A simple pooling technique prevents posterior collapse in sequence VAEs.
problem Posterior collapse in sequence VAEs, causing models to ignore latent variables.
method Proposed a pooling technique to prevent posterior collapse.
result Significantly better data log-likelihood achieved compared to standard sequence VAEs.
Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.
problem Characterizing the Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature.
method Analysis of the Gromov-Hausdorff limit space of orthonormal frame bundles equipped with an almost canonical metric.
result The singular set of the limit space has codimension ≥4 and the complement contains an open and dense C1,α-Riemannian manifold. A triangulation of a 3-manifold can be shown to be homeomorphic to the 3-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a …
The paper proves a fibration theorem for collapsing sequences of Alexandrov spaces.
problem Understanding the structure of collapsing sequences of Alexandrov spaces.
method Constructing almost Lipschitz submersions and proving locally trivial fibrations.
result A locally trivial fibration is established under certain conditions on volumes and singularity weakness.
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
problem Prove that collapsing constant scalar curvature metrics can be perturbed to invariant collapsing constant scalar curvature metrics.
method Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
result Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
This paper examines how skip connections prevent rank collapse in sequence models.
problem Rank collapse in sequence models, leading to reduced expressivity and training instabilities.
method Analytical and ablation studies of lambda-skip connections in SSMs.
result A sufficient condition to prevent rank collapse across various architectures.
The paper improves collapsing Alexandrov spaces results using good coverings.
problem Collapsing Alexandrov spaces with no proper extremal subsets.
method Application of good coverings of Alexandrov spaces.
result Construction of an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups.
We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, u(K). In particular, for knots with u(K)<3, we show that the Lee spectral sequence must collapse at the E_2 page. An immediate corollary is that the Knight Move Conjecture is true when u(K)<3.
Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.
problem Understanding how the first Betti number behaves under manifold collapse with Ricci curvature bounds.
method Analyzing sequences of Riemannian manifolds with lower Ricci curvature bounds.
result The first Betti number cannot drop more than the dimension in collapsing manifolds.
Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild samplin…
The paper studies how spaces collapse to Alexandrov spaces with mild singularities.
problem Understanding how Riemannian manifolds collapse to Alexandrov spaces with isolated singularities.
method Analyzes the structure of locally trivial fibrations over compact Alexandrov spaces.
result Proves that collapsing sequences of Riemannian manifolds admit locally trivial fibrations over the limit space.
Researchers describe a spectral sequence for knots in 3D space.
problem Understanding the Sinha spectral sequence for knots in R^3.
method Explicit description using Fox Neuwirth chain complexes and multicomplex structure.
result A non-trivial third page differential found, contradicting the rational case.
We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.
Proves torus sequences can't collapse to intervals under curvature bounds.
problem Proving torus sequences can't collapse to intervals under curvature bounds.
method Contradiction proof using Yamaguchi fibration theorem and covers.
result Proves tori can't collapse to intervals under curvature constraints.
Special Lagrangian submanifolds emerge from K3 surface collapse.
problem Understanding special Lagrangian submanifolds in K3 surface collapse.
method Lifting affine lines to degenerating sequences of special Lagrangian submanifolds.
result Constructing special Lagrangian two-spheres connecting Taub-NUT bubbles.
We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…
The purpose of this paper is to compare two spectral sequences converging to the cohomology of a configuration space. The collapsing of these spectral sequences is established, in some cases, using Massey products.
We consider the geometric inverse problem of determining a closed Riemannian manifold from measurements of the heat kernel in an open subset of the manifold. In this paper we analyze the stability of this problem in the class of n-dimensional Riemannian manifolds with bounded diameter and sectional curvature. It is w…
Study shows convergence of Yang-Mills connections on K3 surfaces under fiber collapse.
problem Analyzing Yang-Mills connections on K3 surfaces as fibers collapse.
method Proves convergence of a sequence of anti-self-dual connections under specific conditions.
result Connections converge to a limiting flat connection with holomorphic structure.
A new mechanism for GANs improves text generation by evaluating sub-sequences.
problem Exposure bias and mode collapse in GANs for text generation.
method Segmenting the sequence into sub-sequences and evaluating them individually.
result Significant improvement in text generation models on benchmark data.
Levenshtein VAE prevents posterior collapse in text generation models.
problem Posterior collapse in VAEs where generators ignore latent variables.
method Replaces ELBO with a Levenshtein distance-based objective to prevent collapse.
result Levenshtein VAE produces more informative latent representations.
Let X→P1 be an elliptically fibered K3 surface, admitting a sequence ωi of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to ωi. Given the corresponding sequence Ξi of Hermitian-Yang-Mills connections on V, we prove …
This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (Mi of pointed hyperbolic cone-manifolds with topological type (M,Σ), where M is a closed, orientab…
Gravitational instantons collapse to a punctured plane with a special Kahler metric.
problem The collapse of gravitational instantons from a complex structure limit.
method Analysis of a sequence of ALH*-gravitational instantons and their collapse to a punctured plane.
result The moduli space of pointed ALH*-gravitational instantons collapses to a punctured plane with a special Kahler metric.
