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This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka–Łojasiewicz property of exponent being a half for its objective function under a suitable assumption and provide some examples to illustrate that such an assumption is not very restricted which, in particular, involve the zero-norm regularized or constrained piecewise linear–quadratic function, the zero-norm regularized or constrained logistic regression function, the zero-norm regularized or constrained quadratic function over a sphere.

This paper proposes two new algorithms, namely (i) SSReL1Min(CVX)-Scalar-Sign function-based Reweighted L 1 − norm Minimization algorithm combined with Disciplined Convex Programming for a high-performance L 0 − norm Minimization algorithm and (ii) SSReL1Min(MBB) – SSReL1Min algorithm combined with modified Barzilai-Borwein algorithm for a computational fast L 0 − norm Minimization algorithm (without significantly sacrificing the performance). Based on the proposed L 0 − norm minimization algorithm, this paper also presents an upgraded compressed sensing to improve its performance on the recovery of noisy signals. The proposed L 0 − norm minimization algorithm includes a new optimal scalar-sign function-based weighting (in the least squares sense), as well as a new and systematic mapping mechanism in pre- and post-processing, for noisy compressed sensing. This improvement is further confirmed by experimental results. Comparisons with different state-of-the-art solvers are also included, to show that the proposed method outperforms existing ones.

The progressively increasing data rates have induced the research and application of sparse channel model. To obtain the source signal via the signal got from the terminal equipment, we expect to improve the estimation performance such as accuracy by exploiting the sparsity of the channel impulse response. Our work focuses on exploiting training signal to detect and estimate the channel. However, the classical algorithms such as greedy algorithm like Orthogonal Matching Pursuit (OMP), Least Mean Square (LMS), and even the modified Least Mean Square with approximate-zero norm, $l_{1}$ norm or $l_{2}$ norm as cost function, are not satisfying in the efficiency and reliability as we expected, i.e., their performances are expected better in convergent speed and bias of convergent value in our study. To design superior channel estimator, we tried 3 new adaptive algorithms called LMS integrated with Block $l_{1,0}$ norm (BL 10 ) and LMS integrated with Block $l_{2,0}$ norm (BL 20 ). Simulation in this paper will demonstrate the superiority of the proposed methods when dealing with sparse system.

The sparsity-based approaches have demonstrated promising performance in image processing. In this paper, for better preservation of the salient edge structures of images, we propose an ℓ 0 + ℓ 2-norm based analysis model， which requires solving a challenging non-separable ℓ 0-norm related minimization problem, and we also propose an inexact augmented Lagrangian method with proven convergence to a local minimum. Extensive experiments in image smoothing, including texture removal and context smoothing, show that our method achieves better visual results over various sparsity-based models and the CNN method. Also, experiments on sparse view CT reconstruction further validate the advantage of the proposed method.

We reformulate the zero-norm minimization problem as an equivalent mathematical program with equilibrium constraints and establish that its penalty problem, induced by adding the complementarity constraint to the objective, is exact. Then, by the special structure of the exact penalty problem, we propose a decomposition method that can seek a global optimal solution of the zero-norm minimization problem under the null space condition in [M. A. Khajehnejad et al. IEEE Trans. Signal. Process., 59(2011), pp. 1985-2001] by solving a finite number of weighted $l_1$-norm minimization problems. To handle the weighted $l_1$-norm subproblems, we develop a partial proximal point algorithm where the subproblems may be solved approximately with the limited memory BFGS (L-BFGS) or the semismooth Newton-CG. Finally, we apply the exact penalty decomposition method with the weighted $l_1$-norm subproblems solved by combining the L-BFGS with the semismooth Newton-CG to several types of sparse optimization problems, and compare its performance with that of the penalty decomposition method [Z. Lu and Y. Zhang, SIAM J. Optim., 23(2013), pp. 2448- 2478], the iterative support detection method [Y. L. Wang and W. T. Yin, SIAM J. Sci. Comput., 3(2010), pp. 462-491] and the state-of-the-art code FPC_AS [Z. W. Wen et al. SIAM J. Sci. Comput., 32(2010), pp. 1832-1857]. Numerical comparisons indicate that the proposed method is very efficient in terms of the recoverability and the required computing time.

