New bounds show transformers need longer training for length generalization.
problem Understanding when transformers can generalize to longer inputs.
method Analyzing different settings of transformers, providing quantitative bounds.
result Transformers need training data longer than previously thought for length generalization.
Task hinting improves transformer performance on longer tasks.
problem Transformers struggle with length generalization for certain tasks.
method Simultaneously train on a related auxiliary task during main task training.
result Significant improvement in length generalization for sorting tasks.
Constructs non-isometric iso-length-spectral surfaces.
problem Creating non-isometric surfaces with identical geodesic lengths.
method Combining Sunada's construction with amalgams of hyperbolic surfaces.
result Found non-isometric surfaces with the same geodesic lengths.
ReLU networks don't exponentially distort curve lengths as previously thought.
problem Understanding how neural networks distort curve lengths with depth.
method Analyzing expected length distortion of ReLU networks with random initialization.
result Expected length distortion does not grow with depth, and shrinks slightly.
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
Study finds numerical moduli in special 2-flags of length 5.
problem Identifying numerical moduli in local classifications of special multi-flags.
method Analyzing distributions generating special multi-flags, focusing on lengths up to 5.
result Three numerical moduli appear in special 2-flags of length 5.
Randomized positional encodings boost transformer performance on longer sequences.
problem Transformers struggle with generalizing to sequences of arbitrary length.
method Introduced randomized positional encodings that simulate longer sequences and randomly select positions.
result Randomized positional encodings increase test accuracy by 12.0% on average for sequences of unseen length.
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.
problem Understanding when and how Transformers can generalize to different input lengths.
method Proposed a unifying framework and introduced the RASP-Generalization Conjecture.
result Transformers tend to length generalize on tasks if solvable by short RASP programs.
New theorem shows metrics of certain groups are close if their lengths are identical.
problem Identifying metrics of relatively hyperbolic groups from their lengths.
method Proved rigidity for relatively hyperbolic groups using coarse marked length spectrum.
result Metrics of relatively hyperbolic groups are uniformly close if lengths are identical.
New proof shows surfaces can have identical length spectra but not simple ones.
problem Identifying when two covers of a surface have identical length spectra but not simple ones.
method Characterized isomorphism of covers and constructed surfaces with identical spectra but different simple length spectra.
result Found surfaces with identical length spectra but not simple length isospectral covers.
Method calculates systolic length of modular curves.
problem Computing upper bounds on systolic length of Riemann surfaces.
method Using congruence subgroups of hyperbolic triangle groups and traces of generators.
result Systolic length grows logarithmically with genus.
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.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.
This research proves guarantees on sequence models' generalization to longer and novel sequences.
problem Generalization to longer sequences and novel token combinations in sequence models.
method Provable guarantees on length and compositional generalization for various sequence models.
result Limited capacity models achieve both length and compositional generalization with diverse training distributions.
Convex curves evolve into circles over time.
problem Deforming convex curves into circles.
method Generalized length-preserving flow for convex curves.
result Convex curves evolve into circles over time.
Paper trains a Transformer to add numbers of any length.
problem Training Transformers to handle arbitrary-length addition.
method Autoregressive generation from right to left.
result Trains a Transformer to generalize addition of numbers of any length.
New proof shows Fuchsian groups have irrational length spectra.
problem Irrationality of the length spectrum in Fuchsian groups.
method Elementary proof of linear independence of group elements' lengths.
result Non-elementary Fuchsian groups contain elements with linearly independent lengths over Q.
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 C2
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving C2 regularity result Theorem 1.1 in [6] is sharp
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.
problem Identifying surfaces by their geodesic lengths.
method Analyzes metrics on simple, thick negatively curved two-dimensional P-manifolds.
result Piecewise negatively curved Riemannian metrics on simple, thick two-dimensional P-manifolds are uniquely determined 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.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
The paper offers generalization bounds for Transformers that ignore sequence length.
problem Developing generalization bounds for Transformers that are independent of sequence length.
method Covering number approach to upper bound Rademacher complexity of bounded linear transformations.
result Theoretical bounds for Transformer generalization are independent of sequence length.
New inequality linking geodesic length and volume in complex projective plane.
problem Understanding geometric properties of complex projective plane.
method Combining recent results on area minimizers and geodesics with Kronheimer-Mrowka's proof.
result Proved a new inequality relating volume and length of geodesics.
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.
problem Understanding the relationship between metric and causal geometry in Lorentzian spaces.
method Constructing a Lorentzian length space with an orthogonal splitting on a product of an interval and a metric space, and using synthetic time-like Ricci curvature bounds.
result Established sufficient conditions for global hyperbolicity and formulated time-like Ricci curvature bounds without push-up and regularity assumptions.
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.
problem Bias in LLM auto-annotators, particularly length bias.
method Simple regression analysis to control for length difference and other mediators.
result Improved robustness and increased correlation with human preferences.
Transformers learn chain-of-thought reasoning for longer problems, proving length generalization.
problem Challenging problems require deeper reasoning, but how do models generalize this to longer tasks?
method Theoretical analysis of transformers on synthetic state-tracking tasks, proving length generalization through attention concentration.
result Transformers can learn chain-of-thought reasoning for longer problems, proving length generalization.
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…
Study mixed commutator lengths in wreath products and their relation to general ranks.
problem Understanding mixed commutator lengths in wreath products and their relation to general ranks.
method Analyzing wreath products (G,N)=(Z≀Γ,⨁ΓZ) and determining mixed commutator lengths in terms of general rank. result Mixed commutator lengths and ordinary commutator lengths coincide under certain conditions.
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
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.
problem Understanding the translation lengths of monodromies in fibered manifolds.
method Defined the generalized fibered cone and related cones, and proved their properties.
result Proved the generalized fibered cone is a rational slice of Fried's cone, providing bounds for asymptotic translation lengths.
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.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
Let Σ be a surface of negative Euler characteristic and S a generating set for π1(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 π1(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.
problem Detecting steady solitons in Ricci flow.
method Develops a modified length functional and shows it satisfies differential inequalities.
result The length functional generates a distance function that saturates on steady soliton manifolds.
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.
problem Proving a conjecture about trisections with a specific length.
method Examining trisections with Kirby-Thompson length 2 and proving the conjecture.
result Proves the conjecture about length 2 trisection being a 4-manifold with length 0.
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…