MSA compares neural representations' intrinsic geometry for better understanding.
problem Existing similarity measures fail to capture subtle distinctions between neural network solutions.
method Metric similarity analysis (MSA) using Riemannian geometry.
result MSA can disentangle features of neural computations and compare nonlinear dynamics.
The paper explores log-aesthetic curves under similarity geometry and their relation to Euler's elasticae.
problem Characterizing log-aesthetic curves and their relation to Euler's elasticae.
method The approach involves similarity geometry, variational formulation, and solving the Burgers equation.
result Log-aesthetic curves and their generalization are the similarity geometric analogue of Euler's elasticae.
We characterize the contractions that are similar to the backward shift in the Hardy space H2. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.
New method uses Riemannian geometry to quantify molecular shapes.
problem Quantifying molecular similarity for drug discovery.
method Riemannian geometry and Kähler quantization (KQMolSA).
result KQMolSA method compares well to existing shape similarity methods.
Hyperbolic geometry reveals financial network structure and systemic importance.
problem Understanding the structure and importance of financial networks.
method Data from European banking stress tests, hyperbolic geometry analysis.
result Latent dimensions of `popularity' and `similarity' are strongly associated with systemic importance.
The study explores how to infer the geometry of space forms from similarity comparisons.
problem Inferring the geometry of space forms from unreliable similarity measurements.
method Introducing ordinal capacity and spread, proving their relation to space form properties, and using statistical analysis of similarity measurements.
result The statistical behavior of ordinal spread variables can identify the underlying space form.
Fried's theorem proven for symmetric space boundaries.
problem Characterizing manifolds with similarity structures.
method General proof for all rank one symmetric space boundary geometries.
result Closed manifolds are either complete or develop onto Heisenberg-type spaces.
The paper explores similarities in even and odd-dimensional geometry.
problem Understanding the Lagrangian Grassmannian in cosymplectic geometry.
method Study of compatible co-complex structures, Moser's trick, and Weinstein 1-form derivation.
result The de Rham class of the Weinstein 1-form is a co-flux.
Explains differences and similarities of strictly nef and ample vector bundles.
problem Characterizing geometry of projective manifolds with strictly nef bundles.
method Brief exposition on strictly nef and ample vector bundles.
result Differences and similarities between strictly nef and ample vector bundles.
Lecture notes introduce contact complete integrability for odd-dimensional manifolds.
problem Integrability on odd-dimensional manifolds using contact geometry.
method Introduce contact geometry concepts, discuss contact Hamiltonian vector field, Jacobi bracket, and contact complete integrability.
result Two different notions of contact complete integrability coincide.
The paper introduces log-aesthetic curves and their integrable discretization.
problem Characterizing and discretizing log-aesthetic curves.
method Similarity geometry, integrable Burgers equation, variational principles.
result Proposed variational principle and discretization preserving integrable structure.
We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manif…
Geometries and dual field theories linked by AdS/CFT.
problem Understanding the AdS/CFT correspondence.
method Geometric extremization principles informed by physical considerations.
result Key role of Sasaki-Einstein and GK geometry.
New method uses Riemannian geometry to describe molecular shapes.
problem Predicting drug-like molecules using shape similarity.
method Riemannian geometry applied to molecular surfaces.
result RGMolSA method captures molecular shape effectively.
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
Study of pulleys and gears in spherical and hyperbolic geometries.
problem Understanding mechanical systems in non-Euclidean spaces.
method Analysis of pulley and gear systems in spherical and hyperbolic geometries.
result Similar laws governing movement in non-Euclidean geometries.
Geometric mechanism mimics physics' symmetry breaking.
problem Understanding spontaneous symmetry breaking in geometry.
method Analogous to physics, studying symmetry breaking in differential geometry.
result Symmetry breaking can be used to solve geometric problems.
A framework for natural gradient with arbitrary similarity measures.
problem Unclear metric for natural gradient in non-Euclidean spaces.
method Derive a metric for natural gradient given an arbitrary similarity measure.
result General framework for natural gradient in non-Euclidean spaces.
New exponential map for Lie groups connects to sub-Riemannian geometry.
problem Developing a new exponential map for Lie groups.
method Introducing a new exponential map related to sub-Riemannian geometry.
result New exponential map connects to sub-Riemannian geometry.
