Proposes MSN to improve DNN performance and speed.
problem Improving Deep Learning model regularization and performance.
method Empirical approach to study Spectral Normalization (SN) and Mean Spectral Normalization (MSN).
result MSN significantly improves DNN performance and speed.
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
The study of spectral-tightness in Riemannian manifolds and its topological implications.
problem Understanding the spectral properties of Riemannian manifolds and their coverings.
method Analyzing the fundamental group and the Euclidean local de Rham factor to characterize spectral-tightness.
result Spectral-tightness is a topological property of the fundamental group, and it can be characterized by the dimension of the Euclidean local de Rham factor.
This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for…
A new method for spectral barycentre of graph datasets.
problem Creating a summary graph from a set of graphs with community structure.
method Using multiscale spectral distance based on normalized graph Laplacian eigenvalues.
result The barycentre inherits the topological structure of the graphs in the sample dataset.
Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about performance guarantees for spectral clustering. This issue is partly due to the…
New method clusters signed graphs using matrix power means.
problem Clustering signed graphs with positive and negative relations.
method Signed Power Mean Laplacian, defined as matrix power mean of normalized standard and signless Laplacians.
result Signed power mean Laplacian captures ground truth clusters under reasonable settings.
One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to in…
In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold (M,ω) canonically relates the action spectra of different normalized Hamiltonians on {\it arbitrary} symplectic manifolds (M,ω). The natural class of normalized Hamiltonians consists of those w…
Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.
problem Understanding the manifold structures of orbits of normal operators under different norm topologies.
method Unified treatment of unitary and groupoid orbits, using moment maps and conditional expectations.
result Differentiable structures for orbits and necessary spectral conditions for norm closure and submanifold properties.
Maps from spheres and disks to convex shapes via curvature flow.
problem Constructing contractions from spheres and disks to convex shapes.
method Inverse mean curvature flow to create normalized-area-preserving contractions.
result Proves E. Milman's conjecture and gives spectral comparison results.
A new method speeds up spectral normalization for neural nets.
problem Efficiently controlling the spectral norm of convolutional layers.
method Depthwise separable convolutions with spectral normalization.
result Significant reduction in computational and memory costs.
We prove a central limit theorem for the components of the eigenvectors corresponding to the d largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized La…
Spectral normalization stabilizes GANs by controlling gradient explosion and vanishing.
problem Stability and sample quality issues in GAN training.
method Spectral normalization controls gradient explosion and vanishing, improving GAN training stability and sample quality.
result Bidirectional Scaled Spectral Normalization (BSSN) outperforms standard spectral normalization in sample quality and training stability.
Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both th…
New method improves GAN training stability and quality.
problem Improving training stability and sample quality in GANs.
method Proposes a new method for Lipschitz continuity in GANs that is efficient and unbiased.
result Demonstrates the effectiveness of the new method in various GAN training scenarios.
The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence …
Deep neural networks (DNNs) have set benchmarks on a wide array of supervised learning tasks. Trained DNNs, however, often lack robustness to minor adversarial perturbations to the input, which undermines their true practicality. Recent works have increased the robustness of DNNs by fitting networks using adversarially…
Neural network model improves leaf spectral reflectance prediction for grapevines.
problem Inaccurate modeling of grapevine leaf spectral reflectance from traits.
method Multi-head attention neural network trained on grapevine-specific data.
result Model achieved high accuracy (R^2=0.84, NRMSE=1.52%) and outperformed PROSPECT-PRO.
Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.
problem Wiegold problem about groups of normal rank > 1
method Topological argument and intricate construction of left-orders
result Free products of nontrivial left-orderable groups have normal rank > 1
Bayesian model improves traffic prediction with uncertainty estimates.
problem Lack of uncertainty estimates in deep-learning traffic models.
method Proposes a Bayesian recurrent neural network with spectral normalization.
result Spectral normalization improves uncertainty estimates and generalizability.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.
Develops a new representation for constant mean curvature surfaces in hyperbolic 3-space.
problem Finding conformal immersions of constant mean curvature in hyperbolic 3-space.
method Uses a Weierstrass-Kenmotsu type representation based on the Hermitian model, balanced spectral deformation, and Iwasawa splitting of $\SL$.
result Establishes an explicit correspondence with Aiyama and Akutagawa's representation and interprets the construction in terms of Kokubu's adjusted normal Gauss map.
