Optimal Landscape Research Articles

Variational quantum optimization with multibasis encodings

Despite extensive research efforts, few quantum algorithms for classical optimization demonstrate a realizable quantum advantage. The utility of many quantum algorithms is limited by high requisite circuit depth and nonconvex optimization landscapes. We tackle these challenges by introducing a variational quantum algorithm that benefits from two innovations: multibasis graph encodings using single-qubit expectation values and nonlinear activation functions. Our technique results in increased observed optimization performance and a factor-of-two reduction in requisite qubits. While the classical simulation of many qubits with traditional quantum formalism is impossible due to its exponential scaling, we mitigate this limitation with exact circuit representations using factorized tensor rings. In particular, the shallow circuits permitted by our technique, combined with efficient factorized tensor-based simulation, enable us to successfully optimize the MaxCut of the 512-vertex DIMACS library graphs on a single GPU. By improving the performance of quantum optimization algorithms while requiring fewer quantum resources and utilizing shallower, more error-resistant circuits, we offer tangible progress for variational quantum optimization.

Urban Green Space Planning and Design for Sponge City

Urban green space designed based on the concept of sponge city is an important sponge carrier. Guided by the theory of landscape ecology, this article introduces the artificial neural network (ANN) model and uses the research method of landscape pattern index to establish a model for the proportion of urban green space types and the characteristics and dynamic changes of landscape spatial pattern. Based on the optimal landscape pattern index, the theoretical value of the optimal proportion of ecological green space, community park, comprehensive park, roadside green space, protective green space, and strip green space is obtained through the model. Based on this proportion value, combined with the ANN model, the reasonable evaluation results of patch density and average perimeter area ratio, spread degree, and diversity index of planned green space are obtained. Then, according to the influence of various green spaces on each landscape index, the development trend of urban green space construction in the future is predicted.

Mixed spatial scale effects of landscape structure on water quality in the Yellow River

Scientific understanding of the scale effects of landscape structure on water quality is essential for national land planning and water environment management. However, there is no consensus on the optimal spatial landscape scale affecting water quality. To this end, we developed a novel mix scale division method, which is consist of the intersection of sub-basin, stream buffer zone and circular buffer zone. Based on the water quality data from 113 monitoring stations of the Yellow River Basin, we quantitatively identified the spatial scale effects of various landscape indexes on water quality, as well as the nonlinear response of the water quality along the gradient of landscape indicators in the Yellow River, by using the Redundancy analysis (RDA) and Generalized Additive Model (GAM). We show that landscape indexes related to buildup and agricultural lands had significant impact on water quality changes, and this impact was more influential during the rainy season compared with the dry season. We also identified that the optimal scale and landscape pattern corresponding to the main non-point source factors, i.e., CODCr, total nitrogen (TN), and total phosphorus (TP). The variations of these three water quality indicators were predominated by buildup land (B-CA) at sub-basin (with a contribution percentage of 43.7%), agricultural land (A-CA) at the scale of stream buffer (bs) = 1000 m and circular buffer (bc) = 20 km (50.2%), and the largest patch index of agriculture (A-LPI) at the scale of bs = 1000m and bc = 25 km (41.7%), respectively. The B-CA>50000 ha at the sub-basin scale and A-LP = 50% at the mixed scale (bs = 1000m, bc = 25 km) would accelerate the deterioration of water quality. Results implied that water quality could be improved by implementing landscape regulation within an appropriate scale and considering the non-linear effects of landscape structure on water quality.

Optimizing vortex pinning in YBa2Cu3O7-x superconducting films up to high magnetic fields

The magnetic flux pinning capabilities of YBa2Cu3O7-x (YBCO) coated conductors vary strongly across different regions of the magnetic field–temperature phase diagram and with the orientation of the magnetic field θ. Here, we determine the optimal pinning landscape for a given region of the phase diagram by investigating the critical current density Jc(H,θ,T) in the 5–77 K temperature range, from self-field to high magnetic fields of 35 T. Our systematic analysis reveals promising routes for artificially engineering YBCO coated conductors in any region of interest of the phase diagram. In solution-derived nanocomposites, we identify the relevance of coexisting high amounts of short stacking faults, Cu-O vacancy clusters, and segmentation of twin boundaries, in combination with nanoparticles, for enhanced pinning performance at high magnetic fields and low temperatures. Moreover, we demonstrate that twin boundaries preserve a high pinning energy in thick YBCO films, which is beneficial for the pinning performance at high magnetic fields and high temperatures.

