New method designs joint initial noises for diffusion models to improve diversity and alignment.
problem Independent initial noises limit diversity in generated images.
method Coupling of initial noises, maintaining Gaussian distribution while allowing dependence.
result Repulsive Gaussian coupling improves diversity without increasing sampling cost.
Improved neural network training by coupled initialization reduces neuron count.
problem Training neural networks efficiently with fewer neurons.
method Coupled initialization of weights into pairs of identical Gaussian vectors.
result Significantly reduced number of neurons required for network convergence.
End-to-end training of DBMs with improved gradient estimation.
problem Biased gradient estimation in DBMs, especially with high-dimensional states.
method Unbiased contrastive divergence using MH coupling and local mode initialization.
result End-to-end training of DBMs without greedy pretraining, achieving FID score of 10.33 for MNIST.
Study on kinetic Langevin diffusions and their couplings, showing subtle TV bounds and new non-Markovian couplings.
problem Understanding and quantifying the TV distance between solutions of kinetic Langevin diffusions with different initial values.
method Established new non-Markovian couplings for kinetic Langevin diffusions, derived from optimal coalescence trajectories, and analyzed their TV bounds.
result No Markovian coupling can capture the asymptotic decay rate of the TV distance between solutions of kinetic Langevin diffusions with different initial values.
Paper proposes a coupling-based diagnostic for SGD stepsize optimization.
problem Optimizing stepsize for SGD convergence.
method Coupling-based convergence diagnostic for monitoring stationarity.
result Proposed stepsize scheme achieves superior performance across convex and non-convex problems.
New method tackles convergence issues in approximating FBSDEs.
problem Convergence issues in approximating coupled FBSDEs.
method Approximates initial condition for a family of FBSDEs, then uses it to approximate the original FBSDE.
result Method converges even when standard deep BSDE method fails.
The goal of this paper is twofold: we study metric measure spaces (X,d,m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k:X→R we introduce the curvature-dimension condition CD(k,∞) which canonically ex…
Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.
problem Global well-posedness of a wave-Klein-Gordon system with strong couplings in divergence form.
method Constructed an auxiliary system with shifted primitives to handle the strong couplings.
result Established global well-posedness theorem for the wave-Klein-Gordon system.
In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simp…
In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are …
Augmented bridge matching preserves coupling information between distributions.
problem Preserving the original empirical pairing in flow and bridge matching processes.
method Augmenting the velocity field with initial sample point information.
result Simple modification recovers coupling information without losing Markovian property.
We consider an integro-differential equation derived from a system of coupled parabolic PDE and an ODE which describes an European option pricing with liquidity shocks. We study the well-posedness and prove comparison principle for the corresponding initial value problem.
Study Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
We consider G2 structures with torsion coupled with G2-instantons, on a compact 7-dimensional manifold. The coupling is via an equation for 4-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as a…
We study the evolution of wormhole geometries under Ricci flow using numerical methods. Depending on values of initial data parameters, wormhole throats either pinch off or evolve to a monotonically growing state. The transition between these two behaviors exhibits a from of critical phenomena reminiscent of that obser…
Gradient descent with small initialization solves matrix completion without regularization.
problem Symmetric matrix completion from observed entries.
method Vanilla gradient descent with small initialization.
result GD converges to the ground truth matrix without regularization in over-parameterized scenario.
A network-based approach identifies financial factors from asset interactions, explaining market dynamics.
problem Characterizing joint financial asset behavior through underlying drivers.
method Modeling market as coupled iterated maps, where asset returns depend on past returns and interactions.
result Stable patterns of co-movement (financial factors) emerge from asset interactions, explaining asset variance.
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws μ,ν on Rd and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where 0<p≤1, and the dimensio…
We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chruściel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given a Yamabe positive metric g, which is conformally asymptotically cylindrical on ea…
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
BalLOT uses optimal transport for balanced k-means clustering.
problem Balanced k-means clustering of data. method BalLOT is an optimal transport approach to alternating minimization.
result BalLOT provides theoretical guarantees for exact and partial recoveries of planted clusters.
We solve Bartnik's stationary extension problem near Schwarzschild spheres.
problem Existence and uniqueness of asymptotically flat stationary vacuum spacetimes.
method Developed a double geodesic gauge, reducing equations to elliptic and transport-type problems.
result Local well-posedness for Bartnik stationary metric extension problem near Schwarzschild spheres.
Method for initializing Gaussian mixtures for variational inference with multi-modal distributions.
problem Challenges in variational inference with Gaussian mixtures due to multimodality and nonconvex loss functions.
method Optimization to find local maxima, local Gaussian approximations, and constrained least squares regression.
result Robust initialization improves variational inference performance and scalability.
Paper studies asymmetric matrix sensing, proving gradient descent converges to low-rank solutions.
problem Reconstructing asymmetric low-rank matrices from linear measurements.
method Factorized gradient descent with coupling and regularization properties.
result Gradient descent from small random initialization converges to globally optimal and generalizing solutions.
Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formu…
We study the mean field games equations, consisting of the coupled Kolmogorov-Fokker-Planck and Hamilton-Jacobi-Bellman equations. The equations are complemented by initial and terminal conditions. It is shown that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear syste…
Faster GW alignment for incomparable point clouds via low-rank couplings.
problem Aligning points across incomparable point clouds.
method Low-rank couplings and costs to solve Gromov-Wasserstein framework in linear time.
result Linear-time computation of Gromov-Wasserstein distances.
