Study shows LLC correlates with neural network compressibility.
problem Evaluating limits of neural network compression.
method Extended minimum description length principle using singular learning theory.
result Complexity estimates based on LLC are linearly correlated with compressibility.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.
Advances variational Bayesian neural networks using singular learning theory.
problem Discrepancies between predictive performance and variational objective in BNNs.
method Corrected asymptotic form of singular posterior distributions to inform variational family design.
result Improvements in variational free energy and generalization error with proposed normalizing flow.
SLT explains neural network success by closing theory-practice gap.
problem Failure of classical inference and learning theory in modern neural networks.
method Physics-inspired Singular Learning Theory (SLT) applied to neural networks.
result SLT recovers known and novel scaling laws for neural network phase transitions.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
We introduce thermodynamic response functions for singular Bayesian models.
problem Singular Bayesian models violate regular asymptotics due to non-identifiability and degenerate Fisher geometry.
method Posterior tempering induces thermodynamic response functions, linking WAIC, WBIC, and singular fluctuation.
result WAIC, WBIC, and singular fluctuation are unified within a thermodynamic response framework.
PAC-Bayes bounds for Gibbs posteriors derived via singular learning theory.
problem Generalization bounds for overparameterized models with data-dependent priors.
method Explicit non-asymptotic PAC-Bayes bounds using singular learning theory.
result Explicit posterior-averaged risk bounds for overparameterized models.
New class of singular complex manifolds studied with degenerate theory.
problem Understanding singular complex manifolds.
method Developed degenerate Kodaira-Hodge theory for new class.
result New degenerate theory for singular complex manifolds.
Paper reviews algebraic research in machine learning theory.
problem Understanding phase transitions in machine learning models.
method Algebraic approaches in statistical mechanics.
result Algebraic methods are essential for analyzing machine learning models with singularities.
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
Introduces LLC, a new complexity measure for DNNs based on SLT.
problem Lack of effective complexity measures for DNNs.
method Uses Singular Learning Theory to define LLC and proposes scalable estimator.
result Empirical evidence shows LLC provides valuable insights into DNN complexity.
The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.
problem Analyzing vector bundles with singular Hermitian metrics and positivity.
method Develops harmonic theory and extends results from complex geometry.
result Extends Nakano's vanishing theorem to vector bundles with singular metrics.
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical …
Building on author's previous results in singular semi-Riemannian geometry and singular general relativity, the behavior of gauge theory at singularities is analyzed. The usual formulations of the field equations at singularities are accompanied by infinities which block the evolution equations, mainly because the metr…
The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
Quantum statistical models with singularities are studied for state estimation and model selection.
problem Understanding statistical properties of quantum singular models.
method Classical singular learning theory extended to quantum state estimation and model selection using algebraic geometrical methods.
result Asymptotically unbiased estimator (QWAIC) for quantum generalization loss constructed.
Examines learning efficiency in neural networks and related models.
problem Analyzing efficiency in deep learning models with singular learning coefficients.
method Examined learning coefficients in neural networks and three-layer neural networks with ReLU units.
result Extended results to include Softmax function, providing a broader understanding of learning efficiency.
The paper connects orbifold singularities to higher symmetries in SQFTs.
problem Understanding higher symmetries in supersymmetric quantum field theories.
method Cutting and gluing of orbifold singularities to determine symmetries.
result Local orbifold singularities encode 0-form, 1-form, and 2-group symmetries.
The paper introduces contexture theory to characterize representation learning from contexts.
problem Lack of systematic characterization of representation learning methods.
method Characterizes representation learning as learning from the association between input and context variable.
result Contexture theory shows that representations can be approximated by top singular functions of the context.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. New method cleans cross-covariance matrices for better financial forecasting.
problem Asymptotically optimal cross-covariance cleaners fail in real-world, time-varying markets.
method Physics-informed neural network that learns from empirical singular values.
result Trained model outperforms analytical cleaners in out-of-sample cross-covariance prediction.
Paper solves the minimal generating set problem for singular Reidemeister moves.
problem Determine minimal generating sets of oriented singular Reidemeister moves.
method Introduced new invariant for singular links to detect type IV moves and provide obstructions.
result Proved exactly 96 distinct inclusion-minimal generating sets for singular moves.
