Paper proves stability of multi-dimensional rarefaction waves in gas dynamics.
problem Challenges in constructing multi-dimensional rarefaction waves in gas dynamics.
method Geometric Weighted Energy Method (GWEM) to overcome derivative losses.
result Established nonlinear stability of multi-dimensional rarefaction waves for compressible Euler equations.
Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.
problem Proving geometric results for substatic Riemannian manifolds.
method Comparison theory based on a newly discovered conformal connection.
result Sharp, weighted Isoperimetric inequality quantifying boundary minimization.
The paper studies matrix normalization and graph balancing using a new functional and gradient descent.
problem Matrix normalization and graph balancing.
method A new functional called the non-normal energy, and gradient descent.
result Gradient descent of the non-normal energy converges to balanced graphs and preserves spectra and realness of weights.
We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …
New summary measures reveal geometric structure in weighted measures on manifolds.
problem Lack of geometric information in standard weight-only summaries.
method Heat-kernel entropy profiles, tracking nonuniformity across scales.
result Geometric effective sample size discounts nearby or duplicate particles.
New neural method calculates EMD for particle physics data.
problem Metric for particle collider events based on Wasserstein metric.
method Neural network architecture estimating EMD using Kantorovich-Rubinstein duality.
result Differentiable way to calculate EMD for geometric fitting.
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on…
Paper extends positive energy theorem to anti-de Sitter spacetimes.
problem Proving positive energy theorem for weighted anti-de Sitter spacetimes.
method Generalized positive energy theorem for 3D anti-de Sitter initial data sets.
result Positive energy theorem proved for weighted anti-de Sitter spacetimes.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for (−Δ)γ when γ∈(0,1), and both…
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.
Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order 2γ∈(0,2) or 2γ∈(2,4) and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…
New method estimates Nishimori temperature for node classification in weighted graphs.
problem Estimating Nishimori temperature for Bayesian inference.
method Spectral method using eigenvalues of Bethe Hessian matrix.
result Spectral method outperforms existing approaches in node classification.
The paper studies quaternionic Monge-Ampère equations in weighted energy classes.
problem Characterizing the finite energy range of quaternionic Monge-Ampère operator.
method Proving well-definedness and fine property of the operator in weighted energy classes.
result Explicit characterization of the finite energy range of quaternionic Monge-Ampère operator.
Deep Neural Networks (DNNs) are increasingly deployed in highly energy-constrained environments such as autonomous drones and wearable devices while at the same time must operate in real-time. Therefore, reducing the energy consumption has become a major design consideration in DNN training. This paper proposes the fir…
Derives energy-momentum tensor from Standard Model, examines energy conditions.
problem Validating energy conditions in the context of the Standard Model.
method Geometric variational problem on globally hyperbolic manifold, deriving energy-momentum tensor.
result Validates various energy conditions in general relativity.
We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix W∗∈Rm×d, where m is the width of the hidden layer and d≤m is the data dimension. We study the optimization landscape associated with the empirical and the popula…
Uniform small energy regularity for fractional geometric problems proved.
problem Proving regularity for fractional geometric problems.
method Analyzing parabolic boundary reaction Ginzburg-Landau problems and fractional harmonic maps to spheres.
result Uniform small energy regularity results for s∈(0,1), answering a posed question. The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly …
Paper improves DNN accelerator robustness against bit errors with energy savings.
problem Bit errors in quantized DNN weights reduce energy efficiency.
method Combines robust fixed-point quantization, weight clipping, and random bit error training.
result Significantly improves robustness against random bit errors with high energy savings.
Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.
problem Tackles relations between three types of energy functions for cylindrical critical points.
method Uses differential equations and critical point analysis for Willmore-type energies and weighted areas.
result Generating curves coincide for Willmore-type energies and weighted areas, and similar results hold for Willmore-type energies and vertical potential energies.
A new energy-efficient pruning method for federated learning.
problem Energy inefficiency in gradient sparsification for federated learning.
method Formalized energy-constrained projection problem and proposed Cost-Weighted Magnitude Pruning (CWMP).
result CWMP optimally balances performance and energy efficiency in federated learning.
SmartDeal reduces energy and storage costs for deep neural networks.
problem Heavy parameterization of deep neural networks leads to inefficient use of DRAM.
method SmartDeal decomposes weights into a small basis matrix and a structurally sparse coefficient matrix, quantized to power-of-2.
result Up to 2.44x energy efficiency improvement in inference and 10.56x reduction in training energy.
We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…
Proves invariance of weighted extremal Kähler metrics under smooth blowups.
problem Invariance of weighted extremal Kähler metrics under smooth blowups.
method Uniform coercivity estimate for the (relative, weighted) Mabuchi energy on blowups.
result Invariance of weighted extremal Kähler metrics under smooth blowups.