The comment on “Three-dimensional potential field data inversion with L0 quasi-norm sparse constraints” by Vatankhah mainly gives some advice on my paper. Vatankhah pays attention on the L0 norm, which is applied on the potential field data inversion. Last and Kubik (1983) firstly introduced the compact inversion of gravity data. Later, other inversion studies were based on this principle (e.g.: Portniaguine and Zhdanov 1999). The strategy to obtain compact models is similar to the L0 norm constraint. However, they are not same. In my paper, I used the L0 norm which was firstly introduced in the compressive sensing theory. Compressive sensing is the reconstruction of sparse images or signals from very few samples, by means of solving a tractable optimization problem (Baraniuk 2008; Chartrand 2009). It is a great breakthrough. Later, this principle has been applied in many domains (e.g., Mohimani, Babaie-Zadeh and Jutten 2009; Zhang and Tian 2017). In these researches, an approximate zero norm replaces the L0 norm. In Vatankhah’s comment, there may be some inaccurate understanding. Function f 2 σ (m) (equation (3) in Vatankhah’s comment, corresponding to the function used in Meng (2018)) is better suited for potential field data inversion comparing with f 1 σ (m) (equation (1) in Vatankhah’s comment, introduced by Last and Kubik, 1983). In Vatankhah’s Figure 1(a), for large values of σ and for big absolute values of m, f 2 σ (m) has smaller values than f 1 σ (m), and therefore f 2 σ (m) is smoother than f 1 σ (m). For large σ more weight is imposed on the large elements of the parameter vector m in f 1 σ (m), as compared with f 2 σ (m), yielding a larger penalty on these large elements of m. Thus, the f 2 σ (m) can provide smoother solutions. This phenomenon demonstrates that noise has less effect on f 2 σ (m) than on f 1 σ (m), both used to approximate the L0 norm. Therefore, the f 2 σ (m) function has better performance with big values of σ. Even when σ = 0.1 (Vatankhah’s Fig. 1(b)), f 2 σ (m) have smaller values than f 1 σ (m) for absolute values of m lesser than 0.2, and bigger values than f 1 σ (m) for absolute values ofmbetween 0.2 and 0.6. Therefore, equation (3) can give sparser and better inversion results. Vatankhah’s Figure 1 shows this phenomenon clearly. In the potential field data inversion, the depth-weighting function and density constraint function are very important and commonly used. I think that the researchers can focus on the solution on the L0 norm. For example, the Newton gradient method is applied in my paper, which can effectively avoid the calculations of step size. I think that this reply can help the readers to have a better understanding of this topic.

We study scattering of higher spin closed string states at arbitrary mass levels from D-particle in the Regge regime. We extract the infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. In this calculation, we have used an identity proved recently based on a signless Stirling number identity in combinatorial theory. The complete ratios calculated by this indirect method include a subset of ratios calculated previously by direct fixed angle calculation. Moreover, we discover that in spite of the non-factorizability of the closed string D-particle scattering amplitudes, the complete ratios derived for the fixed angle regime are found to be factorized. These ratios are consistent with the decoupling of high-energy zero norm states calculated previously., 19 pages v2. Eqs.(3.19) to (3.21) added. v3. 27 pages. example added in sec.III. final in Prog. Theo. Phys

Aiming at the problem of small infrared target detection, improved structural collaborative sparseness (ISCS) detection algorithm which combined sparseness and edge-preserve smoothing was put forward. Firstly, in order to improve the background, component of bigger gradient was extracted based on zero norm of gradient; Secondly, background sparseness was modeled by row norm; At last, small dim infrared target was located by column norm of error matrix. Against other detection algorithms, results of experiments show that ISCS detection algorithm can explore relationships between pieces of background, restrain clutter and enhance performance of the small dim infrared target detection algorithm against complex environment.

We give an explicit example of an exactly solvable -symmetric Hamiltonian with unbroken symmetry which has one eigenfunction with zero norm. The set of its eigenfunctions is not complete in the corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm–Liouville problem any diagonalizable -symmetric Hamiltonian with unbroken symmetry has a complete set of positive -normalizable eigenfunctions. For non-diagonalizable Hamiltonians, a complete set of -normalizable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are no longer eigenfunctions of the Hamiltonian.

We calculate the NS–R fermionic zero-norm states of the type II string spectrum. The massless and some possible massive zero-norm states are seen to be responsible for the space-time supersymmetry. The existence of other fermionic massive zero-norm states with higher spinor–tensor indices correspond to new enlarged boson–fermion symmetries of the theory at high energy. We also discuss the R–R charges and R–R zero-norm states and justify the idea that the perturbative string does not carry the massless R–R charges.

Recently, the so-called Support Feature Machine (SFM) was proposed as a novel approach to feature selection for classification, based on minimisation of the zero norm of a separating hyper plane. We propose an extension for linearly non-separable datasets that allows a direct trade-off between the number of misclassified data points and the number of dimensions. Results on toy examples as well as real-world datasets demonstrate that this method is able to identify relevant features very effectively.

