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Journal articles on the topic 'Hessian metric'

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Published: 2 June 2025

Last updated: 13 July 2025

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1

García Ariza, M. Á. "Degenerate Hessian structures on radiant manifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (2018): 1850087. http://dx.doi.org/10.1142/s0219887818500871.

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We present a rigorous mathematical treatment of Ruppeiner geometry, by considering degenerate Hessian metrics defined on radiant manifolds. A manifold [Formula: see text] is said to be radiant if it is endowed with a symmetric, flat connection and a global vector field [Formula: see text] whose covariant derivative is the identity mapping. A degenerate Hessian metric on [Formula: see text] is a degenerate metric tensor that can locally be written as the covariant Hessian of a function, called potential. A function on [Formula: see text] is said to be extensive if its Lie derivative with respect to [Formula: see text] is the function itself. We show that the Hessian metrics appearing in equilibrium thermodynamics are necessarily degenerate, owing to the fact that their potentials are extensive (up to an additive constant). Manifolds having degenerate Hessian metrics always contain embedded Hessian submanifolds, which generalize the manifolds defined by constant volume in which Ruppeiner geometry is usually studied. By means of examples, we illustrate that linking scalar curvature to microscopic interactions within a thermodynamic system is inaccurate under this approach. In contrast, thermodynamic critical points seem to arise as geometric singularities.

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2

MOHAMMAD, SAMEER, PRADEEP KUMAR PANDEY, and SUMAN THAKUR. "A NOTE ON THE SILVER HESSIAN STRUCTURE." Journal of Science and Arts 24, no. 2 (2024): 375–88. http://dx.doi.org/10.46939/j.sci.arts-24.2-a12.

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In this paper, we study locally decomposable Silver-Hessian manifolds and holomorphic Silver Norden Hessian manifolds. We investigate that if the smooth function f:M→R is decomposable, then the triplet (M,Θ,∇^2 f ) is a locally decomposable Silver Hessian manifold, where Θ is a Silver structure and ∇ represents the Levi-Civita Connection of g. We obtain some conditions under which the manifold Mis associated with Hessian metric h and complex Silver structure. Moreover, we investigate the twin Norden Silver Hessian metric for a Kaehler-Norden Silver Hessian manifold.

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3

BERCU, GABRIEL, CLAUDIU CORCODEL, and MIHAI POSTOLACHE. "ITERATIVE GEOMETRIC STRUCTURES." International Journal of Geometric Methods in Modern Physics 07, no. 07 (2010): 1103–14. http://dx.doi.org/10.1142/s0219887810004749.

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In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

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4

Beltracchi, T. J., and G. A. Gabriele. "A Hybrid Variable Metric Update for the Recursive Quadratic Programming Method." Journal of Mechanical Design 113, no. 3 (1991): 280–85. http://dx.doi.org/10.1115/1.2912780.

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The Recursive Quadratic Programming (RQP) method has become known as one of the most effective and efficient algorithms for solving engineering optimization problems. The RQP method uses variable metric updates to build approximations of the Hessian of the Lagrangian. If the approximation of the Hessian of the Lagrangian converges to the true Hessian of the Lagrangian, then the RQP method converges quadratically. The choice of a variable metric update has a direct effect on the convergence of the Hessian approximation. Most of the research performed with the RQP method uses some modification of the Broyden-Fletcher-Shanno (BFS) variable metric update. This paper describes a hybrid variable metric update that yields good approximations to the Hessian of the Lagrangian. The hybrid update combines the best features of the Symmetric Rank One and BFS updates, but is less sensitive to inexact line searches than the BFS update, and is more stable than the SR1 update. Testing of the method shows that the efficiency of the RQP method is unaffected by the new update but more accurate Hessian approximations are produced. This should increase the accuracy of the solutions obtained with the RQP method, and more importantly, provide more reliable information for post optimality analyses, such as parameter sensitivity studies.

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5

Li, Wuchen. "Hessian metric via transport information geometry." Journal of Mathematical Physics 62, no. 3 (2021): 033301. http://dx.doi.org/10.1063/5.0012605.

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6

TOTARO, BURT. "THE CURVATURE OF A HESSIAN METRIC." International Journal of Mathematics 15, no. 04 (2004): 369–91. http://dx.doi.org/10.1142/s0129167x04002338.

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Inspired by Wilson's paper on sectional curvatures of Kähler moduli, we consider a natural Riemannian metric on a hypersurface {f=1} in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question posed by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R3 such that the associated metric has constant negative curvature. We ask if our examples, together with one example by Dubrovin, are the only ones with constant negative curvature. This question can be rephrased as an appealing question in classical invariant theory, involving the "Clebsch covariant". We give a positive answer for polynomials of degree at most 4, as well as a partial result in any degree.

