Study shows how certain spacetimes evolve in the future.
problem Analyzing the future behavior of vacuum spacetimes.
method Rescaling analysis of -symmetric spacetimes.
result Universal cover converges to a non-Einstein spacetime.
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Application of Multifractal Measures to Tehran Price Indexphysics.data-an
We report an empirical study of Tehran Price Index (TEPIX). To analyze our data we use various methods like as, rescaled range analysis (), modified rescaled range analysis (Lo's method), Detrended Fluctuation Analysis (DFA) and generalized Hurst exponents analysis. Based on numerical results, the scaling range of…
Proposes an iterative algorithm for optimizing attention mechanisms in large language models.
problem Optimizing attention mechanisms in large language models.
method Iterative algorithm for rescaled hyperbolic functions regression.
result Efficiency and generalizability of the rescaled softmax regression framework.
The paper proves a rescaling principle for quasiregular curves and applies it to hyperbolicity.
problem Proving hyperbolicity for quasiregular curves.
method Rescaling principle for quasiregular curves into calibrated manifolds.
result Equivalence of Brody hyperbolicity and normality of quasiregular curves.
J.J.L. Velzquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled…
We test for departures from normal and independent and identically distributed (NIID) returns, when returns under the alternative hypothesis are self-affine. Self-affine returns are either fractionally integrated and long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moment…
In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.
The paper calculates spectral torsion for rescaled Dirac operators on manifolds.
problem Computing spectral torsion for rescaled Dirac operators.
method Using trilinear Clifford multiplication and functional of differential one-forms.
result Computed spectral torsion for four types of rescaled Dirac operators.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
For a Riemannian manifold , we determine some curvature properties of a tangent bundle equipped with the rescaled metric.The main aim of this paper is to give explicit formulae for the rescaled metric on , and investigate the geodesics on the tangent bundle with respect to the rescaled Sasaki metric.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.
problem Computing the noncommutative residue for a specific Dirac operator on 6D manifolds.
method Calculations and proofs for the rescaled Dirac operator fDh on 6D compact manifolds.
result Proof of the Kastler-Kalau-Walze type theorem for the rescaled Dirac operator on 6D compact manifolds with boundary.
This paper approximates SA iterates using Gaussian distributions for tail bounds.
problem Characterizing the distribution of stochastic approximation iterates in finite time.
method Approximating pre-limit distributions of SA iterates by Gaussian sequences with recursively defined covariances.
result Explicit bounds on the Wasserstein-1 distance between rescaled iterates and Gaussians.
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
Fast algorithm for rescaling vectors with clipping, improving training efficiency.
problem Efficiently rescale vectors to a desired length while maintaining them within a domain after clipping.
method Analytical solution for optimal rescaling using fast and differentiable algorithm.
result Optimal rescaling can be found analytically, improving training efficiency for neural networks.
Study Chen's flow of curves in two settings: closed circles and lines, identifying geometric conditions for global behavior.
problem Understanding the global behavior of Chen's flow of curves in two settings.
method Investigated two settings: closed immersed ω-circles and immersed lines with a cocompactness condition. Analyzed geometric conditions and curvature effects.
result Identified conditions ensuring the flow shrinks every initial curve to a point, including a rescaling method.
A new method to rescale ReLU neural networks based on path-lifting.
problem Lack of principled ways to leverage rescaling symmetries in ReLU neural networks.
method Introduces a geometrically motivated criterion to rescale neural network parameters, aligning a kernel in the path-lifting space with a chosen reference.
result Proposed method can speed up training and aligns a kernel in the path-lifting space with a chosen reference.
An analytic solution for asset allocation with Laplace distribution.
problem Asset allocation with multivariate Laplace distribution.
method Specialization of elliptically symmetric distribution theory to Laplace distribution, accounting for dimensionality and variance rescaling.
result A result consistent with conjecture but with differences due to omitted term and rescaling.
Sharp curvature pinching for mean curvature flow in spheres proved.
problem Proving sharp curvature pinching for mean curvature flow in spheres.
method Using blow-up arguments, codimension and cylindrical estimates, and rescaling.
result Smooth convergence to a totally geodesic limit in infinite time.
This paper tackles non-vacuous generalization bounds in ReLU networks by resolving rescaling invariances.
problem Non-vacuous generalization guarantees for ReLU networks with rescaling invariances.
method Proposes a lifted representation to resolve rescaling invariances and studies KL-based rescaling-invariant PAC-Bayes bounds.
result KL-based rescaling-invariant PAC-Bayes bounds provide tighter guarantees and resolve discrepancies in network complexity.
Localizes Wodzicki residue for logarithm of differential operators.
problem Localizing Wodzicki residue for logarithm of differential operators.
method Localisation formula using rescaled differential operators and spinor bundles.
result Expresses index of Dirac operator in terms of local density involving logarithm.
