Study equiangular surfaces in 3D, extending plane spirals.
problem Understanding 3D surfaces with constant normal-vector angles.
method Investigates three-dimensional extensions of equiangular spirals.
result Identifies self-similar structures in sea shell geometry.
An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to π/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperboli…
Graphs with maximum degree Δ have at most O(1) equiangular lines for λ < 3/sqrt(2).
problem Finding the maximum number of equiangular lines in graphs with a given maximum degree.
method Using eigenfunctions and nodal domains to estimate the multiplicity of eigenvalues.
result The maximum multiplicity of λ as the second largest eigenvalue is O(1) for graphs with maximum degree Δ and cyclomatic number.
We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration spa…
This paper interprets Dropout as creating an equiangular tight frame in autoencoders.
problem Understanding the success of Dropout in deep learning.
method Connecting Dropout to analog channel coding and equiangular tight frames (ETF).
result Optimizing autoencoders with dropout leads to an equiangular tight frame structure.
The paper proves regularity for minimal surfaces near polyhedral cones.
problem Understanding the regularity of minimal surfaces near polyhedral cones.
method Adapting Simon's method and establishing C1,α-regularity for minimal varifolds. result Proves C1,α-regularity for minimal varifolds near polyhedral cones. We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
Six quaternionic lines with optimal angles found in 2D quaternion space.
problem Finding optimal configurations of quaternionic lines in 2D space.
method Simple presentation of lines as orbit of a reflection group, finding other optimal designs.
result Optimal spherical designs of 10, 15, and 20 lines in quaternion space.
New hyperbolic polyhedra with π/3 angles and volumes calculated.
problem Finding new hyperbolic polyhedra with specific dihedral angles.
method Constructed a new sequence of hyperbolic polyhedra with π/3 angles and determined their volumes. result Volumes of some constructed polyhedra determined.
A twisted curve in Euclidean 3-space E^3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E^3 and characterize such curves in terms of their curvature functions. Further, we obtain some re…
New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.
problem Investigate geometric properties of kth Order Preserving Sets and ovals. method Introduce and analyze kth Order Preserving Sets and Midpoint Sets; study geometric properties and isoperimetric inequalities. result Established an isoperimetric-type inequality relating perimeter and area of ovals and their associated sets.
Let C be a smooth, convex curve on either the sphere S2, the hyperbolic plane H2 or the Euclidean plane E2, with the following property: there exists α, and parameterizations x(t),y(t) of C such that for each t, the angle between the chord connecting x(t) to y(t)…
Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training …
We analyze deep neural networks using convex duality to reveal hidden layer structures.
problem Understanding the structure of deep neural networks.
method Introducing a convex analytic framework to characterize hidden layer weights.
result Optimal hidden layer weights align with previous layers via duality.
Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.
problem Understanding and optimizing deep learning training phases.
method Direct measurements on three deepnet architectures across seven datasets.
result Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.
Language models allocate information storage, not collapsing into uniform representations.
problem Incomplete neural collapse in language model representations.
method Analyzing variance and information sharing across 14 models, proving an information floor.
result Within-class variance is allocated information storage, not collapsed into uniform representations.
Redundancy helps speed up slow nodes in distributed learning.
problem Slow nodes (stragglers) bottleneck distributed optimization and learning performance.
method Encode data with redundancy, dynamically exclude stragglers, and compensate losses.
result Optimization algorithms converge to solutions even with straggling nodes.
A new loss function HUG decouples and generalizes neural collapse.
problem Neural collapse limits in deep learning models.
method Hyperspherical uniformity gap (HUG) as a unified framework.
result HUG decouples and generalizes neural collapse, improving model flexibility and robustness.
This paper studies how to compress neural networks while maintaining accuracy.
problem Compressing a two-layer neural network with fewer nodes without losing accuracy.
method Using tools from high-dimensional probability, the authors minimize the L_2 loss between the target and compressed networks.
result The error rate of the approximation is shown as a function of input dimension and network size in the mean-field limit.
This work justifies neural collapse under MSE loss and analyzes the optimization landscape.
problem Understanding neural collapse in deep neural networks under MSE loss.
method Global landscape analysis of vanilla nonconvex MSE loss.
result The only global minimizers are neural collapse solutions.
New model explains neural collapse and limits on minority classes in imbalanced datasets.
problem Understanding and predicting performance limits of deep learning models on imbalanced datasets.
method Layer-Peeled Model, a nonconvex optimization program isolating top layers and applying constraints.
result Reveals a new phenomenon called Minority Collapse that limits deep learning models on minority classes.
Deep linear networks exhibit collapsing features and classifiers across datasets.
problem Understanding the collapse of features and classifiers in deep linear networks.
method Theoretical and empirical analysis of deep linear networks with MSE and CE losses.
result Deep linear networks exhibit NC properties, collapsing features and classifiers to orthogonal vectors.
