Nonnegative Coordinates Research Articles

Embeddings as epistemic states: Limitations on the use of pooling operators for accumulating knowledge

Various neural network architectures rely on pooling operators to aggregate information coming from different sources. It is often implicitly assumed in such contexts that vectors encode epistemic states, i.e. that vectors capture the evidence that has been obtained about some properties of interest, and that pooling these vectors yields a vector that combines this evidence. We study, for a number of standard pooling operators, under what conditions they are compatible with this idea, which we call the epistemic pooling principle. While we find that all the considered pooling operators can satisfy the epistemic pooling principle, this only holds when embeddings are sufficiently high-dimensional and, for most pooling operators, when the embeddings satisfy particular constraints (e.g. having non-negative coordinates). We furthermore show that these constraints have important implications on how the embeddings can be used in practice. In particular, we find that when the epistemic pooling principle is satisfied, in most cases it is impossible to verify the satisfaction of propositional formulas using linear scoring functions, with two exceptions: (i) max-pooling with embeddings that are upper-bounded and (ii) Hadamard pooling with non-negative embeddings. This finding helps to clarify, among others, why Graph Neural Networks sometimes under-perform in reasoning tasks. Finally, we also study an extension of the epistemic pooling principle to weighted epistemic states, which are important in the context of non-monotonic reasoning, where max-pooling emerges as the most suitable operator.

Positive semigroups in lattices and totally real number fields

Abstract Let L be a full-rank lattice in ℝ d and write L + for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L +. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L +, and we investigate the distribution of the gaps of S(X) in L +, i.e. the points in L + ∖ S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L + and of L + ∖ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.

Переходные вероятности марковского процесса на отрезке, лежащем в четверти плоскости

The paper considers a quadratic birth-death Markov process. The points on a line segment located within a quarter-plane represent the states of the random process. We designate the set of vectors that have integer non-negative coordinates as our quarter plane. The process is defined by infinitesimal characteristics, or transition probability densities. These characteristics are determined by a quadratic function of the coordinates at the segment points with integer coordinates. The boundary points of the segment are absorbing; at these points, the random process stops. We investigated a critical case when process jumps are equally probable at the moment of exiting a point. We derived expressions describing transition probabilities of the Markov process as a spectral series. We used a two-dimensional exponential generating function of transition probabilities and a two-dimensional generating function of transition probabilities. The first and second systems of ordinary differential Kolmogorov equations for Markov process transition probabilities are reduced to second-order mixed type partial differential equations for a double generating function. We solve the resulting system of linear equations using separation of variables. The spectrum obtained is discrete. The eigen-functions are expressed in terms of hypergeometric functions. The particular solution constructed is a Fourier series, whose coefficients are derived by means of expo-nential expansion. We employed sums of functional series known in the theory of special functions to construct the exponential expansion required

Anisotropic multivariate deconvolution using projection on the Laguerre basis

We investigate adaptive density estimation in the additive model Z=X+Y, where X and Y are independent d-dimensional random vectors with non-negative coordinates. Our goal is to recover the density of X from independent observations of Z, assuming the density of Y is known. In the d=1 case, an estimation procedure using projection on the Laguerre basis has already been studied. We generalize this procedure in the multivariate case: we establish non-asymptotic upper bounds on the mean integrated squared error of the estimator and we derive convergence rates on anisotropic functional spaces. Moreover, we provide data-driven strategies for selecting the right projection space (for d=1, we improve the previous projection procedure). We illustrate these procedures on simulated data, and in dimension d=1 we compare our procedure with the previous adaptive projection procedure.

ВЕКТОРНАЯ МОДЕЛЬ МНОГОКРИТЕРИАЛЬНОЙ ОЦЕНКИ

To assess the performance of economic systems, criteria and performance indicators are used that show the degree of achievement of the desired performance result. To characterize complex systems in most real cases, one criterion is not enough. Therefore, several criteria are used to comprehensively characterize the system. The multi-criteria problem raises the problem of matching individual criteria when used together. There are many ways to select alternatives in the face of multiple decision criteria. This article discusses the multi-criteria comparison of objects with Pareto dominance, vector estimates with arbitrary non-negative coordinates of vectors. The advantages of vector estimates in comparison with average estimates are considered. In this work, three important theorems of multi-criteria estimation are proved. The theorems define the conditions for the equivalence of three options for comparing objects by a variety of attributes: 1) arithmetic mean values; 2) the lengths of the corresponding vectors; 3) scalar products of vectors. As the research results, the application of the developed methods for aggregating the quality indicators of the educational process into a single assessment is shown. The following indicators of the educational process were used: coefficient of diligence, coefficient of interest, coefficient of difficulty of mastering the studied material, assessment of academic performance. A four-criteria optimization problem is formulated. It is shown that the objective function reaches its extreme value at the point that is as close as possible to the point of the optimal solution of the formulated problem. The issues of multi-criteria assessment developed in the article can find application in the management of the educational process in the formation of individual trajectories of trainees, in the tasks of working with the personnel of firms and organizations (selection, training, retraining), in assessing the quality of products and in other areas of activity.

