Study on rotating surfaces in 4D space with matrices.
problem Understanding rotational surfaces in pseudo-Euclidean 4-space.
method Defined hyperbolic and elliptic rotational surfaces using curves and matrices in 4D semi-Euclidean space.
result Generated rotated surfaces using specific rotation matrices.
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation mat…
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we…
The paper learns pose variations within shape populations using constrained mixtures of factor analyzers.
problem Learning pose variations within a shape population with articulated parts and relative rotations.
method Formulated as mixtures of factor analyzers, segmentation by component posterior probabilities, and constraints on factor loading matrices for rotation matrices.
result Automatic learning of pose variations from shape populations, resulting in smooth and realistic animations.
RotRNN uses rotations to simplify long sequence modelling.
problem Complex initialisation and normalisation schemes in linear recurrent models.
method RotRNN employs rotation matrices to simplify and normalise linear recurrent models.
result RotRNN achieves competitive performance on long sequence modelling datasets.
Generating point clouds, e.g., molecular structures, in arbitrary rotations, translations, and enumerations remains a challenging task. Meanwhile, neural networks utilizing symmetry invariant layers have been shown to be able to optimize their training objective in a data-efficient way. In this spirit, we present an ar…
Innovates rotation index for matrix pairs, solving group action problems.
problem Solving group actions problems, especially Nielsen realization and higher-rank Anosov actions.
method Rotation index and Milnor--Munkres--Novikov pairing applied to Z2 group actions. result Solved specific group action problems using new matrix pair invariant.
Non-orthogonal joint diagonalization (NJD) free of prewhitening has been widely studied in the context of blind source separation (BSS) and array signal processing, etc. However, NJD is used to retrieve the jointly diagonalizable structure for a single set of target matrices which are mostly formulized with a single da…
We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the …
Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.
problem Classifying conformally flat hypersurfaces and characterizing rotational hypersurfaces.
method Derives a sharp inequality relating eigenvalues of trace-free matrices and applies it to hypersurfaces.
result New proof of the classification of conformally flat hypersurfaces and construction of a functional for rotational hypersurfaces.
Study optimizes estimation of orthogonal and rotation matrices from noisy data.
problem Estimating orthogonal and rotation matrices from noisy data.
method Iterative polar decomposition algorithm initialized by spectral methods.
result Algorithm achieves optimal error rate of $(1+o(1)) rac{σ^2 d(d-1)}{2np}$.
We study the problem of approximating orthogonal matrices so that their application is numerically fast and yet accurate. We find an approximation by solving an optimization problem over a set of structured matrices, that we call extended orthogonal Givens transformations, including Givens rotations as a special case. …
We investigate the problem of estimating a given real symmetric signal matrix C from a noisy observation matrix M in the limit of large dimension. We consider the case where the noisy measurement M comes either from an arbitrary additive or multiplicative rotational invariant perturbati…
Study connects covariance cleaning theory to information theory for heavy-tailed distributions.
problem Optimizing covariance matrices for heavy-tailed distributions using information theory.
method Minimizing Frobenius norm and information loss between true and estimated covariance matrices.
result Asymptotic regime of large matrices minimizes information loss for Student's t distributions.
BOOOM optimizes orthonormal matrices without needing gradients.
problem Optimizing over the Stiefel manifold in non-convex, non-smooth settings.
method Global Givens rotation-based parametrization and Recursive Modified Pattern Search.
result BOOOM achieves strong performance across various optimization problems.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Simplifies denoising score matching for manifold learning.
problem Learning distributions on manifolds is computationally intensive.
method Modifies denoising score matching to implicitly account for the manifold.
result Reduces computational burden while maintaining efficiency.
DFRot improves LLMs by reducing outlier and massive activation effects.
problem Reducing outlier and massive activation effects in rotated LLMs.
method Weighted loss function and orthogonal Procrustes transforms for rotation matrix refinement.
result DFRot achieves dual free (Outlier-Free and Massive Activation-Free) with significant improvements in perplexity.
The concepts of unitary evolution matrices and associative memory have boosted the field of Recurrent Neural Networks (RNN) to state-of-the-art performance in a variety of sequential tasks. However, RNN still have a limited capacity to manipulate long-term memory. To bypass this weakness the most successful application…
This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densi…
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
LOCUS separates brain network connectivity matrices efficiently.
problem High dimensionality, latent sources, and spurious findings in analyzing brain connectivity matrices.
method LOCUS: low-rank structure with uniform sparsity, iterative Node-Rotation algorithm.
result LOCUS achieves more efficient and accurate source separation for connectivity matrices.
