Nonnegative Source Research Articles

Biologically plausible single-layer networks for nonnegative independent component analysis.

An important problem in neuroscience is to understand how brains extract relevant signals from mixtures of unknown sources, i.e., perform blind source separation. To model how the brain performs this task, we seek a biologically plausible single-layer neural network implementation of a blind source separation algorithm. For biological plausibility, we require the network to satisfy the following three basic properties of neuronal circuits: (i) the network operates in the online setting; (ii) synaptic learning rules are local; and (iii) neuronal outputs are nonnegative. Closest is the work by Pehlevan et al.(Neural Comput 29:2925-2954, 2017), which considers nonnegative independent component analysis (NICA), a special case of blind source separation that assumes the mixture is a linear combination of uncorrelated, nonnegative sources. They derive an algorithm with a biologically plausible 2-layer network implementation. In this work, we improve upon their result by deriving 2 algorithms for NICA, each with a biologically plausible single-layer network implementation. The first algorithm maps onto a network with indirect lateral connections mediated by interneurons. The second algorithm maps onto a network with direct lateral connections and multi-compartmental output neurons.

Discrete maximum principle for the weak Galerkin method on triangular and rectangular meshes

In this paper, the discrete maximum principle (DMP) of the weak Galerkin (WG) method for the second order elliptic equations is considered. The WG schemes are proposed on triangular and rectangular meshes, respectively. For both schemes, the discrete maximum principle for non-negative source and general source are proved. Numerical experiments are also presented to verify the convergence and the DMP-preserving property.

Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.

Development of [formula omitted]-inverse model by using generalized polynomial chaos

We present a hp-inverse model to estimate a smooth, non-negative source function from a limited number of observations for a two-dimensional linear source inversion problem. A standard least-square inverse model is formulated by using a set of Gaussian radial basis functions (GRBF) on a rectangular mesh system with a uniform grid space. Here, the choice of the mesh system is modeled as a random variable and the generalized polynomial chaos (gPC) expansion is used to represent the random mesh system. It is shown that the convolution of gPC and GRBF provides hierarchical basis functions for the linear source inverse model with the hp-refinement capability. We propose a mixed l1 and l2 regularization to exploit the hierarchical nature of the basis functions to find a sparse solution. The hp-inverse model has an advantage over the standard least-square inverse model when the number of data is limited. It is shown that the hp-inverse model provides a good estimate of the source function even when the number of unknown parameters (m) is much larger the number of data (n), e.g., m∕n>40.

On the maximum principle for a time-fractional diffusion equation

Abstract In this paper, we discuss the maximum principle for a time-fractional diffusion equation ∂ t α u ( x , t ) = ∑ i , j = 1 n ∂ i ( a i j ( x ) ∂ j u ( x , t ) ) + c ( x ) u ( x , t ) + F ( x , t ) , t > 0 , x ∈ Ω ⊂ R n , $$\begin{array}{} \displaystyle \partial_t^{\alpha} u(x,t) = \sum\limits_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t))+ c(x)u(x,t) + F(x,t),\,\, t\gt 0,\,\,x \in \Omega \subset {\mathbb R}^n, \end{array} $$ with the Caputo time-derivative of the order α ∈ (0, 1) in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions F = F(x, t) we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient c = c(x) by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient c = c(x).

Blind Nonnegative Source Separation Using Biological Neural Networks.

Blind source separation-the extraction of independent sources from a mixture-is an important problem for both artificial and natural signal processing. Here, we address a special case of this problem when sources (but not the mixing matrix) are known to be nonnegative-for example, due to the physical nature of the sources. We search for the solution to this problem that can be implemented using biologically plausible neural networks. Specifically, we consider the online setting where the data set is streamed to a neural network. The novelty of our approach is that we formulate blind nonnegative source separation as a similarity matching problem and derive neural networks from the similarity matching objective. Importantly, synaptic weights in our networks are updated according to biologically plausible local learning rules.

