Efficient Minimization Algorithm Research Articles

Minimization of hypersurfaces

Let F ∈ Z [ x 0 , … , x n ] F \in \mathbb {Z}[x_0, \ldots , x_n] be homogeneous of degree d d and assume that F F is not a ‘nullform’, i.e., there is an invariant I I of forms of degree d d in n + 1 n+1 variables such that I ( F ) ≠ 0 I(F) \neq 0 . Equivalently, F F is semistable in the sense of Geometric Invariant Theory. Minimizing F F at a prime p p means to produce T ∈ M a t ( n + 1 , Z ) ∩ G L ( n + 1 , Q ) T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e ∈ Z ≥ 0 e \in \mathbb {Z}_{\ge 0} such that F 1 = p − e F ( [ x 0 , … , x n ] ⋅ T ) F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T) has integral coefficients and v p ( I ( F 1 ) ) v_p(I(F_1)) is minimal among all such F 1 F_1 . Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w ∈ Z ≥ 0 n + 1 w \in \mathbb {Z}_{\ge 0}^{n+1} to F F . We show that for any dimension n n and degree d d , there is a complete set of weight vectors consisting of [ 0 , w 1 , w 2 , … , w n ] [0,w_1,w_2,\dots ,w_n] with 0 ≤ w 1 ≤ w 2 ≤ ⋯ ≤ w n ≤ 2 n d n − 1 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1} . When n = 2 n = 2 , we improve the bound to d d . This answers a question raised by Kollár. These results are valid in a more general context, replacing Z \mathbb {Z} and p p by a PID R R and a prime element of R R . Based on this result and a further study of the minimization process in the planar case n = 2 n = 2 , we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d d . We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.

Efficient Multi-View Clustering via Unified and Discrete Bipartite Graph Learning.

Although previous graph-based multi-view clustering (MVC) algorithms have gained significant progress, most of them are still faced with three limitations. First, they often suffer from high computational complexity, which restricts their applications in large-scale scenarios. Second, they usually perform graph learning either at the single-view level or at the view-consensus level, but often neglect the possibility of the joint learning of single-view and consensus graphs. Third, many of them rely on the k -means for discretization of the spectral embeddings, which lack the ability to directly learn the graph with discrete cluster structure. In light of this, this article presents an efficient MVC approach via unified and discrete bipartite graph learning (UDBGL). Specifically, the anchor-based subspace learning is incorporated to learn the view-specific bipartite graphs from multiple views, upon which the bipartite graph fusion is leveraged to learn a view-consensus bipartite graph with adaptive weight learning. Furthermore, the Laplacian rank constraint is imposed to ensure that the fused bipartite graph has discrete cluster structures (with a specific number of connected components). By simultaneously formulating the view-specific bipartite graph learning, the view-consensus bipartite graph learning, and the discrete cluster structure learning into a unified objective function, an efficient minimization algorithm is then designed to tackle this optimization problem and directly achieve a discrete clustering solution without requiring additional partitioning, which notably has linear time complexity in data size. Experiments on a variety of multi-view datasets demonstrate the robustness and efficiency of our UDBGL approach. The code is available at https://github.com/huangdonghere/UDBGL.

Efficient ADMM Algorithm for Atomic Norm Minimization in SAR Tomography

Efficient ADMM Algorithm for Atomic Norm Minimization in SAR Tomography

Algorithmic Minimization of Uncertain Continuous-Time Markov Chains

The assumption of perfect knowledge of rate parameters in continuous-time Markov chains (CTMCs) is undermined when confronted with reality, where they may be uncertain due to lack of information or because of measurement noise. Here we consider uncertain CTMCs (UCTMCs), where rates are assumed to vary non-deterministically with time from bounded continuous intervals. An uncertain CTMC can be therefore seen as a specific type of Markov decision process for which the analysis is computationally difficult. To tackle this, we develop a theory of minimization which generalizes the notion of lumpability for CTMCs. Our first result is a quantitative and logical characterization of minimization. Specifically, we show that the reduced UCTMC model has a macro-state for each block of a partition of the state space, which preserves value functions and logical formulae whenever rewards are equal within each block. The second result is an efficient minimization algorithm for UCTMCs by means of partition refinement. As application, we show that reductions in a number of CTMCbenchmark models are robust with respect to uncertainties in original rates.