Predict missing movie ratings or graph embeddings with low rank matrices.
problem Predicting missing entries in a ratings matrix or graph embeddings with known linear relations.
method Low rank matrix completion approach applied to graph embeddings.
result Effective methods for predicting missing entries in matrices and graph embeddings.
Paper proposes a new deflation varimax method for vintage factor analysis.
problem Finding a scientifically meaningful low-dimensional representation of data.
method Deflation varimax procedure for orthogonal matrix rotation.
result The proposed method achieves minimax optimal factor loading estimation.
Recent machine learning methods make it possible to model potential energy of atomic configurations with chemical-level accuracy (as calculated from ab-initio calculations) and at speeds suitable for molecular dynam- ics simulation. Best performance is achieved when the known physical constraints are encoded in the mac…
Model tracks structural changes in Brownian particle configurations on a sphere.
problem Tracking structural changes in Brownian particle configurations on a sphere.
method Introduces Frustrated Distance Matrix (FDM) model for dynamic distance matrices on S^2.
result Preserves static BBS template with dynamics as redistributed spectral mass.
We introduce SARR for symmetric object pose estimation, improving CNN performance.
problem Ambiguities in symmetric object orientations hinder deep learning pose estimation.
method Numeric rotation representation using symmetry-derived trigonometric identities.
result SARR enables standard CNNs to achieve state-of-the-art performance.
A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…
OMD monitors stock market dynamics through matrix trajectories and reveals crisis patterns.
problem Understanding and predicting stock market crises and sector rotations.
method Applying OMD to S\&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with distinct sector leadership.
OMD monitors stock market dynamics through matrix trajectories, revealing crisis patterns and sector rotations.
problem Understanding and predicting stock market dynamics during crises.
method Applying OMD to S&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with sector-specific patterns and volatility clustering.
Unified framework analyzes and compares RFF and RoPE PEs for music generation.
problem Efficiently modeling music generation with positional encodings.
method Kernel methods to analyze and compare RFF and RoPE PEs.
result RoPEPool outperforms other methods in melody harmonization.
A new transform links rotating calorons to solutions of a differential equation.
problem Existence and characterization of rotating calorons.
method Formulated a Nahm transform to relate rotating calorons to solutions of a delayed-differential equation.
result Existence of an eight-parameter family of rotating calorons with nontrivial holonomy.
We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…
New design method improves Lasso performance in sparse regression.
problem Sparse linear regression with correlated design columns.
method Introduces partially-rotated designs to improve Lasso's RE constant.
result Lasso achieves better prediction error with high probability.
New approach links 2D fluid dynamics to matrix theory.
problem Understanding swirling patterns in 2D fluids.
method Matrix hydrodynamics linking 2D fluid dynamics to matrix theory.
result Established connections between 2D hydrodynamics and matrix Lie theory.
The paper defines and analyzes homotopic rotation sets for surfaces of higher genus.
problem Defining and analyzing homotopic rotation sets for surfaces of higher genus.
method Developed a definition and proved several results using the theory of Le Calvez and Tal.
result Found that the homotopic rotation set can imply the existence of infinitely many periodic orbits under certain conditions.
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli m…
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Enhanced rotation prediction improves SSL models by capturing both shape and texture information.
problem Rotation prediction misses texture information, limiting model performance.
method Introduces image enhanced rotation prediction (IE-Rot) that combines rotation and image enhancement tasks.
result IE-Rot models outperform Rotation on various benchmarks.
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pau…
General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…
The paper introduces REQNNs for robust 3D point cloud processing.
problem 3D point cloud processing robustness to rotations.
method Revised neural networks using quaternion features for rotation-equivariance.
result REQNNs exhibit higher rotation robustness compared to original networks.
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
RotEqNet preserves rotation symmetry in fluid systems using high-order tensors.
problem Lack of rotational symmetry in machine learning models for fluid systems.
method Introduces RotEqNet, a network that guarantees rotation-equivariance for high-order tensors.
result RotEqNet reduces errors and maintains rotation-equivariance in fluid systems.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.
Convolutional networks are successful due to their equivariance/invariance under translations. However, rotatable data such as images, volumes, shapes, or point clouds require processing with equivariance/invariance under rotations in cases where the rotational orientation of the coordinate system does not affect the m…