2–3 Paths in a Lattice Graph: Random Walks

A lattice graph with 2–3 reachability constraints is considered. The graph’s vertices are the points with integer nonnegative coordinates in the plane. Each vertex has two outgoing edges, one entering its immediate right neighbor and the other entering its immediate upper neighbor. The admissible paths for 2–3 reachability are those in which the numbers of edges in all but the last inclusion-maximal straight-line segments are divisible by 2 for horizontal segments and by 3 for vertical segments. A formula for the number of 2–3 paths from a vertex to a vertex is obtained. A random walk process on the 2–3 paths in the lattice graph is considered. It is proved that this process can locally be reduced to a Markov process on subgraphs determined by the type of the initial vertex. Formulas for the probabilities of vertex-to-vertex transitions along 2–3 paths are obtained.

Cantor Polynomials and the Fueter-Pólya Theorem

A packing polynomial is a polynomial that maps the set N20 of lattice points with nonnegative coordinates bijectively onto N0. Cantor constructed two quadratic packing polynomials, and Fueter and Pólya proved analytically that the Cantor polynomials are the only quadratic packing polynomials. The purpose of this paper is to present a beautiful elementary proof of Vsemirnov of the Fueter–Pólya theorem. It is a century-old conjecture that the Cantor polynomials are the only packing polynomials on N20.

Wright–Fisher diffusion with negative mutation rates

We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher diffusions, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Going-Jaeschke and Yor.

Computation of multivariate normal probabilities with polar coordinate systems

We consider the problem of evaluation of the probability that all elements of a multivariate normally distributed vector have non-negative coordinates; this probability is called the non-centred orthant probability. The necessity for the evaluation of this probability arises frequently in statistics. The probability is defined by the integral of the probability density function. However, direct numerical integration is not practical. In this article, a method is proposed for the computation of the probability. The method involves the evaluation of a measure on a unit sphere surface in p-dimensional space that satisfies conditions derived from a covariance matrix. The required computational time for the p-dimensional problem is proportional to p2·2p−1, and it increases at a rate that is lower than that in the case of the existing method.

Klein polyhedra and relative minima of lattices

We prove that in ℝ3, the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron. We also prove, for almost any lattice in ℝ3, that the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.

Mixed Powers of Generating Functions

Given an integer $m \geq 1$, let $\| \cdot \|$ be a norm in $\mathbb{R}^{m+1}$ and let $\mathbb{S}_+^m$ denote the set of points $\mathbf{d}=(d_0,\ldots,d_m)$ in $\mathbb{R}^{m+1}$ with nonnegative coordinates and such that $\| \mathbf{d} \|=1$. Consider for each $1 \leq j \leq m$ a function $f_j(z)$ that is analytic in an open neighborhood of the point $z=0$ in the complex plane and with possibly negative Taylor coefficients. Given $\mathbf{n}=(n_0,\ldots,n_m)$ in $\mathbb{Z}^{m+1}$ with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of $z^{n_0}$ of the Taylor series of $\prod_{j=1}^m \{f_j(z)\}^{n_j}$, as $\| \mathbf{n} \| \to \infty$. The associated parameter-varying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many $\mathbf{d} \in \mathbb{S}_+^m$, these methods ensure uniform asymptotic expansions for $[z^{n_0}] \prod_{j=1}^m \{f_j(z)\}^{n_j}$ provided that $\mathbf{n}/ \| \mathbf{n} \|$ stays sufficiently close to $\mathbf{d}$ as $\| \mathbf{n} \| \to \infty$. Our method finds applications in studying the asymptotic behavior of the coefficients of a certain multivariable generating functions as well as in problems related to the Lagrange inversion formula for instance in the context random planar maps.

Almost all normal sets are strictly normal

We consider the space Sn of all nonempty bounded closed normal subsets of the cone where is the set of all vectors x ∈ Rn with nonnegative coordinates. We equip the space Sn with the Hausdorff metric and show that most elements of Sn are, in fact, strictly normal. More precisely, we show that the complement of the collection of all stricly normal elements of Sn is a σ-porous subset of Sn.

