The paper provides theoretical guarantees for optimized sampling in compressed sensing, showing error vanishes with more measurements.
problem Theoretical and practical improvements in compressed sensing with optimized sampling schemes.
method Theoretical analysis and empirical experiments with optimized sampling schemes for subsampled unitary matrices.
result The error caused by measurement noise vanishes with an increasing number of measurements for optimized sampling schemes, assuming Gaussian noise.
Develops a method to estimate rare-event probabilities under distributional uncertainty.
problem Distributional uncertainty limits the effectiveness of rare-event simulation techniques.
method Wasserstein distributionally robust rare-event simulation (DRIS) framework.
result DRIS achieves vanishing relative error in estimating rare-event probabilities.
SS-GEN simulates rare events in heavy and light-tailed data.
problem Estimating probabilities of extreme events in multivariate data.
method Self-Similar Generative Estimation (SS-GEN) decomposes tail distribution into radial and angular components.
result SS-GEN generates representative extreme scenarios and estimates rare-event probabilities beyond observed data.
Extends Gromov's theorem with amenable covers.
problem Gromov's Vanishing Theorem limitations.
method Gromov's multicomplex theory, relative bounded cohomology.
result Relative version of Gromov's Vanishing Theorem.
We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also impl…
New results on relative simplicial volume using bounded acyclicity.
problem Understanding relative simplicial volume in bounded cohomology.
method Equivariant nerve pairs, relative classifying spaces, and small relative amenable category.
result Vanishing results for ℓ2-Betti numbers and mapping degrees. The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
Study the properties of SGD in non-vanishing learning rate regime.
problem Understanding the noise and fluctuation in SGD with finite learning rates.
method Derive exact solvable results for discrete-time SGD in quadratic loss functions.
result Fluctuation caused by discrete-time dynamics is larger than continuous-time theory predicts.
Local vanishing theorems for complex spaces with smooth boundaries.
problem Vanishing of cohomology groups for complex spaces with smooth boundaries.
method Local vanishing theorem for Dolbeault cohomology groups.
result Vanishing of L2 and L2,loc Dolbeault cohomology groups for q>0. Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the …
A new framework evaluates HTE estimators using relative error.
problem Lack of robust evaluation methods for HTE estimators.
method Proposes a relative error-based evaluation framework and neural network architecture to estimate nuisance parameters and robustly compare HTE estimators.
result Demonstrates reliable comparisons and improved HTE estimation through the proposed framework and learning algorithm.
In the differential geometry of certain F-structures, the role of W-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing W-tensor have been considered and the existence of Killing and …
We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature K in the Euclidean space R3, which are characterized by the support functions (α)q=∣K∣α for α∈R (Manhart's relative normalizations). All ruled surfaces for…
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space R3. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…
In regression settings where explanatory variables have very low correlations and there are relatively few effects, each of large magnitude, we expect the Lasso to find the important variables with few errors, if any. This paper shows that in a regime of linear sparsity---meaning that the fraction of variables with a n…
In this paper, we discuss the relative K-stability and the modified K-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative K-stability and the properness of modified K-energy. In …
New method identifies vanishing arcs for curve singularities.
problem Characterizing arcs sent to geometric vanishing cycles.
method Introducing geometric variation operator and vanishing arcsets.
result Existence of topological exceptional collections of arcsets.
Recurrent neural networks (RNNs) are particularly well-suited for modeling long-term dependencies in sequential data, but are notoriously hard to train because the error backpropagated in time either vanishes or explodes at an exponential rate. While a number of works attempt to mitigate this effect through gated recur…
Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.
problem Computing Čech cohomology groups of Morse boundaries.
method Analyzing cusped hyperbolic n-manifolds and relatively hyperbolic groups.
result Reduced Čech cohomology vanishes in dimensions ≤ n-3 and does not vanish in dimension n-2.
New decision-theoretic calibration error metric improves prediction reliability.
problem Improving the reliability of predictions for decision-making.
method Proposed Calibration Decision Loss (CDL) and an efficient algorithm to achieve near-optimal CDL.
result Near-optimal CDL guarantees vanishing payoff loss from miscalibration.
MGDL refines deep neural networks by training grades sequentially, improving stability.
problem Training deep neural networks is challenging due to nonconvex optimization landscapes.
method MGDL trains deep networks grade by grade, freezing previously learned grades and training new ones to fit residuals.
result MGDL guarantees vanishing error in a fixed-width multigrade ReLU architecture.
