Standard Simplex Research Articles

Sampling and Change of Measure for Monte Carlo Integration on Simplices

Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices.

Some characterizations of the complex projective space via Ehrhart polynomials

Let [Formula: see text] be the Ehrhart polynomial associated to an integral multiple [Formula: see text] of the standard simplex [Formula: see text]. In this paper, we prove that if [Formula: see text] is an [Formula: see text]-dimensional polarized toric manifold with associated Delzant polytope [Formula: see text] and Ehrhart polynomial [Formula: see text] such that [Formula: see text], for some [Formula: see text], then [Formula: see text] (where [Formula: see text] is the hyperplane bundle on [Formula: see text]) in the following three cases: (1) arbitrary [Formula: see text] and [Formula: see text], (2) [Formula: see text] and [Formula: see text] and (3) [Formula: see text] under the assumption that the polarization [Formula: see text] is asymptotically Chow semistable.

Deformed Graphical Zonotopes

AbstractWe study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph G, consisting of independent equations defining its linear span (in terms of non-cliques of G) and of the inequalities defining its facets (in terms of common neighbors of neighbors in G). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in G form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if G is triangle-free.

Power enhancement and phase transitions for global testing of the mixed membership stochastic block model

The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an n-node symmetric network generated from a K-community MMSBM, we would like to test K=1 versus K>1. We first study the degree-based χ2 test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a “signal” matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the n membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity βn(K,P,h), which depends on the community structure matrix P and the mean vector h of memberships. For each given (K,P,h), a test is called optimal if it distinguishes two hypotheses when βn(K,P,h)→∞. A test is called optimally adaptive if it is optimal for all (K,P,h). We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular (K,P,h), hence, not optimally adaptive.

From the simplex to the sphere: faster constrained optimization using the Hadamard parametrization

Abstract The standard simplex in $\mathbb{R}^{n}$, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. It frequently appears as a constraint in optimization problems that arise in machine learning, statistics, data science, operations research and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that Karush-Kuhn-Tucker points and strict-saddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient and projection-free algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex or $\ell _{1}$-norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach. Open source code is available at https://github.com/DanielMckenzie/HadRGD.

Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of gksl-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system (“toy model”) on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of [Formula: see text]-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.

Standard simplex induced clustering with hierarchical deep dictionary learning

Clustering remains a fundamental but crucial task on unsupervised pattern recognition. A recent novel deep dictionary learning system where hierarchical discriminative dictionary learning layers are embedded within a neural network showed either competitive or state-of-the-art results for image classification. We now propose a new method for image clustering, extending hierarchical deep dictionary learning to the unsupervised learning setting. Clustering is induced in the neural network in a simple way by requesting vector outputs associated to categories to approximately lie in the corners of a standard simplex and by approximating a given expected value on such vectors. This is carried out in a scalable way due to the batch-wise updates as clustering is learned in the deep system. We evaluate our proposal on four benchmark datasets and we either achieve competitive results or outperform state-of-the-art clustering methods. Under the unbalanced scenario clustering is known to be more challenging. For this our method additionally considers Bayesian updates to automatically learn the correct cluster proportions for a small number of clusters.

Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel--Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.

Loss functions for finite sets

This paper studies loss functions for finite sets. For a given finite set S, we give sum-of-square type loss functions of minimum degree. When S is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When S is approximately given by a sample set T, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.

Lonely Points in Simplices

Given a lattice Lsubseteq mathbb Z^m and a subset Asubseteq mathbb R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

A Hierarchy of Standard Polynomial Programming Formulations for the Maximum Clique Problem

The maximum clique problem is a classical optimization problem which finds many important applications. Some of the most theoretically interesting and computationally successful approaches to this problem are based on the Motzkin--Straus formulation, expressing the clique number in terms of the maximal value of a standard quadratic program. This intriguing relationship has also motivated significant developments in quadratic optimization, copositive programming, and complexity in nonlinear optimization. Inspired by such developments, this paper establishes a hierarchy of polynomial programming formulations (\textbfP$^k$), $k\in\{2,\ldots, \omega\}$ for the maximum clique problem, relating the clique number $\omega$ of a given graph to the global maximal value of a multilinear polynomial function $f_k(x)$ of degree $k$ over the standard simplex in $\mathbb{R}^{|V|}$. The case of $k=2$ corresponds to the Motzkin--Straus formulation, and the hierarchical feature is in the fact that the set of local maxima of (\textbfP$^{k+1}$) is a subset of the set of local maxima of (\textbfP$^{k}$), $k\in\{2,\ldots, \omega-1\}$. In particular, every local maximum of (\textbfP$^{\omega}$) is global.

Characterization of Polynomials Whose Large Powers have Fully Positive Coefficients

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of Colin Tan and the author, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

Symmetry Reduction to Optimize a Graph-based Polynomial From Queueing Theory

For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden (arXiv:2111.05777, 2021) asked whether $f_d$, which arises in a model of job-occupancy in redundancy scheduling, attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAM J. Optim. 31 (2021), 2227--2254] proved that $f_d$ is convex over the standard simplex if $d=2$ and $d=3$, implying the desired result for these $d$. We give a symmetry reduction to show that for fixed $d$, the polynomial is convex over the standard simplex (for all $n\geq 2$) if a constant number of constant matrices (with size and coefficients independent of $n$) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial $f_d$ is convex for $d\leq 9$.

