New proof for Gromov's theorem on almost flat manifolds.
problem Proving Gromov's theorem on almost flat manifolds.
method Inductive proof on dimension.
result New proof for Gromov's theorem.
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48 results for “inductive proof”
Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.
problem Overcoming singularities in the Schoen-Yau proof for arbitrary dimensions.
method Inductive scheme combining shielding principle, conformal blow-up, and Cheeger-Naber bound.
result Proof of positive mass theorem in arbitrary dimensions.
New proof shows exotic 4-manifolds exist without complex calculations.
problem Existence of exotic 4-manifolds in topology.
method Used Beliakova and Wehrli's s-invariant for links in and Stošić's induction scheme to simplify computations.
result Existence of exotic compact, orientable 4-manifolds proven without skein lasagna modules.
Dror Bar-Natan and Ofer Ron have questioned the accuracy of the statement on page 453, line 2 minus of my paper "Groups of Ribbon Knots", Topology Vol 37, No. 2, 441-458, that "It follows from the inductive proof that is of index 2." See their article posted on arXiv: math.GT/0309282. We give here full deta…
We construct a new inductive basis of the Birman-Murakami-Wenzl algebra. Using it, we provide a new proof of the existence of the Markov trace on the BMW algebras affording the two-variable Kauffman polynomial. We prove also that all the transverse Markov traces on the BMW algebras are determined by the self-linking nu…
We provide a proof of backpropagation algorithm in matrix notation.
problem The lack of a full induction proof of backpropagation algorithm in matrix notation.
method We provide a full induction proof of the BP algorithm in matrix notation, situating it in the framework of matrix differential calculus.
result We prove the validity of the backpropagation algorithm in inductive form.
We give an alternative proof of that a critical knot of a Morse-Bott function is a graph knot where the critical set of is a link in . Our proof inducts on the number of index-1 critical knots of .
GraIL predicts relations by reasoning over subgraphs, outperforming embeddings.
problem Relation prediction in knowledge graphs using latent representations is limited.
method Graph neural network with inductive bias to learn entity-independent relational semantics.
result GraIL outperforms existing rule-induction baselines in the inductive setting.
Proposes a topological framework to study modular invariants and related concepts.
problem Exploring modular invariants and related concepts in topological quantum field theory.
method Topological paradigm in alterfold topological quantum field theory.
result Establishes a novel integral identity for modular invariance across multiple Morita contexts.
Equivalence proven between two torsion invariants for flat vector bundles.
problem Equivalence of Igusa-Klein and Bismut-Lott torsion invariants for flat vector bundles.
method Reduction to trivial flat line bundles using Artin's induction theorem.
result Igusa-Klein and Bismut-Lott torsion invariants are equivalent for flat vector bundles with finite holonomy.
We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proo…
X.S. Lin and O. Dasbach proved that the sum of the absolute value of the second and penultimate coefficients of the Jones polynomial of an alternating knot is equal to the twist number of the knot. In this paper we give a new proof of their result using Khovanov homology. The proof is by induction on the number of cros…
Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.
problem Constructing ε-splitting maps for geodesic balls with non-negative Ricci curvature.
method Induction and stratified almost Gou-Gu Theorem for finding directional points; error estimates for projections.
result Constructs -splitting maps on concentric geodesic balls with uniformly small radius.
Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.
problem Classifying polynomial growth solutions to drift-harmonic equations on specific types of manifolds.
method Inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
result All drift-harmonic functions with polynomial growth asymptotically separate variables and dimensions of spaces are computed.
New proof found for Khovanov's assertion about Frobenius algebra twists.
problem Proving the isomorphism between chain complexes of twisted Frobenius algebras and link diagrams.
method Detailed analysis of configurations of circles in each state.
result A new proof of Khovanov's assertion with a detailed analysis of configurations.
Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
It is shown that Nambu-Poisson and Nambu-Jacobi brackets can be defined inductively: a n-bracket, n>2, is Nambu-Poisson (resp. Nambu-Jacobi) if and only if fixing an argument we get a (n-1)-Nambu-Poisson (resp. Nambu-Jacobi) bracket. As a by-product we get relatively simple proofs of Darboux-type theorems for these str…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on , and we list the …
Libgober and Wood proved that the Chern number of a -dimensional compact complex manifold can be determined by its Hirzebruch -genus. Inspired by the idea of their proof, we show that, for compact, spin, almost-complex manifolds, more Chern numbers can be determined by the indices of some twist…
Study generalizes matrix completion with side info in low noise settings.
problem Matrix completion with side information in low noise conditions.
method Inductive matrix completion with i.i.d. subgaussian noise, uniform sampling, and side information.
result Generalization bounds with noise scaling, convergence to zero, and logarithmic dependence on matrix size.
Machine learning improves searching for polynomial proofs.
problem Automatically searching for proofs of polynomial inequalities.
method Deep reinforcement learning guiding inference rules in semi-algebraic proof systems.
result Reduces the size of linear programs by several orders of magnitude.
The paper uses transfinite induction to prove existence in analysis.
problem Proving existence of extremal objects in analysis.
method Iterative procedure over ordinals to increase function and index steps.
result Existence can be proved using a countable number of steps.
