Regular Temporal Cost Functions

We derive a general formula of the minimum achievable rate for fixed-to-variable length coding with a regular cost function by allowing the error probability up to a constant $\varepsilon$. For a fixed-to-variable length code, we call the set of source sequences that can be decoded without error the dominant set of source sequences. For any two regular cost functions, it is revealed that the dominant set of source sequences for a code attaining the minimum achievable rate with a cost function is also the dominant set for a code attaining the minimum achievable rate with the other cost function. We also give a general formula of the second-order minimum achievable rate.

This paper deals with a parallel machine scheduling problem whose objective is to minimize a regular step total cost function. A real world application of the problem is presented, and Mixed Integer Linear Programming models are described for the cases with and without release dates, as well as a dedicated constraint generation procedure. Experimental results are reported and discussed to evaluate the relevance of the different approaches on well-known special cases and the general case.

This work studies the problem of designing computer-generated holograms using phase-shifting masks limited to represent only a small number of discrete phase levels. This problem has various applications, notably in the emerging field of optogenetics and lithography. A novel regularized cost function is proposed for the problem at hand that penalizes the unfeasible phase levels. Since the proposed cost function is non-smooth, we derive proper proximal gradient algorithms for its minimization. Simulation results, considering an optogenetics application, demonstrate that the proposed proximal gradient algorithm yields better performance as compared to other algorithms proposed in the literature.

In this paper, we considered an inverse problem of recovering the space-dependent source coefficient in the third-order pseudo-parabolic equation from final over-determination condition. This inverse problem appears extensively in the modelling of various phenomena in physics such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, etc. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the final input may cause relatively significant errors in the output source term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown force term. The third-order pseudo-parabolic problem is discretized using the Cubic B-spline (CB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Both perturbed data and analytical solutions are inverted. Numerical outcomes are reported and discussed. The computational efficiency of the method is investigated by small values of CPU time. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

In this work, a new supervised learning method for single layer neural networks based on a regularized cost function is presented. This method obtains the optimal weights and biases by solving a system of linear equations and therefore it is always guaranteed the global optimum solution. In order to verify the soundness of the proposed learning algorithm and to analyze the effect of the regularization term, two simulations, one for a classification problem and another for a regression problem, were performed. The obtained results demonstrated the validity of the method.

Non-blind image blur removal method based on a Bayesian hierarchical model with hyperparameter priors

While the main conceptual issue related to deposit insurances is the moral hazard risk, the main technical issue is inaccurate calibration of the implied volatility. This issue can raise the risk of generating an arbitrage. In this paper, first, we discuss that by imposing the no-moral-hazard risk, the removal of arbitrage is equivalent to removing the static arbitrage. Then, we propose a simple quadratic model to parameterize implied volatility and remove the static arbitrage. The process of removing the static risk is as follows: Using a machine learning approach with a regularized cost function, we update the parameters in such a way that butterfly arbitrage is ruled out and also implementing a calibration method, we make some conditions on the parameters of each time slice to rule out calendar spread arbitrage. Therefore, eliminating the effects of both butterfly and calendar spread arbitrage make the implied volatility surface free of static arbitrage.

A novel CGH algorithm is proposed for targeted neural stimulation using a few pixel Spatial Light Modulator (SLM) device. This is achieved with the elaboration of standard CGH design algorithms to include a regularized cost function for the discretization of phase and amplitude. The regularized cost function is appropriate for designing holographic masks that can be realized using a few pixel SLM. It is anticipated that these results will contribute to the research efforts towards a portable in vivo optogenetic system.

In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain, Ω. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary, Γ0, and the goal is to complete the data on the other part of the boundary, Γa. Here, Γ0 and Γa represent the side faces of the cylinder Ω. This problem is known to be ill-posed and is formulated as an optimal control problem with a regularized cost function. To directly approximate the missing data on Γa, we employ the method of factorization of elliptic boundary value problems. This technique allows the factorization of a boundary value problem into a product of parabolic problems. It is successfully applied to the optimality system in this work, yielding new and significant results.

