Minimax Formula Research Articles

Covariant bit threads

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a “thread distribution”, and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.

Minimax formula for the replica symmetric free energy of deep restricted Boltzmann machines

Minimax formula for the replica symmetric free energy of deep restricted Boltzmann machines

Operators Birkhoff–James Orthogonal to Spaces of Operators

Let be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace we partially solve the operator version of Birkhoff–James orthogonality problem, if is orthogonal to when does there exist a unit vector x 0 such that and is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for is a L 1-predual space as well as a M-ideal in Y, we show that if is orthogonal to then there is a unit vector x 0 with and is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.

Multi-generator GAN learning disconnected manifolds with mutual information

Original data usually lies on a set of disconnected manifolds rather than a smooth connected manifold. This causes the problem of mode collapse in the training of vanilla Generative Adversarial Network (GAN). There are many existing GAN variants that attempt to address this problem, but they result in limitations. The existing variants either produce simulated instances with low quality or generate identical simulated instances. In this study, we propose a new approach to training GAN utilizing multiple generators, a classifier and a discriminator to address mode collapse. The classifier outputs the statistical probabilities of generated data belonging to a specific category. These probabilities implicitly reflect which manifolds are captured by generators, and the correlation between generators is quantified by mutual information. Our idea views the mutual information values as a constraint to guide generators in learning different manifolds. Specifically, we traverse the generators, calculating the mutual information between each generator and the others. The calculated values are integrated into the generator loss to form a new generator loss and to update the corresponding generator’s parameters, using back-propagation. We minimize the mutual information to reduce the correlation between generators while also minimizing the generator loss. This ensures generators capture different manifolds while updating their parameters. A new minimax formula is established to train our approach in a similar spirit to vanilla GAN. We term our approach Mutual Information Multi-generator GAN (MIM-GAN). We conduct extensive experiments utilizing the MNIST, CIFAR10 and CelebA datasets to demonstrate the significant performance improvement of MIM-GAN in both achieving the highest Inception Scores and producing diverse generated data at different resolutions.

Tackling mode collapse in multi-generator GANs with orthogonal vectors

Generative Adversarial Networks (GANs) have been widely used to generate realistic-looking instances. However, training robust GAN is a non-trivial task due to the problem of mode collapse. Although many GAN variants are proposed to overcome this problem, they have limitations. Those existing studies either generate identical instances or result in negative gradients during training. In this paper, we propose a new approach to training GAN to overcome mode collapse by employing a set of generators, an encoder and a discriminator. A new minimax formula is proposed to simultaneously train all components in a similar spirit to vanilla GAN. The orthogonal vector strategy is employed to guide multiple generators to learn different information in a complementary manner. In this way, we term our approach Multi-Generator Orthogonal GAN (MGO-GAN). Specifically, the synthetic data produced by those generators are fed into the encoder to obtain feature vectors. The orthogonal value is calculated between any two feature vectors, which loyally reflects the correlation between vectors. Such a correlation indicates how different information has been learnt by generators. The lower the orthogonal value is, the more different information the generators learn. We minimize the orthogonal value along with minimizing the generator loss through back-propagation in the training of GAN. The orthogonal value is integrated with the original generator loss to jointly update the corresponding generator’s parameters. We conduct extensive experiments utilizing MNIST, CIFAR10 and CelebA datasets to demonstrate the significant performance improvement of MGO-GAN in terms of generated data quality and diversity at different resolutions.

Matchings with few colors in colored complete graphs and hypergraphs

The t-color Ramsey problem for hypergraph matchings was settled by the well-known result of Alon, Frankl and Lovász (answering a conjecture of Erdős). This result was the last step in a chain of special cases most notably Lovász’s solution to Kneser’s problem. We proposed an extension of the Erdős problem: for given 1≤s≤t, what is the maximum number of vertices that can be covered by a matching having at most s colors in every t-coloring of the edges of the complete graph Kn (or hypergraph Knr).We revisit the first unknown case, r=2,s=2,t=4, where we conjectured that in every 4-coloring of Kn there is a bicolored matching covering at least ⌊3n∕4⌋ vertices. We prove that this is true asymptotically by applying a recent twist of a standard application of the Regularity method: instead of lifting a (bicolored) matching of the reduced graph to regular cluster pairs, we lift a (bicolored) basic 2-matching, a subgraph whose connected components are edges and odd cycles. To find the bicolored basic 2-matching with at least ⌊3n∕4⌋ vertices in every 4-coloring of Kn we use Tutte’s minimax formula.

Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians

Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen’s inequality. The second proof is completely different with elementary calculations. It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". Then we go on to show that the feedback-assisted capacity is non-zero (with constant activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8:475-478]. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL 102:020504] that allows to reflect additional constraints, and which we believe to be of independent interest. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.

Principal eigenvalues of elliptic problems with large potential

This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the zero order term becomes larger and larger. Use is made of a minimax formula for these principal eigenvalues.

A formula for the weight of a minimal filling of a finite metric space

We consider the problem of finding a minimal filling for a finite metric space, that is, a weighted graph of minimal weight joining a given finite metric space. We obtain a minimax formula for the weight of the minimal filling, which we use to prove various properties of minimal fillings.Bibliography: 10 titles.

On variational eigenvalues of the p -Laplacian which are not of Ljusternik-Schnirelmann type

In this paper we demonstrate the fact that the famous Ljusternik–Schnirelmann characterization of some eigenvalues of nonlinear elliptic problems (by a minimax formula) has a global variational character. Indeed, we show that, for some homogeneous quasilinear elliptic eigenvalue problems, there are variational eigenvalues other than those of the Ljusternik–Schnirelmann type. In contrast, these eigenvalues have a local variational character. Such a phenomenon does not occur in typical linear elliptic eigenvalue problems, owing to the Courant–Fischer theorem which is the linear analogue and predecessor of the Ljusternik–Schnirelmann theory.

Induced Graph Packing Problems

Let $${\mathcal{H}}$$be a set of undirected graphs. The induced $${\mathcal{H}}$$-packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of $${\mathcal{H}}$$. In this paper we focus on the case when $${\mathcal{H}}$$consists of factor-critical graphs and a certain family of ‘propellers’. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai–Edmonds type structure theorem and a Berge–Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when $${\mathcal{H}}$$consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.

The Dirichlet problem for the Bellman equation at resonance

We generalize the Donsker–Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance.

A Minimax Formula for the Best Natural C([0, 1])-Approximate by Nondecreasing Functions

A Minimax Formula for the Best Natural <em>C</em>([0, 1])-Approximate by Nondecreasing Functions

Stochastic homogenization of Hamilton-Jacobi-Bellman equations

We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment. © 2005 Wiley Periodicals, Inc.

The Edmonds-Gallai Decomposition for the $k$-Piece Packing Problem

Generalizing Kaneko's long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the $k$-piece packing problem. A $k$-piece is a simple, connected graph with highest degree exactly $k$ so in the case $k=1$ we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula for the $k$-piece packing problem. However, they leave open the question of an Edmonds-Gallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by $k$-piece packings have a certain matroidal structure.

A minimax formula for principal eigenvalues and application to an antimaximum principle

A minimax formula for the principal eigenvalue of a nonselfadjoint elliptic problem was established in [17]. In this paper we extend this formula to the case where an indefinite weight is present. An application is given to the study of the uniformity of the antimaximum principle.

Hamiltonian powers in threshold and arborescent comparability graphs

We examine powers of Hamiltonian paths and cycles as well as Hamiltonian (power) completion problems in several highly structured graph classes. For threshold graphs we give efficient algorithms as well as sufficient and minimax toughness like conditions. For arborescent comparability graphs we have similar results but also show that for one type of completion problem an ·obvious’ minimax condition fails. For cographs we give examples showing that toughness and other ·obvious’ necessary conditions are not sufficient. For threshold graphs we give additional necessary and sufficient conditions in terms of vertex degrees as well as a minimax formula for the length of a longest cycle power.

Graph minors. X. Obstructions to tree-decomposition

Roughly, a graph has small “tree-width” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: 1. (i) a minimax formula relating tree-width with the largest such obstructions 2. (ii) an association between such obstructions and large grid minors of the graph 3. (iii) a “tree-decomposition” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers.

Characterization of the maximum genus of a signed graph

A signed graph is a graph in which each edge is labelled with a sign. A cycle C of a signed graph is balanced if the product of signs on C is positive. By an embedding of a signed graph G on a closed surface S we mean a topological embedding of the corresponding unsigned graph satisfying the following additional condition: A cycle C of G is embedded as an orientation-preserving cycle on S if and only if C is balanced. Let ε M( G) be the maximum Euler genus of a closed surface in which the signed graph G has a 2-cell embedding. We establish a minimax formula for ε M( G). The method used provides a unification of several results in maximum genus of unsigned graphs. For example, Xuong's and Nebeský's formulas for the maximum orientable genus of a graph and the Ringel-Stahl singleface nonorientable embedding theorem are corollaries of the work in this paper.