Unmixing hyperspectral data by using signal subspace sampling

This paper demonstrates how Signal Subspace Sampling (SSS) is an effective pre-processing step for Non-negative Matrix Factorization (NMF) or Vertex Component Analysis (VCA). The approach allows to uniquely extract non-negative source signals which are orthogonal in at least one observation channel, respectively. It is thus well suited for processing hyperspectral images from X-ray microscopy, or other emission spectroscopies, into its non-negative source components. The key idea is to resample the given data so as to satisfy better the necessity and sufficiency conditions for the subsequent NMF or VCA. Results obtained both on an artificial simulation study as well as based on experimental data from electron-microscopy are reported.

Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted \begin{document}$f$\end{document} , must satisfy the inequality \begin{document}$f≤q 1$\end{document} and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the \begin{document}$H$\end{document} -theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with $N$ species, $M$ reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemistry kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.

On the regularity of the free boundary in the optimal partial transport problem for general cost functions

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a C1 cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann [1] regarding cost functions satisfying the Ma–Trudinger–Wang condition (A3): if the non-negative source density is in some Lp(Rn) space for p∈(n+12,∞] and the positive target density is bounded away from zero, then the free boundary is a semiconvex Cloc1,α hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.

A Convex Geometry-Based Blind Source Separation Method for Separating Nonnegative Sources.

This paper presents a convex geometry (CG)-based method for blind separation of nonnegative sources. First, the unaccessible source matrix is normalized to be column-sum-to-one by mapping the available observation matrix. Then, its zero-samples are found by searching the facets of the convex hull spanned by the mapped observations. Considering these zero-samples, a quadratic cost function with respect to each row of the unmixing matrix, together with a linear constraint in relation to the involved variables, is proposed. Upon which, an algorithm is presented to estimate the unmixing matrix by solving a classical convex optimization problem. Unlike the traditional blind source separation (BSS) methods, the CG-based method does not require the independence assumption, nor the uncorrelation assumption. Compared with the BSS methods that are specifically designed to distinguish between nonnegative sources, the proposed method requires a weaker sparsity condition. Provided simulation results illustrate the performance of our method.

Non-Negative Blind Source Separation Algorithm Based on Minimum Aperture Simplicial Cone

We address the problem of Blind Source Separation (BSS) when the hidden sources are Nonnegative (N-BSS). In this case, the scatter plot of the mixed data is contained within the simplicial cone generated by the columns of the mixing matrix. The proposed method, termed SCSA-UNS for Simplicial Cone Shrinking Algorithm for Unmixing Non-negative Sources, aims at estimating the mixing matrix and the sources by fitting a Minimum Aperture Simplicial Cone (MASC) to the cloud of mixed data points. SCSA-UNS is evaluated on both independent and correlated synthetic data and compared to other N-BSS methods. Simulations are also performed on real Liquid Chromatography-Mass Spectrum (LC-MS) data for the metabolomic analysis of a chemical sample, and on real dynamic Positron Emission Tomography (PET) images, in order to study the pharmacokinetics of the [18F]-FDG (FluoroDeoxyGlucose) tracer in the brain.

Non-existence of local solutions for semilinear heat equations of Osgood type

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f. Global (in time) solutions of the scalar ODE v˙=f(v) exist for v(0)>0 if and only if the Osgood-type condition ∫1∞dsf(s)=∞ holds; by comparison this ensures the existence of global classical solutions of ut=Δu+f(u) for bounded initial data u0∈L∞(Rn). It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in Lq(Rn) for some 1⩽q<∞. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in Lloc1(Rn) for t>0.

Projection-pursuit-based method for blind separation of nonnegative sources.

This paper presents a projection pursuit (PP) based method for blind separation of nonnegative sources. First, the available observation matrix is mapped to construct a new mixing model, in which the inaccessible source matrix is normalized to be column-sum-to-1. Then, the PP method is proposed to solve this new model, where the mixing matrix is estimated column by column through tracing the projections to the mapped observations in specified directions, which leads to the recovery of the sources. The proposed method is much faster than Chan's method, which has similar assumptions to ours, due to the usage of optimal projection. It is also more advantageous in separating cross-correlated sources than the independence- and uncorrelation-based methods, as it does not employ any statistical information of the sources. Furthermore, the new method does not require the mixing matrix to be nonnegative. Simulation results demonstrate the superior performance of our method.