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geo…
Training-free source selection for LLM families with shared vocabularies
problem Source selection for LLM families with shared vocabularies
method Fisher alignment at vocabulary scale
result Fisher alignment is a cosine between kernel mean embeddings in the joint activation-error space
Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …
Physics: Similar long-distance properties can mask vastly different short-distance metrics.
problem Classifying homogeneous metrics on group manifolds by long-distance properties.
method Apply universality concept to geometry, focusing on metrics on Lie groups.
result Many metrics on low-dimensional Lie groups have similar long-distance properties despite differing short-distance properties.
Improved VAEs learn flat latent spaces for better data similarity.
problem Measuring data similarity in latent spaces using Euclidean metric.
method Extend VAEs to learn flat latent manifolds using Riemannian geometry and regularisation.
result Improved performance on video-tracking benchmarks, nears supervised methods.
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on s…
The study analyzes neural network predictions of knot invariants and finds that braid representations work best.
problem Understanding and predicting knot invariants using neural networks.
method Investigated different knot representations and invariants, proposed a cosine similarity score.
result Braid representations are best for predicting knot invariants, and some invariants are easier to learn than others.
This paper demostrates a method for analysing almost CR geometries (H,J), by uniquley defining a partially integrable structure (H,K) from the same data. Thus two almost CR geometries (H,J) and (H′,J′) are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries (H,K) …
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
We use the information metric to investigate the moduli space of a U(1) instanton on (anti)self-dual manifolds, finding an AdS geometry similar to that for the moduli space of a Yang-Mills instanton on flat space. We discuss our results from the perspective of gauge/gravity duality.
Paper proves inequality linking capillary surfaces to Finsler geometry.
problem Proving a Heintze-Karcher inequality for capillary hypersurfaces.
method Introduced a Finsler metric for geodesic flow and studied the hypersurface properties.
result Established a new inequality relating capillary surfaces to Finsler geometry.
Proposes a new phylogenetic tree space with biologically principled geometry.
problem Developing a space for statistical analysis of phylogenies with biologically informed assumptions.
method Introduces wald space, a new phylogenetic tree space, and two related geometries based on Fisher information and Gaussian processes.
result Geodesics in wald space are similar to those in the Fisher information geometry, but the two geometries are distinct.
We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…
Warren's metric makes C2 flat, with known geodesics and volumes.
problem Describing the geometry of C2 with a specific metric. method Defined a Kähler metric on C2 and showed its flatness. result Warren's metric makes C2 a flat manifold. The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is gen…
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures…
Introduces a new G2-Hilbert functional in G2-geometry.
problem None explicitly stated; focuses on introducing a new functional.
method Inspired by the Einstein-Hilbert functional, defines a new G2-Hilbert functional on G2-structures. result Torsion-free and nearly G2-structures are saddle critical points of the volume-normalized G2-Hilbert functional. We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane.
The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …
New measures link neural representation geometry to decoding ability.
problem Understanding how neural representations relate to decoding ability.
method Showed that popular similarity measures can be interpreted from a decoding perspective.
result Proved that measures like CKA and CCA quantify alignment between optimal linear readouts.
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also stu…
The study examines nilpotent similarity structures on manifolds and their properties.
problem Characterizing closed manifolds with nilpotent similarity structures.
method Generalizes convexity arguments to geodesic segments in nilpotent Lie groups.
result Closed manifolds with nilpotent similarity structures are either complete or radiant.
In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 ge…
This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the m…
The paper shows how to use hyperplanes and hyperballs interchangeably using inversive geometry.
problem Tackles the interchangeability of hyperplanes and hyperballs in discriminative boundaries.
method Applies inversive geometry to transform Euclidean data into spherical data and back, providing explicit formulae.
result Shows a duality between hyperspherical caps and hyperballs, providing explicit formulae to map between them.
Visualizes functions on hyperbolic geometry surfaces.
problem Visualizing functions on hyperbolic geometry surfaces.
method Reinvented phase plotting for hyperbolic geometry using conformal maps.
result Illustrates direct motions on geodesics.
The seven and nine dimensional geometries associated with certain classes of supersymmetric AdS3 and AdS2 solutions of type IIB and D=11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by intr…