Totally Positive Research Articles

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Total positivity and least squares problems in the Lagrange basis

SummaryThe problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

On the Total Positivity and Accurate Computations of r-Bell Polynomial Bases

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure.

Large N analytical functional bootstrap. Part I. 1D CFTs and total positivity

We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on the 1D O(N) vector bootstrap. We obtain a remarkably simple bootstrap equation from the O(N) vector crossing equations in the large N limit. The numerical conformal bootstrap bound is saturated by the generalized free field theories, while its extremal functional actions do not converge to any non-vanishing limit. We study the analytical extremal functionals of this crossing equation, for which the total positivity of the SL(2, ℝ) conformal block plays a critical role. We prove the SL(2, ℝ) conformal block is totally positive in the limits with large ∆ or small 1 − z and show that the total positivity is violated below a critical value {Delta }_{textrm{TP}}^{ast } ≈ 0.32315626. The SL(2, ℝ) conformal block forms a surprisingly sophisticated mathematical structure, which for instance can violate total positivity at the order 10−5654 for a normal value ∆ = 0.1627! We construct a series of analytical functionals {αM} which satisfy the bootstrap positive conditions up to a range ∆ ⩽ ΛM. The functionals {αM} have a trivial large M limit. However, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high ΛM. Moreover, in the region ∆ ⩽ ΛM, the analytical functional actions are consistent with the numerical bootstrap results, therefore it clarifies the positive structure in the crossing equation analytically. Our result provides a concrete example to illustrate how the analytical properties of the conformal block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N analytical functional bootstrap in higher dimensions.

Total positivity and accurate computations with Gram matrices of Said‐Ball bases

AbstractIn this article, it is proved that Gram matrices of totally positive bases of the space of polynomials of a given degree on a compact interval are totally positive. Conditions to guarantee computations to high relative accuracy with those matrices are also obtained. Furthermore, a fast and accurate algorithm to compute the bidiagonal factorization of Gram matrices of the Said‐Ball bases is obtained and used to compute to high relative accuracy their singular values and inverses, as well as the solution of some linear systems associated with these matrices. Numerical examples are included.

On two notions of total positivity for partial flag varieties

Given integers 1≤k1<⋯<kl≤n−1, let Flk1,…,kl;n denote the type A partial flag variety consisting of all chains of subspaces (Vk1⊂⋯⊂Vkl) inside Rn, where each Vk has dimension k. Lusztig (1994, 1998) introduced the totally positive part Flk1,…,kl;n>0 as the subset of partial flags which can be represented by a totally positive n×n matrix, and defined the totally nonnegative part Flk1,…,kl;n≥0 as the closure of Flk1,…,kl;n>0. On the other hand, following Postnikov (2007), we define Flk1,…,kl;nΔ>0 and Flk1,…,kl;nΔ≥0 as the subsets of Flk1,…,kl;n where all Plücker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1,…,kl are consecutive integers. We give an elementary proof of this result (including for the case of Grassmannians) based on classical results in linear algebra and the theory of total positivity. We also show that the cell decomposition of Flk1,…,kl;n≥0 coincides with its matroid decomposition if and only if k1,…,kl are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and Fl1,3;4. Finally, we determine which notions of positivity are compatible with a natural action of the cyclic group of order n that rotates the index set.

Totally positive kernels, Pólya frequency functions, and their transforms

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

Total positivity and relative convexity of option prices

This paper studies total positivity and relative convexity properties in option pricing models. We introduce these properties in the Black-Scholes setting by showing the following: out-of-the-money calls are totally positive in strike and volatility; out-of-the-money puts have a reverse sign rule property; calls and puts are convex with respect to at-the-money prices; and relative convexity of option prices implies a convexity-in-time property of the underlying. We then extend these properties to other models, including scalar diffusions, mixture models, and certain Lévy processes. We show that relative convexity typically holds in time-homogeneous local volatility models through the Dupire equation. We develop implications of these ideas for empirical option prices, including constraints on the at-the-money skew. We illustrate connections with models studied by Peter Carr, including the variance-gamma, CGMY, Dagum, and logistic density models.

On stochastic orders and total positivity

The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair (X, Y), it is shown that the conditional distributions of Y, given X = x, are increasing in x with respect to the likelihood ratio order if and only if the joint distribution of (X, Y) is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned.

