Ancient solutions to Ricci flow on torus bundles have additional symmetries.
problem Understanding collapsed ancient solutions to the Ricci flow on compact manifolds.
method Algebraic and tameness assumptions on collapsing directions to prove additional torus symmetries.
result Ancient solutions to the Ricci flow on torus bundles converge to an Einstein metric on the base.
The paper proves extremal black holes form at a critical point of gravitational collapse.
problem Formation of extremal black holes in gravitational collapse.
method Constructing smooth families of spherically symmetric solutions to the Einstein-Maxwell-Vlasov system.
result Extremal Reissner-Nordström black holes form at the critical collapse threshold.
Contrastive learning struggles with class collapse and feature suppression, revealing bias towards simpler solutions.
problem Contrastive learning struggles with class collapse and feature suppression, especially in supervised and unsupervised settings.
method Unified theoretical framework to determine which features are learnt by CL, revealing bias towards simpler solutions.
result Bias towards simpler solutions is a key factor in class collapse and feature suppression.
New metrics prevent event collapse in contrast maximization frameworks.
problem Event collapse in contrast maximization frameworks.
method First principles of space-time deformation based on differential geometry and physics.
result Proposed metrics mitigate event collapse and do not harm well-posed warps.
REPAIR mitigates variance collapse to enable linear interpolation between SGD solutions.
problem Linear interpolation between SGD solutions is difficult due to variance collapse in permuted activations.
method REPAIR rescales preactivations of interpolated networks to mitigate variance collapse.
result 60%-100% relative barrier reduction across various architectures and tasks.
The study characterizes and rules out collapsing in convex ancient mean curvature flow.
problem Characterizing and ruling out collapsing in convex ancient mean curvature flow.
method Characterization and counterexamples.
result Collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane.
In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow (n≥2) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetr…
New findings show DNC is not optimal for deep models, revealing a low-rank bias.
problem Theoretical limitations of DNC in non-linear models and multi-class classification.
method Analysis of non-linear models of arbitrary depth in multi-class classification.
result DNC stops being optimal for DUFM when going beyond two layers or two classes, due to a low-rank bias.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
problem Understanding the fundamental groups of ancient Ricci flows.
method Analyzing the structure of ancient Ricci flows and their tangent flows.
result The fundamental group of non-collapsed ancient Ricci flows is finite and a quotient of the regular part's fundamental group.
The paper explores embedding Ricci flow solutions in flag manifolds.
problem Realizing Ricci flow solutions as embedded submanifolds.
method Investigation of invariant metrics in flag manifolds, proving global attractors and non-realizable collapses.
result Certain Ricci flow collapses cannot be embedded in Euclidean spaces.
New collapsing mechanism for G2-manifolds discovered.
problem Understanding collapsing behavior of G2-manifolds.
method Adiabatic description involving weighted maximal submanifold equation; formal power series solutions.
result Existence of formal power series solutions and heuristic discussion of compactification.
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.
This paper examines how different loss functions affect neural network features and performance.
problem Investigating which loss function is best for deep neural networks.
method Examining last-layer features of deep networks and drawing inspiration from the Neural Collapse phenomenon.
result All relevant loss functions (CE, LS, FL, MSE) produce equivalent features and similar performance.
We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.
In this paper it is proven that the volume entropy of a riemannian metric evolving by the Ricci flow, if does not collapse, nondecreases. Therefore, it provides a sufficient condition for a solution to collapse. Then, for the limit solutions of type I or III, the limit entropy is the limit of the entropy as t approac…
We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $…
The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …
Despite excellent progress in recent years, mode collapse remains a major unsolved problem in generative adversarial networks (GANs).In this paper, we present spectral regularization for GANs (SR-GANs), a new and robust method for combating the mode collapse problem in GANs. Theoretical analysis shows that the optimal …
This work explores how feature decorrelation improves self-supervised learning.
problem Complete and dimensional collapse issues in self-supervised learning.
method Study of a concise framework and connection to feature decorrelation.
result Feature decorrelation improves self-supervised learning.
New principles for collapsing law-invariant functionals to means, extending beyond convexity.
problem Conditions for law-invariant functionals to reduce to means.
method Establishing collapse to the mean principles for non-convex functionals.
result General principles apply beyond convexity, including quasiconvex and Choquet integrals.
This study analyzes why attention layers in neural networks can cause signal loss and proposes a solution.
problem Pathological behavior of attention layers in neural networks, leading to signal loss.
method Spectral analysis using Random Matrix Theory to identify and mitigate rank collapse in width.
result A novel solution to mitigate rank collapse in width by removing outlier eigenvalues.
Study on Ricci flow of invariant metrics on spheres, finding new ancient solutions.
problem Analyzing the Ricci flow of Sp(n+1)-invariant metrics on spheres.
method Determine forward and ancient solutions, classify them, and classify their behavior under flow.
result Exhibit a new one-parameter family of ancient solutions on spheres with larger isometry groups.
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first…
The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.
problem Understanding the breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation.
method Analyzing the pseudospherical nature of local and non-local formulations of the Degasperis-Procesi equation.
result Solutions to Cauchy problems with non-trivial initial data define an orthonormal coframe for pseudospherical metrics.