Loss landscapes and optimization in over-parameterized non-linear systems and neural networks

The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. The purpose of this work is to propose a modern view and a general mathematical framework for loss landscapes and efficient optimization in over-parameterized machine learning models and systems of non-linear equations, a setting that includes over-parameterized deep neural networks. Our starting observation is that optimization landscapes corresponding to such systems are generally not convex, even locally around a global minimum, a condition we call essential non-convexity . We argue that instead they satisfy PL ⁎ , a variant of the Polyak-Łojasiewicz condition [32] , [25] on most (but not all) of the parameter space, which guarantees both the existence of solutions and efficient optimization by (stochastic) gradient descent (SGD/GD). The PL ⁎ condition of these systems is closely related to the condition number of the tangent kernel associated to a non-linear system showing how a PL ⁎ -based non-linear theory parallels classical analyses of over-parameterized linear equations. We show that wide neural networks satisfy the PL ⁎ condition, which explains the (S)GD convergence to a global minimum. Finally we propose a relaxation of the PL ⁎ condition applicable to “almost” over-parameterized systems.

Camera trap data suggest uneven predation risk across vegetation types in a mixed farmland landscape.

Ground‐nesting farmland birds such as the grey partridge (Perdix perdix) have been rapidly declining due to a combination of habitat loss, food shortage, and predation. Predator activity is the least understood factor, especially its modulation by landscape composition and complexity. An important question is whether agri‐environment schemes such as flower strips are potentially useful for reducing predation risk, for example, from red fox (Vulpes vulpes). We employed 120 camera traps for two summers in an agricultural landscape in Central Germany to record predator activity (i.e., the number of predator captures) as a proxy for predation risk and used generalized linear mixed models (GLMMs) to investigate how the surrounding landscape affects predator activity in different vegetation types (flower strips, hedges, field margins, winter cereal, and rapeseed fields). Additionally, we used 48 cameras to study the distribution of predator captures within flower strips. Vegetation type was the most important factor determining the number of predator captures and capture rates in flower strips were lower than in hedges or field margins. Red fox capture rates were the highest of all predators in every vegetation type, confirming their importance as a predator for ground‐nesting birds. The number of fox captures increased with woodland area and decreased with structural richness and distance to settlements. In flower strips, capture rates in the center were approximately 9 times lower than at the edge. We conclude that the optimal landscape for ground‐nesting farmland birds seems to be open farmland with broad extensive vegetation elements and a high structural richness. Broad flower blocks provide valuable, comparatively safe nesting habitats, and the predation risk can further be minimized by placing them away from woods and settlements. Our results suggest that adequate landscape management may reduce predation pressure.

Deep learning, stochastic gradient descent and diffusion maps

Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major challenge. It has been observed empirically that most eigenvalues of the Hessian of the loss functions on the loss landscape of over-parametrized deep neural networks are close to zero, while only a small number of eigenvalues are large. Zero eigenvalues indicate zero diffusion along the corresponding directions. This indicates that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian. Although the parameter space is very high-dimensional, these findings seems to indicate that the SGD dynamics may mainly live on a low-dimensional manifold. In this paper, we pursue a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter surface, and in particular, of the landscape traced out by SGD by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration, we use diffusion maps introduced by R. Coifman and coauthors.

Neural Annealing and Visualization of Autoregressive Neural Networks in the Newman–Moore Model

Artificial neural networks have been widely adopted as ansatzes to study classical and quantum systems. However, for some notably hard systems, such as those exhibiting glassiness and frustration, they have mainly achieved unsatisfactory results, despite their representational power and entanglement content, thus suggesting a potential conservation of computational complexity in the learning process. We explore this possibility by implementing the neural annealing method with autoregressive neural networks on a model that exhibits glassy and fractal dynamics: the two-dimensional Newman–Moore model on a triangular lattice. We find that the annealing dynamics is globally unstable because of highly chaotic loss landscapes. Furthermore, even when the correct ground-state energy is found, the neural network generally cannot find degenerate ground-state configurations due to mode collapse. These findings indicate that the glassy dynamics exhibited by the Newman–Moore model caused by the presence of fracton excitations in the configurational space likely manifests itself through trainability issues and mode collapse in the optimization landscape.