The application of one-class machine learning is gaining attention in the computational biology community. Different studies have described the use of two-class machine learning to predict microRNAs (miRNAs) gene target. Most of these methods require the generation of an artificial negative class. However, designation of the negative class can be problematic and if it is not properly done can affect the performance of the classifier dramatically and/or yield a biased estimate of performance. We present study using one-class machine learning for miRNA target discovery and compare one-class to two-class approaches using a zero-norm for feature selection. The usage of this simple feature selection cause an improving of the one-class results, where in some cases reaches the performance of the two-class approach.

We propose the so-called Support Feature Machine (SFM) as a novel approach to feature selection for classification, based on minimisation of the zero norm of a separating hyperplane. Thus, a classifier with inherent feature selection capabilities is obtained within a single training run. Results on toy examples demonstrate that this method is able to identify relevant features very effectively.

For a mean zero norm one sequence (f n )⊂L 2[0, 1], the sequence (f n {nx+y}) is an orthonormal sequence inL 2([0, 1]2); so if $$\sum\limits_{n = 1}^\infty {\left| {c_n } \right|^2 \log ^2 } n< \infty $$ , then $$\sum\limits_{n = 1}^\infty {c_n f_n \{ nx + y\} } $$ converges for a.e. (x, y)∈[0, 1]2 and has a maximal function inL 2([0, 1]2). But for a mean zerof∈L 2[0, 1], it is harder to give necessary and sufficient conditions for theL 2-norm convergence or a.e. convergence of $$\sum\limits_{n = 1}^\infty {c_n f_n \{ nx\} } $$ . Ifc n ≧0 and $$\sum\limits_{n = 1}^\infty {c_n = \infty } $$ , then this series will not converge inL 2-norm on a denseG δ subset of the mean zero functions inL 2[0, 1]. Also, there are mean zerof∈L∞[0, 1] such that $$\sum\limits_{n = 1}^\infty {({1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n})} f\{ nx\} $$ never converges and there is a mean zero continuous functionf with $$\mathop {sup}\limits_N \left| {\sum\limits_{n = 1}^N {(\log {n \mathord{\left/ {\vphantom {n n}} \right. \kern-\nulldelimiterspace} n})} f \{ nx\} } \right| = \infty $$ a.e. However, iff is mean zero and of bounded variation or in some Lip(α) with 1/21/2, then $$\sum\limits_{n = 1}^\infty {c_n } f\{ nx\} $$ converges a.e. and unconditionally inL 2[0, 1]. In addition, for any mean zerof of bounded variation, the series $$\sum\limits_{n = 1}^\infty {({1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n})} f\{ nx\} $$ has its maximal function in allL p[0, 1] with 1≦p

The structure of a relativistic field which contains "ghost" (zero norm) states is investigated. Restrictions are made for the case of free fields, which can be thought of as asymptotic parts of a more complicated theory. According to the interpretation the in- or out-going fields may be eliminated from the physical results. If explicit calculations are to be made in any more concrete theory, it is easier to handle dipole ghosts than exponential ghosts. (B.O.G.)

It was in July 1966 that the government first imposed a statutory pay freeze, which lasted for the remainder of that year. The first half of 1967 was a period of ‘severe restraint’ with a zero norm for pay increases. From July 1968, the government allowed no automatic increases and gave itself the power to delay any increases for seven months. However, it was during this period that two exceptions were allowed — increases could be granted where low pay could be demonstrated and where productivity improvements were negotiated. For the next decade, on and off, productivity improvements became significant in justifying pay increases and were still around in the guise of ‘self-financing productivity schemes’ during the death throes of formal incomes policy in 1978.

Virasoro pointed out that the dual resonance model with trajectory intercept one (αo = 1) has a larger number of linear dependences. Here we construct a linearly independent basis for the spurious states at α(π2) = N, and demonstrate that the remaining real states have the same mass (degeneracy), spin and parity as those states created by the spatial oscillators a n † . The problem of eliminating ghosts (negative norm states resulting from [a o n , a o n † ] = −1) is reformulated, and no ghosts or unanticipated null (‘zero norm’) states are found on the leading nine trajectories. The projection operator onto the real subspace is calculated and the corrections to the single loop diagrams are shown to be (1−x) ∏ n=1 ∞ (1−x n ) and (1+x) ∏ n=1 ∞ (1−(−x) n ) for orientable and non-orientable diagrams respectively. The sum of all single loop diagrams is expressed as a single integral.

tallR =Vb2+c2-ad Proof is given in [1l. There are a few interesting and peculier things to observe. In case a real circle is chosen the norm is positive and for R = 0 we have zero norm. It is easy to show that the zero norm implies R = 0. Thus the norm is zero if and only if R = 0, i. e., the points of the plane have zero norm. Even though later we use an imaginary circle to discuss some properties of circle product we shall not go into discussion of imaginary