In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of t…
Paper establishes bounds for RNN-TPPs, showing four-layer networks can achieve vanishing errors.
problem Understanding theoretical limits of RNN-TPPs.
method Characterized RNN complexity, constructed neural approximations, applied truncation technique.
result Four-layer RNN-TPPs can achieve vanishing generalization errors.
Study shows how leveraging hierarchical similarity graphs improves matrix completion in recommender systems.
problem Improving matrix completion in recommender systems using hierarchical similarity graphs.
method Characterizes the optimal sample complexity using hierarchical stochastic block models and low-rank rating matrices.
result Exploiting hierarchical structure of social graphs significantly reduces the number of observed entries needed for accurate matrix completion.
In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.
Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a…
The CUR matrix decomposition is an important extension of Nyström approximation to a general matrix. It approximates any data matrix in terms of a small number of its columns and rows. In this paper we propose a novel randomized CUR algorithm with an expected relative-error bound. The proposed algorithm has the advanta…
We show that any degree at least g polynomial in descendant or tautological classes vanishes on Mg,n when g≥2. This generalizes a result of Looijenga and proves a version of Getzler's conjecture. The method we use is the study of the relative Gromov-Witten invariants of P1 relative 2 points combined with…
It is well known that the problem of vanishing/exploding gradients is a challenge when training deep networks. In this paper, we describe another phenomenon, called vanishing nodes, that also increases the difficulty of training deep neural networks. As the depth of a neural network increases, the network's hidden node…
We obtain a deterministic characterisation of the \emph{no free lunch with vanishing risk}, the \emph{no generalised arbitrage} and the \emph{no relative arbitrage} conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…
We investigate the use of Deep Neural Networks for the classification of image datasets where texture features are important for generating class-conditional discriminative representations. To this end, we first derive the size of the feature space for some standard textural features extracted from the input dataset an…
We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken position…
In this work we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of the Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes which are non-flat generalizations of the very special relat…
Vanishing gradients hinder reinforcement finetuning of language models.
problem Vanishing gradients impede the optimization of language models using reinforcement finetuning.
method The study identifies vanishing gradients as a fundamental optimization obstacle in reinforcement finetuning and proposes an initial supervised finetuning phase to mitigate this issue.
result An initial supervised finetuning phase is crucial for successful reinforcement finetuning of language models, as it helps prevent vanishing gradients and maximizes rewards.
New method detects black hole horizons using Lie algebra invariants.
problem Detecting black hole horizons in spacetime.
method Scalar relative differential invariants with Lie algebra structure.
result General relative differential invariant vanishes on black hole horizons.
We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to y′′=0 under such transformations. Moreover w…
This work tightens generalization error bounds using Wasserstein distance.
problem Improving expected generalization error bounds in machine learning.
method Introduces bounds based on Wasserstein distance for various settings.
result New, tighter bounds based on relative entropy and other information measures.
This work bounds the generalization error of private algorithms for discrete data.
problem Bounding the generalization error of private algorithms for discrete data.
method Information-theoretic approach using relative entropy and the method of types.
result Explicit upper bounds on the generalization error of stable private algorithms for discrete data.
New approach uses Gaussian processes to learn and track complex systems with guaranteed accuracy.
problem Inaccurate first principle models for complex systems due to data complexity.
method Bayesian prediction error bound for Gaussian process regression, derived from kernel-based data density.
result Achieves vanishing tracking error with increasing data density, providing time-varying accuracy guarantees.
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
Relative Thom polynomials for maps around boundaries established.
problem Understanding singularities in maps around boundaries.
method Introducing and analyzing Thom polynomials relative to prescribed maps around boundaries, establishing structure theorems and correction terms.
result Unified framework for invariants of immersions and singularities of their extensions.
We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…
We conduct mathematical analysis on the effect of batch normalization (BN) on gradient backpropogation in residual network training, which is believed to play a critical role in addressing the gradient vanishing/explosion problem, in this work. By analyzing the mean and variance behavior of the input and the gradient i…
The main aim of this paper is to provide an analysis of gradient descent (GD) algorithms with gradient errors that do not necessarily vanish, asymptotically. In particular, sufficient conditions are presented for both stability (almost sure boundedness of the iterates) and convergence of GD with bounded, (possibly) non…
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…