The Gottschalk Conjecturemath.DG
The central idea of the proof is to show that a minimal flow v on a compact 3-manifold M implies the existence of a codimension one foliation F on it, which is transverse to the flow. If M is the 3-sphere, Novikov's theorem applies to show that one of the leaves of F is a compact surface X. It is now easy to derive a c…
Theoretical analysis of CNNs' inductive biases and their efficiency in approximating functions.
problem Understanding and optimizing the inductive biases in deep CNNs.
method Theoretical analysis combining multichanneling, downsampling, weight sharing, and locality.
result Deep CNNs with depth can approximate any continuous function, and require samples for sparse functions.
Paper explores how knowledge distillation transfers inductive biases between models.
problem Transferring inductive biases between models for tasks with limited data.
method Knowledge distillation applied to models with different inductive biases (LSTMs vs. Transformers, CNNs vs. MLPs).
result Effect of inductive biases is transferred through knowledge distillation, impacting both performance and solution characteristics.
Interpolated-MLPs control inductive bias for better performance in low-compute tasks.
problem Low-compute performance gap between MLPs and CNNs.
method Introduced Interpolated MLP (I-MLP) approach to control inductive bias incrementally.
result Continuous logarithmic relationship between inductive bias and performance in low-compute tasks.
New method quantifies inductive bias for machine learning tasks.
problem Quantifying the amount of inductive bias in machine learning models.
method Estimates inductive bias by modeling loss distribution of random hypotheses.
result Higher dimensional tasks require greater inductive bias.
Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it ex…
One-layer transformers can't solve induction heads task efficiently.
problem Solving the induction heads task efficiently with one-layer transformers.
method Communication complexity argument showing exponential size requirement.
result No one-layer transformer can solve the induction heads task efficiently.
OTI extends OTP for inductive semi-supervised learning.
problem Inductive semi-supervised learning for out-of-sample data.
method Optimal transport-based approach extended to inductive tasks.
result OTI outperforms state-of-the-art methods in experiments.
Strong inductive biases prevent harmless interpolation in overparameterized models.
problem Understanding the conditions under which overparameterized models can interpolate noise without overfitting.
method Theoretical analysis of high-dimensional kernel regression and deep neural networks, focusing on the role of inductive biases.
result The strength of an estimator's inductive bias determines whether interpolation is harmless or requires fitting noise for good generalization.
Deep ResNets favor low bottleneck rank with proper hyperparameters.
problem Understanding the inductive bias of deep neural networks.
method Computed minimum-norm weights of a deep linear ResNet.
result Deep nonlinear ResNets have an inductive bias towards minimizing bottleneck rank.
This research formalizes inductive generalization and proposes a new learning paradigm called Inductive Learning.
problem Generalization from easy to hard tasks, especially out-of-domain generalization.
method Formalizes inductive generalization, introduces Inductive Learning, and outlines steps to adapt techniques for learning model successors.
result A new learning paradigm (Inductive Learning) that emphasizes induction and universal properties of learning and computation.
We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals which we obtain by applying Neumann (or Weber) formula for the mutual inductance of a pair of loops to the case when two loops are identical.
The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension contains a minimal hypersurface with a singular set of Hausdorff dimension at most . This proof avoids the Almgren--Pitts …
We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups…
If is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to , from the Teichmüller space , for , to actually extends to an embedding between the Thurston compactification of the tw…
Novel framework for Bayesian reinforcement learning infers value function distributions.
problem Bayesian reinforcement learning's challenges in inferring value function distributions.
method Inferential Induction framework for Bayesian reinforcement learning, developing Bayesian Backwards Induction algorithm.
result Proposed algorithm is competitive with state-of-the-art methods.
Noise affects the effectiveness of interpolating models, especially those with strong inductive biases.
problem The impact of noise on interpolating models with strong inductive biases.
method Analyzing linear and classification models with sparse ground truths, proving fast rates for interpolators.
result Strong inductive biases can lead to faster but noisier interpolators, contrary to intuition.
Unsupervised machine translation---i.e., not assuming any cross-lingual supervision signal, whether a dictionary, translations, or comparable corpora---seems impossible, but nevertheless, Lample et al. (2018) recently proposed a fully unsupervised machine translation (MT) model. The model relies heavily on an adversari…
New approach relaxes inductive biases of physics-inspired NNs for better performance.
problem Challenges in applying physics-inspired NNs to real-world systems.
method Examined and relaxed inductive biases of Hamiltonian NNs, improving performance on non-conservative systems.
result Improved performance on practical, non-conservative systems by relaxing inductive biases.
Study links neural network inductive bias, feature learning, and generalization on Boolean functions.
problem Understanding how neural networks learn and generalize on Boolean data.
method End-to-end analysis of depth-2 discrete fully connected networks and DNF formulas, using Monte Carlo learning.
result Predictable training dynamics and interpretable features emerge, linking inductive bias and generalization.
Novel approach trains LLMs for inductive reasoning using probabilistic programs.
problem Training LLMs for inductive reasoning with sparse, ambiguous data.
method Program-based Posterior Training (PPT) using probabilistic inference.
result Significant improvement in estimation accuracy and alignment with human judgments.
We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
I-BERT extends Transformer's self-attention to arbitrary input lengths.
problem Transformer models struggle with inductive generalization to unseen input lengths.
method Replaces positional encodings with a recurrent layer.
result I-BERT achieves state-of-the-art results on algorithmic tasks.
New proofs for curvature problems using a viscosity approach.
problem Constant rank theorems for curvature problems in compact and non-compact settings.
method Viscosity approach to prove constant rank theorems for curvature problems.
result Generalization of a differential inequality for subtrace.