Proves conjecture about sphere widths under rotational symmetry.
problem Width stability of rotationally symmetric metrics.
method Proof of conjecture and extensions to higher dimensions.
result Stability of min-max width under rotational symmetry.
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48 results for “width stability”
Develops a new theory for neural systems stability and width effects.
problem Stability and finite-width effects in deep neural systems.
method Gauge-covariant stochastic effective field theory using classical commuting fields.
result Predicts the edge of chaos and low-frequency spectral deformation.
Wider networks improve natural accuracy but worsen perturbation stability, affecting overall robustness.
problem Understanding the tradeoff between natural accuracy and perturbation stability in wider neural networks for adversarial robustness.
method Careful examination of the relationship between network width, robust regularization parameter λ, and perturbation stability using neural tangent kernels.
result Wider networks can achieve better natural accuracy but worse perturbation stability, leading to potentially worse overall model robustness.
Study on disk configurations in strips shows stability patterns.
problem Understanding stability patterns in disk configurations in strips.
method Finite presentation of rational homology groups, representation stability.
result Disk configuration space exhibits first-order representation stability.
Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfercond-mat.dis-nn
The study examines spectral dynamics in deep neural networks, predicting how outliers evolve during training.
problem Understanding spectral evolution in deep neural networks during training.
method Developed a two-level dynamical mean-field theory (DMFT) to track spectral dynamics.
result The theory predicts how outliers evolve with training time, width, output scale, and initialization variance.
Estimates bandwidth for CMC initial data sets.
problem Estimating bandwidth for constant mean curvature (CMC) initial data sets.
method Three independent proofs: stability of null expansion, spacetime harmonic function perturbation, Dirac operator.
result Generalized Gromov's band width estimate to CMC initial data sets.
New scaling framework for MoE architectures ensures stability and optimal performance at scale.
problem Lack of principled understanding of how hyperparameters should scale in MoE architectures.
method Developed a novel Dynamical Mean Field Theory (DMFT) for three scaling regimes of MoE architectures.
result Derived Maximally Scale-Stable Parameterization (MSSP) for SGD and Adam, providing robust learning rate transfer and monotonic improvement with scale.
Fisher width is a geometric measure of complexity on statistical manifolds.
problem Complexity measures on statistical manifolds
method Introducing Fisher width as a Fisher-geometric analogue of Gaussian width
result Fisher width retains key structural features of Gaussian width while capturing anisotropic geometric effects
Gradient methods improve deep network training with tighter bounds and faster convergence.
problem Improving convergence and generalization of gradient methods for neural networks.
method Algorithmic stability analysis and novel bounds on excess risk.
result Gradient descent achieves optimal excess risk for deep nets with polynomial width conditions.
This work studies fluctuation in multilayer neural networks using mean field theory.
problem Understanding fluctuation in multilayer neural networks with mean field training.
method Developed a second-order mean field limit to capture fluctuation, demonstrating stability of gradient descent training.
result Gradient descent training in multilayer networks biases towards minimal fluctuation, even after convergence.
New optimizers control network width scaling, improving stability and transfer across different model sizes.
problem Designing stable optimizers for networks of varying widths.
method Interpreting optimizers as steepest descent under mean-normalized operator norms, enabling layerwise composability and width-independent bounds.
result New optimizers like row normalization and column normalization provide stable learning-rate transfer across different model widths.
Mean-field analysis for heavy ball methods: Dropout-stability, connectivity, and global convergencecs.LG
Paper analyzes SHB method for neural networks, proving stability, connectivity, and global convergence.
problem Theoretical understanding of SHB method for neural networks.
method Mean-field analysis of SHB dynamics related to a partial differential equation.
result SHB method converges to global optimum and exhibits stability and connectivity.
Study on fluctuations in neural network kernels and predictions, focusing on finite width effects.
problem Characterizing fluctuations in finite width neural networks.
method Dynamical mean field theory analysis of wide but finite feature learning neural networks.
result Fluctuations in kernels and predictions are dynamically coupled, leading to reduced variance in feature learning regimes.
New Transformer architecture prevents rank degeneracy in deep attention models.
problem Rank degeneracy in deep attention models.
method Modified Softmax-based attention model with skip connections, centered at identity, and scaled logits.
result Existence of a stable SDE implies well-behaved covariance structure, preventing rank degeneracy.
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of 's. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…
Gradient descent converges linearly in finite-width networks with positive NTK and compatible conditions.
problem Local convergence of gradient descent in finite-width networks.
method Positive Neural Tangent Kernel (NTK), local Polyak-Łojasiewicz inequality, fixed-step containment in Locally Quasi-Convex Region (LQCR).
result Linear convergence achieved under specific conditions.
PCS-UQ framework improves uncertainty quantification for machine learning models.
problem Ensuring trustworthy uncertainty quantification for machine learning models in high-stakes domains.
method PCS-UQ framework based on Predictability, Computability, and Stability principles, integrating prediction-checking, bootstrap samples, and multiplicative calibration.
result PCS-UQ maintains target coverage while outperforming or matching conformal methods in interval width and subgroup coverage.
This work challenges the Neural Tangent Kernel's role in overparameterized neural networks, especially with large width and depth.
problem The Neural Tangent Kernel's behavior in overparameterized neural networks with large width and depth is unclear.
method Experimental and theoretical analysis of ReLU networks with large width and depth.
result The aggregate norm of hidden neuron deviations does not vanish in infinitely-wide ReLU networks, indicating non-trivial behavior.
Gradient descent on shallow neural networks achieves near-optimal generalization error.
problem Optimizing shallow neural networks with minimal width for generalization and stability.
method Gradient descent in the interpolating regime with minimal width.
result Gradient descent achieves near-optimal generalization error with minimal width.