Joint Gramian inversion of gravity, magnetic and gravity gradient tensor data for the modelling of Mesoproterozoic iron oxide deposits, southeast Missouri, USA

SUMMARY Within the iron metallogenic province of southeast Missouri, USA, there are several mines that contain not only economic iron resources, magnetite and/or hematite, but also contain rare earth elements, copper and gold. An area including three major deposits, Pea Ridge, Bourbon and Kratz Spring, was selected for detailed modelling for the upper crustal magnetic susceptibility and density structures. For the study area, ground gravity and high-resolution airborne magnetic and gravity gradiometry data sets are available. An efficient and novel joint inversion algorithm for the simultaneous inversion of these multiple data sets is presented. The Gramian coupling constraint is used to correlate the reconstructed density and magnetic susceptibility models. The implementation relies on the structures of the sensitivity matrices and an efficient minimization algorithm to achieve significant reductions in the memory requirements and computational costs. Consequently, it is feasible to use a laptop computer for the inversion of multiple data sets, each containing thousands of data points, for the recovery of models on the study area, each including approximately one million model parameters. This is the first time that these multiple data sets have been simultaneously inverted for this area. The L1-norm stabilizer is used to provide compact and focused images of the ore deposits. For contrast, independent inversions of each data set are also discussed. In general, our results provide new insights about the concealed ore deposits in the Mesoproterozoic basement rocks of southeast Missouri. Both short- and long-wavelength anomalies exist in the recovered models; these provide a high-resolution image of the subsurface. The geometry and physical properties of the known deposits are determined very well. Additionally, some unknown concealed deposits are revealed; these could be economically valuable and should be considered in future geophysical and geological investigations.

On Minimization and Learning of Deterministic ω-Automata in the Presence of Don’t Care Words

We study minimization problems for deterministic ω-automata in the presence of don’t care words. We prove that the number of priorities in deterministic parity automata can be efficiently minimized under an arbitrary set of don’t care words. We derive that from a more general result from which one also obtains an efficient minimization algorithm for deterministic parity automata with informative right-congruence (without don’t care words). We then analyze languages of don’t care words with a trivial right-congruence. For such sets of don’t care words it is known that weak deterministic Büchi automata (WDBA) have a unique minimal automaton that can be efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give a congruence-based characterization of the corresponding minimal WDBA, and show that the don’t care minimization results for WDBA do not extend to deterministic ω-automata with informative right-congruence: for this class there is no unique minimal automaton for a given don’t care set with trivial right congruence, and the minimization problem is NP-hard. Finally, we extend an active learning algorithm for WDBA (Maler, Pnueli 1995) to the setting with an additional set of don’t care words with trivial right-congruence.

Hard-sphere jamming through the lens of linear optimization.

The jamming transition is ubiquitous. It is present in granular matter, foams, colloids, structural glasses, and many other systems. Yet, it defines a critical point whose properties still need to be fully understood. Recently, a major breakthrough came about when the replica formalism was extended to build a mean-field theory that provides an exact description of the jamming transition of spherical particles in the infinite-dimensional limit. While such theory explains the jamming critical behavior of both soft and hard spheres, investigating the transition in finite-dimensional systems poses very difficult and different problems, in particular from the numerical point of view. Soft particles are modeled by continuous potentials; thus, their jamming point can be reached through efficient energy minimization algorithms. In contrast, the latter methods are inapplicable to hard-sphere (HS) systems since the interaction energy among the particles is always zero by construction. To overcome these difficulties, here we recast the jamming of hard spheres as a constrained optimization problem and introduce the CALiPPSO algorithm, capable of readily producing jammed HS packings without including any effective potential. This algorithm brings a HS configuration of arbitrary dimensions to its jamming point by solving a chain of linear optimization problems. We show that there is a strict correspondence between the force balance conditions of jammed packings and the properties of the optimal solutions of CALiPPSO, whence we prove analytically that our packings are always isostatic and in mechanical equilibrium. Furthermore, using extensive numerical simulations, we show that our algorithm is able to probe the complex structure of the free-energy landscape, finding qualitative agreement with mean-field predictions. We also characterize the algorithmic complexity of CALiPPSO and provide an open-source implementation of it.

