Paper proposes new loss functions for training energy networks.
problem Challenges in computing gradients for training energy networks.
method Proposes generalized Fenchel-Young losses for efficient gradient computation.
result Demonstrates the calibration of excess risk for linear-concave energies.
The energy function associated to harmonic maps between surfaces is convex at critical points.
problem Proving convexity of the energy function for harmonic maps between surfaces.
method Analyzing the energy function on Teichmüller space and proving convexity at critical points.
result The energy function is convex at critical points and strictly convex under certain conditions.
Characterizes complex Hessian equations for bounded energy functions.
problem Understanding degenerate complex Hessian equations for bounded energy functions.
method Proving sublevel set estimates and using Sobolev inequalities.
result Characterization of degenerate complex Hessian equations for bounded (p,m)-energy functions. This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle.
According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the le…
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
We study the energy functional on the set of Lagrangian tori in CP2 . We prove that the value of the energy functional on a certain family of Hamiltonian minimal Lagrangian tori in CP2 is strictly larger than energy of the Clifford torus.
We introduce the "Energy-based Generative Adversarial Network" model (EBGAN) which views the discriminator as an energy function that attributes low energies to the regions near the data manifold and higher energies to other regions. Similar to the probabilistic GANs, a generator is seen as being trained to produce con…
New method trains deep neural networks for non-interacting kinetic-energy functionals in DFT.
problem Lack of exact relationship between electron density and non-interacting kinetic energy.
method Variational principle to regularize machine-learned density functionals.
result Excellent results on kinetic-energy functionals for various systems.
Improved diffusion models using energy distillation and sequential Monte Carlo.
problem Training instability and inferior performance in energy parameterized diffusion models.
method Introduced a novel training regime for energy functions through distillation of pre-trained diffusion models, and cast the sampling procedure as a Feynman Kac model.
result Demonstrated improved performance and new sampling techniques.
Traditional centralized energy systems have the disadvantages of difficult management and insufficient incentives. Blockchain is an emerging technology, which can be utilized in energy systems to enhance their management and control. Integrating token economy and blockchain technology, token economic systems in energy …
Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
Unified perspective on Hopfield networks with attention module.
problem Understanding and optimizing Hopfield networks with attention mechanisms.
method Study of BM counterparts of modern Hopfield networks and their salient properties.
result Introduction of AttnBM with tractable likelihood and gradient.
Energy Transformer integrates attention, energy models, and associative memory.
problem Lack of clear theoretical foundations in attention mechanisms and straightforward design of energy functions in energy-based models.
method Proposes Energy Transformer, a sequence of attention layers with a specifically engineered energy function.
result Obtained strong results on graph anomaly detection and classification tasks.
Geodesics found in a metric space of m-subharmonic functions.
problem Metric structure on energy class of m-subharmonic functions.
method Inspired by Kähler geometry, introduced a metric structure and studied metric convergence.
result Geodesics constructed in a subspace of the complete metric space.
Characterizes CR manifolds as critical points of an energy functional.
problem Understanding homogeneous three-dimensional CR manifolds.
method Uses an energy functional dependent on Webster curvature and torsion.
result Identifies Rossi spheres as a specific type of critical point.
In this paper, energy function is used to investigate the eigen-solutions of −Δu+Vu=λu on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.
Synthetic approach to pluripotential theory measures finite energy.
problem Global pluripotential theory on compact Kähler manifolds and projective Berkovich spaces.
method Definition and study of measures of finite energy, introduction of twisted and free energy functionals.
result Coercivity of energy functionals is an open condition with respect to polarization.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
The paper calculates the second variation of energy functions for families of canonically polarized manifolds.
problem Computing the second variation of energy functions for families of canonically polarized manifolds.
method Analyzing the Dirichlet energy of maps between fibers and using harmonic maps.
result The energy function is plurisubharmonic under certain curvature conditions.
In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.