Stability of cigar solitonmath.DG
In this paper, we shall use the Kähler geometry formulation to study the global behavior of the Ricci flow on . The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classificatio…
New method improves generalization in deep learning models.
problem Improving generalization in overparameterized deep neural networks.
method Stochastic Gauss-Newton method with Levenberg-Marquardt damping and mini-batch sampling.
result Established finite-time convergence and non-asymptotic generalization bounds.
Study of deep Stable neural networks with various activation functions.
problem Characterizing the infinitely wide limits of deep Stable neural networks.
method Investigation of large-width properties of deep Stable NNs with a generalized central limit theorem for heavy tails.
result Extension of characterization to a broader class of activation functions, including sub-linear, asymptotically linear, and super-linear functions.
New method stabilizes deep neural networks by setting Lyapunov exponent to zero.
problem Stability issues in deep neural networks with low width.
method Lyapunov initialization method to set Lyapunov exponent to zero.
result Lyapunov exponent governs stability of deep networks; standard methods fail for low width.
For a convex domain bounded by the hypersurface in a space of constant curvature we give sharp bounds on the width of a spherical shell with radii and that can enclose , provided that normal curvatures of are pinched by two positive constants. Furthermore, in the …
We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …
MGDL refines deep neural networks by training grades sequentially, improving stability.
problem Training deep neural networks is challenging due to nonconvex optimization landscapes.
method MGDL trains deep networks grade by grade, freezing previously learned grades and training new ones to fit residuals.
result MGDL guarantees vanishing error in a fixed-width multigrade ReLU architecture.
An Empirical Analysis of the Advantages of Finite- v.s. Infinite-Width Bayesian Neural Networksstat.ML
Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.
problem Comparing BNNs with different widths due to conflicting model properties and inference intractability.
method Empirical comparison of finite- and infinite-width BNNs, analyzing performance under model mismatch.
result Increasing width can hurt BNN performance when the model is mis-specified, and finite-width BNNs generalize better under model mismatch.
New Gaussian priors for neural networks improve scalability and Bayesian inference stability.
problem Scalability and stability issues in Bayesian neural network inference.
method Introduces a new Gaussian neural network prior with decreasing variance in network width, enabling stable MCMC sampling.
result The new prior enables stable MCMC sampling for Bayesian neural network inference, improving scalability and stability.
The isospectral problem for p-widths is solved using Zoll metrics on S^2.
problem Determine if a Riemannian manifold is uniquely determined by its p-widths.
method Construct counterexamples on S^2 using Zoll metrics and properties of geodesic p-widths.
result Many counterexamples exist on S^2, showing uniqueness is not guaranteed.
Width trees link link invariants and bridge number.
problem Understanding link invariants through geometric structures.
method Associate width trees to links and use their geometric properties to bound link invariants.
result Width trees uniquely realize certain link invariants under specific conditions.
The classical isoperimetric inequality in the Euclidean plane states that for a simple closed curve of the length , enclosing a region of the area , one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…
The study refines known counterexamples in 4D to satisfy certain inequalities.
problem Addressing counterexamples in Gromov's and Rosenberg's conjectures.
method Analyzing simply connected and non-simply connected four manifolds up to homeomorphism.
result Gromov's and Rosenberg's conjectures hold for simply connected four manifolds up to homeomorphism.
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…
Lectures on deep learning properties in infinite and large-width networks.
problem Understanding deep neural networks in extreme width conditions.
method Analysis of random deep neural networks, connections to linear models, kernels, and Gaussian processes, perturbative and non-perturbative treatments.
result Properties and behaviors of deep neural networks in the infinite-width limit and large-width regime.
Computed p-widths for hemisphere, first for manifolds with boundary.
problem Finding p-widths for manifolds with boundary.
method Computed p-widths for the hemisphere.
result First known p-widths for a manifold with boundary.
The p-widths of a polygonmath.DG
Polygon -widths are found via billiard trajectories.
problem Finding -widths of polygons.
method Proved via billiard trajectories and computed specific cases.
result Polygon -widths are achieved by billiard trajectories.
A number of results for C-smooth surfaces of constant width in Euclidean 3-space are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…
The p-widths of math.DG
Computed p-widths for real projective plane.
problem Calculating p-widths for real projective plane.
method Standard metric used to compute p-widths.
result Computed p-widths for real projective plane.
Urysohn Width and Surgeriesmath.MG
Study bounds Urysohn width of manifolds under surgeries.
problem Bounding Urysohn width of manifolds after surgeries.
method Analyzes connected sums and universal covers, applies to general surgeries.
result Optimal constants in estimates of width bounds are shown.
Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width Onecs.LG
Residual networks with block width max(d_x, d_y) approximate all functions.
problem Achieving universal approximation with residual networks.
method Established bounds on block width for different activation functions.
result Minimum block width for universal approximation is max(d_x, d_y) with inner width 1.
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…
New link invariants from diagram colorings match link widths.
problem Defining link widths via diagram colorings.
method Colorings of link diagrams to define invariants and prove their equivalence to link widths.
result Invariants of link widths calculated algorithmically.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.
problem Finding the minimum value of the first Neumann eigenvalue for convex domains.
method Proved the sharp lower bound using diameter and width.
result Sharp lower bound for the first Neumann eigenvalue established.
New method selects neural network architectures without needing data.
problem Choosing efficient deep neural network architectures.
method Developed the deep frame potential to quantify network capacity.
result Deep frame potential correlates with generalization error.
We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.
Wide neural networks can degrade performance, contrary to conventional wisdom.
problem Understanding the limitations of increasing network width in neural networks.
method Using Deep Gaussian Processes to decouple capacity and width, analyzing their effects on representational power and non-Gaussianity.
result Wide neural networks can become less adaptable and more Gaussian, leading to performance degradation.