帮我简写Direct minimization of the classification loss may lead to overfitting. To avoid this, prototype loss is added as regularization to improve the model's generalization ability. The so-called prototype loss, that is, center loss centered on the centroid of the subclasses, is used to determine the class to which the input x belongs to. Then, its decision boundary is the location where the distances to the centers of the subclasses of two adjacent classes are equal.

Directly minimizing classification loss may cause overfitting. To prevent this, prototype loss is added as regularization to enhance model generalization. Prototype loss, also known as center loss, is used to determine the input x's class based on the centroid of its subclasses. The decision boundary is located where the distances to the centers of adjacent subclasses are equal.

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Abstract— Image nonlocal self-similarity (NSS) property has been widely exploited via various sparsity models such as joint sparsity (JS) and group sparse coding (GSC). However, the existing NSS-based sparsity models are either too restrictive, e.g., JS enforces the sparse codes to share the same support, or too general, e.g., GSC imposes only plain sparsity on the group coefficients, which limit their effectiveness for modeling real images. In this paper, we propose a novel NSS-based sparsity model, namely, low-rank regularized group sparse coding (LR-GSC), to bridge the gap between the popular GSC and JS. The proposed LR-GSC model simultaneously exploits the sparsity and low-rankness of the dictionary-domain coefficients for each group of similar patches. An alternating minimization with an adaptive adjusted parameter strategy is developed to solve the proposed optimization problem for different image restoration tasks, including image denoising, image deblocking, image inpainting, and image compressive sensing. Extensive experimental results demonstrate that the proposed LR-GSC algorithm outperforms many popular or state-of-the-art methods in terms of objective and perceptual metrics.翻译

摘要—图像的非局部自相似性（NSS）属性已经被广泛应用于各种稀疏模型中，例如联合稀疏（JS）和群组稀疏编码（GSC）。然而，现有的基于NSS的稀疏模型要么太过严格，例如JS强制使稀疏编码共享相同的支持，要么太过通用，例如GSC仅对群组系数施加简单的稀疏性，这限制了它们对实际图像建模的有效性。本文提出了一种新的NSS-based稀疏模型，即低秩正则化群组稀疏编码（LR-GSC），以弥合流行的GSC和JS之间的差距。所提出的LR-GSC模型同时利用了每组相似补丁的字典域系数的稀疏性和低秩性。开发了一种带有自适应调整参数策略的交替最小化方法，用于解决不同的图像恢复任务，包括图像去噪、图像去块、图像修复和图像压缩感知。广泛的实验结果表明，所提出的LR-GSC算法在客观和感知度量方面优于许多流行或最先进的方法。

解释：def steepest_descent(fun, grad, x0, iterations, tol): """ Minimization of scalar function of one or more variables using the steepest descent algorithm. Parameters ---------- fun : function Objective function. grad : function Gradient function of objective function. x0 : numpy.array, size=9 Initial value of the parameters to be estimated. iterations : int Maximum iterations of optimization algorithms. tol : float Tolerance of optimization algorithms. Returns ------- xk : numpy.array, size=9 Parameters wstimated by optimization algorithms. fval : float Objective function value at xk. grad_val : float Gradient value of objective function at xk. grad_log : numpy.array The record of gradient of objective function of each iteration. """ fval = None grad_val = None x_log = [] y_log = [] grad_log = [] x0 = asarray(x0).flatten() # iterations = len(x0) * 200 old_fval = fun(x0) gfk = grad(x0) k = 0 old_old_fval = old_fval + np.linalg.norm(gfk) / 2 xk = x0 x_log = np.append(x_log, xk.T) y_log = np.append(y_log, fun(xk)) grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) gnorm = np.amax(np.abs(gfk)) while (gnorm > tol) and (k < iterations): pk = -gfk try: alpha, fc, gc, old_fval, old_old_fval, gfkp1 = _line_search_wolfe12(fun, grad, xk, pk, gfk, old_fval, old_old_fval, amin=1e-100, amax=1e100) except _LineSearchError: break xk = xk + alpha * pk k += 1 grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) x_log = np.append(x_log, xk.T) y_log = np.append(y_log, fun(xk)) if (gnorm <= tol): break fval = old_fval grad_val = grad_log[-1] return xk, fval, grad_val, x_log, y_log, grad_log

这是一个使用最速下降法（steepest descent algorithm）来最小化一个一元或多元标量函数的函数。其中，fun是目标函数，grad是目标函数的梯度函数，x0是参数的初始值，iterations是最大迭代次数，tol是优化算法的收敛容忍度。函数的返回值包括参数估计的值xk，目标函数在xk处的值fval，目标函数在xk处的梯度grad_val，记录梯度的numpy数组grad_log，记录参数的numpy数组x_log和记录目标函数值的numpy数组y_log。在函数中，使用了线性搜索（_line_search_wolfe12函数）来确定步长，以最小化目标函数。这个函数的主要思想是在每个迭代步骤中，计算目标函数的梯度，然后以梯度的相反方向为方向进行搜索，计算步长，更新参数值。重复上述步骤，直到达到收敛条件或达到最大迭代次数。