Introduces Causal Energy Minimization to understand Transformer layers.
problem Empirical parameterization of Transformer blocks remains largely unexplored.
method Causal Energy Minimization framework that recasts Transformer layers as optimization steps on conditional energy functions.
result Identifies design space for Transformer layers including weight sharing and energy-based interpretations.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
Unified geometric description of Kepler flow across all energies.
problem Understanding the Kepler flow across different energy levels.
method Revisiting Ligon--Schaaf regularization and identifying geometric origins of anomalies.
result Unified geometric description of Kepler flow for all energies.
EWFM trains continuous flows with only energy evaluations, improving sample quality with fewer computations.
problem Efficiently sampling from complex, high-dimensional Boltzmann distributions using only energy evaluations.
method Energy-Weighted Flow Matching (EWFM) using importance sampling and iterative/annealed training.
result Improved sample quality with up to 3 orders of magnitude fewer energy evaluations compared to existing methods.
L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.
problem Extracting scientific understanding from particle-physics experiments with high precision and efficiency.
method L-GATr, a geometric algebra Transformer, representing data in 4D space-time and being equivariant under Lorentz transformations.
result L-GATr achieves performance comparable to or better than domain-specific baselines on regression, classification, and generative tasks.
The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa…
In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded m-dimensional Lipschitz submanifolds in Rn. It turns out that due to a smoothing effect any seq…
In the present work, torsion energy is defined. Its law of conservation is given. It is shown that this type of energy gives rise to a repulsive force which can be used to interpret supernovae type Ia observations, and consequently the accelerating expansion of the Universe. This interpretation is a pure geometric one …
Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.
problem Proving the Wulff theorem for crystalline integrands.
method Direct approach using Minkowski Theory to exploit convex properties.
result Simpler proof of the Wulff theorem for crystalline shapes.
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
The Palais-Smale condition is proven for various knot energies.
problem Existence and smoothness of minimizing knots in geometric knot theory.
method Proof of the Palais-Smale condition for specific knot energies.
result Existence of minimizing knots and long-time existence of their flows.
In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into Sn is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…
New geometric interpretation of discrete Willmore energy using rolling spheres connection.
problem Discrete formulation of Willmore energy for simplicial surfaces.
method Geometric interpretation of Möbius invariant discrete Willmore energy using rolling spheres connection.
result Clear geometric interpretations of discrete Willmore energy with manifest Möbius invariance.
Study on spin-zero rest-mass fields using conformal geometric method.
problem Wellposedness of Cauchy and Goursat problems for spin-n/2 zero rest-mass equations. method Conformal geometric method, energy equalities, partial conformal compactification.
result Proves wellposedness of Cauchy and Goursat problems and establishes field decays.
Spheres minimize weighted curvature on spheres.
problem Minimizing weighted mean curvature on spheres.
method Solving a limit of Euler-Lagrange equations for Lp problems. result Solutions have at most three mean curvature values.
In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carlema…
Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.
problem Characterizing extremal Kahler and Sasaki metrics using energy coercivity.
method Maximal complex torus, coercive weighted Mabuchi energy, K-polystability.
result Coercive weighted Mabuchi energy implies strict positivity of Donaldson-Futaki invariant and existence of extremal metrics.
The machine learning community has become increasingly interested in the energy efficiency of neural networks. The Spiking Neural Network (SNN) is a promising approach to energy-efficient computing, since its activation levels are quantized into temporally sparse, one-bit values (i.e., "spike" events), which additional…
SRFE clarifies KL divergences without unifying learning frameworks.
problem Inductive biases of KL divergences and their limitations.
method Introducing SRFE, a log-moment-based functional of the likelihood ratio.
result SRFE recovers KL divergences as limits and reveals a mean-variance tradeoff.
The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.
problem Characterizing the null energy condition in Lorentzian manifolds.
method Characterization via convexity of the relative entropy along displacement interpolations on null hypersurfaces.
result The null energy condition is characterized in terms of convexity of the relative entropy.
New method calculates Ricci curvature from distances between weighted volumes.
problem Calculating Ricci curvature for weighted Riemannian manifolds.
method Asymptotic retrieval of generalized Ricci tensor from scaled metric derivatives of Wasserstein 1-distances.
result Limiting coarse curvature of random graphs converges to generalized Ricci tensor.
Establishes convexity and coercivity of K-energy functional for complex tori.
problem Convexity and coercivity of K-energy functional for complex tori.
method Geodesics in finite energy space, cone angle perturbations, stability of coercivity.
result Openness of coercivity under cone angle perturbations and existence of cscK cone metrics.