of data-driven black-box methods, and offering broad prospects for tackling complex real-world problems.

Hypercube quantum search: exact computation of the probability of success in polynomial time

In the emerging domain of quantum algorithms, Grover’s quantum search is certainly one of the most significant. It is relatively simple, performs a useful task and more importantly, does it in an optimal way. However, due to the success of quantum walks in the field, it is logical to study quantum search variants over several kinds of walks. In this paper, we propose an in-depth study of the quantum search over a hypercube layout. First, through the analysis of elementary walk operators restricted to suitable eigenspaces, we show that the acting component of the search algorithm takes place in a small subspace of the Hilbert workspace that grows linearly with the problem size. Subsequently, we exploit this property to predict the exact evolution of the probability of success of the quantum search in polynomial time.

Truncated atomic plane wave method for subband structure calculations of moiré systems

We propose a highly efficient and accurate numerical scheme named the truncated atomic plane wave method to determine the subband structure of twisted bilayer graphene (TBG) inspired by the Bistritzer-MacDonald model. Our method utilizes real-space information of carbon atoms in the moir\'e unit cell and projects the full tight-binding Hamiltonian into a much smaller subspace using atomic plane waves. Using our method, we are able to present accurate electronic band structures of TBG in a wide range of twist angles together with the detailed moir\'e potential and screened Coulomb interaction at the first magic angle. Furthermore, we generalize our formalism to solve the problem of low-frequency moir\'e phonons in TBG.

Machine learning matrix product state ansatz for strongly correlated systems.

Machine learning (ML) has been used to optimize the matrix product state (MPS) ansatz for the wavefunction of strongly correlated systems. The ML optimization of MPS has been tested for the Heisenberg Hamiltonian on one-dimensional and ladder lattices, which correspond to conjugated molecular systems. The input descriptors and output for the supervised ML are lattice configurations and configuration interaction coefficients, respectively. Efficient learning can be achieved from data over the full Hilbert space via exact diagonalization or full configuration interaction, as well as over a much smaller sub-space via Monte Carlo Configuration Interaction. We show that this circumvents the need to calculate energy and operator expectation values and is therefore a computationally efficient alternative to variational optimization.

A truncated Davidson method for the efficient "chemically accurate" calculation of full configuration interaction wavefunctions without any large matrix diagonalization.

This work develops and illustrates a new method of calculating "chemically accurate" electronic wavefunctions (and energies) via a truncated full configuration interaction (CI) procedure, which arguably circumvents the large matrix diagonalization that is the core problem of full CI and is also central to modern selective CI approaches. This is accomplished simply by following the standard/ubiquitous Davidson method in its "direct" form-wherein, in each iteration, the electronic Hamiltonian operator is applied directly in second quantization to the Ritz vector/wavefunction from the prior iteration-except that (in this work) only a small portion of the resultant expansion vector is actually even computed (through the application of only a similarly small portion of the Hamiltonian). Specifically, at each iteration of this truncated Davidson approach, the new expansion vector is taken to be twice as large as that from the prior iteration. In this manner, a small set of highly truncated expansion vectors (say 10-30) of increasing precision is incrementally constructed, forming a small subspace within which diagonalization of the Hamiltonian yields clear, consistent, and monotonically variational convergence to the approximate full CI limit. The good efficiency in which convergence to the level of chemical accuracy (1.6 mhartree) is achieved suggests, at least for the demonstrated problem sizes-Hilbert spaces of 1018 and wavefunctions of 108 determinants-that this truncated Davidson methodology can serve as a replacement of standard CI and complete-active space approaches in circumstances where only a few chemically significant digits of accuracy are required and/or meaningful in view of ever-present basis set limitations.

Error Estimates for Adaptive Spectral Decompositions

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L^2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.