This paper sets lower bounds for scalar curvatures in Ricci flow singularity models.
problem Understanding scalar curvatures in Ricci flow singularity models.
method Developed high-dimensional theory of Hamilton's Ricci flow, including new monotonicity formulas, compactness theorem, and partial regularity theory.
result Obtained a quadratic decay lower bound for the scalar curvature in 4-dimensional non-Ricci-flat steady soliton singularity models.
Refined theorem on linear perturbations with applications in singularity theory and optimization.
problem Linear perturbations and their implications in singularity theory and optimization.
method New perspective of Hausdorff measures for refined transversality theorem.
result Applications in singularity theory and optimization.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
Integrates singular subalgebroids using diffeological groupoids.
problem Integration of singular subalgebroids.
method Definition of integration via diffeological groupoids with specific properties.
result Holonomy groupoids correspond to singular subalgebroids with submersive property.
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
Study the geometry of matrix multiplication in deep neural networks.
problem Understanding the structure of matrix multiplication in deep neural networks.
method Using quiver representations and equivariant cohomology, determine codimension and irreducible components.
result Codimension and number of top-dimensional irreducible components of matrix multiplication are invariant under permutations and have specific log-canonical thresholds.
New insights into symplectic singularities via canonical torus actions.
problem Understanding symplectic singularities and their actions.
method Use of (C∗)r actions and Donaldson-Sun theory. result Symplectic singularities admit canonical (C∗)r actions. We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…
Study symplectic fillings of sandwiched singularities.
problem Contrast deformation theory and symplectic topology of Milnor fibers.
method Develop an analog of de Jong--van Straten's theory in the symplectic setting using spinal open books and nearly Lefschetz fibrations.
result Minimal symplectic fillings of links are generated by certain immersed disk arrangements.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.
The paper extends foam theory to more complex trivalent graphs.
problem Extending foam theory to more complex trivalent graphs.
method Considering foams with singular vertices homeomorphic to cones over more general planar trivalent graphs.
result Modules associated with the dodecahedron graph are free of rank 60.
We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov-Lawson Conjecture for such manifolds in the case of particular type of singularitie…
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
Study branched coverings of singular (G,X)-manifolds, solving open questions.
problem Understanding branched coverings of singular (G,X)-manifolds.
method Developed a Galois theory for branched coverings, constructed developping maps for singular manifolds.
result Solved open questions and constructed new examples related to singular (G,X)-manifolds.
Deformations of singular Cayley submanifolds studied.
problem Constructing fibrations of compact Spin(7) manifolds.
method Deformation theory of conically singular and asymptotically conical Cayley submanifolds.
result Detailed description of the deformation theory.
SLT reveals how grokking occurs via basin selection in training.
problem Understanding grokking in machine learning models.
method Singular Learning Theory (SLT) to analyze the loss landscape and local learning coefficient (LLC).
result LLC ranks basins by statistical preference, leading to grokking.
Researchers study rank two theories with eight supercharges using Lefschetz pencils.
problem Understanding the global Seiberg-Witten geometries for rank two theories with eight supercharges.
method Combining combinatorial methods with geometric analysis of Lefschetz pencils.
result The conjugacy class of mapping class group determines the local singularity, and the global study reduces to questions about MCG.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
The paper proves the existence of singular cscK metrics on smoothable varieties.
problem Existence of singular cscK metrics on smoothable varieties.
method Developing a strong topology of pluripotential theory in families and uniform estimates for cscK metrics.
result Existence of singular cscK metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. Reduces symplectic manifolds with singularities for quantum reduction.
problem Quantization commutes with reduction for singular symplectic manifolds.
method Reduction theory for bm-symplectic manifolds and folded symplectic manifolds under general symmetries. result New constructions of (singular) quasi-Hamiltonian spaces via reduction and fusion product.
Study on mean curvature flow through singularities in 3D and 4D.
problem Understanding mean curvature flow through singular points.
method General introduction and classification of singularities in R3 and R4. result Classification of all noncollapsed singularities in R4. We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the `local stability theory' of an arbitrary Kawamata log…
Estimates LLC for deep linear networks up to 100M parameters.
problem Quantifying model complexity for large-scale deep learning architectures.
method Empirical estimation of LLC using a method developed for DLNs.
result LLC can be accurately measured for DLNs up to 100M parameters.