Totally Positive Research Articles

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ (or $TN_k$), if all minors of size $\leq k$ are positive (or non-negative). It is well-known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that $A$ is $TP_k$ if and only if every submatrix formed from at most $k$ consecutive rows and columns has the sign non-reversal property. In fact this can be strengthened to only consider test vectors in $\mathbb{R}^k$ with alternating signs. We also show a similar characterization for all $TN_k$ matrices - more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing $TP$ matrices by Gantmacher-Krein [Compos. Math. 1937] and $P$-matrices by Gale-Nikaido [Math. Ann. 1965]. As an application, we study the interval hull $\mathbb{I}(A,B)$ of two $m \times n$ matrices $A=(a_{ij})$ and $B = (b_{ij})$. This is the collection of $C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is between $a_{ij}$ and $b_{ij}$. Using the sign non-reversal property, we identify a two-element subset of $\mathbb{I}(A,B)$ that detects the $TP_k$ property for all of $\mathbb{I}(A,B)$ for arbitrary $k \geq 1$. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull $\mathbb{I}(A,B)$.

Doubly constrained totally positive line insertion

AbstractIt is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.

Entrainment to subharmonic trajectories in oscillatory discrete-time systems

A matrix A is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix A is called oscillatory if it is TN and some power of A is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time k is oscillatory. We analyze the properties of n-dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-varying and T-periodic then any trajectory either leaves any compact set or converges to an ((n−1)T)-periodic trajectory, that is, a subharmonic trajectory. These results hold for any dimension n. The analysis of such systems requires establishing that a line integral of the Jacobian of the nonlinear system is an oscillatory matrix. This is non-trivial, as the sum of two oscillatory matrices is not necessarily oscillatory, and this carries over to integrals. We derive several new sufficient conditions guaranteeing that the line integral of a matrix is oscillatory, and demonstrate how this yields interesting classes of discrete-time nonlinear systems that admit a well-ordered behavior.

Optimal rates for estimation of two-dimensional totally positive distributions

We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for distributions with $\beta$-Holder smooth densities where $\beta\in(0,2)$, we observe polynomially faster minimax rates of estimation when, additionally, the total positivity condition is imposed. Moreover, we demonstrate fast algorithms to compute the proposed estimators and corroborate the theoretical rates of estimation by simulation studies.

Total positivity of Toeplitz matrices of recursive hypersequences

We present a new class of totally positive Toeplitz matrices composed of recently introduced hyperfibonacci numbers of the r -th generation. As a consequence, we obtain that all sequences F n ( r ) of hyperfibonacci numbers of r -th generation are log-concave for r ≥ 1 and large enough n .

Maximum likelihood estimation in Gaussian models under total positivity

We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_{2}$). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximum likelihood estimator (MLE) for such distributions exists based on $n\geq2$ observations, irrespective of the underlying dimension. Slawski and Hein [Linear Algebra Appl. 473 (2015) 145–179], who first proved this result, also provided empirical evidence showing that the $\mathrm{MTP}_{2}$ constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that we can find an upper bound for the ML graph by adding edges corresponding to correlations in excess of those explained by the maximum weight spanning forest of the correlation matrix. Moreover, we provide globally convergent coordinate descent algorithms for calculating the MLE under the $\mathrm{MTP}_{2}$ constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed $\mathrm{MTP}_{2}$ distributions.

Total positivity of a kind of generalized Toric-Bernstein basis

The normalized totally positive bases are widely used in many fields. Based on the generalized Vandermonde determinant, the normalized total positivity of a kind of generalized toric-Bernstein basis is proved, which is defined on a set of real points. By this result, then the progressive iterative approximation property of the generalized toric-Bézier curve is obtained and some examples are provided to illustrate this property.

Total nonnegativity of GCD matrices and kernels

Let X=(x1,…,xn) be a vector of distinct positive integers. The n×n matrix S=S(X):=(gcd(xi,xj))i,j=1n, where gcd(xi,xj) denotes the greatest common divisor of xi and xj, is called the greatest common divisor (GCD) matrix on X. By a surprising result of Beslin and Ligh (1989) [7], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when n≥3, and is TN if and only if it is TN2, i.e., all its 2×2 minors are nonnegative. We next demonstrate that a GCD matrix is TN2 if and only if the exponents of each prime divisor in the prime factorization of the xi's form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel K(x,y)=gcd(x,y) is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices.

Loading monotonicity of weighted premiums, and total positivity properties of weight functions

We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of generalized weighted premiums; in particular, we deduce for weight functions that are totally positive of order three a monotonicity property of the variance-to-mean ratio, or index of dispersion, of the loss variable. We derive the higher order total positivity properties of some ratios that arise in actuarial and insurance analysis of combined risks. Further, we examine seven classes of weight functions that have appeared in the literature and we ascertain the higher order total positivity properties of those functions.

Revisiting totally positive differential systems: A tutorial and new results

A matrix is called totally nonnegative (TN) [totally positive (TP)] if all its minors are nonnegative [positive]. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, Binyamin Schwarz (1970) considered matrices whose exponentials are TN or TP. He also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system. His work, however, considered only linear systems.In a seemingly different line of research, Smillie (1984), Smith (1991), and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer-valued Lyapunov function.We show that these two research topics are intimately related. This allows to derive important generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. These generalizations are particularly relevant in the context of control systems. Also, the results by Smillie and Smith provide sufficient conditions for analyzing stability based on the number of sign changes in the vector of derivatives and the connection to the work of Schwarz allows to show in what sense these results are also necessary. We describe several new and interesting research directions arising from this new connection.

Total positivity in Markov structures

We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semi-graphoid which is upward-stable and singletontransitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular, we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t≥r−2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.