SML improves pancreatic mass diagnosis accuracy using CT images.
problem Improving accuracy in pancreatic mass screening using CT imaging.
method Spectral machine learning method trained on 30,000 images, choosing fundamental images based on eigenvectors and removing irrelevant pixels.
result Achieved 94.6% test accuracy in diagnosing 113 patients.
Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
Generative models create paintings that match training data.
problem Creating realistic paintings using machine learning.
method Used Spectral Normalization GAN (SN-GAN) and SN-GAN with Gradient Penalty to generate paintings.
result SN-GAN produced paintings most comparable to the training dataset.
Unified approach to trend-following systems, deriving exact relationships and expected returns.
problem Designing and understanding trend-following systems in financial markets.
method Derive exact relationships, analyze expected returns, and use fractional ARFIMA processes.
result Profitability of trend-following systems depends on positive long-term autocorrelation and excess spectral mass at low frequencies.
The paper proves geometric and spectral alignment for deep neural networks.
problem Understanding the singular spectra of deep neural network layers.
method Proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors.
result Exact power-law spectra form a trace-normalized Cartan orbit under Frobenius normalization.
An important form of prior information in clustering comes in form of cannot-link and must-link constraints. We present a generalization of the popular spectral clustering technique which integrates such constraints. Motivated by the recently proposed 1-spectral clustering for the unconstrained problem, our method is…
We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (call…
We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. Th…
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in \cite{MS3}.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
This work uses neural density estimation to analyze laser-induced breakdown spectroscopy data, enabling accurate predictions and uncertainty quantification.
problem Inference of probability densities in high-dimensional spectral data is often intractable.
method Normalizing flows on structured spectral latent spaces for density estimation and uncertainty quantification.
result The approach enables generation of realistic spectral samples and accurate prediction of state vectors with well-calibrated uncertainties.
This study examines clustering of correlated random variables using k-means and spectral methods.
problem Clustering of correlated random variables.
method Used k-means and spectral algorithms, analyzed different similarity measures.
result Impact of initial points on k-means efficiency was analyzed.
For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of…
Method selects number of communities in weighted networks.
problem Selecting the number of communities in weighted networks.
method Proposes a novel weighted DCSBM and uses a sequential testing framework with spectral clustering and matrix scaling.
result Method is consistent in estimating the true number of communities under mild conditions.
ULES embeds dynamic networks with stability guarantees.
problem Stability of time-varying node embeddings in evolving networks.
method Unfolded Laplacian Spectral Embedding (ULSE) using normalized Laplacian operators.
result ULES satisfies cross-sectional and longitudinal stability under dynamic stochastic block model.
Develops spectral estimators for network structure with nodal covariates.
problem Identifying observed and unobserved factors affecting network structure.
method Spectral estimators for unobserved blocks and covariates in stochastic blockmodels.
result Asymptotic normality of estimators and superior performance compared to existing methods.
Estimates Gaussian location model with ridge regularization, comparing variational and spectral methods.
problem Estimating parameters in Gaussian location model with regularization.
method Ridge-regularized log-density-ratio estimation, variational and spectral approaches.
result Regularized variational estimator has lower risk with many observations, spectral estimator with fewer observations.
Bayesian models offer great flexibility for clustering applications---Bayesian nonparametrics can be used for modeling infinite mixtures, and hierarchical Bayesian models can be utilized for sharing clusters across multiple data sets. For the most part, such flexibility is lacking in classical clustering methods such a…
Study on the geometric Dyson Brownian motion of non-square matrix products.
problem Understanding the spectrum of a product of non-square random matrices.
method Proportional depth-width limit followed by mean-field limit, solving Burgers equation.
result Free log-normal law is obtained in the identity-start case.
New complexity measure shows similar generalization bounds for CNNs and non-CNNs.
problem Understanding why CNNs generalize well despite fitting random labels.
method Theoretical and empirical investigation of spectral complexity measure insensitivity to CNN invariances.
result Spectral complexity measure results in the same upper bound complexity estimates for CNNs and non-CNNs, contradicting common intuition.
Improved spectral clustering guarantees for dynamic stochastic block models.
problem Analyzing Spectral Clustering in dynamic stochastic block models.
method Extending guarantees to sparse and smooth DSBM, linking sparsity and smoothness.
result Improved error bounds for consistent recovery in dynamic DSBM.
The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in S3 result when these spectral curves satisfy periodicity conditions. We prove that the spectra…
Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula …