Nonnegative Function Research Articles

Nonconvex low-rank tensor approximation with graph and consistent regularizations for multi-view subspace learning

Multi-view clustering is widely used to improve clustering performance. Recently, the subspace clustering tensor learning method based on Markov chain is a crucial branch of multi-view clustering. Tensor learning is commonly used to apply tensor low-rank approximation to represent the relationships between data samples. However, most of the current tensor learning methods have the following shortcomings: the information of the local graph is not taken into account, the relationships between different views are not shown, and the existing tensor low-rank representation takes a biased tensor rank function for estimation. Therefore, a nonconvex low-rank tensor approximation with graph and consistent regularizations (NLRTGC) model is proposed for multi-view subspace learning. NLRTGC retains the local manifold information through graph regularization, and adopts a consistent regularization between multi-views to keep the diagonal block structure of representation matrices. Furthermore, a nonnegative nonconvex low-rank tensor kernel function is used to replace the existing classical tensor nuclear norm via tensor-singular value decomposition (t-SVD), so as to reduce the deviation from rank. Then, an alternating direction method of multipliers (ADMM) which makes the objective function monotonically non-increasing is proposed to solve NLRTGC. Finally, the effectiveness and superiority of the NLRTGC are shown through abundant comparative experiments with various state-of-the-art algorithms on noisy datasets and real world datasets.

Blind separation of noisy piecewise-stationary mixtures via probability measure transform

In this paper, we consider the problem of blind source separation (BSS) under non-Gaussian impulsive noise. We consider the case of overdetermined instantaneous-linear-mixtures of piecewise-stationary signals. These are corrupted by additive stationary noise. Under this framework, we propose a two-stage separation method, called measure-transformed BSS (MT-BSS), that applies a transform to the probability distribution associated with each data segment. The generating function of the transform at hand is a non-negative function, called MT-function, that weights the data points. We show that proper choice of the involved MT-functions can lead to enhanced separation performance. The performance advantage of MT-BSS over alternative BSS techniques is illustrated in simulation examples. In these studies, we consider synthetic data and real audio signals.

Persistence probabilities of weighted sums of stationary Gaussian sequences

With {ξi}i≥0 being a centered stationary Gaussian sequence with non-negative correlation function ρ(i)≔E[ξ0ξi] and {σ(i)}i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑i=1ℓσ(i)ξi, ℓ≥1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non-summable ρ, even for polynomial weight functions σ(i)∼ip the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(H,p) and study its properties as a function of (p,H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations – e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero – that might be of independent interest.

Two new methods to construct fuzzy metrics from metrics

In the last years, the interest in the notion of fuzzy metric has been growing in such a way that many works have focused their efforts on the study of their topological properties and their applications to Engineering problems. However, the applicability of fuzzy metrics is limited due to lack of examples in the literature. Motivated, on the one hand, by these facts and, on the other hand, by the fact that most of the instances of fuzzy metrics in the literature are constructed from classical metrics, in this paper we introduce two new techniques which allow us to construct systematically fuzzy metrics from metrics in such a way that the celebrated classical method for constructing indistinguishability operators from metrics is retrieved as a particular case. Hence, we construct strong fuzzy metrics from a given classical one considering continuous Archimedean t-norms and the pseudo-inverse of their additive generators acting on the metric modified by a positive real function. Moreover, we extend this technique tackling the particular case of the minimum t-norm, which is continuous but non-Archimedean. In such a construction, two non-negative real functions are now involved in order to modify the classical metric and one of them must be superadditive. In this case, the fuzzy metric obtained is not strong in general. Furthermore, the new methods are illustrated by means of different examples which, in addition, show that some celebrated examples of fuzzy metrics can be retrieved as a particular case through them. Finally, in the light of the developed theory, an open problem about strong fuzzy metrics is solved completing the partial solutions that can be found in the literature.

Failure of the local chain rule for the fractional variation

We prove that the local version of the chain rule cannot hold for the fractional variation defined in in our previous article (2019). In the case n = 1 , we prove a stronger result, exhibiting a function f \in BV^{\alpha}(\mathbb{R}) such that \vert f \vert \notin BV^{\alpha}(\mathbb{R}) . The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019–2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (2020) and we prove a fractional version of the closely related Meyers–Ziemer trace inequality.

Convex sets with obstacle and gradient constraints

For 1<p<∞, given g and ψ non-negative functions belonging to L∞(Ω) and W01,p(Ω)∩C(Ω‾), respectively, we show that there exists a pseudo-metric L‾g defined in Ω‾ such that u(x):=miny∈Ω‾⁡{ψ(y)+L‾g(x,y)} is a subsolution of the Hamilton-Jacobi equation with obstacle max⁡{|∇u|−g,u−ψ}=0. Besides, for Kg,ψ={v∈W01,p(Ω):|∇v|≤g,v≤ψ}, we have u(x)=⋁{v(x):v∈Kg,ψ}.As a consequence, we prove the Mosco convergence of Kgn,ψn to Kg,ψ, as long as gn converges to g in L∞(Ω) and ψn to ψ in C(Ω‾). As an application, we prove a stability result for the solutions un of variational inequalities defined in the convex sets Kgn,ψn.

On the finiteness of strong maximal functions associated to functions whose integrals are strongly differentiable

Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral of f is strongly differentiable. On the other hand, Papoulis proved the existence of an integrable function on R2 (taking on both positive and negative values) whose integral is strongly differentiable but whose associated strong maximal function is infinite on a set of positive measure. In this paper, we prove that if n≥2 and if f is a measurable nonnegative function on Rn whose integral is strongly differentiable and moreover such that f(1+log+⁡f)n−2 is integrable, then MSf is finite a.e. We also show this result is sharp by proving that, if φ is a continuous increasing function on [0,∞) such that φ(0)=0 and with φ(u)=o(u(1+log+⁡u)n−2)(u→∞), then there exists a nonnegative measurable function f on Rn such that φ(f) is integrable on Rn and the integral of f is strongly differentiable, although MSf is infinite almost everywhere.

Spreading speed in a fractional attraction–repulsion chemotaxis system with logistic source

This paper studies an attraction-repulsion chemotaxis system with logistic source and a fractional diffusion of order α∈(0,1) on RN: (0.1)ut=−(−Δ)αu−χ1∇⋅(u∇v1)+χ2∇⋅(u∇v2)+u(a−bu),x∈RN,t>0,0=(Δ−λ1I)v1+μ1u,x∈RN,t>0,0=(Δ−λ2I)v2+μ2u,x∈RN,t>0,where α, χi, μi, λi(i=1,2), a and b are constants. We show that (0.1) has a unique global bounded solution and investigate the asymptotic behavior of the global classical solutions under some assumptions on the parameters. Particularly, we consider the spreading properties of the global classical solutions. Under some conditions for the nonnegative initial function u0, we prove that (0.2)lim inft→∞inf|x|≤ectu(x,t;u0)>0,for all0<c<aN+2αand (0.3)limt→∞sup|x|≥ectu(x,t;u0)=0,for allc>aN+2α.This fact shows that the spreading speed of (0.1) is exponential in time, which is different from the linear spreading speed obtained in Salako and Shen (2019).

Sums of squares II: Matrix functions

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω-monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C2,δ functions for some δ>0, namely that f is ω-monotone for some Hölder modulus of continuity ω. Counterexamples were provided for any larger modulus of continuity.In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M=[akj]k,j=1n can be written as a finite sum of squares of C2,δ vector fields if the diagonal entries akk are ω-monotone for some Hölder modulus of continuity ω, and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed.Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series.

Quantitative description on initial trace for the solvability to heat equation with exponential boundary in unbounded domain

In this paper, we study the solvability of heat equation in R+N with flux ∂∂νu=exp⁡(pu) on ∂R+N for any constant p>0. The initial trace μ(x) is a Radon measure in RN with suppμ⊂R+N‾ or a nonnegative measurable function in R+N. We at first show the necessary conditions on the initial trace for the existence of solutions for every positive p, and then we divide into three cases to give sufficient conditions on the solvability of solutions, where all of the descriptions on the initial trace are determined quantitatively. For some cases, we identify the strongest singularity on the initial trace to guarantee the solvability of this nonlinear parabolic problem.

Multiple Solutions to p-Biharmonic Equations of Kirchhoff Type with Vanishing Potential

In this paper, we study the p-biharmonic equation of Kirchhoff type where is a positive parameter, is the p-Laplacian operator and is the p-biharmonic operator, V, K, g are nonnegative functions, V is vanishing at infinity in the sense that When the nonlinear term satisfies some suitable conditions, we prove that the above problem has at least two nontrivial solutions using the mountain pass theorem combined with the Ekeland variational principle.

Probability distributions and the maximum entropy principle

It is shown that every probability distribution with finite entropy can be characterized as the minimum relative entropy distribution respect to a given non-negative function within a non-trivial collection of probability distributions. This result is extended to families of distributions. We also study sufficient conditions to guarantee the existence and uniqueness of a distribution with maximum entropy on certain families of distributions. Also several examples are presented of how the general results can be applied.

