arXiv research

Max flow/min cut theorem extended to currents and topology.

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for net…

New convex programs solve minimal-area problems on Riemann surfaces.

We propose a faster and more accurate method for learning classification trees.

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow…

A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This gives a relative version of Stallings' Theorem on the structure of groups with mo…

Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.

In this paper it is shown that for any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair $s, t$ where each of $s, t$ is either a vertex or an end in the original network. A Max-Flow Min-Cut Theorem is proved for any net…

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, …

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of …

New algorithm clusters Gaussian mixtures with unknown covariance efficiently.

In this paper we investigate the behavioural differences between mobile phone customers with prepaid and postpaid subscriptions. Our study reveals that (a) postpaid customers are more active in terms of service usage and (b) there are strong structural correlations in the mobile phone call network as connections betwee…

Max-Cut decision tree improves classification accuracy and reduces computation time.

A new reinforcement learning method improves Max-Cut solutions without needing training data.

New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.

The maximum a posteriori (MAP) configuration of binary variable models with submodular graph-structured energy functions can be found efficiently and exactly by graph cuts. Max-product belief propagation (MP) has been shown to be suboptimal on this class of energy functions by a canonical counterexample where MP conver…

This paper presents a method to summarize directed graphs while preserving edge information.

A new algorithm solves semidefinite programs using Langevin diffusion.

Infinite $G$-invariant minimal hypersurfaces found in Riemannian manifolds.

New findings on maximizing noise stability in partitions of Gaussian space.

New methods optimize complex optimization problems with improved efficiency.

The study provides theoretical foundations for using smaller instances to predict algorithm performance on larger ones.

New method solves complex optimization problems faster.

Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow \cite{KZ:PAMI04} has had significant impact in computer vision \cite{BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04}. In this paper we address the important class of sum-of-submodular (SoS) function…

In this paper we relate the partition function to the max-statistics of random variables. In particular, we provide a novel framework for approximating and bounding the partition function using MAP inference on randomly perturbed models. As a result, we can use efficient MAP solvers such as graph-cuts to evaluate the c…

Novel method for bilevel optimization with convex lower-level problem.

Three-candidate plurality voting is stable for small correlations.

New algorithm solves large SDPs with provable convergence guarantees.

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely, a distribution derived from the so-called Mittag-Lef…

New framework analyzes effectiveness of neural network-based combinatorial problem solvers.

AMP algorithms can be efficiently simulated by SDPs even with corrupted data.

NeuralCut learns to select cutting planes by looking ahead, outperforming traditional methods.

Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.

NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.

Paper connects probability density cuts to graph theory eigenfunctions.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …

Stability of cut locus under metric perturbations in compact Riemannian manifolds.

In this paper we study output coding for multi-label prediction. For a multi-label output coding to be discriminative, it is important that codewords for different label vectors are significantly different from each other. In the meantime, unlike in traditional coding theory, codewords in output coding are to be predic…

New equivalence relation for links using cut-diagrams.

Study shows convergence rates for Cheeger cuts on data clouds.

Unified framework for differentiable graph partitioning with probabilistic cuts.

Stochastic cutting planes improve data-driven optimization speed.

Study of Randers metrics on spheres with simple cut loci.