The Planar Ising Model and Total Positivity

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let a_1,dots ,a_k,b_k,dots ,b_1 be vertices placed in a counterclockwise order on the outer face of G. We show that the ktimes k matrix of the two-point spin correlation functions Mi,j=⟨σaiσbj⟩\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{i,j} = \\langle \\sigma _{a_i} \\sigma _{b_j} \\rangle \\end{aligned}$$\\end{document}is totally nonnegative. Moreover, det M > 0 if and only if there exist k pairwise vertex-disjoint paths that connect a_i with b_i. We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between a_i and b_i in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37].

Isospectral matrix flow maintaining staircase structure and total positivity of an initial matrix

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given bydAdt=[Au−Al,A],A(0)=A0, where A is a real n×n matrix (not necessarily symmetric), [A,B]=AB−BA is the matrix commutator (also known as the Lie bracket), Au is the strictly upper triangular part of A and Al is the strictly lower triangular part of A. We prove that if the initial matrix A0 is staircase, so is A(t). Moreover, we prove that this flow preserves the certain positivity properties of A0. Also we prove that if the initial matrix A0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A(t) converges to a diagonal matrix while preserving the spectrum of A0. Some simulations are provided to confirm the convergence properties.

A note on faithfulness and total positivity

The concept of faithfulness is important for statistical learning of graphical models. In this paper, we show that a faithful positive binary multivariate totally positive of order two (MTP2) distribution exists for every undirected graph, and generalize this result to the case of undirected graphs and positive discrete MTP2 distributions with fixed sample space.

Positivities of Vandermonde tensors

We continue our work on Vandermonde tensors. An mth order n-dimensional real tensor is called a Vandermonde tensor associated with a vector if for every element ,Vandermonde tensors are the generalization of Vandermonde matrices. A real tensor is called a totally positive (non-negative) tensor if each of its hypercubic subtensors has a positive (non-negative) (hyper)determinant. We first show that the product of a Vandermonde tensor with its associated Vandermonde matrix is a Hankel tensor. Then, we investigate the positivities of Vandermonde tensors, including their positive definiteness and total positivity. We also show that a Vandermonde tensor generated by a positive increasing ordered vector is totally positive.

Direct effects of estrogens on cholinergic primary neurons from the human fetal nucleus basalis of Meynert

Epidemiological studies have indicated that Alzheimer’s disease (AD) is more common in females and that post-menopausal women are at increased risk than their male counterpart, thus suggesting that estrogens could play a protective role to counteract neurodegenerative processes (1). However, the mechanisms underlying this association remain to be clarified. Since the nucleus basalis of Meynert (nbM) is the major source of cholinergic innervation selectively vulnerable to degeneration in AD, our study is aimed at investigating the effects of estrogens on human cholinergic primary neurons (hfCNs) isolated from the nbM of 12-week old fetuses. The primary culture obtained was immunophenotyped with flow cytometry and resulted almost totally positive (97±2 %) for the neuronal marker MAP2 and for the choline acetyltransferase (ChAT). We demonstrated that hfCNs express receptors for hormones of the reproductive axis (ERs, LHR, GnRHR). In particular, besides to classical estrogen receptors (ERa and ERb), hfCNs express the transmembrane receptor GPR30, which is known to mediate rapid non-genomic estrogen actions. Increasing concentrations of 17-β estradiol (E2, 0.1-100 nM) determined a dose-dependent significant increase in cell number after 24h exposure, which was antagonized by tamoxifen treatment. In addition, E2 exposure determined a significant increase in ChAT expression, thus indicating a direct positive effect of E2 on cholinergic phenotype. Given that substantial evidence now indicates that estrogens exert an anti-inflammatory activity even in the central nervous system (2), we exposed hfCN cells to the proinflammatory cytokine TNFα. E2 treatment (1nM) was able to significantly counteract the TNFα-induced nuclear NF-kB p65 translocation. Interestingly, this effect was mimicked by G1, a GPR30 agonist, and abolished by pretreating cells with the GPR30 antagonist G15, but not by tamoxifen, which usually antagonizes classical ERs. Overall, our results indicate that estrogens exert direct neuroprotective mechanisms on hfCNs through the activation of either classical (trophic) and non-classical (anti-inflammatory) receptors.

Bruhat interval polytopes

Let u and v be permutations on n letters, with u≤v in Bruhat order. A Bruhat interval polytopeQu,v is the convex hull of all permutation vectors z=(z(1),z(2),…,z(n)) with u≤z≤v. Note that when u=e and v=w0 are the shortest and longest elements of the symmetric group, Qe,w0 is the classical permutohedron. Bruhat interval polytopes were studied recently in [15] by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety.In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and R-polynomials, we also give a generalization of the standard recurrence for R-polynomials. Finally, we define a more general class of polytopes called Bruhat interval polytopes forG/P, which are moment map images of (closures of) totally positive cells in (G/P)≥0, and are a special class of Coxeter matroid polytopes. Using tools from total positivity and the Gelfand–Serganova stratification, we show that the face of any Bruhat interval polytope for G/P is again a Bruhat interval polytope for G/P.

Exploring the total positivity of yields correlations

We test the plausibility of the total positivity assumption of interest rates changes recently introduced in order to justify the presence of shift, slope and curvature for yield curves. To this aim, we introduce and discuss a test of total positivity of order for covariance and correlation matrices. The explicit expressions of the test statistics are given for Gaussian samples and an extension to a distribution-free framework is made via a bootstrap method. After exploring with simulation the robustness of such tests, we show using real data how it is realistic to assume that correlation matrices of interest rates changes are totally positive of order two. Conclusions on total positivity of order three are more controversial.

Zeros of the Zak Transform of Totally Positive Functions

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.