In this article, for regression problems, we proposed shrinkage methods in training a machine in terms of a regularized cost function. The machine considered in this article is represented by a linear combination of fixed basis functions, in which the number of basis functions, or equivalently, the number of adjustable weights is identical to the number of training data. This setting can be viewed as a nonparametric regression method in statistics. In the regularized cost function employed in this article, the error function is defined by the sum of squared errors and the regularization term is defined by the quadratic form of the weight vector. By assuming i.i.d. Gaussian noise, we proposed three thresholding methods for the orthogonal components which are obtained by eigendecomposition of the Gram matrix of the vectors of basis function outputs. The final weight values are obtained by a linear transformation of the thresholded orthogonal components and are shrinkage estimators. The proposed methods are quite simple and automatic, in which the regularization parameter is enough to be fixed for a small constant value. Simple numerical experiments showed that, by comparing the leave-one-out cross validation method, the computational costs of the proposed methods are strictly low and the generalization capabilities of the trained machines are comparable when the number of data is relatively large.

We address the problem of scheduling n jobs on a batching machine to minimize regular scheduling criteria that are non-decreasing in the job completion times. A batching machine is a machine that can handle up to b jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is equal to the largest processing time of any job in the batch. We analyse two variants: the unbounded model, where b¿n; and the bounded model, where b1; for the case with m different processing times, we give a dynamic programming algorithm that requires O(b2m22m) time. Moreover, we prove that due date based scheduling criteria give rise to NP-hard problems. Finally, we show that an arbitrary regular cost function can be minimized in polynomial time for a fixed number of batches.

Cost functions are defined as mappings from a domain like words or trees to \(\mathbb{N} \cup \left\{{\infty}\right\}\), modulo an equivalence relation ≈ which ignores exact values but preserves boundedness properties. Cost logics, in particular cost monadic second-order logic, and cost automata, are different ways to define such functions. These logics and automata have been studied by Colcombet et al. as part of a “theory of regular cost functions”, an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability. We develop this theory over infinite words, and show that the classical results FO = LTL and MSO = WMSO also hold in this cost setting (where the equivalence is now up to ≈). We also describe connections with forms of weak alternating automata with counters.

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), pp. 851–883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), pp. 251–267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225–241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

The least squares based cost functions are sensitive to outliers in the measured data. The presence of outliers is considered as impulsive noise. In practical scenarios, the co channel interference, saturation effects, non linearity of the measuring instruments, atmospheric conditions and malfunction of sensors will result in outliers or impulsive noise. The robust function obtained by considering the error as the linear combination of sign preserving basis functions is found to be robust against outliers in the desired data. In many practical applications, the parameter to be estimated can be sparse in nature, i.e. only a few elements are large values and the rest are insignificantly small. In such sparse systems, if the prior information about the sparsity is known, then the known information can be incorporated in the cost function as a regularization function. A robust sparse diffusion algorithm is proposed in this work, which is robust against outliers in the desired data and performs better than the existing algorithms in sparsity underlying systems. Simulations performed for different cases of outliers conditions and sparsity conditions validate that the proposed method outperforms the state of the art methods.

A multiplicative regularization scheme with edge-preserving characteristics is applied to the inversion of electrical impedance tomography (EIT) data. This scheme employs a multiplicative cost function of a weighted L2-norm regularization function and the data misfit function. It avoids the use of a weighting factor when the regularization term is added to the cost function and allows an adaptive weighting between data misfit and the regularization function. Gauss-Newton method is used to minimize the multiplicative cost function. In this work, we extend the weighted L2-norm regularization scheme onto a triangular grid with an updated formula for gradient and divergence operators. This scheme is tested using synthetic data. The reconstructed images show good piecewise constant characteristics and noise-resistance performance.