Optimization landscape in the simplest constrained random least-square problem

We analyze statistical features of the ‘optimization landscape’ in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of M linear equations in N unknowns: ( a k , x ) = b k , k = 1, …, M on the N-sphere x 2 = N. We treat both the N-component vectors a k and parameters b k as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case M > N in the framework of the Kac-Rice approach combined with the random matrix theory for Wishart ensemble. Then we perform its asymptotic analysis as N → ∞ at a fixed α = M/N > 1 in various regimes. In particular, this analysis allows to extract the large deviation function for the density of the smallest Lagrange multiplier λ min associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value of the loss function as N → ∞. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the large deviation function for the density of at N ≫ 1 and any fixed ratio α = M/N > 0. As a by-product, we find the compatibility threshold α c < 1 which is the value of α beyond which a large random linear system on the N-sphere becomes typically incompatible.

Modularized Information Fusion Design of Urban Garden Landscape in Big Data Background

Traditional information fusion model has the problem of low efficiency in urban landscape design. In addition, using the current method to design urban commercial landscape public facilities, there are problems of large regional space occupation and unsatisfactory design effect. This paper designs a new modular information fusion model for urban landscape design process in view of genetic back propagation. On the basis of preprocessing sensor images, a digital elevation model is created using an ordered numerical sequence. Then, the stereo orthophoto image pair is obtained through the artificial parallax assistance mechanism, and the 3D garden landscape is generated by combining with the ant colony algorithm. The positive feedback mechanism of the ant colony algorithm is used to make the processing process converge continuously, and the optimal 3D garden landscape is finally generated by obtaining stereo orthophoto pairs through the artificial parallax-assisted mechanism. At the same time, the strong robustness and fault tolerance of neural network and parallel processing mechanism are utilized for fast information fusion. The scale and resources of garden design are described by the process dimension and the context dimension, and a modular garden landscape with distinct main body is built. Finally, the initial weight is optimized in the genetic real number coding algorithm, and the appropriate learning factor is selected to train the neural network so as to make the information fusion task. Experimental results show that the above model fusion process has good stability and low energy consumption for information fusion, which can promote the efficient construction of garden landscapes.

Fast gradient algorithm with toral decomposition for complex ICA

The article introduces the new gradient descent/ascent algorithm for independent component analysis for complex valued data (cICA). This algorithm uses toral decomposition of skew-Hermitian gradient matrix and very simple and computationally inexpensive method of projection based on Lie structure of optimization landscape. Toral decomposition is the diagonalization of the gradient matrix. In this stage a very fast Schur decomposition was used. The algorithm was tested for complex communication signals and compared to several classes of algorithms as gradient descent, quasi-Newton and complex JADE. The simulations showed very good results in terms of speed of operation and separation quality, as well as the highest versatility of applications in the context of the cost functions used among all the tested algorithms. The specificity of gradient methods characterized by quick response to changes in the signal indicates the practical usage potential of this method in on-line applications.

Early-time measure in eternal inflation

In a situation like eternal inflation, where our data is replicated at infinitely-many other space-time events, it is necessary to make a prior assumption about our location to extract predictions. The principle of mediocrity entails that we live at asymptotic late times, when the occupational probabilities of vacua has settled to a near-equilibrium distribution. In this paper we further develop the idea that we instead exist during the approach to equilibrium, much earlier than the exponentially-long mixing time. In this case we are most likely to reside in vacua that are easily accessed dynamically. Using first-passage statistics, we prove that vacua that maximize their space-time volume at early times have: 1. maximal ever-hitting probability; 2. minimal mean first-passage time; and 3. minimal decay rate. These requirements are succinctly captured by an early-time measure. The idea that we live at early times is a predictive guiding principle, with many phenomenological implications. First, our vacuum should lie deep in a funneled region, akin to folding energy landscapes of proteins. Second, optimal landscape regions are characterized by relatively short-lived vacua, with lifetime of order the de Sitter Page time. For our vacuum, this lifetime is ∼ 10130 years, which is consistent with the Standard Model estimate due to Higgs metastability. Third, the measure favors vacua with small, positive vacuum energy. This can address the cosmological constant problem, provided there are sufficiently many vacua in the entire ensemble of funnels. As a concrete example, we study the Bousso-Polchinski lattice of flux vacua, and find that the early-time measure favors lattices with the fewest number of flux dimensions. This favors compactifications with a large hierarchy between the lightest modulus and all other Kähler and complex structure moduli.

Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry

We present a new coil design paradigm for magnetic confinement in stellarators. Our approach directly optimizes coil shapes and coil currents to produce a vacuum quasi-symmetric magnetic field with a target rotational transform on the magnetic axis. This approach differs from the traditional two-stage approach in which first a magnetic configuration with desirable physics properties is found, and then coils to approximately realize this magnetic configuration are designed. The proposed single-stage approach allows us to find a compromise between confinement and engineering requirements, i.e., find easy-to-build coils with good confinement properties. Using forward and adjoint sensitivities, we derive derivatives of the physical quantities in the objective, which is constrained by a nonlinear periodic differential equation. In two numerical examples, we compare different gradient-based descent algorithms and find that incorporating approximate second-order derivative information through a quasi-Newton method is crucial for convergence. We also explore the optimization landscape in the neighborhood of a minimizer and find many directions in which the objective is mostly flat, indicating ample freedom to find simple and thus easy-to-build coils.

Exploring changes in landscape ecological risk in the Yangtze River Economic Belt from a spatiotemporal perspective

Landscape ecological risk reflects the extent to which ecosystems are threatened by human activities and environmental changes and is increasingly seen as the basis for decision-making in regional ecosystem management. Although the Yangtze River Economic Belt (YREB) has experienced drastic land use changes affected by human activities, the spatiotemporal heterogeneity of ecological risk in the region has not been thoroughly investigated. This study develops and applies an ecological risk assessment framework that integrates landscape pattern characteristics and landscape vulnerability dynamics to analyze spatiotemporal variations in landscape ecological risk in the YREB from 2000 to 2018. The results show moderate risk levels across most of the YREB during the study period, but risk was notably higher in the western and northern regions. Due to the gradual improvement in regional policies and the implementation of ecosystem restoration projects, there is a clear trend of risk reduction, and the area previously designated as high or medium–high risk was reduced by more than 150,000 km2 over the study period. Approximately 45% of the study area, where the risks are more difficult to mitigate or maintain at lower levels, was identified as a key area for future risk management. Significant spatiotemporal differences in ecological risks underscore the necessity of implementing spatially differentiated risk management strategies and long-term dynamic monitoring. This study provides a reference for future land use optimization and sustainable landscape management in the YREB.

Hyperparameter on-line learning of stochastic resonance based threshold networks

Aiming at training the feed-forward threshold neural network consisting of nondifferentiable activation functions, the approach of noise injection forms a stochastic resonance based threshold network that can be optimized by various gradient-based optimizers. The introduction of injected noise extends the noise level into the parameter space of the designed threshold network, but leads to a highly non-convex optimization landscape of the loss function. Thus, the hyperparameter on-line learning procedure with respective to network weights and noise levels becomes of challenge. It is shown that the Adam optimizer, as an adaptive variant of stochastic gradient descent, manifests its superior learning ability in training the stochastic resonance based threshold network effectively. Experimental results demonstrate the significant improvement of performance of the designed threshold network trained by the Adam optimizer for function approximation and image classification.

Constraint optimization and quantum control landscapes

We develop the embedded gradient vector field method, introduced in [8, 9], for the case of the special unitary group regarded as a constraint submanifold of the unitary group . The optimization problem associated to the trace fidelity cost function defined on that appears in the context of quantum control landscapes is completely solved using the embedded gradient vector field method. We prove that for N ⩾ 5, the landscape is not -trap free, there are always kinematic local extrema that are not global extrema.

Energy Landscape of State-Specific Electronic Structure Theory.

State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree–Fock and excited-state mean-field representations of singlet H2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.

