Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the grou…
A new method studies symplectic configurations in rational 4-manifolds using computer-aided techniques.
problem Understanding symplectic configurations in rational 4-manifolds.
method Computer-aided approach combining Cremona transformations and pseudoholomorphic curves.
result Nonexistence of Fano planes in the symplectic category.
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show …
Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of Kähler groups on infinite dimensional re…
Study length functions on various groups and prove homomorphisms to finite groups.
problem Understanding length functions on different types of groups.
method Analyzing length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups, and Cremona groups.
result Prove that homomorphisms to certain groups must have finite images.
Elliptic curve governs Hopf linking in symmetric tensegrity.
problem Understanding Hopf linking in A4-symmetric tensegrity.
method Parametrized by points on the elliptic curve 30a2.
result Only one rational point corresponds to a stable configuration.
I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the r…
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancel…
Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis o…
Study extends geodesic ray transform results to orientable surfaces.
problem Characterize and stabilize mixed and transverse ray transforms on surfaces.
method Algebraic arguments applied to various geometries and ray transforms.
result Characterization of kernel and stability for mixed and transverse ray transforms on orientable surfaces.
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a …
Introduces pseudo-codecomposition of transformation groups.
problem Understanding and categorizing transformation groups.
method Introduces pseudo-codecomposition and analyzes properties of transformation groups.
result The class of pseudo-codecomposable transformation groups is a proper intermediate class.
This paper investigates efficient Transformers and finds they scale with problem size.
problem Finding suitable replacements for standard Transformers in large-scale tasks.
method Modeling efficient Transformers (Sparse and Linear) as Dynamic Programming problems and analyzing their reasoning capabilities.
result Efficient Transformers scale with problem size, but can be more efficient for certain DP problems.
The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the know…
Transformation Equivariant Representations (TERs) aim to capture the intrinsic visual structures that equivary to various transformations by expanding the notion of {\em translation} equivariance underlying the success of Convolutional Neural Networks (CNNs). For this purpose, we present both deterministic AutoEncoding…
Data is said to follow the transform (or analysis) sparsity model if it becomes sparse when acted on by a linear operator called a sparsifying transform. Several algorithms have been designed to learn such a transform directly from data, and data-adaptive sparsifying transforms have demonstrated excellent performance i…
Transforms classical connections using pushforwards and gauge transformations.
problem Transforming classical connections in categorical settings.
method Constructing pushforwards and applying gauge transformations to decorated path spaces.
result Combines traditional gauge transformation with affine translation.
Transformers interpret as probabilistic mixtures, offering new insights.
problem Understanding Transformers from a probabilistic perspective.
method Modeling Transformers as mixtures of Gaussian models.
result Transformers can be seen as maximum posterior probability estimators.
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.
Transformer-MGK replaces redundant heads with Gaussian key mixtures, improving efficiency and performance.
problem Redundant attention heads in transformers degrade performance and efficiency.
method Transformer-MGK replaces redundant heads with a mixture of Gaussian keys.
result Transformer-MGK accelerates training and inference, reduces parameters and FLOPs, and achieves comparable or better accuracy.
B-cos transformers explain Vision Transformers' decisions.
problem Lack of holistic explanations for transformer outputs.
method Formulate each component as dynamic linear, allowing a single linear transform for summarization.
result Bcos-ViTs are highly interpretable and competitive on ImageNet.
Novel power transform unifies various mathematical functions.
problem Normalizing and standardizing datasets.
method Presented a novel power transform.
result Unified various mathematical functions.
Algorithm finds optimal affine transformation to minimize overall distortion.
problem Minimizing distortion in affine transformations.
method Riemannian geometry approach to define and minimize distortion.
result Mean distorting transformation found for minimizing overall distortion.
XR-Transformer accelerates XMC by recursively fine-tuning on multi-resolution objectives.
problem Efficiently classifying texts with large label sets.
method Recursive multi-resolution fine-tuning of transformers.
result XR-Transformer achieves 20x faster training time and 54% Precision@1 on Amazon-3M.
We study the dynamics of the discrete bicycle (Darboux, Backlund) transformation of polygons in n-dimensional Euclidean space. This transformation is a discretization of the continuous bicycle transformation, recently studied by Foote, Levi, and Tabachnikov. We prove that the respective monodromy is a Moebius transform…
Data augmentation (DA) is fundamental against overfitting in large convolutional neural networks, especially with a limited training dataset. In images, DA is usually based on heuristic transformations, like geometric or color transformations. Instead of using predefined transformations, our work learns data augmentati…
We propose a new class of transforms that we call {\it Lehmer Transform} which is motivated by the {\it Lehmer mean function}. The proposed {\it Lehmer transform} decomposes a function of a sample into their constituting statistical moments. Theoretical properties of the proposed transform are presented. This transform…
Transformers struggle to approximate smooth functions, relying on piecewise constant approximations.
problem Understanding the expressivity of Transformers for function approximation.
method Theoretical analysis and experimental validation of Transformer's ability to approximate smooth functions.
result Transformers cannot reliably approximate smooth functions, relying on piecewise constant approximations.
The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform …
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by usin…
One of the most challenging problems in the domain of 2-D image or 3-D shape is to handle the non-rigid deformation. From the perspective of transformation groups, the conformal transformation is a key part of the diffeomorphism. According to the Liouville Theorem, an important part of the conformal transformation is t…
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
New geometric transformations link discrete and continuous curve motions.
problem Establishing a connection between discrete and continuous curve motions.
method Infinitesimal Darboux transformations of smooth curves.
result Alternate geometric interpretation for semi-discrete mKdV equation.
SPTN uses invertible transformations to improve sum-product networks.
problem Improving inference efficiency and tractability in sum-product networks.
method Integrates invertible transformations into sum-product networks (SPNs).
result SPTNs with Gaussian leaves and affine transformations are as tractable as SPNs.
FEDformer combines Transformer with seasonal-trend decomposition for efficient long-term forecasting.
problem Transformer's inefficiency and inability to capture global time series views.
method Combines seasonal-trend decomposition with Transformer, exploiting Fourier basis for frequency enhancement.
result Reduces prediction error by 14.8% and 22.6% for multivariate and univariate time series, respectively.
One-layer transformers can't solve induction heads task efficiently.
problem Solving the induction heads task efficiently with one-layer transformers.
method Communication complexity argument showing exponential size requirement.
result No one-layer transformer can solve the induction heads task efficiently.
BoostTransformer uses boosting to improve transformer efficiency and accuracy.
problem Heavy computational resources and hyperparameter tuning in transformer architectures.
method Augments transformers with boosting principles through subgrid token selection and importance-weighted sampling, incorporating a least square boosting objective directly into the pipeline.
result BoostTransformer demonstrates faster convergence and higher accuracy compared to standard transformers.
We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutabili…
We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere, when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached all Darboux transforms converge to the original …
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…
W-Transformers use wavelets to improve time series forecasting.
problem Forecasting non-stationary time series with long-range dependencies.
method Wavelet-based transformer architecture.
result W-Transformers outperform baseline models on various time series datasets.
Paper proves Fourier transform for valuations, simplifying previous work.
problem Existence of isomorphism for translation-invariant smooth valuations.
method Directly describes Alesker's isomorphism in terms of Fourier transform on functions.
result Simple proofs of Alesker's Fourier transform properties, including a previously conjectured result.
Survey on broken ray transforms and open problems.
problem Understanding broken ray transforms and their applications.
method Survey of recent developments and open problems.
result Discussion of open problems in broken ray transforms.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.