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.
New methods improve clustering accuracy in noisy data sets.
problem Improving clustering accuracy in data sets with noise features.
method Feature rescaling factors to enhance clustering validity indexes.
result Our methods increase the likelihood of estimating the true number of clusters.
New method learns optimal prediction strategies in adversarial games.
problem Learning optimal prediction procedures in uncertain data environments.
method Adversarial Monte Carlo approach with neural network architecture.
result Optimal strategy is equivariant and invariant to various transformations.
This work bridges two views of feature learning in neural networks.
problem The relationship between kernel scale changes and data-adaptive feature learning in neural networks remains unresolved.
method Using statistical mechanics, the work derives analytical expressions for network output statistics across scaling regimes.
result Kernel adaptation can be reduced to an effective kernel rescaling, but multi-scale adaptive approach provides richer insights.
Unsupervised preprocessing can bias cross-validation estimates in regression models.
problem Bias in cross-validation estimates due to unsupervised preprocessing.
method Analysis of three preprocessing procedures: feature selection, grouping, and rescaling.
result Unsupervised preprocessing can introduce substantial bias into cross-validation estimates.
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.
A new method clusters malware data more effectively.
problem Difficult clustering of drive-by-download malware data.
method Iterative data rescaling method to enhance cluster separation.
result Improved separation between malware clusters, higher silhouette width.
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 M, 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 TM, 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.
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.
Local index theorem for manifolds with Lie structure at infinity using rescaling and renormalized supertrace.
problem Proving an Atiyah-Singer type index theorem for manifolds with Lie structure at infinity.
method Rescaling technique similar to Getzler's, integrating Lie groupoid, functional calculus, heat kernel expansion, Lichnerowicz theorem.
result Proof of a local index theorem for Dirac operators on Lie manifolds.
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.
This work justifies neural collapse under MSE loss and analyzes the optimization landscape.
problem Understanding neural collapse in deep neural networks under MSE loss.
method Global landscape analysis of vanilla nonconvex MSE loss.
result The only global minimizers are neural collapse solutions.
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.
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.
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.
Generative model controls heterophily in graph signals.
problem Controlling heterophily in graph signals for better model effectiveness.
method Combines graphon-based generator with spectral filtering of Gaussian node features.
result Establishes theoretical guarantees for heterophily control and convergence.
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.
RESCAL fails to encode transitive relations in knowledge bases.
problem Link prediction in noisy knowledge graphs.
method Analysis of RESCAL model for transitive relations.
result RESCAL cannot encode asymmetric transitive relations.
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.
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 (M,g) be an n−dimensional Riemannian manifold and T11(M) be its (1,1)−tensor bundle equipped with the rescaled Sasaki type metric which rescale the horizontal part by a nonzero differentiable function f. In the present paper, we discuss curvature properties of the Levi-Civita connectio…
New algorithms predict without tuning for varying feature scales.
problem Learning linear models online with unknown feature scales.
method Adaptive scale-invariant online algorithms without tuning parameters.
result Achieves regret bounds matching OGD with optimal tuning, comparable runtime.
The paper studies how convex shapes evolve over time based on their curvature.
problem Understanding how convex shapes change over time based on their curvature.
method Analyzes the evolution of strictly convex hypersurfaces in \(\mathbb{R}^{n+1}\) moving with a specific speed.
result The flow converges to a round sphere after rescaling for \(α > \frac{1}{n+2}\), and to an ellipsoid for \(α = \frac{1}{n+2}\).
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.
Paper analyzes Velázquez's solution to a singularity in mean curvature flow.
problem Analyzing a type II singularity in mean curvature flow.
method Degree theory and time-dependent rescaling to study convergence.
result Rescaled flow converges locally smoothly to a minimal hypersurface.
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…
Symbol calculus extended for foliations' transverse geometry.
problem Understanding index theory of transversely elliptic operators on foliations.
method Constructing Getzler rescaling calculus and Block-Fox calculus of asymptotic operators.
result Composition of AΨDOs is again an AΨDO, with a leading symbol formula. 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 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.
Study shows how certain spacetimes evolve in the future.
problem Analyzing the future behavior of vacuum spacetimes.
method Rescaling analysis of T2-symmetric spacetimes. result Universal cover converges to a non-Einstein spacetime.
Gradient descent reshapes the function space of neural networks.
problem Understanding how feature learning affects the function space of neural networks.
method Characterized the evolution of the feature space during training using a two-layer neural network.
result Gradient descent induces a data-adaptive deformation that selectively enhances signal-aligned directions.
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.
Study small-time fluctuations for sub-Riemannian diffusion loops.
problem Analyzing fluctuations of sub-Riemannian diffusion processes.
method Analyzes small-time fluctuations of diffusion processes with sub-Riemannian structure, identifying degenerate and non-degenerate covariance matrices.
result Rescaled fluctuations converge to a non-degenerate limiting diffusion loop.