Ozeki and Takeuchi \cite[I]{OT} introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will…
A new method extends GLRAM for matrices, preserving feature relationships and reducing overfitting.
problem Overfitting and loss of feature relationships in matrix data.
method Generalized Low-Rank Approximation of Matrices (GLRAM) extension with multiple-pairs of transformations.
result The proposed method reduces overfitting and preserves feature relationships.
We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair (7,8) is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated…
In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-Münzner \cite{FKM}, referred to collectively as of OT-…
The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair {4,5},{6,9} or {7,8} in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypers…
A large class of trading strategies focus on opportunities offered by the yield curve. In particular, a set of yield curve trading strategies are based on the view that the yield curve mean-reverts. Based on these strategies' positive performance, a multiple pairs trading strategy on major currency pairs was implemente…
A new method uses preference relations to reconcile contradictory trading signals from multiple securities.
problem Difficulty in exploiting multiple pairs trading signals due to contradictions.
method Proposes a portfolio construction method based on preference relation graphs to reconcile contradictory signals.
result Portfolios based on preference relations exhibit robust returns even with high transaction costs and improve with more securities considered.
BLOCCS improves sparse CCA for better interpretation of multi-omics data.
problem Improving interpretation of multi-omics data.
method Block Sparse Canonical Correlation Analysis (BLOCCS) using a bi-convex objective and gradient descent.
result BLOCCS provides more interpretable solutions with improved orthogonality of sparse directions.
Efficiently predicts optimal transport plans using sliced potentials.
problem Predicting optimal transport plans across multiple measure pairs efficiently.
method Regression-based and objective-based amortization strategies using sliced optimal transport potentials.
result Efficient and accurate prediction of optimal transport plans for various tasks.
Annealing Double-Head calibrates deep neural networks during training.
problem Overestimation or underestimation of predictive confidence in deep neural networks.
method An additional calibration head and Annealing technique to dynamically scale logits.
result State-of-the-art model calibration performance achieved without post-processing.
A method for fast estimation of Wasserstein distances using sliced Wasserstein distances.
problem Efficiently computing Wasserstein distances for multiple pairs of distributions.
method Regression on sliced Wasserstein distances to predict true Wasserstein distances.
result The proposed method provides a better approximation of Wasserstein distance than state-of-the-art models, especially in low-data regimes.
EnScale learns to downscale climate models efficiently, capturing both spatial and temporal consistency.
problem Downscaling climate models from coarse to high-resolution data is computationally expensive and challenging.
method EnScale uses generative models and proper scoring rules to map GCM data to RCM data, reducing computational cost.
result EnScale achieves competitive performance and computational efficiency in downscaling multiple climate variables.
A novel graphical matching approach improves pairs trading by reducing portfolio variance and risk-adjusted returns.
problem Common pairs trading methods lead to high portfolio variance and low risk-adjusted returns due to focusing on highly cointegrated assets.
method Model all assets and their cointegration levels with a weighted graph. Select pairs as a maximum weighted matching to ensure no shared assets and lower portfolio variance.
result The matching-based strategy shows a significant improvement in risk-adjusted performance, with a gross Sharpe ratio of 1.23.
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.
The paper introduces models to learn generalized transformation equivariant representations.
problem Capturing intrinsic visual structures equivariant to various transformations.
method Deterministic and probabilistic AutoEncoding Transformations (AET and AVT) models trained to learn visual representations from generic groups of transformations.
result Generalized TERs (GTERs) that are equivariant to transformations in a more general fashion.
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 …
Integration procedure for Lie groupoid natural transformations.
problem Infinitesimal counterpart of natural transformations in Lie groupoids.
method Integration procedure for Lie groupoid morphisms.
result Provides smooth natural transformations between Lie groupoid morphisms.
Proposes differential and integral invariants under Mobius transformation.
problem Handling non-rigid deformation in 2-D and 3-D shapes.
method Focuses on Mobius transformation, proposes differential and integral invariants.
result Proposes differential and integral invariants under Mobius transformation.
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…
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…
Paper improves tensor completion using unitary transforms.
problem Robust tensor completion for various datasets.
method Transformed tensor SVD with unitary matrices.
result Recovered images have better PSNR than traditional methods.
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.
Paper introduces graph-based transforms for video compression.
problem Efficiently represent video signals for compression.
method Develops two techniques for designing graph-based transforms (GL-GBTs and EA-GBTs).
result Graph-based transforms outperform traditional KLT in video compression.
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.
Adversarial learning improves image augmentation for neural networks.
problem Improving data augmentation for neural networks with limited data.
method Adversarial learning using an encoder-decoder architecture with a spatial transformer network.
result Our approach outperforms previous generative data augmentation methods.
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.
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.
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.
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…
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.
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…
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…
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.
ETs improve model robustness to transformations in images.
problem Improving model robustness to predefined transformations.
method Equivariant Transformers (ETs) incorporating functions equivariant to continuous transformation groups.
result ETs achieve up to 15% relative improvement in error rate on image classification tasks.
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.
Proves upper bounds for Bäcklund transformations in hyperbolic systems.
problem Bounding the generality of Bäcklund transformations in hyperbolic systems.
method Cartan's Method of Equivalence, classification results for specific symmetry groups.
result Obtains classification results and new examples of auto-Bäcklund transformations.
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…