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7

Zhang, Jun, та Ting-Kam Leonard Wong. "λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature". Entropy 24, № 2 (2022): 193. http://dx.doi.org/10.3390/e24020193.

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This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

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8

Dogan, Keremcan. "Statistical geometry and Hessian structures on pre-Leibniz algebroids." Journal of Physics: Conference Series 2191, no. 1 (2022): 012011. http://dx.doi.org/10.1088/1742-6596/2191/1/012011.

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Abstract We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to include many physically motivated algebroids such as Lie, Courant, metric and higher-Courant algebroids. They create a natural framework for generalizations of differential geometric structures on a smooth manifold. The symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid satisfies a certain property depending on a choice of an equivalence class of connections which are called admissible. These admissible connections are shown to be necessary to generalize aforementioned structures on pre-Leibniz algebroids. Consequently, we prove that, provided certain conditions are met, statistical and conjugate connection structures are equivalent when defined for admissible connections. Moreover, we also show that for ‘projected-torsion-free’ connections, one can generalize Hessian metrics and Hessian structures. We prove that any Hessian structure yields a statistical structure, where these results are completely parallel to the ones in the manifold setting. We also prove a mild generalization of the fundamental theorem of statistical geometry. Moreover, we generalize a-connections, strongly conjugate connections and relative torsion operator, and prove some analogous results.

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9

Feng, Guanhua, Weifeng Liu, Dapeng Tao, and Yicong Zhou. "Hessian Regularized Distance Metric Learning for People Re-Identification." Neural Processing Letters 50, no. 3 (2019): 2087–100. http://dx.doi.org/10.1007/s11063-019-10000-4.

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10

Kumar, Rakesh, Garima Gupta, and Rachna Rani. "Adapted connections on Kaehler–Norden Golden manifolds and harmonicity." International Journal of Geometric Methods in Modern Physics 17, no. 02 (2020): 2050027. http://dx.doi.org/10.1142/s0219887820500279.

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We study almost complex Norden Golden manifolds and Kaehler–Norden Golden manifolds. We derive connections adapted to almost complex Norden Golden structure of an almost complex Norden Golden manifold and of a Kaehler–Norden Golden manifold. We also set up a necessary and sufficient condition for the integrability of almost complex Norden Golden structure. We define twin Norden Golden Hessian metric for a Kaehler–Norden Golden Hessian manifold. Finally, we prove that a complex Norden Golden map between Kaehler–Norden Golden manifolds is a harmonic map.

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11

Bach, Annika. "Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals". ESAIM: Control, Optimisation and Calculus of Variations 24, № 3 (2018): 1107–39. http://dx.doi.org/10.1051/cocv/2017027.

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We provide an approximation result for free-discontinuity functionals of the form 𝓕(u) = ∫Ωf(x, u, ∇u)dx + ∫Su∩Ωθ(x, νu)d𝓗n−1, u ∈ SBV2(Ω), where f is quadratic in the gradient-variable and θ is an arbitrary smooth Finsler metric. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian of the edge variable through a suitable nonhomogeneous metric ϕ.

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12

Grandjean, Vincent. "On Hessian Limit Directions along Gradient Trajectories." Canadian Journal of Mathematics 65, no. 4 (2013): 808–22. http://dx.doi.org/10.4153/cjm-2012-021-6.

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AbstractGiven a non-oscillating gradient trajectory |γ|of a real analytic function f, we show that the limit v of the secants at the limit point 0of |γ|along the trajectory |γ| is an eigenvector of the limit of the direction of the Hessian matrix Hess(f) at 0along |γ|. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.

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13

Boyom, Michel Nguiffo, and Robert A. Wolak. "Transversely Hessian foliations and information geometry." International Journal of Mathematics 27, no. 11 (2016): 1650092. http://dx.doi.org/10.1142/s0129167x16500920.

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A family of probability distributions parametrized by an open domain [Formula: see text] in [Formula: see text] defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry, the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannian metric on [Formula: see text]. It seems to be quite a strong condition. In general, not much can be said about the Fisher information matrix tensor. To develop a more general theory, we weaken the assumption and replace “positive definite” by the existence of a suitable torsion-free connection. It permits us to define naturally a foliation with a transversely Hessian structure. We develop the theory of transversely Hessian foliations along the lines of the classical theory.

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14

Agouzal, A., K. Lipnikov, and Yu Vassilevski. "Hessian-free metric-based mesh adaptation via geometry of interpolation error." Computational Mathematics and Mathematical Physics 50, no. 1 (2010): 124–38. http://dx.doi.org/10.1134/s0965542510010112.