Study on Vietoris-Rips complexes of regular polygons, revealing complex homotopy types.
problem Understanding the homotopy types and persistent homology of Vietoris-Rips complexes of regular polygons.
method Use of persistent homology, cyclic graphs, and winding fractions.
result Characterization of homotopy types and persistent homology of Vietoris-Rips complexes of Pn up to a scale parameter. This paper extends neural collapse to class-imbalanced datasets using an unconstrained ReLU feature model.
problem Understanding neural collapse in class-imbalanced datasets with cross-entropy loss.
method Generalized neural collapse to class-imbalanced settings using an unconstrained ReLU feature model.
result Class-means converge to orthogonal vectors with different lengths, and classifier weights align to these vectors.
Affine λ-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.
problem Reconstruction and area estimates for affine λ-equidistants of convex polygons with parallel opposite sides. method Using Wigner caustics and centre symmetry sets.
result Proving a discrete version of the improved isoperimetric inequality.
Pipeline combines ETF preprocessing with tabular model for cross-modal inference.
problem Transferability of tabular models across different modalities.
method Fixed comparison object, ETF preprocessing, in-context inference.
result Pipeline is broadly competitive, runs faster, and produces well-calibrated probabilities.
This paper investigates how large language models achieve neural collapse, a phenomenon linked to generalization.
problem Neural collapse in large language models under imbalanced and token-rich conditions.
method Empirical investigation of scaling and regularization effects on CLMs' progression towards neural collapse.
result Neural collapse properties develop with scale and regularization, linked to generalization in language modeling.
We analyze neural collapse in neural networks, showing that features collapse to vertices of a Simplex ETF.
problem Understanding and optimizing the features learned in the last layer of neural networks during training.
method Simplified unconstrained feature model, studying the global optimization landscape of cross-entropy loss with weight decay.
result The global minimizers of the loss are Simplex ETFs, and other critical points are strict saddles with negative curvature.
Deep nets trained with MSE loss exhibit Neural Collapse, collapsing features and classifiers to class means.
problem Understanding Neural Collapse in MSE-trained deep nets.
method Developed a new MSE loss decomposition and introduced the central path concept.
result Exact dynamics of Neural Collapse along the central path can be predicted.
The paper uncovers symmetries in large language models through layer-peeled optimization.
problem Understanding geometric structure in large language model weights and context embeddings.
method Constrained layer-peeled optimization program to analyze symmetries in next-token distributions.
result Symmetries in target next-token distributions are transferred to optimal model weights and context embeddings.
This paper extends neural collapse to imbalanced data under cross-entropy loss.
problem Analyzing neural collapse in deep networks with imbalanced data.
method Using the unconstrained feature model and cross-entropy loss, the paper studies neural collapse in imbalanced datasets.
result Feature vectors within the same class collapse to a single mean vector, but angles between them depend on sample size.
This work explains neural collapse in shallow neural networks and its impact on generalization.
problem Understanding neural collapse in shallow neural networks and its effect on generalization.
method Analysis of two and three-layer ReLU neural networks, focusing on data dimension, sample size, and signal-to-noise ratio.
result Neural collapse occurs in shallow ReLU networks under certain conditions related to data properties and network architecture.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
The study characterizes polynomial conserved quantities for Lie applicable surfaces.
problem Characterizing polynomial conserved quantities for Lie applicable surfaces.
method Gauge theoretic approach for Lie applicable surfaces, including isothermic, Guichard, and L-isothermic surfaces. result Induced transformations of Lie applicable surfaces for well-known transformations and new Bäcklund-type transformation for linear Weingarten surfaces.
Study on focal surfaces of tubular surfaces in 3D space, focusing on their flatness and asymptotic properties.
problem Characterizing and understanding focal surfaces of tubular surfaces in 3D space.
method Defined tubular surfaces using Frenet and Darboux frames, analyzed their focal surfaces, and derived conditions for flatness.
result No minimal focal surface exists in 3D space for tubular surfaces.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
Study on dual surfaces of rotational minimal and maximal in Euclidean and Lorentz-Minkowski spaces.
problem Investigating the duality between minimal and maximal surfaces in different spaces.
method Analysis of rotational surfaces and use of one-parameter group of rotations.
result Family of Bonnet minimal and maximal surfaces emerge in the duality process.
Stable-ribbon surface-links have unique handle-irreducible summands.
problem Characterize handle-irreducible summands of stable-ribbon surface-links.
method Combining old results on stably trivial surface-links and surface-knots with infinite cyclic fundamental groups.
result Every handle-irreducible summand of a stably trivial surface-link is a trivial 2-link.