Proves Hadamard states for Dirac fields on manifolds with timelike boundaries.
problem Existence of Hadamard states for Dirac fields with MIT boundary conditions.
method Introducing geometric Møller operator to implement unitary isomorphism between spaces of initial data.
result Existence of Hadamard states for Dirac fields with MIT boundary conditions.
Study wormholes in Einstein-Yang-Mills theory with a phantom field.
problem Existence of wormholes in Einstein-Yang-Mills theory with a phantom scalar field.
method Rigorous mathematical proof and numerical analysis of wormhole solutions.
result Existence of an infinite sequence of symmetric wormhole solutions.
We recently outlined the vision of "Learning Everywhere" which captures the possibility and impact of how learning methods and traditional HPC methods can be coupled together. A primary driver of such coupling is the promise that Machine Learning (ML) will give major performance improvements for traditional HPC simulat…
Extracts intrinsic spatial coordinates for complex agent systems to learn PDEs.
problem Modeling collective dynamics of heterogeneous agents.
method Data-driven extraction of intrinsic spatial coordinates, learning PDEs in emergent space.
result Collective dynamics can be approximated through learned PDEs in emergent coordinates.
New study on No-U-Turn Sampler for accelerated mixing in Hamiltonian Monte Carlo.
problem Achieving accelerated convergence in Hamiltonian Monte Carlo.
method Combining concentration of measure and coupling analysis for mixing.
result Rigorous mixing guarantees for the No-U-Turn Sampler in certain Gaussian distributions.
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
Optimizes angular velocity transfers for rigid bodies under deadline constraints.
problem Stochastic guidance of spin states of rigid bodies over a hard deadline.
method Structural analysis of Kantorovich optimal coupling formulation for nonlinear dynamics.
result Derives the ground cost for optimal transport of angular velocity.
NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniquene…
Study well-poses Dirac operator problem with APS boundary conditions.
problem Well-posedness of Cauchy problem for Dirac operator on Lorentzian manifolds.
method Derived energy estimates, established uniqueness and existence of weak solutions, introduced mollifier operators.
result Well-posedness of Cauchy problem for Dirac operator with APS boundary conditions.
Given a metric measure space (X,d,m) and a lower semicontinuous, lower bounded function k:X→R, we prove the equivalence of the synthetic approaches to Ricci curvature at x∈X being bounded from below by k(x) in terms of ∙ the Bakry-Émery estimate $ΔΓ(f)/2 - Γ(f,Δf)…
Elvet solves differential equations and variational problems with neural networks.
problem Solving complex differential and variational equations with arbitrary conditions.
method Machine learning, specifically neural networks, to represent and solve equations.
result Elvet can solve a wide range of differential and variational problems.
Study on hyperbolic elastic flow, proving convergence and quantifying singularities.
problem Understanding singularities and convergence of elastic flow in hyperbolic plane.
method Analyzes closed and open curves with clamped boundary conditions, proving convergence without small energy assumption.
result Each singularity carries an energy cost of at least 8, and blow-ups are explicitly classified.
Physics-constrained deep learning predicts geophysical dynamics with boundedness.
problem Forecasting geophysical systems with hidden variables and incomplete observations.
method Physics-constrained neural ordinary differential equation (NODE) representations with boundedness constraints.
result The approach generalizes learned dynamics to arbitrary initial conditions.
Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström and coupled Kähler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields…
As is well known, both Weyl and Weitzenböck spacetimes were initially used as attempts to geometrize the electromagnetic field. In this letter, we prove that this field can also be regarded as a geometrical quantity in an extended version of the Weitzenböck spacetime. The new geometry encompasses features of both Weyl …
We replace the Hidden Markov Model (HMM) which is traditionally used in in continuous speech recognition with a bi-directional recurrent neural network encoder coupled to a recurrent neural network decoder that directly emits a stream of phonemes. The alignment between the input and output sequences is established usin…
Paper analyzes Langevin dynamics for multimodal Gaussian mixtures, controlling errors across dimensions.
problem Challenges in obtaining stable diffusion-based samplers in high- and infinite-dimensional settings.
method Study of preconditioned Annealed Langevin Dynamics (ALD) for Gaussian mixtures, focusing on Euler-Maruyama (EM) and exponential-integrator schemes.
result Proves dimension-uniform KL bounds for the exponential-integrator scheme, allowing arbitrarily small divergence with dimension.
Defines coupled embeddability for maps on products of spaces, generating examples and nonexamples.
problem Understanding when maps on products of spaces can be embedded.
method Uses known results for nonsingular biskew and bilinear maps, studies genericity properties, extends Whitney embedding theorems, and relates to Z/2-coindex of embedding spaces. result Generates strong obstructions to coupled embeddability in terms of combinatorics of triangulations.
DGP learns speech recognition by modeling complex relationships between utterances.
problem Modeling complex relationships in speech recognition without relational data.
method Bayesian nonparametric deep learning method (DGP) that generates infinite probabilistic graphs.
result DGP successfully infers relationships among utterances without relational data during training.
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.