In this paper, we show that Generative Adversarial Networks (GANs) suffer from catastrophic forgetting even when they are trained to approximate a single target distribution. We show that GAN training is a continual learning problem in which the sequence of changing model distributions is the sequence of tasks to the d…
A new method improves molecule generation accuracy and efficiency.
problem Posterior collapse in VAEs for molecule sequence generation.
method Re-balancing reconstruction loss to avoid posterior collapse.
result Our method achieves state-of-the-art reconstruction accuracy and competitive validity.
Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.
problem Understanding the fundamental groups of aspherical manifolds under certain curvature conditions.
method Analyzing the collapsing behavior of manifolds with bounded Ricci curvature and diameter.
result The fundamental groups of such manifolds have non-trivial finitely generated abelian normal subgroups.
CF-VAE models capture multi-modal distributions for better structured sequence prediction.
problem Challenges in capturing multi-modality of future states in latent variable models.
method Conditional Flow Variational Autoencoders (CF-VAE) with conditional normalizing flows.
result CF-VAE achieves state-of-the-art results on multi-modal structured sequence prediction datasets.
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with …
Study shows dimension constraints for isometry groups in non-collapsed Riemannian manifolds.
problem Dimension constraints for isometry groups in non-collapsed Riemannian manifolds.
method Equivariant Gromov--Hausdorff convergence and lower Ricci curvature bounds.
result Dimension of isometry group is at least the limit superior of dimensions of subgroups.
New data accumulation prevents model collapse in generative models.
problem Model collapse in generative models trained on their own outputs.
method Empirical study of language models, diffusion models, and variational autoencoders; analytically tractable framework for linear models.
result Accumulating synthetic data alongside real data avoids model collapse, preventing performance degradation.
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
problem Characterizing limits of inradius collapsed manifolds with lower Ricci curvature bounds.
method Analyzing families of manifolds with specific curvature and boundary conditions, proving properties of the limits.
result Limits of inradius collapsed manifolds have at most two boundary components and a lower Ricci curvature bound.
Deep learning predicts nuclear equation of state from rotating core collapse GW signals.
problem Classifying the nuclear equation of state from rotating core collapse gravitational wave signals.
method Employed deep convolutional neural networks to classify visual and temporal patterns in GW signals.
result Up to 97% correct classifications of nuclear equation of state in the test set.
Given a positive function u∈W1,n, we define its John-Nirenberg radius at point x to be the supreme of the radius such that ∫Bt∣∇logu∣n<ε0n when n>2, and ∫Bt∣∇u∣2<ε02 when n=2. We will show that for a collapsing sequence in a fixed conformal class under some curvature c…
We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alter…
We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral …
Study rigidifies torus bundles under first Betti number constraints.
problem Understanding the structure of torus fibrations under first Betti number restrictions.
method Established rigidity results and necessary/sufficient conditions for topological splitting.
result Classification of torus bundles under specific Betti number constraints.
The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.
problem The study of limits of sub-Riemannian Heisenberg manifolds and their properties.
method Analysis of Gromov-Hausdorff limits with upper and lower bounds on diameter and Popp's measure.
result The non-collapsed limits of sub-Riemannian Heisenberg manifolds converge to a compact Heisenberg manifold.
This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (M_{i},p_{i}) of pointed hyperbolic cone-manifolds with topological type (M,Σ), where M is a closed, orientable and irr…
This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not nec…
New analysis shows how attention masks and LayerNorm prevent rank collapse in transformers.
problem Rank collapse in transformer models with increasing depth.
method General analysis of rank collapse under self-attention, considering attention masks and LayerNorm.
result Self-attention with LayerNorm can prevent rank collapse and maintain a rich set of equilibria.
We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed t…
Compactness theory for biharmonic maps on degenerating Einstein manifolds.
problem Analyzing biharmonic maps on degenerating Einstein manifolds.
method Developed a compactness theory using asymptotic analysis over degenerating neck regions.
result Established a compactness theory for biharmonic maps with finitely many bubbles.
Study diverging sequences of unit volume metrics with bounded curvature on homogeneous spaces.
problem Understanding 1-parameter families of invariant metrics with bounded curvature.
method Analyzing diverging sequences in the space of G-invariant, unit volume metrics on compact homogeneous spaces. result Prove structure results for diverging sequences with bounded curvature.
Khovanov homology for pro-tangles and spectral sequences
problem Developing a framework for Khovanov homology for pro-tangles and spectral sequences
method Using pro-tangles, simplicial presheaves, and spectral sequences
result Establishing a fully faithful embedding and an algebraic spectral sequence for pro-tangles