Unified analysis of MPLE for Ising models with bounded operator norm or infinity norm.
problem Estimating Ising models in Total Variation distance with limited samples.
method Maximum Pseudo-Likelihood Estimator (MPLE) for two general classes of Ising models.
result Unified framework for polynomial-time estimation in TV distance for two general classes of Ising models.
Gradient descent-based adversarial training converges to robust classifiers on linearly separable data.
problem Understanding the inductive bias of adversarial training for robustness.
method Gradient descent on binary classification tasks with linearly separable data, focusing on inductive bias and convergence rates.
result Gradient descent-based adversarial training converges to the maximum margin classifier at a faster rate than clean data training.
Paper tackles partial label learning with self-guided retraining.
problem Dealing with partially labeled examples where each instance has a set of candidate labels.
method Unified formulation with constraints for joint training and pseudo-labeling; maximum infinity norm regularization for automatic differentiation; convex-concave optimization problem; upper-bound surrogate objective function.
result Significantly outperforms state-of-the-art partial label learning approaches.
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
We identify the Variational Principle governing inifinity-Harmonic maps, that is solutions to the Infinity-Laplacian. The system was first derived in the limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in [K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued Calculus of Variat…
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
The paper examines partial regularity of Lipschitz solutions to minimal surface system.
problem Understanding the regularity of solutions to the minimal surface system.
method Investigation of stationary, integral weak, and viscosity solutions; interior gradient estimate using maximum principle.
result Partial regularity results for Lipschitz solutions, including interior gradient estimate.
In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over …
Paper addresses LSTM stability for thermal systems using infinity-norm.
problem Stability of LSTM networks in thermal systems.
method Derived ISS∞ condition for LSTM, developed training strategy. result ISS∞-promoted LSTM outperforms other models in thermal system case study. We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann. Global Anal. Geom. 20, 325-343 2001] and De Lima and Meeks [Indiana Univ. Math. Journal 53 5, 1211-1223 2004], for disjoints hypersurfaces of Rn+1 with bounded mean curvature without restriction…
New algorithms estimate matrix norms without matrix multiplication.
problem Estimating matrix norms efficiently in a matrix-free setting.
method Randomized algorithms based on Hutchinson's estimator modifications.
result Oracle complexity bounds for two-to-infinity and one-to-two norms.
Study shows distance to boundary is always attained on varifolds with bounded curvature.
problem Understanding varifolds with bounded mean curvature in Riemannian manifolds.
method Proves a barrier principle at infinity using sharp maximum principles.
result Distance to boundary is always attained on varifolds with bounded curvature.
The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
Study potential theory to detect completeness of Finsler manifolds.
problem Detecting completeness of Finsler manifolds via potential theory.
method Potential theoretic aspects of eikonal and infinity Laplace operator, Liouville properties, maximum principles at infinity, viscosity solutions.
result Forward completeness of Finsler manifolds can be detected using Liouville properties and maximum principles at infinity.
This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-…
Early stopping improves logistic regression's calibration and consistency in high dimensions.
problem Improving the statistical performance of gradient descent in overparameterized logistic regression.
method Investigates the effects of early stopping on gradient descent in logistic regression.
result Early-stopped gradient descent is well-calibrated and statistically consistent, while asymptotic gradient descent is not.
Norm-range partition improves MIPS search efficiency by reducing query complexity.
problem Efficiently searching for maximum inner product in large datasets.
method Norm-range partition technique that divides datasets into sub-datasets with similar norms and builds independent hash indexes.
result Significantly reduces the number of probed buckets for LSH-based MIPS algorithms.
Mirror flow optimizes separable data problems, converging to a maximum margin classifier.
problem Optimizing classification problems with separable data using mirror flow.
method Examine mirror flow on linearly separable classification problems, focusing on the horizon function of the mirror potential.
result Mirror flow converges to a maximum margin classifier for separable data under certain conditions.
Study shows how networks converge to minimum norm solutions with regularization.
problem Interpolating between known regions in shallow ReLU networks.
method Investigates empirical risk minimizers and weight decay regularizers.
result Empirical risk minimizers converge to minimum norm interpolants under specific conditions.
We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
This work analyzes the maximum-margin bias in quasi-homogeneous neural networks.
problem Analyzing the maximum-margin bias in quasi-homogeneous neural networks.
method Geometric analysis of gradient dynamics for quasi-homogeneous models.
result Gradient flow implicitly favors a subset of parameters, leading to asymmetric norm minimization.
