Study improves oracle inequality for tree graphs using total variation regularization.
problem Improving oracle inequality for tree graphs with total variation regularization.
method Generalized Fused Lasso result to tree graphs, using harmonic mean of distances.
result Proved a lower bound on compatibility constant for total variation penalty.
The paper derives oracle inequalities for estimators with fast and slow rates.
problem Developing fast and slow oracle inequalities for estimators.
method Direct study of analysis estimator and adaptation of Dalalyan, Hebiri and Lederer's arguments.
result Constant-friendly rates for (square root) total variation regularized estimators over graphs.
Method segments graphs to estimate network models using power graph fused lasso.
problem Estimating non-parametric network models from noisy data.
method Power graph fused lasso (PGFL) for graph segmentation.
result PGFL achieves optimal error rate for graphon estimation under subGaussian noise.
Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
problem Understanding the relationship between graph curvature and expansion properties.
method Proving an inequality linking isoperimetric profiles to total variation decay of random walks.
result Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
Hybrid system combines NMF and TV graph for song recommendations.
problem Improving song recommendation systems.
method Formulates a matrix completion problem using NMF and TV on graphs.
result Outperforms models using only low-rank or graph-based information.
Paper proposes a new method to minimize submodular functions with fewer calls to simpler oracles.
problem Minimizing the sum of submodular set functions with limited information.
method Introduces a modified convex problem requiring constrained total variation oracles that can be solved with fewer calls to minimization oracles.
result Shows significant reduction in the number of calls to minimization oracles.
Random walk sampling recovers smooth graph signals from few samples.
problem Efficiently sampling graph signals from large networks.
method Random walk sampling strategy based on network nullspace property.
result Graph signals can be accurately recovered from few samples.
Social-sparsity brain decoders improve speed and interpretability.
problem Computational cost and interpretability in brain decoding models.
method Introduced social-sparsity, a structured shrinkage operator.
result Social-sparsity performs almost as well as total-variation models and better than graph-net, with a fraction of the computational cost.
The paper develops estimators for variance in graph structures using fused lasso.
problem Variance estimation in graph-structured problems.
method Developed linear time estimator for homoscedastic case and total variation regularization estimator for heteroscedastic case.
result Minimax rates and consistency for variance estimation in various graph structures.
Estimates parameters of interconnected linear systems using total variation penalization.
problem Joint estimation of parameters in interconnected linear dynamical systems.
method Total variation penalized least-squares estimator.
result The MSE goes to zero as the number of systems increases, even with constant trajectory length.
Network Lasso clusters sparse graph clusters efficiently.
problem Local graph clustering of sparse and chain-like clusters.
method Network Lasso minimizes total variation of cluster indicator signals.
result Network Lasso handles sparse clusters difficult for spectral clustering.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of availa…
Method uses TV minimization for semi-supervised learning on network data.
problem Semi-supervised learning from partially-labeled network data.
method Graph signal recovery interpretation, total variation minimization, primal-dual method for non-smooth convex optimization.
result TV minimization recovers clusters in the empirical graph of the data under certain network conditions.
Proposes a method for semi-supervised learning from unlabeled data.
problem Learning labels from partially labeled network data.
method Formulates as a non-smooth convex optimization problem balancing empirical loss and graph signal smoothness.
result Sparse label propagation algorithm for scalable learning.
Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning frame…
Paper defends sensitive attributes in GNNs from inference attacks.
problem Protecting sensitive attributes in GNNs from inference attacks.
method Proposes adversarial training with TV and Wasserstein distance to locally filter sensitive attributes.
result Framework creates strong defense against inference attacks with minimal performance loss.
We present a graph-based variational algorithm for multiclass classification of high-dimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves s…
Develops method for learning signed graphs from smooth signals.
problem Learning signed graphs from observed data, especially in contexts with both positive and negative interactions.
method Uses net Laplacian as graph shift operator and minimizes total variation of observed signals with ADMM.
result Theoretical proofs of convergence and estimation error bound provided.
Efficiently test and learn causal Bayesian networks with interventions and samples.
problem Testing and learning causal Bayesian networks with interventions and samples.
method Interventions and samples to distinguish causal Bayesian networks, with algorithms for testing and learning.
result Efficient algorithms for testing and learning causal Bayesian networks, with subadditivity inequality for squared Hellinger distance.
We present a graph-based variational algorithm for classification of high-dimensional data, generalizing the binary diffuse interface model to the case of multiple classes. Motivated by total variation techniques, the method involves minimizing an energy functional made up of three terms. The first two terms promote a …
In this paper we study heat kernels associated to a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics $σ_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on G as $\e\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the…
ReAPR simplifies hard unknots by reembedding and rerouting, revealing hidden simplifications.
problem Training AI to recognize knots, especially hard unknots, is challenging.
method Alternates pass-move reduction with geometric re-embedding, minimizing total variation of a height function.
result ReAPR successfully simplifies hard unknots, including Kauffman's challenge unknots, in under 30 seconds.
The paper tackles sparse model fitting in distributed machine learning with graph-structured data.
problem Sparse model fitting across a distributed collection of heterogeneous data sets.
method Basis Pursuit Denoising with a total variation penalty, using ADMM for distributed methods.
result Recovery is successful with fewer samples than solving problems independently, or using methods with large overlap in signal supports.
