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.
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.
New method extends invariant reduction to rescaled geometric structures.
problem Computing invariant geometric structures under symmetries.
method Extends invariant reduction to rescaled structures using shift rule.
result Emergence and loss of invariance in reductions.
Symmetry in loss functions constrains model parameters, leading to specific learning outcomes.
problem Understanding and leveraging symmetries in neural networks to improve learning outcomes.
method Analyzing the impact of loss function symmetries on model parameters and learning behavior.
result Mirror-reflection symmetries in loss functions lead to constraints on model parameters, influencing learning outcomes.
Paper constructs flows converging to cones and foliations.
problem Understanding mean curvature flow convergence to cones and foliations.
method Constructs a family of mean curvature flows converging to cones and foliations under specific conditions.
result Flow converges to area minimizing, strictly stable hypercone and Hardt-Simon foliation of the cone.
"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…
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…
We show that the action functional of the nonlinear sigma model with gravitino considered in a previous article [18] is invariant under rescaled conformal transformations, super Weyl transformations and diffeomorphisms. We give a careful geometric explanation how a variation of the metric leads to the corresponding var…
A new optimizer DDC improves deep learning models by respecting symmetries.
problem Deep networks' loss is invariant to continuous symmetries, leading to optimization issues.
method DDC builds a Dead-Direction Conditioner that lifts a base optimizer into a G-equivariant one, preserving the quotient geometry.
result DDCAdam and DDCMuon outperform standard optimizers in various tasks, improving validation-train loss gaps and learning dynamics.
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.
Flow taxes and stock taxes preserve portfolio neutrality under specific conditions.
problem Analyzing the impact of different types of taxes on portfolio choice.
method Extending the neutrality result to a full system of ownership taxes, showing how each tax modifies the drift of the wealth process.
result The combined system of taxes preserves portfolio neutrality under three conditions, and the drift-shift symmetry generalizes to a drift-shift-and-rescale symmetry.
We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>n+21 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…
Study on stability of cylindrical singularities in MCF of finite codimensions.
problem Stability of cylindrical singularities in mean curvature flow.
method Construction of stable manifold, explicit solutions, asymptotic analysis.
result Asymptotic stability of cylindrical singularities under generic perturbations.
We define risk-free portfolios using three gauge invariant differential operators that require such portfolios to be insensitive to price changes, to be self-financing, and to produce a zero real return so there are no risk-free profits. This definition identifies the risk-free rate as the return of an infinitely diver…
Noise balance theory explains SGD's behavior in neural networks.
problem Understanding SGD's navigation in neural network loss landscapes.
method Analyzes minibatch noise and loss function symmetries.
result Derives the stationary distribution of SGD for deep networks.
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Study of Lagrangian mean curvature flow with equivariant symmetry.
problem Understanding singularities in Lagrangian mean curvature flow.
method Structural theorems about blowups of finite-time singularities.
result Classification of singularities in equivariant case.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
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 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.
For each integer k≥2, we apply gluing methods to construct sequences of minimal surfaces embedded in the round 3-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of k Clifford tori intersecting with …
New surfaces described that are symmetric and solve a specific equation.
problem Understanding symmetric shapes of membranes.
method Characterized and described axially symmetric Helfrich spheres using the reduced membrane equation.
result These surfaces are symmetric and belong to a specific family.
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.
We reformulate wealth taxation using Fokker-Planck equations to ensure tax neutrality.
problem Ensuring tax neutrality in wealth taxation frameworks.
method Reformulating the neutral wealth tax framework using stochastic dynamics and statistical physics, specifically Fokker-Planck equations.
result The framework clarifies when wealth taxation is a benign rescaling of dynamics and when it introduces new physics.
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…
B List has proposed a geometric flow whose fixed points correspond to solutions of the static Einstein equations of general relativity. This flow is now known to be a certain Hamilton-DeTurck flow (the pullback of a Ricci flow by an evolving diffeomorphism) on RxM^n. We study the SO(n) rotationally symmetric case of Li…
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)…
The curve shortening flow transforms figure-eight curves into bowties.
problem Transforming figure-eight curves into a specific shape under curve shortening flow.
method Applied curve shortening flow to figure-eight curves with specific properties, proving convergence to a quadrilateral.
result The renormalized limit of the flow converges to a quadrilateral called a bowtie.
Models for 3D harmonic 1-forms and spinors near singular points.
problem Constructing models for Z/2 harmonic 1-forms and spinors in 3D near singular points. method Using symmetries of tetrahedron, octahedron, and icosahedron to construct local models on R3. result Local models are Z/2 harmonic 1-forms or spinors on R3 with zero locus consisting of rays from the origin. We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…
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 has been empirically observed that the flatness of minima obtained from training deep networks seems to correlate with better generalization. However, for deep networks with positively homogeneous activations, most measures of sharpness/flatness are not invariant to rescaling of the network parameters, corresponding…
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.
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…
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.
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…
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…