An alogrithm for monotonic global optimization problems*

We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We eastablish the con vergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited

Structure of optimal trajectories of convex processes

In this paper we study the dynamic properties of optimal trajectories of convex processes where K⊂ R nis a closed convex cone. We show that optimal trajectories of a convex process Gspend most of the time in a small neighborhood of a von Neumann path. The turnpike theorem that we obtain generalizes the result of Rubinov [10] which was established for a convex process where is the cone of the elements of the Euclidean space R nthat have nonnegative coordinates. Also, we show that the turnpike phenomenon is stable under small pertubations of a convex process G.

Edge reduction for weakly non-negative quadratic forms

Let \( q\colon \Bbb Z^n\to \Bbb Z \) be an integral quadratic form of the shape¶¶\( q(x)=\sum\limits\limits ^n_{i=1}q_ix^2_i+\sum\limits\limits _{i\le j}q_{ij}x_ix_j \)¶¶with \( q_i\le 1 \), for every \( 1\le i\le n \). Several procedures have been introduced to study these forms. In this paper we consider the edge reduction procedure introduced in [7]. ¶A form \( q\colon \Bbb Z^n\to \Bbb Z \) is weakly non-negative if \( q(x)\ge 0 \) for every vector x with non-negative coordinates. Let \( q^\prime \colon \Bbb Z^{n+1}\to \Bbb Z \) be obtained from q by edge reduction, then q is weakly non-negative if and only if so is \( q^\prime \). We propose an algorithm to decide if q is weakly non-negative.

Asymptotic stationarity of multichannel queues

The paper deals with asymptotic stationarity of the process where is a vector in with non-negative coordinates, is an -valued process, S is a separable metric space and all operations in are meant in the coordinate-wise sense. It is shown that a type of asymptotic stationarity of (X, Y), together with some conditions, implies the same type of asymptotic stationarity of (w, X, Y). This result is applied to analyze asymptotic stationarity of multichannel queues. It may also be used to analyze asymptotic stationarity of series of multichannel queues.

Asymptotic stationarity of multichannel queues

The paper deals with asymptotic stationarity of the process where is a vector in with non-negative coordinates, is an -valued process, S is a separable metric space and all operations in are meant in the coordinate-wise sense. It is shown that a type of asymptotic stationarity of (X, Y), together with some conditions, implies the same type of asymptotic stationarity of (w, X, Y). This result is applied to analyze asymptotic stationarity of multichannel queues. It may also be used to analyze asymptotic stationarity of series of multichannel queues.

A characterization of semilinear sets

Introduction. The recent interest in the structure of programming languages has led to the study of their mathematical properties. Characterizations of bounded context-free languages (also called bounded ALGOL-like languages) [1] and bounded regular sets [3] have been given in terms of certain semilinear subsets of Nn. Semilinear sets have been extensively studied as subsets of lattice points in n-space which are finite unions of cosets of finitely generated subsemigroups of the set of all lattice points with nonnegative coordinates and which are also shown to be equivalent to the family of sets defined by modified Presburger formulas [2]. In this note we give a characterization and discuss decision procedures for semilinear sets of words (hereafter called semilinear sets) [4] which include bounded context-free languages and hence bounded regular sets.

The solution of positive-symmetric hyperbolic systems by difference methods

where u and f are m-vector-valued functions, Ai and B are m Xrm matrix-valued functions, and x = (X1, X2, . , x.n) varies over all points of the real Euclidean n-space En which have non-negative coordinates. The system (1.1) is called positive-symmetric hyperbolic if the matrices Ai are symmetric and positive-definite. Such systems are included in the general class of symmetric hyperbolic systems introduced by Friedrichs [1]. Symmetric systems are of particular interest because they possess an energy inequality; that is, the square integral of the solution over a space-like hyperplane can be estimated in terms of square integrals of initial values of the solution and the nonhomogeneous term in the differential equation. We shall not consider the Cauchy problem for (1.1), but rather, we prescribe the initial values on the coordinate hyperplanes xi= 0, i.e.,

Analysis of the Two-Mode-of-Fission Hypothesis

Mass yields for gamma fission of ${\mathrm{U}}^{238}$, proton fission of ${\mathrm{Th}}^{232}$, alpha fission of ${\mathrm{U}}^{238}$, and deuteron fission of normal uranium have been examined in terms of vector spaces for consistency with the hypothesis that for each target and projectile there are two and only two modes of fission. The first three cases are on the whole consistent with the hypothesis. The fourth is not consistent with the hypothesis; the measured yields include some yields from neutron fission. Some consequences of the hypothesis with the added requirement of non-negative yields and non-negative coordinates are derived.