Discrete diffusion models improve data generation for discrete data like language and graphs.
problem Adapting diffusion models to discrete state spaces for better data generation.
method Formulated as CTMCs, used uniformization of continuous Markov chains for sampling.
result Derive guarantees for sampling from any distribution on a hypercube, aligning with state-of-the-art achievements.
Knot theory is the study of isotopy classes of embeddings of the circle S1 into a 3-manifold, specifically R3. The Fáry-Milnor Theorem says that any curve in R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topologica…
New model for shape graph registration with partial matching constraints.
problem Shape graph registration with topological inconsistencies and partial matching.
method Higher order invariant Sobolev metrics, varifolds, inexact variational formulation, SFISTA algorithm.
result Existence of minimizers for variational problem with TV regularization.
Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a g…
Total variation denoising improves image quality adaptively.
problem Improving image quality from noisy data.
method Total variation regularization for image denoising.
result Denoised images converge to true images at a parametric rate.
The paper studies curves in Riemannian manifolds using total variation flow.
problem Analyzing the evolution of curves in Riemannian manifolds using total variation.
method Defining and proving the existence of strong solutions to the flow equations, showing variational equality, and proving convergence.
result Strong solutions converge to a constant map in finite time for non-positive sectional curvature.
Optimal pre-processing reduces disparate impact by minimizing total variation distance.
problem Achieving fairness in data outputs based on protected attributes.
method Using pre-processing to enforce fairness, minimizing total variation distance between pre-processed and original data distributions.
result The problem of fairness can be formulated as a linear program, efficiently solvable.
Planar graphs' curvature total is a multiple of 1/12.
problem Total curvature of planar graphs with nonnegative combinatorial curvature.
method Proved using combinatorial curvature.
result Total curvature is an integral multiple of 1/12.
New minimax rates for total variation denoising in higher dimensions, showing limitations of linear smoothers.
problem Estimating functions with bounded total variation over high-dimensional grids.
method Minimax analysis, focusing on linear and non-linear estimators.
result Linear estimators like Laplacian smoothing and eigenmaps are suboptimal for total variation denoising in higher dimensions.
We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $σ_\e$ collapsing to a subRiemannian metric σ0 as $\e\to 0$. We establish Ck,α estimates for this…
Study variational formulas for Q-prime curvature on complex manifolds.
problem Derive variational formulas for Q-prime curvature.
method Deformation of strictly pseudoconvex domains in complex manifolds, study CR invariant differential operators.
result Total Q-prime curvature agrees with renormalized volume for certain metrics.
Snake solves large graph optimization problems with fast proximal steps.
problem Optimization over large unstructured graphs with graph-specific regularization.
method Snake algorithm using random simple paths for proximal gradient steps.
result Convergence proven for the Snake algorithm.
This work provides guaranteed bounds on the total variation distance for univariate mixtures.
problem Lack of closed-form expressions for total variation distance between mixtures.
method Two methods: information monotonicity for lower bounds and geometric envelopes for upper bounds.
result Demonstrated tightness of bounds on Gaussian, Gamma, and Rayleigh mixtures.
We show a very simple and general total second variation formula for Perelman's W-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for P…
We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph …
Total variation minimization clusters partially labeled data points.
problem Clustering partially labeled data points in stochastic block models.
method Total variation minimization as a clustering method.
result Total variation minimization allows for accurate clustering under certain model parameters.
Network Lasso improves graph signal learning from few samples.
problem Ensuring network Lasso accuracy for graph signal learning.
method Compressed sensing concepts applied to network Lasso.
result Precise conditions for network Lasso accuracy quantified.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
problem Extending graph signatures to Klein graphs and foams.
method Developed an analogy of Murasugi's bounds and used signatures to lower bound knot properties.
result Lower bounds on negative orbifold Euler characteristics and unknotting numbers.
Efficiently infers interventional distributions from observational data.
problem Inferring interventional distributions from limited observational data.
method Polynomial-time algorithm for causal Bayesian networks.
result Efficient algorithm for generating close approximations to interventional distributions.
Minimal surfaces in H^2xR have infinite total curvature under certain conditions.
problem Understanding total curvature concentration in minimal surfaces in H^2xR.
method Analyzing stable minimal surfaces with specific asymptotic boundaries.
result Minimal surfaces in H^2xR have infinite total curvature under certain geometric conditions.
We find the maximum regularization parameter for total-variation denoising.
problem Finding the maximum regularization parameter for anisotropic total-variation denoising.
method Established a closed form expression for the one-dimensional case and an upper-bound for the two-dimensional case using the pseudo-inverse of the divergence.
result The maximum regularization parameter is crucial for optimal parameter tuning and can be computed efficiently.
The study shows that random surfaces built from polygons converge to a Poisson-Dirichlet partition.
problem Understanding geometric properties of random surfaces constructed from polygons.
method Uniformly pairing polygon sides to form surfaces, analyzing degree sequences and geometric properties using probabilistic techniques.
result Several geometric properties of the graph are universal, converging to a Poisson-Dirichlet partition as no∞. We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…
Study variational formulas for distribution geometry, finding critical metrics.
problem Analyzing the total mixed scalar curvature of a distribution.
method Developed variational formulas for extrinsic geometry, solved Euler-Lagrange equations.
result Found critical metrics related to various geometric properties.