Study tackles criterion collapse in learning criteria, showing conditions for loss minimization.
problem Criterion collapse in optimization, focusing on error probability minimizers.
method Analyzes various learning criteria, including DRO, OCE risks, and non-monotonic criteria.
result Non-monotonic criteria can avoid collapse, while monotonic ones cannot.
Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.
problem Characterize minimal volume entropy for aspherical simplicial complexes with these groups as fundamental groups.
method Algebraic and geometric characterization, using fiber π1-growth collapse and non-collapsing assumptions. result Provide bounds and criteria for minimal volume entropy in aspherical simplicial complexes.
I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…
Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smoo…
Vacuum gravity shows black holes can form without collapse.
problem Formation of black holes in vacuum gravity.
method Cauchy--double-null framework, Yau's boundary criterion.
result Black holes form from boundary effects in vacuum spacetime.
Proof confirms cosmic censorship for charged gravitational collapse.
problem Cosmic censorship for Einstein-Maxwell-Charged Scalar Field system.
method Systematic approach to incorporate charge and complex scalar field, new trapped surface criterion, modified BV area estimates, and new instability theorems.
result Cosmic censorship holds for charged gravitational collapse.
The paper extends Nakamaye's theorem to non-closed forms on complex manifolds.
problem Analyzing non-closed (1,1)-forms on compact complex manifolds. method Developed analytic technique by Collins and Tosatti to study non-Hermitian loci.
result Non-Hermitian locus equals union of positive-dimensional null subvarieties.
For gravitational collapse, we observe a correspondence between region close to past null infinity and region close to central singularity. In line with this philosophy, we construct a new ansatz, with which we first present a 40-page self-contained proof of trapped surface formation in a far-from-center region. A syst…
We study the Kähler-Ricci flow on a class of projective bundles P(OΣ⊕L) over compact Kähler-Einstein manifold Σn. Assuming the initial Kähler metric ω0 admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple (Σ,L,[ω0]), under which the $\mathbb{P…
Essential spectrum of differential forms on curved manifolds is connected.
problem Understanding the essential spectrum of differential forms on curved manifolds.
method Using Gromov-Hausdorff convergence and Weyl criterion, the authors show the essential spectrum is a connected interval.
result The essential spectrum of the Hodge Laplacian on differential forms is a connected interval over complete manifolds with vanishing curvature at infinity.
A criterion for training-free time-lagged spectral embeddings of multivariate time series
problem Applicability of fixed-length descriptors for multivariate time series
method Using a stationary Gaussian VAR(1) model and cosine similarity to classify descriptors
result D(τ) separates two classes when signals are approximately stationary and cross-channel temporal coupling is present
Bayesian deep learning faces posterior collapse due to likelihood vs. prior competition.
problem Posterior collapse in Bayesian deep learning models.
method Identified competition between likelihood and prior regularization in a linear latent variable model.
result Posterior collapse is related to neural and dimensional collapse, suggesting a broader learning issue.
Proves weakly non-collapsed RCD spaces are strongly non-collapsed.
problem Proving the equivalence of weakly non-collapsed and strongly non-collapsed RCD spaces.
method Analyzes properties of RCD spaces and uses auxiliary results.
result Confirms conjecture about RCD spaces being strongly non-collapsed.
Study on Neural Collapse limits in deep learning.
problem Understanding the limits of Neural Collapse in deep learning.
method Investigated Neural Collapse in the context of generalization and feature learning, refining conjectures and conducting experiments.
result Neural Collapse primarily occurs on the train set and not on the test set, suggesting it is an optimization phenomenon with unclear connections to generalization.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
LLMs generate answers under incomplete context, and their uncertainty should scale with missing information.
problem Evaluating the quality of LLM answers under incomplete context.
method A controlled framework with varying context availability, and two uncertainty measures (sampling-based confidence and response entropy) evaluated on SQuAD.
result Response entropy increases with context removal and explains more variance in accuracy than confidence, suggesting it is a more responsive uncertainty measure.
Mathematical analysis shows annealing prevents mode collapse in Gaussian mixtures.
problem Mode collapse in variational inference for multimodal distributions.
method Analyzed annealing strategies for Gaussian mixtures, derived formulas, and tested on neural networks.
result Appropriately chosen annealing schemes can robustly prevent mode collapse.
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.
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 …
New method controls posterior collapse in VAEs without network architecture constraints.
problem Posterior collapse in VAEs reduces diversity of generated samples.
method Introduces Latent Reconstruction (LR) loss to control posterior collapse.
result Controls posterior collapse on various datasets without architectural constraints.
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
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.
Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated group…
The torus cannot collapse to a segment under certain curvature conditions.
problem Impossibility of codimension-one collapse for surfaces of negative Euler characteristic.
method Analysis of Gaussian curvature and homology loops.
result The torus cannot collapse to a segment under similar conditions to surfaces of negative Euler characteristic.
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
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.
We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelma…
Estimate collapsibility of causal effects in CPDAGs via strong d-convex hulls.
problem Estimate causal effects in CPDAGs.
method Use strong d-convex hulls to characterize minimal collapsible sets.
result Efficient algorithm for obtaining collapsible sets in DAGs and CPDAGs.
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…
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.
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.
Study on collapsing Calabi-Yau manifolds and their metrics.
problem Understanding degenerations of Calabi-Yau manifolds with Ricci-flat Kahler metrics.
method Survey of recent developments, focusing on volume collapsing metrics.
result New insights into the behavior of Calabi-Yau manifolds under volume collapse.
This is an expositiry article on collapsing theory in Riemannian geometry written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applicat…
In this paper we extend the works of Tancer and of Malgouyres and Francés, showing that (d,k)-collapsibility is NP-complete for d≥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 study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds w…
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.
Survey on collapsing manifolds using group actions and foliations.
problem Collapsing manifolds with controlled curvature.
method Using group actions and singular Riemannian foliations.
result Recent extensions to singular Riemannian foliations.
Enhances Ricci flow theorem with scalar curvature bound.
problem Improving no-local-collapsing theorem of Ricci flow.
method Derives improved theorem under scalar curvature bound condition.
result Refines Perelman's no-local-collapsing theorem.
The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.
problem The study of collapsing non-positive immersion for 2-complex pairs and its implications.
method Introduced a relative version of collapsing non-positive immersion for 2-complex pairs (L,K) and proved a transitivity law under certain conditions. result Under certain conditions, a transitivity law holds: If (L,K) has relative collapsing non-positive immersion and K has collapsing non-positive immersion, then L has collapsing non-positive immersion. ContraNorm prevents dimensional collapse in GNNs and Transformers.
problem Dimensional collapse in Graph Neural Networks and Transformers.
method Proposes ContraNorm, a novel normalization layer inspired by contrastive learning.
result Proves ContraNorm alleviates both complete and dimensional collapse under certain conditions.
Topology of non-orientable spaces without boundary is studied.
problem Topology of non-collapsed RCD spaces without boundary.
method Studied the stability of non-orientability and topology under Gromov-Hausdorff convergence.
result Non-orientable spaces without boundary have a stable ramified double cover.
We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibratio…
Cryptos remained resilient after SVB's collapse, contrary to expectations.
problem Impact of SVB collapse on crypto markets.
method Factual summary, sentiment analysis, and market performance examination.
result Cryptocurrencies showed resilience after SVB's collapse.
Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.
problem Characterizing when a polyhedron can be continuously shrunk onto a subpolyhedron.
method Piecewise-linear free deformation retraction and metric considerations.
result A polyhedron collapses to a subpolyhedron if and only if it admits a free deformation retraction onto that subpolyhedron.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …
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.