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15

Li, Yanlin, Aydin Gezer, and Erkan Karakas. "Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections." Mathematics 12, no. 13 (2024): 2101. http://dx.doi.org/10.3390/math12132101.

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In this study, we investigate the tangent bundle TM of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇˜. Our primary goal is to establish the necessary and sufficient conditions for TM to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for TM to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to TM, alongside specific constraints on the conformal factor λ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve λ and the Hessian of the potential function. Similarly, for TM to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of λ, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry.

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16

Beltracchi, T. J., and G. A. Gabriele. "A Recursive Quadratic Programming Based Method for Estimating Parameter Sensitivity Derivatives." Journal of Mechanical Design 113, no. 4 (1991): 487–94. http://dx.doi.org/10.1115/1.2912809.

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Parameter sensitivity analysis is defined as the estimation of changes in the modeling functions and design point due to small changes in the fixed parameters of the formulation. There are currently several methods for estimating parameter sensitivities which either require second order information, or do not return reliable estimates for the derivatives. This paper presents a method based on the use of the recursive quadratic programming method in conjunction with differencing formulas to estimate parameter sensitivity derivatives without the need to calculate second order information. In addition, a modified variable metric method for estimating the Hessian of the Lagrangian function is presented that is used to increase the accuracy of the sensitivity derivatives. Testing is performed on a set of problems with Hessians obtained analytically, and on a set of engineering related problems whose derivatives must be estimated numerically. The results indicate that the method provides good estimates of the parameter sensitivity derivatives on both test sets.

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17

AIDA, SHIGEKI. "WITTEN LAPLACIAN ON PINNED PATH GROUP AND ITS EXPECTED SEMICLASSICAL BEHAVIOR." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, supp01 (2003): 103–14. http://dx.doi.org/10.1142/s0219025703001274.

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A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure νλ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □λ on L2(νλ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □λ as λ→∞ by a formal semiclassical analysis.

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18

Chaczko, Zenon, Germano Resconi, Christopher Chiu, and Shahrazad Aslanzadeh. "N-Body Potential Interaction as a Cost Function in the Elastic Model for SANET Cloud Computing." International Journal of Electronics and Telecommunications 58, no. 1 (2012): 63–70. http://dx.doi.org/10.2478/v10177-012-0009-3.

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N-Body Potential Interaction as a Cost Function in the Elastic Model for SANET Cloud ComputingGiven a connection graph of entities that send and receive a flow of data controlled by effort and given the parameters, the metric tensor is computed that is in the elastic relational flow to effort. The metric tensor can be represented by the Hessian of the interaction potential. Now the interaction potential or cost function can be among two entities: 3 entities or ‘N’ entities and can be separated into two main parts. The first part is the repulsion potential the entities move further from the others to obtain minimum cost, the second part is the attraction potential for which the entities move near to others to obtain the minimum cost. For Pauli's model [1], the attraction potential is a functional set of parameters given from the environment (all the elements that have an influence in the module can be the attraction of one entity to another). Now the cost function can be created in a space of macro-variables or macro-states that is less of all possible variables. Any macro-variable collect a set of micro-variables or microstates. Now from the hessian of the macro-variables, the Hessian is computed of the micro-variables in the singular points as stable or unstable only by matrix calculus without any analytical computation - possible when the macro-states are distant among entities. Trivially, the same method can be obtained by a general definition of the macro-variable or macro-states and micro-states or variables. As cloud computing for Sensor-Actor Networks (SANETS) is based on the bonding concept for complex interrelated systems; the bond valence or couple corresponds to the minimum of the interaction potential V and in the SANET cloud as the minimum cost.

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19

Wang, Zijun. "Gradient and Hessian estimates for an elliptic equation on smooth metric measure spaces." Journal of Mathematical Analysis and Applications 509, no. 2 (2022): 125980. http://dx.doi.org/10.1016/j.jmaa.2021.125980.

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20

Maheshkumar Kankarej, Manisha. "Different Types of Curvature and Their Vanishing Conditions." Academic Journal of Applied Mathematical Sciences, no. 73 (May 2, 2021): 143–48. http://dx.doi.org/10.32861/ajams.73.143.148.

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In the present paper, I studied different types of Curvature like Riemannian Curvature, Concircular Curvature, Weyl Curvature, and Projective Curvature in Quarter Symmetric non-Metric Connection in P-Sasakian manifold. A comparative study of a manifold with a Riemannian connection is done with a P-Sasakian Manifold. Conditions for vanishing for different types of curvature are also a part of the study. Some necessary properties of the Hessian operator are discussed with respect to all curvatures as well.