A Hybrid Multi-GPU Implementation of Simplex Algorithm with CPU Collaboration

The simplex algorithm has been successfully used for many years in solving linear programming (LP) problems. Due to the intensive computations required (especially for the solution of large LP problems), parallel approaches have also extensively been studied. The computational power provided by the modern GPUs as well as the rapid development of multicore CPU systems have led OpenMP and CUDA programming models to the top preferences during the last years. However, the desired efficient collaboration between CPU and GPU through the combined use of the above programming models is still considered a hard research problem. In the above context, we demonstrate here an excessively efficient implementation of standard simplex, targeting to the best possible exploitation of the concurrent use of all the computing resources, on a multicore platform with multiple CUDA-enabled GPUs. More concretely, we present a novel hybrid collaboration scheme which is based on the concurrent execution of suitably spread CPU-assigned (via multithreading) and GPU-offloaded computations. The experimental results extracted through the cooperative use of OpenMP and CUDA over a notably powerful modern hybrid platform (consisting of 32 cores and two high-spec GPUs – Titan Rtx and Rtx 2080Ti) highlight that the performance of the presented here hybrid GPU/CPU collaboration scheme is clearly superior to the GPU-only implementation under almost all conditions. The corresponding measurements validate the value of using all resources concurrently, even in the case of a multi-GPU configuration platform. Furthermore, the given implementations are completely comparable (and slightly superior in most cases) to other related attempts in the bibliography, and clearly superior to the native CPU implementation with 32 cores. Keywords—Parallel Computing; Linear Programming; Simplex Method; Multicore Platform; Hardware Acceleration, GPGPU; OpenMP; CUDA

Algorithmic Symplectic Packing

In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to Z n , and we require the embeddings to induce isomorphisms between first homology groups. In this case, Miller Maley, Mastrangeli, and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.

Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmetics

Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic, a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The systematic tests conducted up to dimension $n$ = 100 on the cube, the isotropic and the standard simplex show that HMC significantly outperforms HAR both in terms of accuracy and running time. Additional tests show that calculations may be handled up to dimension $n$ = 500. These tests also establish that multiprecision is mandatory to avoid exits from the polytope.

Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

Quadratic non-stochastic operators: examples of splitted chaos

There is one-to-one correspondence between quadratic operators (mapping \({\mathbb {R}}^m\) to itself) and cubic matrices. It is known that any quadratic operator corresponding to a stochastic (in a fixed sense) cubic matrix preserves the standard simplex. In this paper we find conditions on the (non-stochastic) cubic matrix ensuring that corresponding quadratic operator preserves simplex. Moreover, we construct several quadratic non-stochastic operators which generate chaotic dynamical systems on the simplex. These chaotic behaviors are splitted meaning that the simplex is partitioned into uncountably many invariant (with respect to quadratic operator) subsets and the restriction of the dynamical system on each invariant set is chaos in the sense of Devaney.

Two-stage stochastic standard quadratic optimization

Two-stage stochastic decision problems are characterized by the property that in stage one part of the decision has to be made before relevant data are observed and the rest of the decision can be made in stage two after data are available. It is generally assumed that while the data missing in stage one are not known, their probability distribution is known. In this paper, we consider a special form of such problems where the objective function is quadratic in the first and second stage decisions and the goal is to minimize the expected quadratic cost function. We assume that the decisions must lie in a standard simplex, but do not require convexity of the uncertain objective. It is well known that such quadratic optimization programs are NP-complete and thus belong to the most complex optimization problems. A viable method to attack such problems is to find bounds for the original complicated problem by solving easier approximate problems. We describe such approximations and show that by exploiting their properties, good but not necessarily optimal solutions can be found. We also present possible applications, such as selecting invited speakers for conferences with the aim to generate a community with high coherence. Finally, systematic numerical results are provided.

Escort Function Definitions for Feasible Region Flexibility in Applications of Replicatorlike Dynamics With Constraints

This articleproposes an extension to a family of evolutionary game dynamics called escort replicator dynamics (ERD), defined by (Harper, 2011) based on information-geometric concepts. Specifically, the objective of this article is to propose three types of convenient so-called escort functionsand evaluate the stability of ERD under the use of these functions. Since it is proven that ERD is stable using the proposed definitions, ERD is able to confine trajectories of the state vector in regions defined by upper and lower boundaries which can be, but are not required to be, nonnegative. From an engineering application perspective, these proposed escort functions are convenient since they allow to extend replicatorlike dynamics applications to engineering problems with feasible regions more general than the standard simplex or the nonnegative orthant. From a population dynamics perspective the proposed escort functions are useful to represent upper and lower limits on the share of a population that is allowed to play a given pure strategy of the underlying symmetric game.