Quantifying brain tumor tissue abundance in HR-MAS spectra using non-negative blind source separation techniques

Given high-resolution magic angle spinning (HR-MAS) spectra from several glial tumor subjects, our goal is to differentiate between tumor tissue types by separating the different sources that contribute to the profile of each spectrum. Blind source separation techniques are applied for obtaining characteristic profiles for necrosis, highly cellular tumor and border tumor tissue and providing the contribution (abundance) of each of these tumor tissue types to the profile of each spectrum. The problem is formulated as a non-negative source separation problem. Non-negative matrix factorization, convex analysis of non-negative sources and non-negative independent component analysis methods are considered. The results are in agreement with the pathology obtained by the histopathological examination that succeeded the HR-MAS measurements. Furthermore, an analysis to verify to which extent the dimension of the input space, the input features and the number of sources to be extracted from the HR-MAS data could influence the performance of the source separation is presented. Copyright © 2012 John Wiley & Sons, Ltd.

A geometric approach to blind separation of nonnegative and dependent source signals

Blind source separation (BSS) consists in processing a set of observed mixed signals to separate them into a set of original components. Most of the current blind separation methods assumes that the source signals are “as statistically independent as possible” given the observed data. In many real-world situations, however, this hypothesis does not hold. In order to cope with such signals, a first geometric method was proposed that separates statistically dependent signals, provided that they are nonnegative and locally orthogonal.This paper presents a new geometric method for the separation of nonnegative source signals which relies on a working assumption that is weaker than local orthogonality. The separation problem is expressed as the identification of relevant facets of the data cone. After a rigorous proof of the proposed method, the details of the separation algorithm are given. Experiments on signals from various origins clearly show the efficiency of the new procedure.

Maximum contrast analysis for nonnegative blind source separation

In this paper, we propose a maximum contrast analysis (MCA) method for nonnegative blind source separation, where both the mixing matrix and the source signals are nonnegative. We first show that the contrast degree of the source signals is greater than that of the mixed signals. Motivated by this observation, we propose an MCA-based cost function. It is further shown that the separation matrix can be obtained by maximizing the proposed cost function. Then we derive an iterative determinant maximization algorithm for estimating the separation matrix. In the case of two sources, a closed-form solution exists and is derived. Unlike most existing blind source separation methods, the proposed MCA method needs neither the independence assumption, nor the sparseness requirement of the sources. The effectiveness of the new method is illustrated by experiments using X-ray images, remote sensing images, infrared spectral images, and real-world fluorescence microscopy images.

Blind source separation of images based on general cross correlation of linear operators

Blind source separation is a process in which mixed signals, obtained as a linear combination of various source signals, are decomposed into their original sources. The source signals and their mixture weights are unknown, but a priori information about their statistical behavior and mixing model is available. In this paper, a new algorithm based on generalized cross correlation linear-operator set is proposed. This algorithm significantly improves source-separation quality compared to several other well-known algorithms, such as subband decomposition independent component analysis, block Gaussian likelihood, and convex analysis of mixtures of non-negative sources.

Superharmonic functions are locally renormalized solutions

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.

Underdetermined Sparse Blind Source Separation of Nonnegative and Partially Overlapped Data

We study the solvability of sparse blind separation of n nonnegative sources from m linear mixtures in the underdetermined regime $m<n$. The geometric properties of the mixture matrix and the sparseness structure of the source matrix are closely related to the identification of the mixing matrix. We first illustrate and establish necessary and sufficient conditions for the unique separation for the case of m mixtures and $m+1$ sources, and develop a novel algorithm based on data geometry, source sparseness, and $\ell_1$ minimization. Then we extend the results to any order $m\times n$, $3\leq m<n$, based on the degree of degeneracy of the columns of the mixing matrix. Numerical results substantiate the proposed solvability conditions and show satisfactory performance of our approach.