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21

NENCKA, H., and R. F. STREATER. "INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 03 (1999): 441–60. http://dx.doi.org/10.1142/s0219025799000254.

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For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.

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22

BERCU, GABRIEL. "2D RICCI FLAT GRADIENT SOLITONS ARISING FROM REMARKABLE MODELS IN PHYSICS." International Journal of Geometric Methods in Modern Physics 10, no. 10 (2013): 1350059. http://dx.doi.org/10.1142/s021988781350059x.

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On a pseudo-Riemannian manifold (M, g) we consider ∇ the Levi-Cività connection associated to metric g and a function f : M → ℝ whose pseudo-Riemannian Hessian [Formula: see text] is non-degenerate and with constant signature. We study properties of the pseudo-Riemannian manifold (M, h), [Formula: see text] in terms of local computation. Investigating the conditions for existence of the equal connections [Formula: see text], produced by g and h, we determine classes of explicit Ricci flat gradient solitons for some particular forms of g, arising from remarkable physics models.

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23

Chen, Xiuxiong. "Obstruction to the existence of metric whose curvature has umbilical Hessian in a $K$-surface." Communications in Analysis and Geometry 8, no. 2 (2000): 267–99. http://dx.doi.org/10.4310/cag.2000.v8.n2.a2.

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24

Khan, Suhail, Amjad Mahmood, and Ahmad T. Ali. "Concircular vector fields for Kantowski–Sachs and Bianchi type-III spacetimes." International Journal of Geometric Methods in Modern Physics 15, no. 08 (2018): 1850126. http://dx.doi.org/10.1142/s0219887818501268.

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This paper intends to obtain concircular vector fields (CVFs) of Kantowski–Sachs and Bianch type-III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields (CKVFs) are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. It is shown that Kantowski–Sachs and Bianchi type-III spacetimes admit four-, six-, or fifteen-dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.

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25

Fang, Shun, Jiaxin Wu, and Shiqian Wu. "A Content-Aware Non-Local Means Method for Image Denoising." Electronics 11, no. 18 (2022): 2898. http://dx.doi.org/10.3390/electronics11182898.

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Parameter setting and information redundancy are essential issues for the non-local means (NLM) algorithm. This paper introduces a new factor based on the Hessian matrix to adapt the smoothing parameter. Then, a strategy is proposed to implement the NLM by representing patches in terms of features, which uses the 2D histogram and summed-area table. Compared with other methods, the metric for patch similarity in this paper is based on statistical features of patches instead of Euclidean distance. More importantly, not many predefined thresholds are needed. Experimental results show that the proposed algorithm obtains better visual quality and numerical results, especially for images with rich contents and high noise.

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26

Lamboni, Matieyendou. "Derivative Formulas and Gradient of Functions with Non-Independent Variables." Axioms 12, no. 9 (2023): 845. http://dx.doi.org/10.3390/axioms12090845.

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Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables.

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Xia, Yu, and Hao Wu. "Convergence Rate of Greedy SR1 With Trust Region Method." Highlights in Science, Engineering and Technology 115 (October 28, 2024): 468–81. http://dx.doi.org/10.54097/axh66s95.

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Recently, Greedy Quasi-Newton methods have attracted wide interests of some researchers for their explicit superlinear convergence rate. These algorithms which achieve rapid local convergence construct a matrix sequence to approximate the Hessian matrix of the objective function iteratively. This paper proposes an algorithm, GR-SR1-TR, which incorporate Greedy SR1 method with Trust Region framework and employs a new correction technique. We prove that the approximation matrix sequence has a linear descent property under the Frobenius-norm metric. Further, both the global convergence and an explicit superlinear rate are established. The effectiveness of GR-SR1-TR has been verified by preliminary numerical experiments.

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Remaki, L., and W. G. Habashi. "A posteriori error estimate improvement in mesh adaptation for computer fluid dynamics applications." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 223, no. 5 (2008): 1117–26. http://dx.doi.org/10.1243/09544062jmes1165.

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The objective of this work is to study, in a first part, the impact of mesh adaptation on computational estimates of lift and drag coefficients. The convergence of these quantities with successive adaptation is demonstrated by comparing with experimental results. As a second part, optimization of relevant adaptation parameters to accelerate the convergence is investigated. A combination of adaptation variables is proposed to better capture some physical features that are poorly represented when a single variable is used. On the other hand, a new error metric is derived from both Hessian and gradient that are used separately in the framework of mesh adaptation in general. The impact of the proposed improvements is demonstrated through test cases.

