Fundamental weight systems identified as quantum states.
problem Identifying which weight systems are quantum states.
method Analyzing the Cayley distance kernel on the symmetric group and its positivity.
result All fundamental gl(n)-weight systems are quantum states.
Study pressure metrics for cusped Hitchin representations.
problem Characterize cusped Hitchin representations of Fuchsian groups.
method Develop pressure metrics associated to fundamental weights and roots.
result New pressure metrics for Hilbert length when d=3. In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces Σ⊂Rn+1. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of pro…
Lower bounds for delta invariant of weighted hypersurfaces proved for K-stability.
problem Proving K-stability of weighted hypersurfaces.
method Abban-Zhuang method and study of linear systems on flags of weighted hypersurfaces.
result Proves K-stability of a large class of quasi-smooth Fano hypersurfaces and all smooth Fano weighted hypersurfaces.
The paper generalizes a rigidity theorem for hypersurfaces with constant weighted mean curvature.
problem Classifying hypersurfaces with constant weighted mean curvature.
method Using polynomial volume growth and specific curvature conditions, the authors prove rigidity theorems.
result Hypersurfaces with constant weighted mean curvature must be either a hyperplane or a generalized cylinder under certain conditions.
Survey on importance weighting in machine learning applications.
problem Distribution shift in supervised learning.
method Weighting objective function or probability distribution based on instance importance.
result Importance weighting can guarantee desirable statistical properties in distribution shift scenarios.
Study on Einstein solitons with bounds and asymptotic behavior.
problem Understanding the properties of Einstein solitons.
method Computed lower bounds for scalar curvature, established asymptotic behavior, proved finiteness of fundamental group and weighted volume.
result Established finiteness of fundamental group and weighted volume for gradient shrinking Einstein solitons.
Study concavity of solutions to elliptic equations under conformal deformations.
problem Establish concavity estimates for the principle eigenfunction of weighted Schrödinger operators.
method Analyzing the Dirichlet problem for the weighted Schrödinger operator \[-Δu + Vu = λρu\] with conformal connections.
result Partial resolution of Nguyen's conjecture on fundamental gap of horoconvex domains and power convexity estimate for solutions in spherical geometry.
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-li…
The paper proves rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.
problem Proving rigidity for self-shrinkers and surfaces with parallel weighted mean curvature.
method Using a new generalization of Cauchy's Theorem in complex analysis.
result Rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.
The weighted k-nearest neighbors algorithm is one of the most fundamental non-parametric methods in pattern recognition and machine learning. The question of setting the optimal number of neighbors as well as the optimal weights has received much attention throughout the years, nevertheless this problem seems to have r…
Study uses graph techniques to understand meromorphic quadratic differential strata.
problem Understanding the topology of meromorphic quadratic differential strata.
method Exchange graph techniques to study fundamental groups; generalizes relations for mixed-angulations.
result Explicit presentations of fundamental groups in genus-zero case with four singularities.
In this paper, we obtain results on rigidity of complete Riemannian manifolds with weighted Poincaré inequality. As an application, we prove that if M is a complete nn−2-stable minimal hypersurface in Rn+1 with n≥3 and has bounded norm of the second fundamental form, then M must eithe…
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
New examples of 3-manifolds not obtained by surgery on knots.
problem Identifying 3-manifolds not obtained by Dehn surgery.
method Combining Furuta's 10/8-theorem with combinatorial arguments.
result Found new 3-manifolds with weight one fundamental group not obtained by surgery.
This paper tackles negative transfer in multi-task learning by introducing class-wise weights.
problem Negative transfer hampers function from achieving optimality in multi-task learning.
method Introduces class-wise weights to drive positive transfer and suppress negative transfer.
result Demonstrates improved performance in multi-task learning by reducing negative transfer.
We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted L2 condition on the norm of the second fundamental form. Our approach adopt the …
We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in S3. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the SU(2) character variety of the fundame…
This work characterizes the fundamental limit of network pruning using statistical dimension and convex geometry.
problem The fundamental limit of network pruning is still lacking, especially for deep neural networks.
method Directly imposing sparsity constraint on the loss function and using statistical dimension in convex geometry.
result Characterizes the sharp phase transition point as the fundamental limit of pruning ratio.
We show that if B is an aspherical 2-orbifold in one of the families known to have orbifold fundamental groups of weight 1 then B is the base of a Seifert fibration of a 2-knot manifold M(K).
We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for…
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
Proves Schoen's conjecture on tori with specific conditions.
problem Proving Schoen's conjecture on tori with non-negative scalar curvature.
method Uses weighted scalar curvature and the relative index theorem.
result If the fundamental group of the singular set is not surjective, the metric extends to a smooth flat metric.