We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in s…
New Lipschitz bound for ReLU networks resists weight rescaling.
problem Lack of robustness guarantees for ReLU networks under weight perturbations.
method Rescaling-invariant Lipschitz bound based on path-metrics.
result The new bound applies to various ReLU-DAG architectures and resists neuron-wise rescalings.
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
Correcting bias in least squares regression with volume-rescaled sampling.
problem Bias in linear least squares solutions without distributional assumptions.
method Volume-rescaled sampling to correct bias in i.i.d. samples.
result Combined sample becomes unbiased with rescaled volume additional sample.
We introduce a new test for detection of power-law cross-correlations among a pair of time series - the rescaled covariance test. The test is based on a power-law divergence of the covariance of the partial sums of the long-range cross-correlated processes. Utilizing a heteroskedasticity and auto-correlation robust est…
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
The paper constructs bundles and recovers Kirillov character formula.
problem Constructing smooth vector bundles over deformation to the normal cone.
method Rescaling of vector bundles and equivariant constructions.
result Recovery of Kirillov character formula for equivariant index.
Study proves existence and uniqueness of ancient flows from cones.
problem Existence and uniqueness of ancient rescaled mean curvature flows.
method Proved existence and uniqueness using strong uniqueness theorem.
result Proved existence and uniqueness of ancient flows from cones.
It is now known that an extended Gaussian process model equipped with rescaling can adapt to different smoothness levels of a function valued parameter in many nonparametric Bayesian analyses, offering a posterior convergence rate that is optimal (up to logarithmic factors) for the smoothness class the true function be…
Method determines latent dimensionality in international trade flows.
problem Finding meaningful low-dimensional latent features in high-dimensional international trade data.
method Proposes a latent dimension determination method based on clustering of nonnegative RESCAL decompositions.
result Validates the latent features against empirical economic facts.
Improved LLM pre-training performance through better weight and variance control.
problem Improper weight and variance control in LLM pre-training affects downstream task performance.
method Introduced Layer Index Rescaling (LIR) and Target Variance Rescaling (TVR) techniques.
result Substantial improvements in downstream task performance (up to 4.6%) and reduced extreme activation values.
"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrica…
Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…
A new method to improve deep neural networks using weight rescaling.
problem Overfitting and sensitivity to hyperparameters in weight decay.
method Weight rescaling (WRS) to control weight norm and prevent overfitting.
result WRS outperforms weight decay and other methods in various applications.
Let be an dimensional Riemannian manifold and be its tensor bundle equipped with the rescaled Sasaki type metric which rescale the horizontal part by a nonzero differentiable function . In the present paper, we discuss curvature properties of the Levi-Civita connectio…
We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective str…
The abstract discusses nonuniqueness results for specific Riemannian invariants.
problem Identifying conditions for nonhomothetic conformal rescalings with constant Riemannian invariants.
method Identifying sufficient conditions for finite and infinite geometrically distinct periodic conformal rescalings.
result Improves and establishes nonuniqueness results for various Riemannian invariants.
New flatness measure for deep networks invariant to scaling.
problem Lack of invariant flatness measures for deep networks under parameter rescaling.
method Introduced a quotient manifold structure and Hessian-based invariant flatness measure.
result Confirms that Large-Batch SGD minima are sharper than Small-Batch SGD minima.
We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…
Paper introduces normalized flat minima to address scale dependence in neural network optimization.
problem Scale dependence in existing flat minima definitions affects generalization studies.
method PAC-Bayesian analysis to introduce normalized flat minima, free from scale dependence.
result Normalized flat minima provides better hierarchy in hypothesis class and improved generalization.
We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…
We establish a Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.
Rescaled Asynchronous SGD: Optimal Distributed Optimization under Data and System Heterogeneitycs.LG
Rescaled ASGD optimizes distributed learning under heterogeneous data.
problem Vanilla ASGD biases towards a frequency-weighted average of local objectives.
method Rescale worker stepsizes by their computation times.
result Rescaled ASGD converges to the correct global objective in fixed-computation model.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Let $\1$ and $\2$ be $\s$ domains in $\Cn$ and $f: \1 \rt \2$ an isometry for the Kobayashi or Carathéodory metrics. Suppose that extends as a map to $ \bar \om_1$. We then prove that $f|_{\partial \1}: \partial \1 \rt \partial \2$ is a CR or anti-CR diffeomorphism. It follows that $\1$ and $\2$ must be bihol…
The paper proves a new theorem about paths on nilmanifolds.
problem Understanding the asymptotic behavior of paths on nilmanifolds.
method Analytic mechanism involving uniform convergence and layer-by-layer convergence of nilpotent developments.
result Recovering and extending previous theories to arbitrary nilpotent steps.
Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.
problem Analyzing singularity formation and asymptotic behavior of null mean curvature flow.
method Rescaling procedure to relate to 2d-Ricci flow, singularity analysis, asymptotic behavior study.
result Ancient solutions to the flow can be understood in terms of 2d-Ricci flow.