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唐, 清艳. "The Hessian Metric of Two Classes of Homogeneous Polynomials on R<sup>3</sup>." Pure Mathematics 15, no. 04 (2025): 162–70. https://doi.org/10.12677/pm.2025.154119.

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Gabriele, G. A., and T. J. Beltracchi. "An Investigation of Pshenichnyi’s Recursive Quadratic Programming Method for Engineering Optimization." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 2 (1987): 248–53. http://dx.doi.org/10.1115/1.3267445.

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This paper discusses Pshenichnyi’s recursive quadratic programming algorithm for use in engineering optimization problems. An evaluation of the original algorithm is offered and several modifications are presented. The modifications include; addition of a variable metric update of the Hessian, an improved active set criterion, direct inclusion of the variable bounds, a divergence control mechanism, and updating schemes for the algorithm parameters. Implementations of the original algorithm and the modified algorithm were tested against the Sandgren test set of 23 engineering optimization problems. The results indicate that the modified algorithm was able to solve 20 of the 23 test problems while the original algorithm solved only 11. The modified algorithm was more efficient than the original on all the test problems.

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Karakida, Ryo, Shotaro Akaho, and Shun-ichi Amari. "Pathological Spectra of the Fisher Information Metric and Its Variants in Deep Neural Networks." Neural Computation 33, no. 8 (2021): 2274–307. http://dx.doi.org/10.1162/neco_a_01411.

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The Fisher information matrix (FIM) plays an essential role in statistics and machine learning as a Riemannian metric tensor or a component of the Hessian matrix of loss functions. Focusing on the FIM and its variants in deep neural networks (DNNs), we reveal their characteristic scale dependence on the network width, depth, and sample size when the network has random weights and is sufficiently wide. This study covers two widely used FIMs for regression with linear output and for classification with softmax output. Both FIMs asymptotically show pathological eigenvalue spectra in the sense that a small number of eigenvalues become large outliers depending on the width or sample size, while the others are much smaller. It implies that the local shape of the parameter space or loss landscape is very sharp in a few specific directions while almost flat in the other directions. In particular, the softmax output disperses the outliers and makes a tail of the eigenvalue density spread from the bulk. We also show that pathological spectra appear in other variants of FIMs: one is the neural tangent kernel; another is a metric for the input signal and feature space that arises from feedforward signal propagation. Thus, we provide a unified perspective on the FIM and its variants that will lead to more quantitative understanding of learning in large-scale DNNs.

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Yao, Guobiao, Alper Yilmaz, Li Zhang, Fei Meng, Haibin Ai, and Fengxiang Jin. "Matching Large Baseline Oblique Stereo Images Using an End-to-End Convolutional Neural Network." Remote Sensing 13, no. 2 (2021): 274. http://dx.doi.org/10.3390/rs13020274.

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The available stereo matching algorithms produce large number of false positive matches or only produce a few true-positives across oblique stereo images with large baseline. This undesired result happens due to the complex perspective deformation and radiometric distortion across the images. To address this problem, we propose a novel affine invariant feature matching algorithm with subpixel accuracy based on an end-to-end convolutional neural network (CNN). In our method, we adopt and modify a Hessian affine network, which we refer to as IHesAffNet, to obtain affine invariant Hessian regions using deep learning framework. To improve the correlation between corresponding features, we introduce an empirical weighted loss function (EWLF) based on the negative samples using K nearest neighbors, and then generate deep learning-based descriptors with high discrimination that is realized with our multiple hard network structure (MTHardNets). Following this step, the conjugate features are produced by using the Euclidean distance ratio as the matching metric, and the accuracy of matches are optimized through the deep learning transform based least square matching (DLT-LSM). Finally, experiments on Large baseline oblique stereo images acquired by ground close-range and unmanned aerial vehicle (UAV) verify the effectiveness of the proposed approach, and comprehensive comparisons demonstrate that our matching algorithm outperforms the state-of-art methods in terms of accuracy, distribution and correct ratio. The main contributions of this article are: (i) our proposed MTHardNets can generate high quality descriptors; and (ii) the IHesAffNet can produce substantial affine invariant corresponding features with reliable transform parameters.

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Otis, Richard, Brandon Bocklund, and Zi‐Kui Liu. "Sensitivity estimation for calculated phase equilibria." Journal of Materials Research 36, no. 1 (2021): 140–50. http://dx.doi.org/10.1557/s43578-020-00073-6.