A recent analysis of a model of iterative neural network in Hilbert spaces established fundamental properties of such networks, such as existence of the fixed points sets, convergence analysis, and Lipschitz continuity. Building on these results, we show that under a single mild condition on the weights of the network,…
Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of or…
We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed …
The moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0,w is homotopic to a wedge sum of spheres, possi…
Analyzes minima of deep linear networks with weight decay.
problem Understanding the loss landscape of deep neural networks.
method Analytical solutions for global minima with weight decay and stochastic neurons.
result The origin is a special point with qualitatively different minima in networks with more than 1 hidden layer.
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…
This work investigates fundamental questions related to learning features in convolutional neural networks (CNN). Empirical findings across multiple architectures such as VGG, ResNet, Inception, DenseNet and MobileNet indicate that weights near the center of a filter are larger than weights on the outside. Current regu…
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at …
The paper tightens bounds on covering numbers for deep ReLU networks.
problem Characterizing the capacity and performance of deep ReLU networks.
method Derives tight lower and upper bounds on metric entropy of ReLU networks.
result Establishes optimality in nonparametric regression via deep networks.
New method makes neural networks transparent, revealing learning modes.
problem Lack of interpretability in neural networks.
method Weight pathway analysis (WPA) to decompose neural networks into subnetworks.
result Neural networks store and utilize information holographically, with linear and nonlinear learning modes.
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
Improves transfer learning by weighting importance based on test-over-training density.
problem Distribution shift in training and test data.
method Joint and dynamic importance-predictor estimation, causal mechanism transfer.
result Enhanced transfer learning performance in complex, high-dimensional tasks.
WBCP improves conformal prediction for distribution shifts using weighted Dirichlet posteriors.
problem Handling distribution shifts in conformal prediction.
method Generalizes Bayesian Quadrature Conformal Prediction (BQ-CP) to arbitrary importance-weighted settings.
result WBCP maintains coverage guarantees while providing richer uncertainty information.
Flow on weighted graphs sharpens Bakry-Émery curvature.
problem Sharp curvature in weighted graphs.
method Bakry-Émery curvature flow on mixed weighted graphs.
result Limits of curvature flow are curvature sharp.
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
This work explores how overparametrization and priors affect Bayesian neural network posteriors.
problem Symmetries, non-identifiabilities, and weight-space priors fragment and inflate BNN posteriors.
method We study the interplay between overparametrization and priors in BNN posteriors, deriving key phenomena and validating through experiments.
result Overparametrization induces structured, prior-aligned weight posterior distributions.
Book covers tools for zeroth-order convex optimisation.
problem Zeroth-order convex optimisation.
method Cutting plane methods, interior point methods, continuous exponential weights, gradient descent, online Newton step.
result Improved existing bounds and algorithms.
Regularizers change the geometric properties of loss functions in neural networks.
problem Understanding how different regularizers affect the geometric properties of loss functions in neural networks.
method Examined several regularizers, including weight decay, to determine if the regularized loss function becomes Morse.
result For certain regularizers, the regularized loss function becomes Morse, indicating a change in geometric properties.
Deep fundamental factor models are developed to automatically capture non-linearity and interaction effects in factor modeling. Uncertainty quantification provides interpretability with interval estimation, ranking of factor importances and estimation of interaction effects. With no hidden layers we recover a linear fa…
In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds N×R, submanifolds with bounded φ-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…
Optimizes kernel density ratios for better predictions and information measures.
problem Improving accuracy of kernel density estimates for density ratios.
method Derives an optimal weight function using calculus of variations.
result Reduces bias in kernel density estimates, leading to improved prediction posteriors and information-theoretic measures.
The paper resolves fundamental groups for three exceptional surface singularity families.
problem Determining the fundamental groups for three exceptional families of surface singularities.
method New explicit constructions and the Pinkham method for some families.
result Fundamental groups for three exceptional families are determined.
Adam's generalization performance is improved by batch size and weight decay in neural networks.
problem Understanding how batch size and weight decay affect Adam's generalization in neural networks.
method Theoretical analysis of two-layer over-parameterized CNNs on image data.
result Adam's mini-batch variants can achieve near-zero test error, unlike full-batch Adam.
Facing the FRTB, banks need to allocate their capital to each business units or risk positions to evaluate the capital efficiency of their strategies. This paper proposes two computationally efficient allocation methods which are weighted according to liquidity horizon. Both methods provide more stable and less negativ…