New algorithm solves Schrödinger bridge problem with mismatched channels.
problem Solving Schrödinger bridge problem with input and noise channel mismatch.
method Design of a Sinkhorn recursion with memory for nonlinear PDEs.
result Demonstrates solving control-affine Schrödinger bridge problem.
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
New algorithm solves complex mean-field Schrödinger bridge problem.
problem Designing a controller for diffusion processes with nonlocal interaction.
method Generalized Hopf-Cole transform and Sinkhorn-type algorithm.
result Convergence guarantees for the proposed algorithm under mild assumptions.
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.
Transformers improve Finnish language modeling, achieving lower perplexity scores.
problem Improving language modeling for Finnish using deep learning models.
method Used BERT and Transformer-XL models in a sub-word setting, compared to LSTM.
result Transformer-XL outperforms LSTM, achieving a 27% better perplexity score.
Geometric transformations on null curves in AdS induce KdV solutions.
problem Transforming null curves in AdS to solve KdV equations.
method Geometric transformations that induce Bäcklund transformations for KdV.
result Satisfies permutability theorem for null curves with constant bending.
Classifies conformal transformations in spacetimes without observer horizons.
problem Understanding conformal transformations in spacetimes without observer horizons.
method Proves classification of conformal transformations into two types: escaping and non-escaping.
result Conformal transformations of Einstein's static universe are classified.
Transformers can learn optimal variable selection in group-sparse classification.
problem Understanding how transformers leverage attention to select relevant variables in group-sparse classification.
method Training a one-layer transformer using gradient descent to select variables from one group of input variables.
result A one-layer transformer can correctly leverage the attention mechanism to select variables, disregarding irrelevant ones.
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
The paper explores geometric aspects of Miura transformations in integrable systems.
problem Relating different integrable equations and classifying bi-Hamiltonian structures.
method Construction of generalized Miura transformations under algebraic and geometric settings.
result Miura transformations relate integrable curve flows in different geometries and induce moving frame transitions.
Develops support theorem for analytic transforms in tomography.
problem Analytic wave front set resolution for integral transforms.
method Microlocal analysis, double fibration framework, wave packet transforms.
result Uniqueness and support theorems for analytic transforms.
We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the …