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AbstractThe development of a consistent framework for Calphad model sensitivity is necessary for the rational reduction of uncertainty via new models and experiments. In the present work, a sensitivity theory for Calphad was developed, and a closed‐form expression for the log‐likelihood gradient and Hessian of a multi‐phase equilibrium measurement was presented. The inherent locality of the defined sensitivity metric was mitigated through the use of Monte Carlo averaging. A case study of the Cr–Ni system was used to demonstrate visualizations and analyses enabled by the developed theory. Criteria based on the classical Cramér–Rao bound were shown to be a useful diagnostic in assessing the accuracy of parameter covariance estimates from Markov Chain Monte Carlo. The developed sensitivity framework was applied to estimate the statistical value of phase equilibria measurements in comparison with thermochemical measurements, with implications for Calphad model uncertainty reduction.

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Bono, G., and A. M. Awruch. "A MESH ADAPTION METHOD BY NODE RE-ALLOCATION USING AN EDGE-BASED ERROR MEASURE." Revista de Engenharia Térmica 4, no. 2 (2005): 145. http://dx.doi.org/10.5380/reterm.v4i2.5416.

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A mesh adaptation technique implemented in an algorithm to simulate compressible flows characterized by the presence of strong shocks, using the finite element method (FEM), is presented in this work. The initial mesh is continuously adapted during the solution process using a node movement technique, keeping as much as possible mesh smoothness and local orthogonality with an unconstrained optimization method. The error is estimated as a function of the Hessian tensor, containing second derivatives of the specific mass, and a Riemann metric projected on the element edges is obtained in order to determine node movements. Time and spatial discretization of the governing equations are carried out using an explicit Taylor-Galerkin scheme and an isoparametric hexahedrical element with eight nodes. An Arbitrary Lagrangean Eulerian (ALE) description is used to take into account mesh movement. Finally, some two-dimensional examples involving transonic and supersonic flows are presented to validate the algorithm.

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Zhang, Ran, Thomas Auzinger, and Bernd Bickel. "Computational Design of Planar Multistable Compliant Structures." ACM Transactions on Graphics 40, no. 5 (2021): 1–16. http://dx.doi.org/10.1145/3453477.

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This article presents a method for designing planar multistable compliant structures. Given a sequence of desired stable states and the corresponding poses of the structure, we identify the topology and geometric realization of a mechanism—consisting of bars and joints—that is able to physically reproduce the desired multistable behavior. In order to solve this problem efficiently, we build on insights from minimally rigid graph theory to identify simple but effective topologies for the mechanism. We then optimize its geometric parameters, such as joint positions and bar lengths, to obtain correct transitions between the given poses. Simultaneously, we ensure adequate stability of each pose based on an effective approximate error metric related to the elastic energy Hessian of the bars in the mechanism. As demonstrated by our results, we obtain functional multistable mechanisms of manageable complexity that can be fabricated using 3D printing. Further, we evaluated the effectiveness of our method on a large number of examples in the simulation and fabricated several physical prototypes.

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QUAPP, WOLFGANG. "THE GROWING STRING METHOD FOR FLOWS OF NEWTON TRAJECTORIES BY A SECOND-ORDER METHOD." Journal of Theoretical and Computational Chemistry 08, no. 01 (2009): 101–17. http://dx.doi.org/10.1142/s0219633609004575.

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The reaction path is an important concept of theoretical chemistry. We use a definition with a reduced gradient (see Quapp et al., Theor Chem Acc100:285, 1998), also named Newton trajectory (NT). To follow a reaction path, we design a numerical scheme for a method for finding a transition state between reactant and product on the potential energy surface: the growing string (GS) method. We extend the method (see W. Quapp, J Chem Phys122:174106, 2005) by a second-order scheme for the corrector step, which includes the use of the Hessian matrix. A dramatic performance enhancement for the exactness to follow the NTs, and a dramatic reduction of the number of corrector steps are to report. Hence, we can calculate flows of NTs. The method works in nonredundant internal coordinates. The corresponding metric to work with is curvilinear. The GS calculation is interfaced with the GamessUS package (we have provided this algorithm on ). Examples for applications are the HCN isomerization pathway and NTs for the isomerization C7ax ↔ C5 of alanine dipeptide.

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Boucetta, Mohamed. "On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric." Complex Manifolds 9, no. 1 (2022): 18–51. http://dx.doi.org/10.1515/coma-2021-0128.

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Abstract Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM) k ≥1 the sequence of tangent bundles given by TkM = T(Tk −1 M) and T 1 M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk , gk ) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk , gk , ∇k ) are left invariant. It is well-known that (TM, J 1, g 1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x 1, . . ., xn ), 〈 ∂ x i , ∂ x j 〉 = ∂ 2 φ ∂ x i ∂ x j \left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle = {{{\partial ^2}\phi } \over {{\partial _x}_{_i}{\partial _x}_j}} . Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J 1, g 1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk , gk ) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk , gk ) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.