Nonsmooth Convex Optimization to Estimate the Covid-19 Reproduction Number Space-Time Evolution With Robustness Against Low Quality Data

Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design counter-measures. In an earlier work, we proposed to formulate the estimation of the reproduction number as an optimization problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that first formulation significantly lacks robustness against the Covid-19 data low quality (irrelevant or missing counts, pseudo-seasonalities,...) stemming from the emergency and crisis context, which significantly impairs accurate pandemic evolution assessments. The present work aims to overcome these limitations by carefully crafting a functional permitting to estimate jointly, in a single step, the reproduction number and outliers defined to model low quality data. This functional also enforces epidemiology-driven regularity properties for the reproduction number estimates, while preserving convexity, thus permitting the design of efficient minimization algorithms, based on proximity operators that are derived analytically. The explicit convergence of the proposed algorithm is proven theoretically. Its relevance is quantified on real Covid-19 data, consisting of daily new infection counts for 200+ countries and for the 96 metropolitan France counties, publicly available at Johns Hopkins University and Santé-Publique-France. The procedure permits automated daily updates of these estimates, reported via animated and interactive maps. Open-source estimation procedures will be made publicly available.

On the Power of Automata Minimization in Reactive Synthesis

Temporal logic is often used to describe temporal properties in AI applications. The most popular language for doing so is Linear Temporal Logic (LTL). Recently, LTL on finite traces, LTLf, has been investigated in several contexts. In order to reason about LTLf, formulas are typically compiled into deterministic finite automata (DFA), as the intermediate semantic representation. Moreover, due to the fact that DFAs have canonical representation, efficient minimization algorithms can be applied to maximally reduce DFA size, helping to speed up subsequent computations. Here, we present a thorough investigation on two classical minimization algorithms, namely, the Hopcroft and Brzozowski algorithms. More specifically, we show how to apply these algorithms to semi-symbolic (explicit states, symbolic transition functions) automata representation. We then compare the two algorithms in the context of an LTLf-synthesis framework, starting from LTLf formulas. While earlier studies on comparing the two algorithms starting from randomly-generated automata concluded that neither algorithm dominates, our results suggest that starting from LTLf formulas, Hopcroft's algorithm is the best choice in the context of reactive synthesis. Deeper analysis explains why the supposed advantage of Brzozowski's algorithm does not materialize in practice.

Poisson noisy image restoration via overlapping group sparse and nonconvex second-order total variation priors.

The restoration of the Poisson noisy images is an essential task in many imaging applications due to the uncertainty of the number of discrete particles incident on the image sensor. In this paper, we consider utilizing a hybrid regularizer for Poisson noisy image restoration. The proposed regularizer, which combines the overlapping group sparse (OGS) total variation with the high-order nonconvex total variation, can alleviate the staircase artifacts while preserving the original sharp edges. We use the framework of the alternating direction method of multipliers to design an efficient minimization algorithm for the proposed model. Since the objective function is the sum of the non-quadratic log-likelihood and nonconvex nondifferentiable regularizer, we propose to solve the intractable subproblems by the majorization-minimization (MM) method and the iteratively reweighted least squares (IRLS) algorithm, respectively. Numerical experiments show the efficiency of the proposed method for Poissonian image restoration including denoising and deblurring.

Tensor Field Graph-Cut for Image Segmentation: A Non-Convex Perspective

Image segmentation is a key component of image analysis, which refers to the process of partitioning the image into multiple segments. Graph cut is widely used in image segmentation by constructing a graph that the minimal cut of this graph would lead to partition the corresponding pixels of the different objects. In this paper, we reconstruct the graph cut problem as a special non-convex optimization problem instead of the traditional maximum flow problem. We extend this non-convex problem to the hypergraph method and combine it with a tensor field based on a directional bilateral filter bank to achieve segmentation in grayscale images. Accordingly, an efficient minimization algorithm is proposed to solve this non-convex problem with global convergence. Furthermore, we have selected the data of BSDS300 and BSDS500 as tests. Experimental results and evaluation index tests further demonstrate the superiority of the proposed method.

