arXiv research

New bounds show transformers need longer training for length generalization.

Task hinting improves transformer performance on longer tasks.

Constructs non-isometric iso-length-spectral surfaces.

ReLU networks don't exponentially distort curve lengths as previously thought.

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

Study finds numerical moduli in special 2-flags of length 5.

Randomized positional encodings boost transformer performance on longer sequences.

We show that certain families of iso-length spectral hyperbolic surfaces obtained via the Sunada construction are not generally simple iso-length spectral.

Study shows Transformers can generalize to varying task lengths.

New theorem shows metrics of certain groups are close if their lengths are identical.

New proof shows surfaces can have identical length spectra but not simple ones.

Method calculates systolic length of modular curves.

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…

Generalizes Toponogov theorem to Alexandrov spaces.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

This research proves guarantees on sequence models' generalization to longer and novel sequences.

Convex curves evolve into circles over time.

Paper trains a Transformer to add numbers of any length.

New proof shows Fuchsian groups have irrational length spectra.

Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equ…

Sharp proof of sub-Riemannian length-minimizing curves being at least $C^2$

Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on the surface, the length of the geodesic in one class is equal to the length of the geodesic in the other class. We show that there are elements in th…

In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our…

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from be…

Study approximate marked length spectrum rigidity in non-positively curved groups.

The paper offers generalization bounds for Transformers that ignore sequence length.

We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves' lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We genera…

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surfa…

A simple method reduces bias in LLM auto-evaluators by controlling output length.

In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to…

Transformers learn chain-of-thought reasoning for longer problems, proving length generalization.

Study mixed commutator lengths in wreath products and their relation to general ranks.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …

The paper studies translation lengths on sphere complexes and related cones.

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This prope…

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…

Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.

Let $Σ$ be a surface of negative Euler characteristic and $S$ a generating set for $π_1(Σ,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $π_1(Σ,p)$ is given by its intersection number with a well-chosen collection of curves an…

Introduces a new length functional for Ricci flow to detect steady solitons.

This paper introduces a new method for model selection and more generally hyperparameter selection in machine learning. Minimum description length (MDL) is an established method for model selection, which is however not directly aimed at minimizing generalization error, which is often the primary goal in machine learni…

This paper proves a conjecture about trisections with a specific length.

In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length o…