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Pelissier, Ugo, Augustin Parret-Fréaud, Felipe Bordeu, and Youssef Mesri. "Graph Neural Networks for Mesh Generation and Adaptation in Structural and Fluid Mechanics." Mathematics 12, no. 18 (2024): 2933. http://dx.doi.org/10.3390/math12182933.

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The finite element discretization of computational physics problems frequently involves the manual generation of an initial mesh and the application of adaptive mesh refinement (AMR). This approach is employed to selectively enhance the accuracy of resolution in regions that encompass significant features throughout the simulation process. In this paper, we introduce Adaptnet, a Graph Neural Networks (GNNs) framework for learning mesh generation and adaptation. The model is composed of two GNNs: the first one, Meshnet, learns mesh parameters commonly used in open-source mesh generators, to generate an initial mesh from a Computer Aided Design (CAD) file; while the second one, Graphnet, learns mesh-based simulations to predict the components of an Hessian-based metric to perform anisotropic mesh adaptation. Our approach is tested on structural (Deforming plate–Linear elasticity) and fluid mechanics (Flow around cylinders–steady-state Stokes) problems. Our findings demonstrate the model’s ability to precisely predict the dynamics of the system and adapt the mesh as needed. The adaptability of the model enables learning resolution-independent mesh-based simulations during training, allowing it to scale effectively to more intricate state spaces during inference.

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Cacuci. "Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, with Applications to Uncertainty Quantification and Predictive Modeling." Energies 12, no. 21 (2019): 4216. http://dx.doi.org/10.3390/en12214216.

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This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space.

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GEZER, AYDIN, and CAGRI KARAMAN. "ON DUAL HOLOMORPHIC B-TYPE HESSIAN METRICS." International Journal of Geometric Methods in Modern Physics 10, no. 02 (2012): 1220026. http://dx.doi.org/10.1142/s0219887812200265.

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Laurian-Ioan, Pişcoran, Akram Ali, Barbu Cătălin та Ali H. Alkhaldi. "The χ-Hessian Quotient for Riemannian Metrics". Axioms 10, № 2 (2021): 69. http://dx.doi.org/10.3390/axioms10020069.

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Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper.

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42

Vaisman, Izu. "Hessian Geometry on Lagrange Spaces." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/793473.

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Schlarb, Markus. "A multi-parameter family of metrics on stiefel manifolds and applications." Journal of Geometric Mechanics 15, no. 1 (2023): 147–87. http://dx.doi.org/10.3934/jgm.2023008.

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<abstract><p>The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.</p></abstract>

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Bakker, Ryan, and Keith T. Poole. "Bayesian Metric Multidimensional Scaling." Political Analysis 21, no. 1 (2013): 125–40. http://dx.doi.org/10.1093/pan/mps039.

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In this article, we show how to apply Bayesian methods to noisy ratio scale distances for both the classical similarities problem as well as the unfolding problem. Bayesian methods produce essentially the same point estimates as the classical methods, but are superior in that they provide more accurate measures of uncertainty in the data. Identification is nontrivial for this class of problems because a configuration of points that reproduces the distances is identified only up to a choice of origin, angles of rotation, and sign flips on the dimensions. We prove that fixing the origin and rotation is sufficient to identify a configuration in the sense that the corresponding maxima/minima are inflection points with full-rank Hessians. However, an unavoidable result is multiple posterior distributions that are mirror images of one another. This poses a problem for Markov chain Monte Carlo (MCMC) methods. The approach we take is to find the optimal solution using standard optimizers. The configuration of points from the optimizers is then used to isolate a single Bayesian posterior that can then be easily analyzed with standard MCMC methods.

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Guo, Xiaoyu, Jiajun Hu, Tong Lu, Guoyin Li, and Ruoxiu Xiao. "A novel vessel enhancement method based on Hessian matrix eigenvalues using multilayer perceptron." Bio-Medical Materials and Engineering 36, no. 2 (2025): 83–97. https://doi.org/10.1177/09592989241296431.