Differential network inference via the fused D-trace loss with cross variables

Detecting the change of biological interaction networks is of great importance in biological and medical research. We proposed a simple loss function, named as CrossFDTL, to identify the network change or differential network by estimating the difference between two precision matrices under Gaussian assumption. The CrossFDTL is a natural fusion of the D-trace loss for the considered two networks by imposing the $\ell _{1}$ penalty to the differential matrix to ensure sparsity. The key point of our method is to utilize the cross variables, which correspond to the sum and difference of two precision matrices instead of using their original forms. Moreover, we developed an efficient minimization algorithm for the proposed loss function and further rigorously proved its convergence. Numerical results showed that our method outperforms the existing methods in both accuracy and convergence speed for the simulated and real data.

Fast, Scalable, and Interactive Software for Landau-de Gennes Numerical Modeling of Nematic Topological Defects

Numerical modeling of nematic liquid crystals using the tensorial Landau-de Gennes (LdG) theory provides detailed insights into the structure and energetics of the enormous variety of possible topological defect configurations that may arise when the liquid crystal is in contact with colloidal inclusions or structured boundaries. However, these methods can be computationally expensive, making it challenging to predict (meta)stable configurations involving several colloidal particles, and they are often restricted to system sizes well below the experimental scale. Here we present an open-source software package that exploits the embarrassingly parallel structure of the lattice discretization of the LdG approach. Our implementation, combining CUDA/C++ and OpenMPI, allows users to accelerate simulations using both CPU and GPU resources in either single- or multiple-core configurations. We make use of an efficient minimization algorithm, the Fast Inertial Relaxation Engine (FIRE) method, that is well-suited to large-scale parallelization, requiring little additional memory or computational cost while offering performance competitive with other commonly used methods. In multi-core operation we are able to scale simulations up to supra-micron length scales of experimental relevance, and in single-core operation the simulation package includes a user-friendly GUI environment for rapid prototyping of interfacial features and the multifarious defect states they can promote. To demonstrate this software package, we examine in detail the competition between curvilinear disclinations and point-like hedgehog defects as size scale, material properties, and geometric features are varied. We also study the effects of an interface patterned with an array of topological point-defects.

A convex-nonconvex variational method for the additive decomposition of functions on surfaces

We present a convex-nonconvex variational approach for the additive decomposition of noisy scalar fields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a total variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non-convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components.

To optimize load of hybrid P2P cloud data-center using efficient load optimization and resource minimization algorithm

Cloud is the most trending technology used at almost in every part of the business and in every field of business. Cloud provides number of services in wider spectrum to cloud users from anywhere at any time. But, it achieved through several parameters like deployment model, resource optimization, load optimization etc. Nowadays, Load optimization is playing crucial role in cloud computing behalf system performance. The best optimization technique goal is to fulfill the user requirement efficiently with minimal resources and processing time. Parallel task processing is highly demanded in cloud application. The CPU resources are needed to move for parallelism growth due to communication and synchronization of parallel job arrival in cloud. It is difficult but highly demanded for a data center to response arrived task in parallel way. The objective of work is to design Efficient Load Optimization and Resource Minimization (ELORM) algorithm for optimizing the tasks at different Hybrid P2P Cloud data center zones and different users in cloud environment. The works provides an effective way to distribute the resources based on load prediction in the data centers for resource optimization. It enhances the load optimization, by maintaining the reliability and stability between the user base and data center during data transmission process. It also reduces the resource utilization and response time of the proposed algorithm compared than conventional methods. Proposed ELORM reduces 83.13 s Task completion time, 20.82 $ Virtual machine cost, 6.68% load balancing compare than conventional methods.

Reconstructing a 3D heart surface with stereo-endoscope by learning eigen-shapes.

