TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
Expanding neural networks improves their learning from noisy data.
problem Improving neural network performance in noisy conditions.
method Sparse expansion of neural network inputs, followed by pruning, enhances generalization.
result Sparse expansion of neural networks improves generalization performance, even after pruning.
We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening…
We present MorphNet, an approach to automate the design of neural network structures. MorphNet iteratively shrinks and expands a network, shrinking via a resource-weighted sparsifying regularizer on activations and expanding via a uniform multiplicative factor on all layers. In contrast to previous approaches, our meth…
EGP uses expander graphs to improve GNN performance.
problem Challenges in deploying GNNs on graph tasks, including bottlenecks and oversquashing.
method Proposes EGP model based on expander graph propagation.
result EGP addresses challenges without bottlenecks or oversquashing, with linear complexity.
Enhanced GNN with expanded attention window and partially random embeddings.
problem Limited expressivity of traditional GNNs in distinguishing non-isomorphic graphs.
method Graph attention network with expanding attention window and partially random initial embeddings. Head dropout for regularization.
result Improved ability to differentiate between non-isomorphic graphs.
We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these a…
Explains neural network properties with a simple operation.
problem Understanding neural network performance and generalization.
method Analyze a single operation: expand-and-sparsify.
result Clarifies phenomena like Lottery Ticket Hypothesis and overparameterized models.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. Online random forests improve Q-learning performance in specific gym environments.
problem Improving Q-learning performance in reinforcement learning tasks.
method Proposed online random forests as Q-function approximators and growing them as learning progresses.
result Improved performance over state-of-the-art Deep Q-Networks in specific gym environments.
We explore the energy landscape of a simple neural network. In particular, we expand upon previous work demonstrating that the empirical complexity of fitted neural networks is vastly less than a naive parameter count would suggest and that this implicit regularization is actually beneficial for generalization from fit…
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Unique expanders found with vanishing entropy.
problem Finding unique expanders with specific entropy properties.
method Adapting White's work and using Bernstein-Wang results.
result Generic uniqueness of expanders with vanishing relative entropy.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
TGR rewires temporal graphs to improve TGNN performance.
problem Temporal graphs in evolving networks can suffer from under-reaching and over-squashing issues.
method TGR uses expander graph propagation to create message-passing highways between temporally distant nodes.
result TGR achieves state-of-the-art results on temporal graph benchmarks.
Sharp curvature estimates for expanding Ricci solitons in various dimensions.
problem Estimating curvature bounds for expanding Ricci solitons.
method Sharp lower and upper bounds derived for scalar curvature under specific conditions.
result Sharp curvature estimates provided for expanding Ricci solitons in dimensions three and four.
Study expanding solitons on complex Lie groups with specific algebraic structures.
problem Investigate expanding solitons on complex Lie groups with left-invariant metrics.
method Analyze the algebraic structure of complex Lie groups and their decompositions.
result Show that the Lie algebras of complex Lie groups decompose into semidirect products.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
SOC-ICNN expands neural network representational capacity by using conic optimization.
problem Restrictive representational capacity of ReLU-based ICNNs.
method Proposes SOC-ICNN architecture that uses Second-Order Cone Programming.
result SOC-ICNN strictly expands representational space without increasing complexity.
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.
problem Classifying complete Lagrangian self-expanders in complex 2-space.
method Obtained a classification theorem.
result A classification of 2D complete Lagrangian self-expanders with constant squared norm of the second fundamental form.
Expands pre-trained deep networks to classify new classes with minimal data.
problem Learning new classes with limited data.
method Hard distillation with a compact generative model.
result Low-shot network expansion is possible with minimal memory and data.
Small entropy self-expanders are topologically unique.
problem Uniqueness of self-expanders with small entropy.
method Analysis of mean curvature flow and cone asymptotics.
result All solutions are in the same isotopy class.
ARMA nets expand receptive fields for dense prediction tasks.
problem Global information in dense prediction problems is challenging for traditional convolutional layers.
method ARMA layers with adjustable autoregressive coefficients replace traditional convolutions.
result ARMA networks improve dense prediction tasks including video prediction and semantic segmentation.
TFiLM expands convolutional models' receptive field with minimal overhead.
problem Capturing long-range dependencies in sequential data.
method A novel architectural component using a recurrent neural network to modulate convolutional model activations.
result TFiLM significantly improves learning speed and accuracy on various tasks.
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work to demonstrate features of trained neural networks. Standard network initializat…
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …
We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain …
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to O(n)-invariant co…
Paper proves uniqueness of expanding solutions for harmonic map flow.
problem Uniqueness of expanding solutions for harmonic map flow.
method Introduces a relative entropy for 0-homogeneous maps and uses blow-up/blow-down processes.
result Generic uniqueness of expanding solutions proven for the same 0-homogeneous map.
Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
K-StoNet improves neural networks by avoiding local minima and assessing uncertainty.
problem Local minima and prediction uncertainty in deep neural networks.
method Combines SVR with latent variable model, using RBF kernel for feature space mapping and IRO algorithm for training.
result The model asymptotically converges to the global optimum and assesses prediction uncertainty easily.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function ∣A∣2/∣H∣2 and ∣Aξ∣2/∣H∣2 to understand the structure of self-expanders. result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.