We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …
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48 results for “scale space theory”
Let be a group acting properly and by isometries on a metric space ; it follows that the quotient or orbit space is also a metric space. We study the Vietoris-Rips and Čech complexes of . Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to z…
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.
problem Proving contractibility of Vietoris-Rips complexes for .
method Used Bestvina-Brady discrete Morse theory to provide a short and improved proof.
result Contractible Vietoris-Rips complexes at large scales for .
We introduce deep scale-spaces (DSS), a generalization of convolutional neural networks, exploiting the scale symmetry structure of conventional image recognition tasks. Put plainly, the class of an image is invariant to the scale at which it is viewed. We construct scale equivariant cross-correlations based on a princ…
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.
problem Understanding properties of sections in metric spaces.
method Definition and investigation of intrinsically quasi-isometric sections in metric spaces.
result Properties of sections, including Ahlfors-David regularity and convexity, are defined and investigated.
Theory of ends of spaces using linear algebra.
problem Understanding ends of spaces at infinity.
method Developing a theory using scale, sub-Boolean algebras, and linear algebra.
result All known types of ends are special cases of a linear algebraic process.
Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop …
The paper (in French) exemplifies graphically a solution of the heat equation which is a 1-dimensional unfolding of an elliptic umbilic catastrophe. The example is due to James Damon and adapts Thom-Mather's singularity theory to multiscale models of scale-space analysis in image processing.
Using ideas of the Dowker duality we prove that the Rips complex at scale is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. Thes…
Paper solves robust multi-dimensional scaling with accelerated projections.
problem Localize point locations from noisy pairwise distances.
method Alternating projections with tangent space acceleration.
result Linear convergence of reconstructed points to original points.
In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
Non-Asymptotic Analysis of Tangent Space Perturbationphysics.data-an
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as…
SageMath package diffstrata calculates intersection theory on abelian differentials.
problem Computing intersection theory on the boundary of strata of abelian differentials.
method Explicit combinatorial description of the boundary, implemented algorithms in SageMath.
result Computes the Euler characteristic of strata using intersection theory.
Stochastic Covariant Derivatives in a (Curved) Space-Time: a Glimpse into the Fractoid Spacesmath.PR
Study on stochastic covariant derivatives in curved space-time.
problem Analyzing covariant derivatives in curved space-time under stochastic processes.
method Using Itô-Wiener processes and stochastic calculus, including Besov spaces, Schrödinger operators, and white noise.
result Developed a framework for stochastic geodesics and white noise in fractoid spaces.
The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spacesmath.MG
Develops a new framework for large-scale geometry.
problem Characterizing large-scale models of metric spaces.
method Categorical framework for metric Rips filtration and universal quasigeodesic cones.
result Establishes universal properties and adjointness of the Rips colimit.
The moduli space of isometry classes of Riemannian structures on a smooth manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum gravity. A natural question concerning it is: What is a natural topology on such moduli space that reflects best quantum fluctuations of the geometries within the Planc…
This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…
Theory predicts neural scaling exponents from language statistics.
problem No existing theory could quantitatively predict neural scaling exponents.
method Isolated two key statistical properties of language.
result Derives a simple formula predicting neural scaling exponents.
We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…
We investigate the local geometry on the moduli space of G_2 structures that arises in compactifications of M-theory on holonomy G_2 manifolds. In particular, we determine the homogeneity properties of couplings of the associated N=1, D=4 supergravity under the scaling of moduli space coordinates. We then find some bra…
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …
The study explains transformer scaling laws using statistical and approximation theories.
problem Understanding why transformer scaling laws exist for large models trained on low-dimensional data.
method Established statistical estimation and mathematical approximation theories for transformers on low-dimensional manifolds.
result Predicted a power law between generalization error and model and data sizes, with power depending on intrinsic data dimension.
The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.
Gravity derived from thermodynamics via optimal transport.
problem Equivalence between gravity and thermodynamics.
method Linking optimal transport to Raychaudhuri equation in warped-product spacetimes.
result Equivalence between gravity and concavity of entropy under time evolution.
These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another, approximated on all scales by well behaved spaces, typically just Euclidean space its…
New method selects diffusion scales for graph wavelets.
problem Choosing optimal diffusion scales for graph wavelets.
method Proposes an unsupervised method using information theory.
result Method selects diffusion scales for graph wavelets.
Develops a new method for neural network significance testing without strict constraints.
problem Testing neural networks without bounded weights or specific architectural constraints.
method Uses Rademacher complexity bounds, weakened Sobolev space membership conditions, and a modified sieve space construction.
result Achieves optimal convergence rates and valid asymptotic distributions for test statistics.
Two types of differentials are shown equivalent for compactifying moduli spaces.
problem Compactifying moduli spaces of curves with prescribed orders of zeros and poles.
method Equivalence of multi-scale and logarithmic differentials, isomorphism of moduli stacks, explicit blowups.
result Multi-scale and logarithmic differentials are equivalent and isomorphic.
Simplifies deep learning scaling analysis without sacrificing accuracy.
problem Interpreting feature learning mechanisms and determining network implicit bias in high-dimensional settings.
method Developed a heuristic approach for predicting data and width scales of feature learning patterns.
result Predictions align with known results and extend to complex architectures.
Field theory for optimal signal propagation in ResNetscond-mat.dis-nn
Field theory explains optimal scaling in ResNets for signal propagation.
problem Understanding optimal scaling parameter for ResNet performance.
method Finite-size field theory for ResNets to study signal propagation and scaling.
result Analytical expressions for optimal scaling parameter, independent of other hyperparameters.
Neural networks' performance scales with data size, explained by data manifold dimensionality.
problem Understanding the scaling of neural network performance with the number of parameters.
method Explained by the intrinsic dimension of the data manifold, confirmed through teacher/student framework and various datasets.
result The scaling exponent α is approximately 4 divided by the intrinsic dimension d of the data manifold.
We generalize Penrose's notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Li…
Kernel-based methods improve policy evaluation in MRP models.
problem Estimating value functions in infinite-horizon discounted MRP models.
method Kernel-based temporal difference methods using reproducing kernel Hilbert spaces.
result Optimal error bounds derived for the kernel-based LSTD estimate.
In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, …
The study analyzes a three-layer neural network's training dynamics using a functional-space mean-field theory.
problem Understanding the training dynamics of partially-trained three-layer neural networks.
method Generalized mean-field theory to functional spaces, proving convergence and feature learning.
result The training loss of the model decays to zero at a linear rate in the regression setting.
Extends Bayesian theory to handle complex interdependencies in multidimensional event spaces.
problem Complex interdependencies between events and hypotheses sets in real-world systems.
method Developed a mathematical formalism for modeling complex relationships through rigorous derivation and validated using analytical proofs, simulations, and case studies.
result MDSE theory improves prediction accuracy by 15-20% compared to standard Bayesian methods in high interdimensionality datasets.
Study presents MMC model for better fitting multiple choice data.
problem Improving accuracy of latent trait estimates in IRT models.
method Fit autoencoders to MMC model, demonstrating better fit than nominal response model.
result MMC model outperforms traditional IRT models in fit.
New neural scaling law found for simple quadratic function.
problem Neural scaling laws and their predictions for model performance.
method Analysis of neural networks, lottery ticket ensembling, statistical interpretation.
result Found a new scaling law () for a simple quadratic function, contradicting previous theories.
Knot probabilities in equilateral random polygonscond-mat.stat-mech
Study reveals a universal formula for knotting in random equilateral polygons.
problem Probability of knotting in equilateral random polygons.
method Extensive Monte Carlo simulations with improved algorithms and knot invariants.
result A universal scaling formula for knotting probability with number of edges, involving exponential and power law factors.
EKH adds metrics to knot theory, enabling more detailed analysis.
problem Lack of quantitative data in knot theory.
method Integrates metric into knot theory with evolutionary Khovanov homology (EKH).
result EKH reveals non-trivial knot invariants at appropriate scales.
New scaling framework for MoE architectures ensures stability and optimal performance at scale.
problem Lack of principled understanding of how hyperparameters should scale in MoE architectures.
method Developed a novel Dynamical Mean Field Theory (DMFT) for three scaling regimes of MoE architectures.
result Derived Maximally Scale-Stable Parameterization (MSSP) for SGD and Adam, providing robust learning rate transfer and monotonic improvement with scale.
Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.
problem Spectral number variance convergence for twisted Laplacians and Dirac operators.
method Extends Rudnick's approach to Gaussian ensembles for twisted Laplacians and Dirac operators.
result Convergence to Gaussian ensembles for twisted Laplacians and Dirac operators.
Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic s…
We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nont…
We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime geometry. An inertial observer, usually described by a geodesic in spacetime, becomes instead a choice of ways to reverse the conformal compactification of a Euclidean vector…