Survey on Kähler-Einstein and weighted solitons on Fano manifolds.
problem Existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds.
method Generalization of algebraic conditions for K-polystability.
result Existence of coupled Kähler-Einstein metrics and weighted solitons is equivalent to algebraic conditions.
We investigate robustness of deep feed-forward neural networks when input data are subject to random uncertainties. More specifically, we consider regularization of the network by its Lipschitz constant and emphasize its role. We highlight the fact that this regularization is not only a way to control the magnitude of …
We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled Kähler-Einstein metric in the sense of Hultgren-Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give …
In this paper, we prove that the set of solutions of constraint equations for coupled Einstein and scalar fields in classical general relativity possesses Hilbert manifold structure. We follow the work of R. Bartnik [2] and use weighted Sobolev spaces and Implicit Function Theorem to prove our results.
Optimal coupling among random vectors with known statistics and correlation structure found using minimum spanning tree over measure-valued vertices.
problem Finding the optimal coupling among random vectors with known statistics and correlation structure.
method Formulating the problem as a minimum spanning tree over measure-valued vertices and solving it in two steps.
result Optimal coupling found using the minimum spanning tree approach.
Unified SVD compression fails in practical tasks, highlighting the importance of per layer activation reconstruction.
problem The failure of a unified SVD compression method in practical tasks like perplexity and accuracy.
method Unified optimization problem for SVD based compression methods, focusing on cross-layer coupling.
result Downstream metrics like perplexity and accuracy degrade severely compared to standard per layer SVD LLM.
Study on feature learning dynamics in infinite-depth neural networks, focusing on ResNets.
problem Understanding how features evolve during training in deep neural networks, especially in the large-depth limit.
method Conditional Gaussian representations and SDE system with decoupled backward weights.
result Depth-induced suppression of forward-backward coupling in infinite-depth networks, leading to a decoupled forward-backward SDE system.
We consider multi-task regression models where the observations are assumed to be a linear combination of several latent node functions and weight functions, which are both drawn from Gaussian process priors. Driven by the problem of developing scalable methods for forecasting distributed solar and other renewable powe…
The paper maps time-series onto networks to reveal hidden joint information.
problem Extract hidden joint information from uncorrelated time-series.
method Discretize time-series amplitudes, map onto networks, measure coupling deviations, and compare with Gaussian distributions.
result Markets may possess joint patterns even if initially uncorrelated.
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. A new method for approximating softmax and Gaussian kernels with reduced error.
problem Approximating softmax and Gaussian kernels with low error.
method Simplex Random Features (SimRFs) and SimRFs+.
result SimRFs provide the smallest MSE among weight-independent geometrically-coupled PRF mechanisms.
New theory predicts deep neural networks can operate in an extended critical regime without fine-tuning.
problem Understanding the dynamics and computational principles of deep neural networks.
method Combining theories of heavy-tailed random matrices and non-equilibrium statistical physics.
result Deep neural networks can operate in an extended critical regime without fine-tuning parameters.
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.
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
Artificial neural networks (ANNs) may not be worth their computational/memory costs when used in mobile phones or embedded devices. Parameter-pruning algorithms combat these costs, with some algorithms capable of removing over 90% of an ANN's weights without harming the ANN's performance. Removing weights from an ANN i…
NetOTC compares and aligns directed or undirected networks via random walk transitions.
problem Comparing and aligning networks of different types and sizes.
method NetOTC uses a transport-based approach to find optimal transition couplings of random walks.
result NetOTC quantifies network differences and provides vertex and edge alignments.
Partial Differential Equations are infinite dimensional encoded representations of physical processes. However, imbibing multiple observation data towards a coupled representation presents significant challenges. We present a fully convolutional architecture that captures the invariant structure of the domain to recons…
CPFM integrates dimensionality reduction and reconstruction with flow networks.
problem Learning coupled continuous flows for data and embeddings.
method Coupled flow matching framework with Gromov-Wasserstein objective and dual-conditional flow network.
result CPFM preserves and recovers residual information in latent space.
We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern clas…
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.
New stability criterion for Fano manifolds using anticanonically balanced metrics.
problem Stability conditions for Fano manifolds and their invariant δm. method Proof of equivalence between stability condition and anticanonically balanced metrics.
result Established a Hilbert-Mumford type criterion for δm>1. Improved VAE models avoid posterior collapse in text modeling.
problem Posterior collapse in VAEs leads to poor data manifold parameterization.
method Coupled-VAE couples a VAE with a deterministic autoencoder to improve encoder and decoder parameterizations.
result Coupled-VAE consistently improves results in probability estimation and latent space richness.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
We examine several aspects of explicability of a classification system built from neural networks. The first aspect is the pairwise explicability, which is the ability to provide the most accurate prediction when the range of possibilities is narrowed to just two. Next we consider explicability in development, which me…
Study mean curvature flow into evolving manifold with coupled flows.
problem Analyzing mean curvature flow in evolving Riemannian manifolds.
method Coupling Ricci flow and harmonic map heat flow, calculating variations, and using Harnack expressions.
result Obtained a Huisken monotonicity-type formula for mean curvature flow.