Continuity with respect to the speed for optimal ship forms based on Michell's formula

&lt;p style='text-indent:20px;'&gt;We consider a ship hull design problem based on Michell's wave resistance. The half hull is represented by a nonnegative function and we seek the function whose domain of definition has a given area and which minimizes the total resistance for a given speed and a given volume. We show that the optimal hull depends only on two parameters without dimension, the viscous drag coefficient and the Froude number of the area of the support. We prove that, up to uniqueness, the optimal hull depends continuously on these two parameters. Moreover, the contribution of Michell's wave resistance vanishes as either the Froude number or the drag coefficient goes to infinity. Numerical simulations confirm the theoretical results for large Froude numbers. For Froude numbers typically smaller than &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the famous bulbous bow is numerically recovered. For intermediate Froude numbers, a "sinking" phenomenon occurs. It can be related to the nonexistence of a minimizer.&lt;/p&gt;

Least-Squares Methods for Nonnegative Matrix Factorization Over Rational Functions

Nonnegative matrix factorization (NMF) models are widely used to analyze linearly mixed nonnegative data. When the data is made of samplings of continuous signals, the factors in NMF can be constrained to be samples of nonnegative rational functions. This leads to a fairly general model referred to as NMF using rational functions (R-NMF). We first show that, unlike NMF, R-NMF possesses an essentially unique factorization under mild assumptions, which is crucial in applications where the ground-truth factors need to be recovered, as in blind source separation problems. Then we present different approaches to solve R-NMF : the R-HANLS, R-ANLS and R-NLS methods. In our tests, no method significantly outperforms the others in all cases, and all three methods offer a different trade-off between solution accuracy and computational requirements. Indeed, while R-HANLS is fast and accurate for large problems, R-ANLS is more accurate, but also more resources demanding, both in time and memory and R-NLS is even more accurate but only for small problems. Then, crucially we show that R-NMF models outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.

Refined Berezin number inequalities via superquadratic and convex functions

In this paper, we generalize and refine some Berezin number inequalities for Hilbert space operators. Namely, we refine the Hermite-Hadamard inequality and some other recent results by using the concept of superquadraticity and convexity. Then we extend these inequalities for the Berezin number. Among other inequalities, it is shown that if S, T ? L(H(?)) such that ber(T) ? ber(|S|) and f is a nonnegative superquadratic function, then f (ber (T)) ? ber(f (|S|)) ? ?ber (f (||S| ? ber (T)|)).

Общая задача гарантированного оценивания для многошаговых систем

Questions of the guaranteed estimation of nonlinear multistep systems are considered, for which part of coordinates is not observable. For disturbances we set a priori restrictions with non-negative semi-continuous functions that includes geometrical restrictions as well. As the general formulas for creation of information sets and their specification in special cases are provided. Two-dimensional logistics systems and the equations of Lotka-Volterra are considered as examples. The case of the linear equations where basic functions of convex sets are used is separately considered. Under geometrical restrictions on disturbances the rough procedure of estimation with possible use of functions of distance to the set is given.

A chaotic discrete-time continuous-state Hopfield network with piecewise-affine activation functions

We construct a chaotic discrete-time continuous-state Hopfield networkwith piecewise-affine nonnegative activation functions and weight matrix with small positive entries. More precisely, there exists a Cantor set $ C $ in the state space such that the network has sensitive dependence on initial conditions at initial states in $ C $ and the network orbit of each initial state in $ C $ has $ C $ as its $ \omega $-limit set. The approach we use is based on tools developed and employed recently in the study of the topological dynamics of piecewise-contractions. The parameters of the chaotic network are explicitly given.

Elliptic Harnack inequality for ℤd

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.

Complicated Boundaries of the Attraction Basin in a Class of Three-Dimensional Polynomial Systems

Even if a system has only one stable equilibrium point, its dynamics can still be complicated. In the case of the equilibrium point with a complicated boundary of the attraction basin, it is probably difficult to predict the long-term behavior of the trajectory starting from a point outside but near the boundary. The exploration of such systems is helpful for deepening our understanding of the dynamics of complex systems. This paper studies a class of three-dimensional polynomial systems with a nonelementary singularity. With parameters satisfying some conditions, the asymptotic stability of the origin is proved and the complexity of the attraction basin is investigated. It is demonstrated that the boundary of the attraction basin of the origin has a fractal structure in the following sense: An invariant set homeomorphic to the well-known Lorenz strange attractor is contained in the basin boundary. Based on the non-negative function we constructed, how fast a trajectory of the system tends to the asymptotically stable nonelementary singularity is measured.

SOME RESULTS IN CONNECTION WITH SUM AND PRODUCT THEOREMS RELATED TO GENERALIZED RELATIVE ORDER (α, β) AND GENERALIZED RELATIVE TYPE (α, β) OF ENTIRE FUNCTIONS IN THE UNIT DISC

Orders and types of entire functions have been actively investigated by many authors. In this paper, we investigate some basic properties in connection with sum and product of generalized relative order (α, β), generalized relative type (α, β) and generalized relative weak type (α, β) of entire functions in the unit disc D with respect to another entire function where α, β are continuous non-negative functions on (−∞,+∞).