Application of Landscape-Ecological Approach for Greenways Planning in Rural Agricultural Landscape

This article presents an innovative approach to the concept of facilitating greenways into sustainable landscape planning. The greenways can be planned by application of a landscape-ecological concept, including analysis and synthesis of selected abiotic, biotic, and socio-economic landscape-ecological factors and recreation conditions, and by reviewing the current landscape structure and condition of linear components of green infrastructure. Determining the landscape ecological stability, visual impact of agricultural lands, potential erosion risks and real erosion processes, and identifying the natural, cultural, and historical values in the landscape, contributes to the design of ideal greenways placement and other linear components of green infrastructure. Applying these proposals to the agricultural landscape would increase the ecological stability and connectivity, decrease soil and water erosion risks, eliminate visual impact, and develop recreational infrastructure. In this way, greenways planning brings about a synergy between sustainable rural development, landscape and nature protection, and landscape aesthetics, which provides optimal landscape utilization and may encourage tourism and economic prosperity in the study area. Finally, in addition to the researched ecological benefits, our greenways proposal represents an alternative connection of settlements in rural agricultural landscapes, and so it can stimulate sustainable mobility and recreation as well as physical activity, health, and well-being.

Effects of landscape complexity and local management on bee pollinator diversity in apple orchards in Changping District, Beijing, China.

Pollinators provide important ecosystem services for crop production and food security. With the development of agricultural economy and the increasing intensity of land-use, a large number of natural or semi-natural habitats have been converted to croplands. Landscape homogenization and intensive management lead to the decline of wild bee diversity and threaten the sustainable agricultural production. In this study, we investigated the effects of landscape complexity (proportion of semi-natural habitats), local management practices (local flowering plant diversity and soil total nitrogen), and their interactions on diversity of bee pollinators in apple orchard in Changping District, Beijing. A total of 8642 bee individuals were captured, including 5125 honey bees and 3517 wild bees from 5 families, 14 genera, and 49 species. The optimal landscape scale for the response of bee diversity to landscape complexity and local management intensity was 500 m. Within 500 m radius of the site, the abundance of overall bees and wild bees significantly increased with increasing proportion of semi-natural habitats. The landscape complexity interacting with local flowering plant diversity significantly affected the richness of overall bee and wild bee. When the proportion of semi-natural habitats surrounding the apple orchards was low (≤29.9%), we found a positive effect of flowering plant diversity on the richness of overall bee and wild bee, whereas a reversed trend was found when the proportion of semi-natural habitats surrounding the apple orchards was high (>29.9%). In addition, the abundance of honey bees significantly increased with the increase of local flowering plant diversity and soil total nitrogen. The soil total nitrogen interacting with local flowering plant diversity significantly affected the honey bee abundance. At low levels of soil total nitrogen (≤1.9 g·kg-1), there was a positive effect of flowering plant diversity on honey bee abundance; whereas this trend was reversed at high levels of soil total nitrogen (>1.9 g·kg-1). Increasing the proportion of semi-natural habitats in agricultural landscape was beneficial to the increase of wild bee abundance, and flowering plant diversity could promote bee diversity but depending on landscape scale (proportion of semi-natural habitats) and local scale (nitrogen application). Therefore, multi-scale factors should be considered to develop conservation strategies to maintain the diversity of wild bees in agricultural landscape. Maintaining a higher proportion of cultivated land as much as possible is still a long-term requirement for production, while maintaining intermediate landscape complexity, increasing the diversity of flowering plants on the ground, and reducing the application of nitrogen fertilizer would be effective ways to promote the diversity of pollinating bees in apple orchards.

A dual number abstraction for static analysis of Clarke Jacobians

We present a novel abstraction for bounding the Clarke Jacobian of a Lipschitz continuous, but not necessarily differentiable function over a local input region. To do so, we leverage a novel abstract domain built upon dual numbers, adapted to soundly over-approximate all first derivatives needed to compute the Clarke Jacobian. We formally prove that our novel forward-mode dual interval evaluation produces a sound, interval domain-based over-approximation of the true Clarke Jacobian for a given input region. Due to the generality of our formalism, we can compute and analyze interval Clarke Jacobians for a broader class of functions than previous works supported – specifically, arbitrary compositions of neural networks with Lipschitz, but non-differentiable perturbations. We implement our technique in a tool called DeepJ and evaluate it on multiple deep neural networks and non-differentiable input perturbations to showcase both the generality and scalability of our analysis. Concretely, we can obtain interval Clarke Jacobians to analyze Lipschitz robustness and local optimization landscapes of both fully-connected and convolutional neural networks for rotational, contrast variation, and haze perturbations, as well as their compositions.