It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the f…
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…
A method to prevent image representation collapse through data-dependent augmentation.
problem Representation collapse due to image augmentations that damage information.
method Formalizing a stochastic encoding process with a tug-of-war between corruption and preserved information, using infoMax objective.
result Learning a data-dependent distribution of augmentations to avoid representation collapse.
Theoretical work on mode collapse in variational inference models.
problem Mode collapse in variational inference models, where models focus on a few modes instead of all possible ones.
method Theoretical investigation of mode collapse in Gaussian mixture models, identifying key low-dimensional statistics and equations governing their evolution.
result Mode collapse is present even in favorable scenarios, driven by mean alignment and vanishing weight mechanisms.
RL enhances LLM planning but introduces spurious solutions and diversity collapse.
problem Theoretical understanding of RL's benefits and limitations in LLM planning.
method Graph-based abstraction, policy gradient, Q-learning, supervised fine-tuning.
result RL's exploration is crucial for generalization, but PG suffers from diversity collapse.
A fast geometric regularizer improves event camera performance.
problem Event collapse in contrast maximization framework.
method Geometric regularizer to mitigate overfitting.
result State-of-the-art accuracy with reduced computational complexity.
This paper presents a general solution for a recent model by Keen for endogenous money creation. The solution provides an analytic framework that explains all significant dynamical features of Keen's model and their parametric dependence, including an exact result for both the period and subsidence rate of the Great Mo…
New findings suggest non-contrastive learning has many bad minima, not just collapsed ones.
problem The effectiveness of non-contrastive learning in unsupervised feature learning.
method Theoretical analysis and controlled experiments on simple data models.
result Non-contrastive losses have a preponderance of non-collapsed bad minima, and these minima are not avoided during training.
Wide neural networks with weight decay exhibit neural collapse.
problem Proving neural collapse in wide neural networks trained with weight decay.
method Generic guarantees on neural collapse for wide networks with weight decay, proving low training error and balancedness, and bounded conditioning.
result First proof of neural collapse in end-to-end training of wide neural networks with weight decay.
UCPO improves diversity in reinforcement learning models, maintaining high accuracy.
problem RLVR objectives often lead to diversity collapse, reducing coverage of correct solutions.
method UCPO adds a conditional uniformity penalty to GRPO, redistributing probability mass.
result UCPO improves Pass@K and diversity while maintaining competitive Pass@1 accuracy.
We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as t→−∞ and we give their precise asymptotic description. This description applies …
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.
New families of Ricci solitons found with collapsing volume.
problem Finding new Ricci solitons with specific volume behavior.
method Reduced soliton equation to Monge-Ampère equation coupled with ODEs.
result Explicit complete expanding solitons and existence results for other types.
The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.
problem Understanding the singularities and behavior of the Kähler-Ricci flow.
method Li-Yau type and Harnack estimates for weighted Ricci potential functions.
result Finite time singularities are shown to sub-converge to ancient solutions on analytic normal varieties.
We study the mean curvature flow of hypersurfaces in Rn+1, with initial surfaces sufficiently close to the standard n-dimensional sphere. The closeness is in the Sobolev norm with the index greater than 2n+1 and therefore it does not impose restrictions of the mean curvature of the initial surface. W…
This study investigates abrupt learning dynamics in Transformers, revealing plateau formation and internal representation collapse.
problem Abrupt learning in Transformers, particularly during the loss plateau.
method Investigates mechanisms of abrupt learning in shallow Transformers, focusing on attention maps and hidden states.
result Reveals plateau formation, internal representation collapse, and strong repetition bias in outputs.
Feature normalization prevents collapse in non-contrastive learning dynamics.
problem Non-contrastive learning can collapse into a single point due to lack of repulsive force.
method Extended previous theory based on L2 loss to cosine loss, considering feature normalization.
result Cosine loss induces stable equilibrium, preventing collapse even with insufficient repulsive force.
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.
Neural collapse occurs in normalized features over a Riemannian manifold.
problem Understanding neural collapse in normalized feature models.
method Simplified multi-class classification task to a nonconvex optimization problem over the Riemannian manifold, analyzing the landscape of critical points.
result The only global minimizers are neural collapse solutions, with all other critical points being strict saddles.
Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cyli…
We construct a compact, convex ancient solution of mean curvature flow in Rn+1 with O(1)×O(n) symmetry that lies in a slab of width π. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, O(n)-invariant ancient solution that lies …
New method prevents posterior collapse in iVAE models.
problem Posterior collapse in iVAE models where observations and ICs are independent given covariates.
method Developed CI-iVAE by considering a mixture of encoder and posterior distributions in the objective function.
result Prevents posterior collapse, resulting in latent representations with more information of the observations.
Ancient solutions found on flag manifolds from invariant Einstein metrics.
problem Understanding the behavior of Ricci flow on flag manifolds.
method Global study of the dynamical system induced by the Ricci flow, using invariant Einstein metrics and Poincaré compactification.
result Non-collapsed ancient solutions emerge from invariant Einstein metrics, with a Type I singularity in finite time.