Convolutions of totally positive distributions with applications to kernel density estimation

In this work we study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a random vector implies a strong form of positive dependence between its coordinates and, in particular, it implies positive association. Since estimating a totally positive density is a non-parametric problem, we take on a (modified) kernel density estimation approach. Our main result is that the sum of scaled standard Gaussian bumps centered at a min-max closed set provably yields a totally positive distribution. Hence, our strategy for producing a totally positive estimator is to form the min-max closure of the set of samples, and output a sum of Gaussian bumps centered at the points in this set. We can frame this sum as a convolution between the uniform distribution on a min-max closed set and a scaled standard Gaussian. We further conjecture that convolving any totally positive density with a standard Gaussian remains totally positive.

Higher secondary polytopes and regular plabic graphs

Given a configuration A of n points in Rd−1, we introduce the higher secondary polytopes▪, which have the property that ▪ agrees with the secondary polytope of Gelfand–Kapranov–Zelevinsky, while the Minkowski sum of these polytopes agrees with Billera–Sturmfels' fiber zonotope associated with (a lift of) A. In a special case when d=3, we refer to our polytopes as higher associahedra. They turn out to be related to the theory of total positivity, specifically, to certain combinatorial objects called plabic graphs, introduced by the second author in his study of the totally positive Grassmannian. We define a subclass of regular plabic graphs and show that they correspond to the vertices of the higher associahedron ▪, while square moves connecting them correspond to the edges of ▪. Finally we connect our polytopes to soliton graphs, the contour plots of soliton solutions to the KP equation, which were recently studied by Kodama and the third author. In particular, we confirm their conjecture that when the higher times evolve, soliton graphs change according to the moves for plabic graphs.

Scaffoldings of totally positive matrices and line insertion

Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work, we revisit this problem. First, we show that every totally positive matrix can be associated with a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.

Proof of a conjecture on the total positivity of amazing matrices

Let n and b be positive integers. Define the amazing matrix Pn,b=[P(i,j)]i,j=0n−1 to be an n×n matrix with entriesP(i,j)=1bn∑r≥0(−1)r(n+1r)(n−1−i+(j+1−r)bn). Diaconis and Fulman conjectured that the amazing matrix is totally positive. We give an affirmative answer to this conjecture.

Characterizing total positivity: Single vector tests via linear complementarity, sign non‐reversal and variation diminution

A matrix A $A$ is called totally positive (or totally non-negative) of order k $k$ , denoted by T P k $TP_k$ (or T N k $TN_k$ ), if all minors of size at most k $k$ are positive (or non-negative). These matrices have featured in diverse areas in mathematics, including algebra, analysis, combinatorics, differential equations and probability theory. The goal of this article is to provide a novel connection between total positivity and optimization/game theory. Specifically, we draw a relationship between totally positive matrices and the linear complementarity problem (LCP), which generalizes and unifies linear and quadratic programming problems and bimatrix games — this connection is unexplored, to the best of our knowledge. We show that A $A$ is T P k $TP_k$ if and only if for every submatrix A r $A_r$ of A $A$ formed from r $r$ consecutive rows and r $r$ consecutive columns (with r ⩽ k $r\leqslant k$ ), LCP ( A r , q ) $\mathrm{LCP}(A_r,q)$ has a unique solution for each vector q < 0 $q<0$ . In fact this can be strengthened to check the solution set of the LCP at a single vector for each such square submatrix. These novel characterizations are in the spirit of classical results characterizing T P $TP$ matrices by Gantmacher–Krein [Compos. Math. 1937] and P $P$ -matrices by Ingleton [Proc. London Math. Soc. 1966]. Our work contains two other contributions, both of which characterize total positivity using single test vectors whose coordinates have alternating signs — that is, lie in a certain open bi-orthant. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., 2021], which provided a novel characterization of T P k $TP_k$ matrices using sign non-reversal phenomena. We further improve on a classical characterization of total positivity by Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 1981] (following Gantmacher–Krein, 1950) involving the variation diminishing property. Finally, we use a Pólya frequency function of Karlin [Trans. Amer. Math. Soc. 1964] to show that our aforementioned characterizations of total positivity, involving (single) test-vectors drawn from the ‘alternating’ bi-orthant, do not work if these vectors are drawn from any other open orthant.

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m! , phi _m for each vertex with m proper children. We show that if the weight sequence varvec{phi } is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

Yet another criterion for the total positivity of Riordan arrays

Let R=R(d(t),h(t)) be a Riordan array, where d(t)=∑n≥0dntn and h(t)=∑n≥0hntn. We show that if the matrix[d0h000⋯d1h1h00d2h2h1h0⋮⋮⋱] is totally positive, then so is the Riordan array R.

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if both of the sign patterns $(1,-1)$ and $(-1,1)$ are represented in the sample for every pair of variables; in particular, this implies that the MLE may exist with $n=d$ observations, in stark contrast to unrestricted binary exponential families where $2^d$ observations are required. Finally, we provide a novel and globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders.