arXiv research

The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.

New bounds for convex clustering under graph connectivity.

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

This work studies the denoising of piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness over a graph, where the value at each node can be vector-valued. We extend the graph trend filtering framework to denoising vector-valued graph signals with a family of non-convex regularizers, which exhibi…

Minimal graph theorem proven for convex domains.

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable…

New analysis shows D-SGD can generalize well regardless of graph connectivity.

Unified analysis of multi-attribute graph learning with non-convex penalties.

We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometr…

This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on gra…

Efficient algorithm for self-directed learning of convex clusters on graphs.

Study efficient graph optimization with noisy data.

The paper studies curve shortening flows on non-convex surfaces.

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a $n$-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the $k$-Hessian measures are ca…

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certa…

Traditionally, most complex intelligence architectures are extremely non-convex, which could not be well performed by convex optimization. However, this paper decomposes complex structures into three types of nodes: operators, algorithms and functions. Iteratively, propagating from node to node along edge, we prove tha…

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

Paper estimates differences in multi-attribute Gaussian graphical models using non-convex penalties.

We study the Dirichlet problem for a graph $Σ$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $Γ=\partial Ω$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $Γ$ satisfies $h\geq H$, also suffices when $Ω$ is strictly c…

Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to converge in many cases o…

Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.

Paper solves graph matching problem using convex relaxation to the simplex.

Unified analysis for graph learning from multi-attribute Gaussian time series.

Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a grap…

We study the problem of prediction for evolving graph data. We formulate the problem as the minimization of a convex objective encouraging sparsity and low-rank of the solution, that reflect natural graph properties. The convex formulation allows to obtain oracle inequalities and efficient solvers. We provide empirical…

The paper extends NUP representations to factor graphs for better estimation.

In this paper, we prove a half-space theorem with respect to constant mean curvature $1/2$ entire graphs in $\mathbb{E(-1,τ)}$. If $Σ$ is such an entire graph and $Σ'$ is a properly immersed constant mean curvature $1/2$ surface included in the mean convex side of $Σ$ then $Σ'$ is a vertical translate of $Σ$. We also h…

The paper tackles finding stationary points in stochastic convex optimization problems.

New findings on strong convexity in triangulations of convex polygons.

Characterizes geometric actions on graphs with flexible stabilizers.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

We construct geometric barriers for minimal graphs in H^n xR. We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in H^n extending continuously to the interior of each face, taking infinite boundary data on one face and zero boundary value data on …

Unified framework for graph coarsening using node features and graph matrices.

Estimate collapsibility of causal effects in CPDAGs via strong d-convex hulls.

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. Non-standard multilayer graph clustering methods are needed for assigning clusters to a common multilayer node set and for combining information from each layer. This paper present…

Algorithm learns graph ARMA processes for missing signal estimation.

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

New method improves MAP inference for CGMs on path graphs, avoiding approximation and maintaining integrality.

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

Unified framework infers time-varying graphs from incomplete signals.

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $Φ$ is given by a power $β\geq 1$ of a monotone symmetric and homogeneous of degree one function $F$ of the principal curvatures. Under the assumption that $F$ …

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being $k$-partite), which are desirable for so…

Symmetric graphs flow without singularities on their axis.

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. New challenges arise in multilayer graph clustering for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a theo…

Push-SAGA is a decentralized algorithm for directed graphs that converges linearly.