Study complex Monge-Ampère operator on weighted pluricomplex energy classes.
problem Characterize the range of the Complex Monge-Ampère Operator on weighted pluricomplex energy classes.
method Characterizations and a priori estimates on sub-level sets of solutions.
result A non-negative Borel measure is the Monge-Ampère of a unique function in \(\mathcal E_χ\) if and only if \(χ(\mathcal E_χ) \subset L^1(dμ)\).
We investigate probability measures with finite pluricomplex energy. We give criteria insuring that a given measure has finite energy and test these on various examples. We show that this notion is a biholomorphic but not a bimeromorphic invariant.
New probabilistic constructions for Kähler-Einstein metrics.
problem Finding Kähler-Einstein metrics on complex algebraic varieties.
method Microcanonical measures and maximum entropy principles.
result Novel characterizations and evolution equations.
We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate…
We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary qua…
Paper introduces a new Poisson kernel for strongly pseudoconvex domains.
problem Developing a new mathematical tool for strongly pseudoconvex domains.
method Introducing a maximal plurisubharmonic function called the pluricomplex Poisson kernel.
result The pluricomplex Poisson kernel shares properties with the classical Poisson kernel and reproduces pluriharmonic functions.
Paper investigates pluripotential theory on Teichmüller space using new methods.
problem Understanding pluripotential theory on Teichmüller space.
method Alternative approach to Krushkal formula, natural stratified structure, Levi form description.
result Natural stratified structure and description of Levi form.
It is shown that, on a compact Kahler manifold with boundary, the singularities of the pluricomplex Green's function with multiple poles can be prescribed to be of the form log∑j=1n∣fj(z)∣2 at each pole, where fj(z) are arbitrary local holomorphic functions with the pole as their only common zero. The pr…
Paper proves C1,1 regularity for complex Monge-Ampère equations.
problem Complex Monge-Ampère equations on compact almost Hermitian manifolds.
method Proves C1,1 estimate and uses it to show existence of solutions. result Proves C1,1 regularity for geodesics in Sasakian metrics. 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.
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…
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.
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.
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.
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…
Deep k-Means compresses CNNs by clustering weights and re-training, reducing energy consumption.
problem High energy consumption and large parameter count in deep convolutions.
method Applying k-means clustering on convolutional layer weights, sharing K cluster centers, and re-training with hard assignments. result Significant reduction in energy consumption and compression ratio without accuracy loss.
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.
Novel spectral embedding considers node weights for graph analysis.
problem Graph node importance quantification.
method Normalized Laplacian eigenvectors for low-energy configurations.
result Weighted embeddings improve graph configurations.
Improved community detection in heterogeneous SBM with side information.
problem Misclassification in community detection with noisy labels.
method Optimal weighted message passing and minimum energy flow.
result Optimal weighting improves misclassification rate in heterogeneous SBM.
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.
Paper proposes energy-efficient DNN training methods.
problem Energy-constrained deployment of deep neural networks.
method Weighted sparse projection and layer input masking integrated into DNN training.
result Framework provides higher accuracy with same or lower energy budgets.
Study eigenvalues of Laplacian with weighted functions on manifolds.
problem Investigate how eigenvalues of the Laplacian change with weight functions.
method Analyze eigenvalues of weighted Dirichlet energy functional on compact Riemannian manifolds.
result Eigenvalues of the weighted operator are related to the Laplacian with Neumann conditions.
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.
New technique trains DNNs with fewer weights, saving memory and energy.
problem Training deep neural networks requires many weights, increasing memory and energy costs.
method Constrain weight updates to those with highest gradients, regenerating others.
result Pruned networks maintain accuracy while significantly reducing weight count and memory usage.
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.
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.
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.
Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.
Optimal energy trading strategy for intraday markets using Hawkes processes.
problem Optimal execution in intraday energy markets with specific trading patterns.
method Calibrated Hawkes process model with transient price impact.
result Substantial cost reductions in TWAP and VWAP benchmarks.
No finite-time singularities in Yang-Mills flow in 4D.
problem Finite-time singularities in Yang-Mills flow.
method Weighted energy identity and sharp decay estimates.
result Long-time existence of Yang-Mills flow in 4D.
Study on a metric space derived from Kähler manifolds.
problem Understanding the geometry of low energy classes on Kähler manifolds.
method Introduced a metric dψ on the low energy space Eψ of a Kähler manifold (X,ω). result Demonstrated that the triangle inequality holds for the metric dψ. 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.
Let $X\hookrightarrow \cpn $ be a smooth complex projective variety of dimension n. Let λ be an algebraic one parameter subgroup of $G:=\gc$. Let 0≤l≤n+1. We associate to the coefficients Fl(λ) of the normalized weight of λ on the mth Hilbert point of X new energies $F_{\om,l}(\vp)$. The (loga…
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.
Synthetic proof of Gannon-Lee theorem for spacetimes.
problem Proving incompleteness in globally hyperbolic spacetimes.
method Synthetic null energy condition and synthetically asymptotically regular trappedness condition.
result Generalized classical incompleteness theorem to weighted spacetimes.
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
Paper proposes a method to train ML on sPlot background data without negative weights.
problem Training machine learning on data with sPlot background subtraction leads to negative weights and algorithm divergence.
method Proposes a rigorous mathematical approach to handle negative weights in sPlot background data.
result Allows the use of any machine learning method on sPlot background data samples without encountering negative weights.
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.
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.
Improved SNNs with quantized activations outperform traditional networks.
problem Maintaining SotA accuracy in SNNs with limited bit precision.
method Interpolating between non-spiking and spiking regimes using signal processing tools.
result First hybrid SNN outperforms traditional RNNs in accuracy with reduced bit precision.
New geometric conditions ensure compactness of ∂ˉ-Neumann problem.
problem Compactness of ∂ˉ-Neumann operator on specific domains. method Introduced new geometric conditions for a class of domains, proving compactness equivalence to boundary properties.
result Compactness of ∂ˉ-Neumann operator equivalent to boundary lack of analytic varieties. SmartExchange trades memory for computation in neural networks.
problem Energy-efficient inference of deep neural networks.
method Algorithm-hardware co-design framework integrating sparsification, decomposition, and quantization.
result SmartExchange reduces energy consumption and improves performance.