Neyshabur and Srebro proposed Simple-LSH, which is the state-of-the-art hashing method for maximum inner product search (MIPS) with performance guarantee. We found that the performance of Simple-LSH, in both theory and practice, suffers from long tails in the 2-norm distribution of real datasets. We propose Norm-rangin…
Improved GP bandit algorithms for noiseless, varying noise, and RKHS norms.
problem Minimizing regret in Gaussian process bandits with unknown reward functions.
method New upper bound on maximum posterior variance, refined MVR and PE algorithms.
result Optimal regret bounds for noiseless, varying noise, and RKHS norms.
Researchers prove a conjecture about a specific type of 3D space.
problem Guilloux's conjecture about the Borel function on a hyperbolic 3-manifold.
method Proved Guilloux's conjecture for a particular reflection group.
result The Borel function is rigid at infinity for the tetrahedral reflection lattice.
The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.
problem Characterizing properties of minimal hypersurfaces in higher-dimensional Riemannian manifolds.
method Maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces.
result Two disjoint properly embedded minimal hypersurfaces bound a slab in specific conditions.
We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in Cn, we provide estimates for the norms of these automorphic forms and we find asymptotics of…
Paper studies MCCR models with scale parameters tending to zero, revealing optimal learning rate and comparing robustness.
problem Analyzing MCCR models with scale parameters approaching zero.
method Investigates MCCR models with scale parameters tending to zero, revealing optimal learning rate and comparing robustness.
result Optimal learning rate of MCCR models is O(n−1) in the asymptotic sense. The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.
problem Stability properties of bounded cohomology in mapping class groups and symplectic groups.
method Utilizes results from Bestvina and Fujiwara, calculates norms of signature classes, and estimates cohomology norms.
result The bounded cohomology of mapping class groups does not stabilize, while that of symplectic groups does not stabilize via isometries.
This paper studies the problem of estimating the covariance of a collection of vectors using only highly compressed measurements of each vector. An estimator based on back-projections of these compressive samples is proposed and analyzed. A distribution-free analysis shows that by observing just a single linear measure…
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …
New insights into optimization and generalization for linear models.
problem Understanding the implicit regularization of optimization methods for linear models.
method Investigating the norms minimized by interpolating solutions and using projections to move between solutions.
result Proving that for over-parameterized linear classification, projections onto the data-span enable the use of under-parameterized techniques.
We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the ca…
New proof shows norms can't explain deep learning's implicit regularization.
problem Understanding the implicit regularization in deep learning.
method Mathematical proof on matrix factorization problems.
result Implicit regularization drives norms towards infinity, suggesting rank minimization is key.
We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…
Study improves bounds on p-covectors and proves stable systolic inequalities.
problem Improving bounds on p-covectors and proving stable systolic inequalities.
method Analyzing Euclidean norms and fundamental cohomology classes.
result Improved upper bounds and stable systolic inequalities proved.
Sum-of-norms clustering recovers mixtures of Gaussians even with infinite samples.
problem Recovering a mixture of Gaussians from a large number of samples.
method Sum-of-norms clustering with equal weights, convex optimization.
result Sum-of-norms clustering can recover mixtures of Gaussians even as the number of samples tends to infinity.
We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many a…
We consider in this paper the problem of noisy 1-bit matrix completion under a general non-uniform sampling distribution using the max-norm as a convex relaxation for the rank. A max-norm constrained maximum likelihood estimate is introduced and studied. The rate of convergence for the estimate is obtained. Information…
Optimal financial strategies minimize risk under uncertain models.
problem Maximizing utility in financial markets with model uncertainty.
method Optimized strategies converge to those with minimal norm as uncertainty increases.
result Optimal strategies with minimal norm emerge as uncertainty grows.
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used …
Study of 3D vacuum static spaces with specific curvature properties.
problem Classifying 3D vacuum static spaces with certain curvature conditions.
method Used generalized maximum principle to classify 3D spaces.
result Gave a complete classification of 3D complete vacuum static spaces.
In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-line…
Researchers found the maximum number of holes in polyominoes grows proportionally to the dimension.
problem Finding the maximum number of holes in polyominoes of varying dimensions.
method Used concepts from error-correcting codes and dynamical systems.
result Proved that fd(n)/no(d−1)/d as n goes to infinity for all d≥2. Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up a…