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.
Study of collapsed manifolds with bounded Ricci curvature and non-collapsed universal cover.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Ricci flow techniques applied to non-collapsed universal cover.
result Partial extension of nilpotent structural results to global Ricci bounded covering geometry.
SR-GANs combat mode collapse in GANs by monitoring and compensating spectral distributions.
problem Mode collapse in GANs.
method Spectral regularization (SR-GANs) to combat spectral collapse.
result SR-GANs prevent mode collapse and outperform SN-GANs in experiments.
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.
Study flat manifolds and their collapse using Teichmüller theory.
problem Understanding the collapse of flat manifolds and orbifolds.
method Algebraic description of Teichmüller and moduli spaces, study of boundaries.
result Every closed flat orbifold can be obtained by collapsing closed flat manifolds, and collapsed limits of 3-manifolds are classified.
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.
Proving NP-completeness of (d,k)-collapsibility for d≥k+2 except (2,0).
problem Determining if a d-dimensional simplicial complex can be collapsed to a k-dimensional subcomplex. method Extending previous works to show NP-completeness for d≥k+2. result NP-completeness of (d,k)-collapsibility for d≥k+2 except (2,0). 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.
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.
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
The study finds bounds on Ricci curvature for possibly collapsed spaces.
problem Bounding Ricci curvature in collapsed spaces.
method Lower bound on Ricci curvature, volume collapse consideration.
result Existence of bounds on local Lp norm of Ricci curvature. 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.
New differential operator helps characterize non-collapsed RCD spaces.
problem Characterizing non-collapsed compact RCD spaces.
method Explicit formula of Laplacian associated with pull-back Riemannian metric via heat kernel.
result Confirms conjecture on implication from weakly non-collapsed to non-collapsed condition.
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…
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 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.
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.
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 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 compares two GAN architectures to reduce mode collapse.
problem Reducing mode collapse in Generative Adversarial Networks (GANs).
method Explains and compares PacGAN and VEEGAN models.
result PacGAN performs slightly better than vanilla GAN in terms of mode collapse.
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 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. Researchers use gluing method to describe collapsing Calabi-Yau metrics on K3 fibred 3-folds.
problem Describing collapsing Calabi-Yau metrics on K3 fibred 3-folds.
method Gluing method
result Refined description of collapsing Calabi-Yau metrics.
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.
3D spaces collapse to 8 Thurston geometries.
problem Characterizing collapsed 3D Alexandrov spaces.
method Proving Alexandrov spaces modeled on Thurston geometries.
result Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric.
Develops an efficient approximation for collapsed Gibbs sampling in complex models.
problem Intractability of integrating out variables in collapsed Gibbs sampling for complex models.
method Uses expectation propagation to approximate collapsed Gibbs integrals.
result Approximate sampler enables a runtime-accuracy tradeoff in sampling complex models.
PacGAN tackles mode collapse in GANs by using multiple samples.
problem GANs produce samples with little diversity, known as mode collapse.
method PacGAN modifies the discriminator to make decisions based on multiple samples.
result Packing naturally penalizes generators with mode collapse, favoring less mode collapse.
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…
Deep and narrow neural networks can collapse to incorrect states.
problem The collapse of deep and narrow neural networks to incorrect states.
method Numerical and theoretical analysis of deep and narrow neural networks with ReLU activation.
result Deep and narrow neural networks can converge to erroneous mean or median states with high probability.
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.
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.
Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
problem Understanding the collapsibility of 2-complexes with specific curvature properties.
method Analyzing the fundamental groups and sectional curvatures of 2-complexes.
result Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
Study identifies topologies of 3D spaces with boundary.
problem Understanding the topologies of compact Alexandrov spaces with boundary.
method Continuation of previous work, determining topologies through analysis.
result Identified topologies of collapsing 3D Alexandrov spaces with boundary.
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
New metric space concept and quasi-isometry properties explored.
problem Exploring new metric spaces and quasi-isometry properties.
method Introducing (b,c)-metric and defining collapsing maps.
result Collapsing maps preserve quasi-isometry properties.