An efficient approach to dynamically reconstruct a region of interest (ROI) on a beating heart from stereo-endoscopic video is developed. A ROI is first pre-reconstructed with a decoupled high-rank thin plate spline model. Eigen-shapes are learned from the pre-reconstructed data by using principal component analysis (PCA) to build a low-rank statistical deformable model for reconstructing subsequent frames. The linear transferability of PCA is proved, which allows fast eigen-shape learning. A general dynamic reconstruction framework is developed that formulates ROI reconstruction as an optimization problem of model parameters, and an efficient second-order minimization algorithm is derived to iteratively solve it. The performance of the proposed method is finally validated on stereo-endoscopic videos recorded by da Vinci robots.

Efficient and Robust Direct Image Registration Based on Joint Geometric and Photometric Lie Algebra.

This paper considers the joint geometric and photometric image registration problem. The inverse compositional (IC) algorithm and the efficient second-order minimization (ESM) algorithm are two typical efficient methods applied to the geometric registration problem. Their efficiency stems from the utilization of the group structure of geometric transformations. To allow for photometric variations, the dual IC algorithm (DIC) proposed by Bartoli performs joint geometric and photometric image registration by extending the IC algorithm. The group structures of both geometric and photometric transformations are exploited. Despite the robustness to large photometric variations, DIC is vulnerable to large geometric deformations. The ESM algorithm is extended by Silveira et al. to address photometric variations. In their approach, the photometric transformations are modeled in Euclidean space. Their approach is robust to relatively large geometric and photometric transformations; however, it is not efficient for large photometric variations. We propose a new efficient and robust image registration method by exploiting the non-Euclidean Lie group structure of joint geometric and photometric transformations for both grayscale and color images. The image registration is formulated as a nonlinear least squares problem. In our method, the geometric and photometric transformations are jointly parameterized by their corresponding Lie algebras. Based on this parameterization approach, the second-order approximation strategy of ESM is employed to optimize the joint geometric and photometric parameters. The error function in the nonlinear least squares problem is approximated by a second-order Taylor expansion with respect to joint geometric and photometric parameters without computing the Hessian matrix. For further efficiency, independent convergence criteria for geometric and photometric parameters are used in the iterative optimization process. The superiority of our proposed method over the previous methods, in terms of efficiency, accuracy, and robustness, is demonstrated through extensive experiments on synthetic and real data.

Scaling, proximity, and optimization of integrally convex functions

In discrete convex analysis, the scaling and proximity properties for the class of L $$^{\natural }$$ -convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when $$n \le 2$$ , while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.

Piecewise Linear Approximation of Vector-Valued Images and Curves via Second-Order Variational Model.

Variational models are known to work well for addressing image restoration/regularization problems. However, most of the methods proposed in the literature are defined for scalar inputs and are used on multiband images (such as RGB or multispectral imagery) by the composition of a simple band-wise processing. This involves suboptimal results and may introduce artifacts. Only in a few cases, variational models are extended to the case of vector-valued inputs. However, the known implementations are restricted to the first-order models, while the second-order models are never considered. Thus, typical problems of the first-order models, such as the staircasing effect cannot be overtaken. This paper considers a second-order functional model to function approximation with free discontinuities given by Blake-Zisserman (BZ) and proposes an efficient minimization algorithm in the case of vector-valued inputs. In the BZ model, the Hessian of the solution is penalized outside a set of finite length, therefore the solution is forced to be piecewise linear. Moreover, the model allows the formation of free discontinuities and free gradient discontinuities. The proposed algorithm is applied to difficult color image restoration/regularization problems and to piecewise linear approximation of curves in space.

An Efficient Energy Minimization for Conformal Parameterizations

Surface parameterizations have been widely applied to digital geometry processing. In this paper, we propose an efficient conformal energy minimization (CEM) algorithm for computing conformal parameterizations of simply-connected open surfaces with a very small angular distortion and a highly improved computational efficiency. In addition, we generalize the proposed CEM algorithm to computing conformal parameterizations of multiply-connected surfaces. Furthermore, we prove the existence of a nontrivial accumulation point of the proposed CEM algorithm under some mild conditions. Several numerical results show the efficiency and robustness of the CEM algorithm comparing to the existing state-of-the-art algorithms. An application of the CEM on the surface morphing between simply-connected open surfaces is demonstrated thereafter. Thanks to the CEM algorithm, the whole computations for the surface morphing can be performed efficiently and robustly.