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Background: Vessel segmentation is a critical aspect of medical image processing, often involving vessel enhancement as a preprocessing step. Existing vessel enhancement methods based on eigenvalues of Hessian matrix face challenges such as inconsistent parameter settings and suboptimal enhancement effects across different datasets. Objective: This paper aims to introduce a novel vessel enhancement algorithm that overcomes the limitations of traditional methods by leveraging a multilayer perceptron to fit a vessel enhancement filter function using eigenvalues of Hessian matrix. The primary goal is to simplify parameter tuning while enhancing the effectiveness and generalizability of vessel enhancement. Methods: The proposed algorithm utilizes eigenvalues of Hessian matrix as input for training the multilayer perceptron-based vessel enhancement filter function. The diameter of the largest blood vessel in the dataset is the only parameter to be set. Results: Experiments were conducted on public datasets such as DRIVE, STARE, and IRCAD. Additionally, optimal parameter acquisition methods for traditional Frangi and Jerman filters are introduced and quantitatively compared with the novel approach. Performance metrics such as AUROC, AUPRC, and DSC show that the proposed algorithm outperforms traditional filters in enhancing vessel features. Conclusion: The findings of this study highlight the superiority of the proposed vessel enhancement algorithm in comparison to traditional methods. By simplifying parameter settings, improving enhancement effects, and showcasing superior performance metrics, the algorithm offers a promising solution for enhancing vessel parts in medical image analysis applications.

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PITEA, ARIANA. "A GEOMETRIC STUDY OF SOME EQUATIONS OF MATHEMATICAL PHYSICS." International Journal of Geometric Methods in Modern Physics 09, no. 04 (2012): 1250030. http://dx.doi.org/10.1142/s0219887812500302.

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We introduce geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry, and find geometrical characteristics associated to equations of Mathematical Physics. Also, we introduce a geometric study of some boundary problems. Throughout this work, as main tool we employed an adequate Riemannian Hessian structure, suggested in [Int. J. Geom. Meth. Mod. Phys.7(7) (2010) 1104–1113].

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Prischmann-Voldseth, Deirdre A., Tülin Özsisli, Laura Aldrich-Wolfe, Kirk Anderson, and Marion O. Harris. "Microbial Inoculants Differentially Influence Plant Growth and Biomass Allocation in Wheat Attacked by Gall-Inducing Hessian Fly (Diptera: Cecidomyiidae)." Environmental Entomology 49, no. 5 (2020): 1214–25. http://dx.doi.org/10.1093/ee/nvaa102.

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Abstract Beneficial root microbes may mitigate negative effects of crop pests by enhancing plant tolerance or resistance. We used a greenhouse experiment to investigate impacts of commercially available microbial root inoculants on growth and biomass allocation of wheat (Triticum aestivum L. [Cyperales: Poaceae]) and on survival and growth of the gall-inducing wheat pest Hessian fly, Mayetiola destructor (Say). A factorial design was used, with two near-isogenic wheat lines (one susceptible to Hessian fly, the other resistant), two levels of insect infestation (present, absent), and four inoculants containing: 1) Azospirillum brasilense Tarrand et al. (Rhodospirillales: Azospirillaceae), a plant growth-promoting bacterium, 2) Rhizophagus intraradices (N.C. Schenck & G.S. Sm.) (Glomerales: Glomeraceae), an arbuscular mycorrhizal fungus, 3) A. brasilense + R. intraradices, and 4) control, no inoculant. Larval feeding stunted susceptible wheat shoots and roots. Plants had heavier roots and allocated a greater proportion of biomass to roots when plants received the inoculant with R. intraradices, regardless of wheat genotype or insect infestation. Plants receiving the inoculant containing A. brasilense (alone or with R. intraradices) had comparable numbers of tillers between infested and noninsect-infested plants and, if plants were susceptible, a greater proportion of aboveground biomass was allocated to tillers. However, inoculants did not impact density or performance of Hessian fly immatures or metrics associated with adult fitness. Larvae survived and grew normally on susceptible plants and mortality was 100% on resistant plants irrespective of inoculants. This initial study suggests that by influencing plant biomass allocation, microbial inoculants may offset negative impacts of Hessian flies, with inoculant identity impacting whether tolerance is related to root or tiller growth.

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V. Kolesnikov, Alexander. "Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures." Discrete & Continuous Dynamical Systems - A 34, no. 4 (2014): 1511–32. http://dx.doi.org/10.3934/dcds.2014.34.1511.

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Stein, M. "Quality metrics for the acute care of subarachnoid hemorrhage: the Hessian stroke project." Brain and Spine 1 (2021): 100342. http://dx.doi.org/10.1016/j.bas.2021.100342.

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Boyom, Michel Nguiffo. "Linear Gauge and the Linearization Problem for Webs." Journal of the Tensor Society 8, no. 01 (2007): 1–16. http://dx.doi.org/10.56424/jts.v8i01.10562.

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Regarding the theory of foliation, some aspects of the theory of Riemannian foliations have been brought in completion by the Molino theory. Such a structure is defined by some finite dimensional Lie subalgebra of the Lie algebra of transverse vector fields. The problem I am interested in is more modeste. It is to get sufficient conditions for a smooth manifold admitting foliations with transverse (pseudo) Riemannian metrics. The investigation is inspired by both methods of information geometry and the Hessian geometry.

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