arXiv research

The study characterizes and rules out collapsing in convex ancient mean curvature flow.

New principles for collapsing law-invariant functionals to means, extending beyond convexity.

Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.

Soap films collapse only if their bulk has negative pressure, forming convex shapes.

We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains tr…

Deep nets trained with MSE loss exhibit Neural Collapse, collapsing features and classifiers to class means.

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore it does not impose restrictions of the mean curvature of the initial surface. W…

Bayesian deep learning faces posterior collapse due to likelihood vs. prior competition.

Sphere theorems for specific manifolds with curvature constraints.

MFVI mode collapse explained; RoVI proposed to mitigate.

This paper extends neural collapse to class-imbalanced datasets using an unconstrained ReLU feature model.

New bifurcation found in perturbations of non-generic closed self-shrinkers.

New method uses geometric mean to avoid non-collapsibility in case-control studies.

Deep linear networks exhibit collapsing features and classifiers across datasets.

The paper studies how certain submanifolds collapse to lower-dimensional ones.

New metrics improve scRNA-seq perturbation modeling by reducing mode collapse.

In this paper we extend the works of Tancer and of Malgouyres and Francés, showing that $(d,k)$-collapsibility is NP-complete for $d\geq k+2$ except $(2,0)$. By $(d,k)$-collapsibility we mean the following problem: determine whether a given $d$-dimensional simplicial complex can be collapsed to some $k$-dimensional sub…

In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontall…

Theoretical work on mode collapse in variational inference models.

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…

Study shows 'Ordinal Neural Collapse' in deep OR tasks, revealing simple geometric relationships.

This work justifies neural collapse under MSE loss and analyzes the optimization landscape.

This paper extends neural collapse to imbalanced data under cross-entropy loss.

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) 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…

In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.

Neural collapse occurs in normalized features over a Riemannian manifold.

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show th…

This paper examines how different loss functions affect neural network features and performance.

We present a general method for deriving collapsed variational inference algo- rithms for probabilistic models in the conjugate exponential family. Our method unifies many existing approaches to collapsed variational inference. Our collapsed variational inference leads to a new lower bound on the marginal likelihood. W…

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the…

Variational autoencoders often collapse, showing latent variables are non-identifiable.

Deep Gaussian Processes with polynomial kernels can collapse rapidly without proper hyperparameter tuning.

We study theoretical and empirical aspects of the mean exit time of financial time series. The theoretical modeling is done within the framework of continuous time random walk. We empirically verify that the mean exit time follows a quadratic scaling law and it has associated a pre-factor which is specific to the analy…

Our research proves neural collapse in deep ResNets and transformers is globally optimal.

Wide neural networks with weight decay exhibit neural collapse.

New findings on how certain functionals behave in random variable spaces.

Formula connects curvature to volume in special geometric spaces.

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

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 prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in $\mathbb{C}^m$ that evolve by this reparametrized …

The paper explains how continuous language models can produce discrete, interpretable meanings.

For regular particle filter algorithm or Sequential Monte Carlo (SMC) methods, the initial weights are traditionally dependent on the proposed distribution, the posterior distribution at the current timestamp in the sampled sequence, and the target is the posterior distribution of the previous timestamp. This is techni…

New data accumulation prevents model collapse in generative models.

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametr…

Group factor analysis (GFA) methods have been widely used to infer the common structure and the group-specific signals from multiple related datasets in various fields including systems biology and neuroimaging. To date, most available GFA models require Gibbs sampling or slice sampling to perform inference, which prev…

Study shows translators can have non-removable singularities at infinity but eventually converge to unique planes.

Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$", with respect to initial data. We employ an Angenent-like neck-pinching argument to…