Energy functional for Legendrian knots in Heisenberg group, invariant under PU(2,1).
problem Energy functional for Legendrian knots in Heisenberg group.
method Regularization of divergent integral with Korányi distance, invariant under PU(2,1).
result Characterization of minimizers and Heisenberg analog of Doyle-Schramm cosine formula.
The paper analyses the extrema of p-energy functional on a Finsler space with constant curvature.
Proves weight polytope matches with energy vectors in toric varieties.
problem Understanding the relationship between weight polytopes and energy functionals in toric varieties.
method Combines two slope formulas of K-energy in the toric setting.
result Weight polytope of Hurwitz form matches with convex hull of characteristic vectors.
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …
Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.
problem Extending the geometry of gradient Ricci solitons to non-gradient Ricci solitons.
method Using energy function E to study the geometry. result A non-steady Ricci soliton with symmetric covariant derivative is gradient.
We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compac…
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …
Strict plurisubharmonicity proven for Teichmüller energy on Hitchin representations.
problem Proving strict plurisubharmonicity of Teichmüller energy for Hitchin representations.
method Analyzing energy functional E on Teichmüller space associated to Hitchin representations. result Strict plurisubharmonicity of energy functional E proven. The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
Structured Prediction Energy Networks (SPENs) are a simple, yet expressive family of structured prediction models (Belanger and McCallum, 2016). An energy function over candidate structured outputs is given by a deep network, and predictions are formed by gradient-based optimization. This paper presents end-to-end lear…
Improves energy efficiency of neuromorphic hardware by optimizing memory organization and encoding schemes.
problem Energy inefficiency in neuromorphic hardware, especially in digital accelerators.
method Synthesized controller and memory for different encoding schemes, introduced functional encoding for structured connectivity.
result Functional encoding offers a 58% reduction in energy for weight updates in convolutional layers.
Probabilistic models can be defined by an energy function, where the probability of each state is proportional to the exponential of the state's negative energy. This paper considers a generalization of energy-based models in which the probability of a state is proportional to an arbitrary positive, strictly decreasing…
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For H2-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower s…
A new loss function ED simplifies training energy-based models without scores.
problem Training energy-based models is computationally expensive.
method Energy Discrepancy (ED) loss function that does not rely on scores or MCMC.
result ED effectively interpolates between score matching and negative log-likelihood.
The paper examines conditions for the equator map to be minimizing or unstable for higher order energy functionals.
problem Conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
method Analyzes the extrinsic k-energy functional and the equator map to establish conditions for minimization or instability.
result Establishes necessary and sufficient conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
EBM predicts protein conformations at atomic scale using crystallized data.
problem Predicting the conformation of a side chain from its context within a protein structure.
method Energy-based model trained on crystallized protein data, evaluating performance on rotamer recovery task.
result EBM achieves performance close to state-of-the-art methods, including Rosetta energy function.
New energy functional bounds Ricci flows on ancient spaces.
problem Bounding Ricci flows on ancient spaces.
method Introducing a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows.
result Provides an upper bound for the ordinary λ-functional.
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plan…
We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.
A new method trains discrete EBMs without sampling.
problem Training EBMs on discrete spaces is hard.
method Energy Discrepancy (ED), a contrastive loss.
result ED offers theoretical guarantees for various perturbation types.
Lowered regularity assumption for a phase-dependent Helfrich energy equation.
problem Analyzing the phase separation line of the Helfrich energy.
method Used a carefully chosen test function with a signed distance function.
result Regularity assumption lowered from C2 to C1,1 for the phase separation line. Minimal submanifolds are found as energy concentration sets in variational problems.
problem Understanding the structure of minimal submanifolds in codimension two.
method Purely variational approach, extending previous work on geodesics.
result Non-degenerate minimal submanifolds can be derived from critical maps of the Ginzburg-Landau functional.
Extends K-energy to complexified Kähler classes for scalar curvature study.
problem Scalar curvature equation with B-field on complexified Kähler classes.
method Extended K-energy functional, convex along geodesics.
result Uniqueness of solutions in some cases.