This work aims at solving the problems with intractable sparsity-inducing norms that are often encountered in various machine learning tasks, such as multi-task learning, subspace clustering, feature selection, robust principal component analysis, and so on. Specifically, an Iteratively Re-Weighted method (IRW) with so…
Lo-Hp decouples weight generation into local and global policies to improve flexibility and efficiency.
problem Over-coupling and long-horizon issues in current optimization methods.
method Hybrid-Policy Sub-Trajectory Balance objective.
result Learning local optimization policies addresses long-horizon issues and enhances global weight generation.
Paper finds solutions to a complex equation on surfaces with boundary conditions.
problem Existence of solutions to a super-Liouville equation on compact Riemannian surfaces with boundary.
method Introduced a weighted Dirac operator and constructed a Nehari manifold to show existence of non-trivial solutions.
result Existence of non-trivial solutions to the super-Liouville equation.
Unified framework for optimizing portfolios with distributions over weights, returns, and parameters.
problem Traditional portfolio optimization treats expected returns, covariances, and allocations as fixed. Modern practice replaces at least one with a distribution.
method Unified framework using Gamma_theta(dw,dr) coupling to organize Bayesian, robust, chance-constrained, stochastic-allocation, and distributional reinforcement-learning methods.
result Synthetic and structural contributions, including a portfolio specialization of Wasserstein-CVaR duality and a static no-randomization theorem.
RSO uses random weight perturbations to train deep networks without gradients.
problem Training deep neural networks efficiently and without gradient information.
method RSO is a gradient-free Markov Chain Monte Carlo approach that updates weights based on mini-batch loss reduction.
result RSO achieves high accuracy (99.1% on MNIST) with significantly fewer updates than traditional methods.
Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and sho…
Paper defines a new distance metric for comparing learning tasks.
problem Comparing difficulty of learning tasks between source and target.
method Information geometry, optimal transport, coupled transfer distance.
result Coupled transfer distance correlates with fine-tuning difficulty.
Introduces Exponentially Weighted Signature for better path representation.
problem Uniform treatment of historical information in signatures.
method Generalizes EFM signature to bounded linear operators, enabling contextualised temporal weighting.
result EWS is the unique solution to a linear controlled differential equation and generalizes state-space models.
Sharp eigenvalue bounds and splitting for modified Ricci flow.
problem Eigenvalue bounds and splitting in modified Ricci flow.
method Sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow.
result Splitting theorem in the case of equality.
Guyon-Lekeufack model accurately predicts market volatility.
problem Modeling and predicting market volatility accurately.
method Path-dependent volatility model with weighted past price returns and squared volatility.
result Wellposedness of the coupled system of stochastic differential equations for all parameter values.
Artifical Neural Networks are a particular class of learning systems modeled after biological neural functions with an interesting penchant for Hebbian learning, that is "neurons that wire together, fire together". However, unlike their natural counterparts, artificial neural networks have a close and stringent couplin…
New analysis shows how cross-entropy training shapes attention in transformers.
problem Understanding how gradient-based learning creates the required internal geometry in transformers.
method Developed a first-order analysis of cross-entropy training effects on attention scores and values in a transformer attention head.
result Introduced an advantage-based routing law and responsibility-weighted update for attention scores and values, respectively.
K-Nearest neighbor classifier (k-NNC) is simple to use and has little design time like finding k values in k-nearest neighbor classifier, hence these are suitable to work with dynamically varying data-sets. There exists some fundamental improvements over the basic k-NNC, like weighted k-nearest neighbors classifier (wh…
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.
Decoupled GCN is shown to be equivalent to label propagation.
problem Improving semi-supervised node classification in graph learning.
method The paper proves the equivalence of decoupled GCN and label propagation, and proposes a new method named PTA.
result Decoupled GCN is equivalent to two-step label propagation and can automatically assign weights to pseudo-labels.
Convolutional Neural Networks (CNNs) have become indispensable for solving machine learning tasks in speech recognition, computer vision, and other areas that involve high-dimensional data. A CNN filters the input feature using a network containing spatial convolution operators with compactly supported stencils. In pra…
We propose a sample-efficient alternative for importance weighting for situations where one only has sample access to the probability distribution that generates the observations. Our new method, called Geometric Resampling (GR), is described and analyzed in the context of online combinatorial optimization under semi-b…
Develops a control framework for systemic risk under uncertainty.
problem Systemic risk under model uncertainty.
method Linear-quadratic mean-field control framework with viscosity solutions and verification theorems.
result Explicit feedback controls derived from a coupled Riccati system, preserving analytical tractability.
New method improves accuracy of quantized neural networks.
problem Accuracy drop in quantized neural networks, especially MobileNet family.
method Weight equalizing shift scaler, binary shifting to recover output range.
result Top-1 accuracy improved from 0.1% to 69.78% ~ 70.96% in MobileNets.
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…
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.
New framework optimizes label shift adaptation using aligned distribution mixture.
problem Label shift where source and target label distributions differ.
method Aligned Distribution Mixture (ADM) framework, incorporating insights from generalization theory.
result The ADM framework improves four typical label shift methods and introduces a one-step approach.
The study analyzes prediction errors in systems with memory kernels, providing bounds and stability results.
problem Prediction errors in stochastic dynamical systems with memory kernels.
method Analysis of generalized Langevin equations (GLEs) with Volterra equations, integrating synchronized noise coupling and weighted norms.
result Prediction discrepancies decay at a rate determined by the memory kernel's decay, quantitatively bounded by kernel estimation errors.