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Given -∞ <λ<Λ<∞, f: E → Rn finite, and f : E [λ, Λ] how can we extend f to a Cm(Rn) function F such that λ ≤ F ≤ Λ and ... is within a constant multiple of the least possible, with the constant depending only on m and n? In this paper, we provide the solution to the problem for the case m D 2. Specifi- cally, we construct a (parameter-dependent, nonlinear) C²(Rn) extension operator that preserves the range [λ, Λ], and we provide an efficient algorithm to compute such an extension using O(N log N) operations, where N = #(E). [ABSTRACT FROM AUTHOR]

Objective. Overcomplete dictionaries are widely used in compressed sensing (CS) to improve the quality of signal reconstruction. However, dictionary learning under the â„" 0 -norm or â„" 1 -norm constraint inevitably produces dictionary atoms that are negatively correlated with the original signal; meanwhile, when we use a sparse linear combination of dictionary atoms to represent a signal, it is suboptimal for the dictionary atoms to “cancel each other out” by addition and subtraction to approximate the sample. In this paper, we propose a non-negative constrained dictionary learning (NCDL) algorithm to improve the reconstruction performance of CS with electrocardiogram (ECG) signals. Approach. Non-NCDL was divided into an encoding stage and a dictionary learning stage. In the encoding stage, non-negative constraints were imposed on the encoding coefficients and obtained the sparse solution using the alternating direction method of multipliers. At the same time, a penalty term was integrated into the objective function in order to remove small coding coefficients and achieve the effect of sparse coding. In the dictionary learning stage, the block coordinate descent algorithm was utilized to update the dictionary with a view to obtaining an overcomplete dictionary. Results. The performance of the proposed NCDL algorithm was evaluated using the standard MIT-BIH database. Quantitative performance metrics, such as percent root mean square difference (PRD1) and root mean square error, were compared with existing CS approaches to quantify the efficacy of the proposed scheme. For a PRD1 value of 9%, the compression ratio (CR) of the NCDL approach was around 2.78. When CR ranged from 1.05 to 2.78, the proposed NCDL approach outperformed the method of optimal direction, k-means singular value decomposition, and online dictionary learning approaches in ECG signal reconstruction based on CS. Significance. This promising preliminary result demonstrates the capability and feasibility of the proposed bioimpedance method and may open up a new direction for this application. The non-NCDL method proposed in this paper can be used to obtain a sparse basis and improve the performance of CS reconstruction. [ABSTRACT FROM AUTHOR]

Compressed sensing has recently received considerable attention in signal and image processing, applied mathematics, and statistics. In this paper, the problem of sparse signal restoration in non-negative environment is studied. We propose a greedy algorithm for solving non-negative structure of sparse vector and analyze its theoretical performance based on mutual coherence. The feasibility of the proposed algorithm is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Tensor decomposition is widely used in a variety of applications such as data mining, biomedical informatics, neuroscience, and signal processing. In this paper, we propose a deep non-negative tensor factorization (DNTF) model to learn intrinsic and hierarchical structures from multi-way data. The DNTF model takes the Tucker tensor decomposition as a building block to stack up a multi-layer structure. In such a way, we can gradually learn the more abstract structures in a higher layer. The benefit is that it helps to mine intrinsic correlations and hierarchical structures from multi-way data. The non-negative constraints allow for clustering interpretation of the extracted data-dependent components. The objective of DNTF is to minimize the total reconstruction loss resulting from using the core tensor in the highest layer and the mode matrices in each layer to reconstruct the data tensor. Then, a deep decomposition algorithm based on multiplicative update rules is proposed to solve the optimization problem. It first conducts layer-wise tensor factorization and then fine-tunes the weights of all layers to reduce the total reconstruction loss. The experimental results on biosignal sensor data demonstrate the effectiveness and robustness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a non-negative feature selection/feature grouping (nnFSG) method for general sign-constrained high-dimensional regression problems that allows regression coefficients to be disjointly homogeneous, with sparsity as a special case. To solve the resulting non-convex optimization problem, we provide an algorithm that incorporates the difference of convex programming, augmented Lagrange and coordinate descent methods. Furthermore, we show that the aforementioned nnFSG method recovers the oracle estimate consistently, and that the mean-squared errors are bounded. Additionally, we examine the performance of our method using finite sample simulations and applying it to a real protein mass spectrum dataset. [ABSTRACT FROM AUTHOR]

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

The sum of forecasts of disaggregated time series is often required to equal the forecast of the aggregate, giving a set of coherent forecasts. The least squares solution for finding coherent forecasts uses a reconciliation approach known as MinT, proposed by Wickramasuriya, Athanasopoulos, and Hyndman (2019). The MinT approach and its variants do not guarantee that the coherent forecasts are non-negative, even when all of the original forecasts are non-negative in nature. This has become a serious issue in applications that are inherently non-negative such as with sales data or tourism numbers. While overcoming this difficulty, we reconsider the least squares minimization problem with non-negativity constraints to ensure that the coherent forecasts are strictly non-negative. The constrained quadratic programming problem is solved using three algorithms. They are the block principal pivoting (BPV) algorithm, projected conjugate gradient (PCG) algorithm, and scaled gradient projection algorithm. A Monte Carlo simulation is performed to evaluate the computational performances of these algorithms as the number of time series increases. The results demonstrate that the BPV algorithm clearly outperforms the rest, and PCG is the second best. The superior performance of the BPV algorithm can be partially attributed to the alternative representation of the weight matrix in the MinT approach. An empirical investigation is carried out to assess the impact of imposing non-negativity constraints on forecast reconciliation over the unconstrained method. It is observed that slight gains in forecast accuracy have occurred at the most disaggregated level. At the aggregated level, slight losses are also observed. Although the gains or losses are negligible, the procedure plays an important role in decision and policy implementation processes. [ABSTRACT FROM AUTHOR]

Clustering has been extensively explored in pattern recognition and data mining in order to facilitate various applications. Due to the presence of data noise, traditional clustering approaches may become vulnerable and unreliable, thereby degrading clustering performance. In this paper, we propose a robust spectral clustering approach, termed Non-negative Low-rank Self-reconstruction (NLS), which simultaneously a) explores the nonnegative low-rank properties of data correlation as well as b) adaptively models the structural sparsity of data noise. Specifically, in order to discover the intrinsic correlation among data, we devise a self-reconstruction approach to jointly consider the nonnegativity and low-rank property of data correlation matrix. Meanwhile, we propose to model data noise via a structural norm, i.e., ℓp,2-norm, which not only naturally conforms to genuine patterns of data noise in real-world situations, but also provides more adaptivity and flexibility to different noise levels. Extensive experiments on various real-world datasets illustrate the advantage of the proposed robust spectral clustering approach compared to existing clustering methods. [ABSTRACT FROM AUTHOR]

Abstract Adaptive filters are useful solutions for system identification problem where an optimization problem is used to formulated the estimation of the unknown model coefficients. The nonnegativity constraint is one of the most frequently used constraint which can be imposed to avoid physically unreasonable solutions and to comply with physical characteristics. In this letter, we propose a new variant of non-negative least mean square (NNLMS) that has a less mean square error (MSE) value and faster convergence rate. We provide both mean weight behavior and transient excess mean-square error analysis for proposed algorithm. Simulation results validate the theoretical analysis and show the effectiveness of our proposed algorithm. [ABSTRACT FROM AUTHOR]

From a linear program and its asymmetric dual, invariant. primal and dual problems are constructed. Regular mappings are defined between the solution spaces of the original and invariant problems. The notion of centrality is introduced and subsets of regular mappings are shown to be inversely related surjections of central elements, thus representing the original problems as invariant problems. A fixed-point problem involving an idempotent symmetric matrix is constructed from the invariant problems and the notion of centrality carried over to it; the non-negative central fixed-points are shown to map one-to-one to the central solutions to the invariant problems, thus representing the invariant problems as a fixed-point problem and, by transitivity, the original problems as a fixed-point problem. [ABSTRACT FROM AUTHOR]

Background: In diffuse optical tomography (DOT), the image reconstruction is often an ill-posed inverse problem, which is even more severe for breast DOT since there are considerably increasing unknowns to reconstruct with regard to the achievable number of measurements. One common way to address this ill-posedness is to introduce various regularization methods. There has been extensive research regarding constructing and optimizing objective functions. However, although these algorithms dramatically improved reconstruction images, few of them have designed an essentially differentiable objective function whose full gradient is easy to obtain to accelerate the optimization process.Methods: This paper introduces a new kind of non-negative prior information, designing differentiable objective functions for cases of L1-norm, Lp (0 < p < 1)-norm and L0-norm. Incorporating this non-negative prior information, it is easy to obtain the gradient of these differentiable objective functions, which is useful to guide the optimization process.Results: Performance analyses are conducted using both numerical and phantom experiments. In terms of spatial resolution, quantitativeness, gray resolution and execution time, the proposed methods perform better than the conventional regularization methods without this non-negative prior information.Conclusions: The proposed methods improves the reconstruction images using the introduced non-negative prior information. Furthermore, the non-negative constraint facilitates the gradient computation, accelerating the minimization of the objective functions. [ABSTRACT FROM AUTHOR]

This paper proposed a non-negative tensor decomposition algorithm based on low-rank representation to improve the accuracy of image classification. As the extension and the development of compressed sensing theory, the low-rank representation denoted that the rank of the matrix could be used as a measurement of sparsity. Since the rank of a matrix reflected the inherent property of the matrix, the low-rank analysis could effectively analyze and process the matrix data. This paper introduced the low-rank representation into tensor model, namely to introduce it into non-negative tensor decomposition algorithm and to further expand the non-negative tensor decomposition algorithm. Experimental results show that the classification accuracy of the algorithms proposed in this paper is better compared to other existing algorithms. [ABSTRACT FROM AUTHOR]

Highlights: [•] We use non-negative sparse coding with max pooling to represent images. [•] Use correlation constrained low-rank matrix recovery to decompose image features. [•] Locality-constrained linear coding is used to recode image representation. [•] We achieve the state-of-the-art performances on several public datasets. [Copyright &y& Elsevier]

A typical problem of estimation principles of variance and covariance components is that they do not produce positive variances in general. This caveat is due, in particular, to a variety of reasons: (1) a badly chosen set of initial variance components, namely initial value problem (IVP), (2) low redundancy in functional model, (3) an improper stochastic model, and (4) data's possibility of containing outliers. Accordingly, a lot of effort has been made in order to design non-negative estimates of variance components. However, the desires on non-negative and unbiased estimation can seldom be met simultaneously. Likewise, in order to search for a practical non-negative estimator, one has to give up the condition on unbiasedness, which implies that the estimator will be biased. On the other hand, unlike the variance components, the covariance components can be negative, so the methods for obtaining non-negative estimates of variance components are not applicable. This study presents an alternative method to non-negative estimation of variance components such that non-negativity of the variance components is automatically supported. The idea is based upon the use of the functions whose range is the set of all positive real numbers, namely positive-valued functions (PVFs), for unknown variance components in stochastic model instead of using variance components themselves. Using the PVF could eliminate the effect of IVP on the estimation process. This concept is reparameterized on the restricted maximum likelihood with no effect on the unbiasedness of the scheme. The numerical results show the successful estimation of non-negativity estimation of variance components (as positive values) as well as covariance components (as negative or positive values). [ABSTRACT FROM AUTHOR]

In this paper, we discuss some fuzzy norms on BCK-algebras, and we find several conditions for norms to be non-negative in algebras. Finally we discuss fuzzy stable norms on several algebras. [ABSTRACT FROM AUTHOR]

Abstract: Many applications in computer vision, biomedical informatics, and graphics deal with data in the matrix or tensor form. Non-negative matrix and tensor factorization, which extract data-dependent non-negative basis functions, have been commonly applied for the analysis of such data for data compression, visualization, and detection of hidden information (factors). In this paper, we present a fast and flexible algorithm for sparse non-negative tensor factorization (SNTF) based on columnwise coordinate descent (CCD). Different from the traditional coordinate descent which updates one element at a time, CCD updates one column vector simultaneously. Our empirical results on higher-mode images, such as brain MRI images, gene expression images, and hyperspectral images show that the proposed algorithm is 1–2 orders of magnitude faster than several state-of-the-art algorithms. [Copyright &y& Elsevier]

Images can be coded accurately using a sparse set of vectors from a learned overcomplete dictionary, with potential applications in image compression and feature selection for pattern recognition. We present a survey of algorithms that perform dictionary learning and sparse coding and make three contributions. First, we compare our overcomplete dictionary learning algorithm (FOCUSS-CNDL) with overcomplete independent component analysis (ICA). Second, noting that once a dictionary has been learned in a given domain the problem becomes one of choosing the vectors to form an accurate, sparse representation, we compare a recently developed algorithm (sparse Bayesian learning with adjustable variance Gaussians, SBL-AVG) to well known methods of subset selection: matching pursuit and FOCUSS. Third, noting that in some cases it may be necessary to find a non-negative sparse coding, we present a modified version of the FOCUSS algorithm that can find such non-negative codings. Efficient parallel implementations in VLSI could make these algorithms more practical for many applications. [ABSTRACT FROM AUTHOR]

We show that, under mild conditions, a semigroup of non-negative operators on Lp(X,μ) (for 1p<∞) of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces. Also, in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup. [Copyright &y& Elsevier]

Background: In diffuse optical tomography (DOT), the image reconstruction is often an ill-posed inverse problem, which is even more severe for breast DOT since there are considerably increasing unknowns to reconstruct with regard to the achievable number of measurements. One common way to address this ill-posedness is to introduce various regularization methods. There has been extensive research regarding constructing and optimizing objective functions. However, although these algorithms dramatically improved reconstruction images, few of them have designed an essentially differentiable objective function whose full gradient is easy to obtain to accelerate the optimization process., Methods: This paper introduces a new kind of non-negative prior information, designing differentiable objective functions for cases of L 1 -norm, L p (0 < p < 1)-norm and L 0 -norm. Incorporating this non-negative prior information, it is easy to obtain the gradient of these differentiable objective functions, which is useful to guide the optimization process., Results: Performance analyses are conducted using both numerical and phantom experiments. In terms of spatial resolution, quantitativeness, gray resolution and execution time, the proposed methods perform better than the conventional regularization methods without this non-negative prior information., Conclusions: The proposed methods improves the reconstruction images using the introduced non-negative prior information. Furthermore, the non-negative constraint facilitates the gradient computation, accelerating the minimization of the objective functions.