lect11 Convex Inference Relaxations. LP Relaxations

conv2d中dilation参数摘要：conv2d中dilation参数的介绍与运用一、conv2d的基本概念与原理1.卷积层的定义与作用2.卷积运算的基本过程二、dilation参数的定义与作用1.dilation的概念解释2.dilation参数在卷积运算中的作用三、dilation参数的设置与优化1.不同dilation值对卷积结果的影响2.如何选择合适的dilation值四、dilation在实际应用中的案例1.图像处理中的应用2.计算机视觉任务中的应用五、dilation参数的实战技巧与建议1.如何在训练过程中调整dilation参数2.不同任务中dilation参数的取值规律正文：conv2d是深度学习中卷积运算的基本层，广泛应用于图像处理、计算机视觉等领域。

在conv2d层中，dilation参数是一个重要的超参数，对卷积结果有着显著的影响。

本文将从基本概念、dilation参数的定义与作用、实战技巧等方面进行详细介绍。

一、conv2d的基本概念与原理conv2d层是深度学习中处理图像等数据的常用层，通过对输入数据与卷积核进行点乘并求和，实现对输入数据的特征提取。

conv2d层的主要作用是降低特征图的维度，保留有用信息，从而减少计算量。

卷积运算的基本过程包括：1）将卷积核在输入数据上滑动，2）对每个位置的卷积核与输入数据进行点乘并求和，3）将结果存入输出特征图。

这一过程可以通过编程实现，也可以使用现成的深度学习框架（如TensorFlow、PyTorch等）进行操作。

二、dilation参数的定义与作用在conv2d层中，dilation参数表示卷积核在滑动过程中，每个位置的间隔距离。

换句话说，dilation决定了卷积核在输入数据上移动的步长。

dilation 值越大，卷积核移动的步长越长，对应的感受野越大。

dilation参数在卷积运算中的作用主要体现在以下几点：1.控制卷积核的覆盖范围。

onnx convolution参数-概述说明以及解释1.引言1.1 概述概述部分：卷积神经网络(Convolutional Neural Network, CNN)是一种常用于计算机视觉领域的深度学习算法。

在CNN中，卷积(Convolution)是其中最为重要的操作之一。

在ONNX(Open Neural Network Exchange)规范中，卷积操作是模型中的一个重要组件，它通过一定的参数配置，能够实现图像的特征提取、分类和目标检测等任务。

本文将聚焦于研究ONNX卷积操作的三个重要参数，分别为参数A、参数B和参数C。

通过对这些参数的深入研究，我们将探讨它们在卷积操作中的作用和意义，并探索如何根据具体的问题场景进行参数选择。

参数A是卷积操作中的一个关键参数，它决定了卷积核的大小和形状。

卷积核是一种特殊的矩阵，通过与输入图像进行卷积运算，可以提取图像中的不同特征。

参数A的选择不仅会影响到特征提取的效果，还会直接关系到卷积操作的计算复杂度和模型的大小。

因此，对于参数A的选择需要根据具体的应用场景和需求进行权衡和选择。

参数B是卷积操作中的另一个关键参数，它决定了卷积操作的步长大小。

步长的大小决定了输出特征图的尺寸和数量。

较大的步长可以降低特征图的尺寸和数量，从而减少计算复杂度，但也会对特征提取的精度产生一定的影响。

因此，参数B的选择需要根据具体的应用需求和性能要求进行权衡。

参数C是卷积操作中的还一个关键参数，它决定了卷积操作的填充方式。

填充是指在输入特征图的边缘周围添加一定数量的像素值，它可以使得输出特征图的尺寸和输入特征图的尺寸一致。

参数C的选择不仅会影响到特征提取的精度，还会直接影响到卷积操作的计算效率。

因此，对于参数C的选择需要结合具体的问题和需求进行综合考虑。

本文将通过对参数A、参数B和参数C的详细解析和讨论，旨在帮助读者深入理解ONNX卷积操作的原理和实现机制，并为读者在实际应用中选择合适的参数提供参考和指导。

convlstm的参数（实用版）目录1.ConvLSTM 的概述2.ConvLSTM 的参数分类3.ConvLSTM 的主要参数及其作用4.参数调整与优化5.结论正文一、ConvLSTM 的概述ConvLSTM（卷积长短时记忆网络）是一种结合了卷积神经网络（CNN）和长短时记忆网络（LSTM）的深度学习模型。

它继承了 LSTM 在处理序列数据时的优秀性能，同时具有 CNN 的空间局部感知能力。

ConvLSTM 广泛应用于自然语言处理、语音识别等序列数据的建模任务。

二、ConvLSTM 的参数分类ConvLSTM 的参数主要分为以下几类：1.LSTM 参数：包括输入门、遗忘门、输出门和记忆细胞的激活函数、权重和偏置。

2.卷积参数：包括卷积核的大小、步长和填充。

3.池化参数：包括池化核的大小和步长。

4.序列到序列（seq2seq）模型参数：包括编码器和解码器的参数。

5.损失函数参数：包括交叉熵损失和其他正则化损失。

6.优化器参数：包括学习率、迭代次数等。

三、ConvLSTM 的主要参数及其作用1.LSTM 参数（1）输入门、遗忘门、输出门：决定信息在记忆细胞间的流动。

（2）激活函数：常用的有 sigmoid、tanh 等，用于引入非线性。

（3）权重和偏置：用于调整记忆细胞的状态更新。

2.卷积参数（1）卷积核大小：决定卷积操作的感受野范围。

（2）步长：决定卷积核在序列上的移动距离。

（3）填充：为了使输入序列与卷积核的对齐，需要在序列前后填充一定数量的零。

3.池化参数（1）池化核大小：决定池化操作的感受野范围。

（2）步长：决定池化核在序列上的移动距离。

4.序列到序列（seq2seq）模型参数（1）编码器参数：包括编码器的权重和偏置，用于编码输入序列。

（2）解码器参数：包括解码器的权重和偏置，用于生成输出序列。

5.损失函数参数（1）交叉熵损失：衡量模型输出序列与真实序列之间的差距。

（2）正则化损失：包括 L1、L2 正则化等，用于防止模型过拟合。

halcon convexity计算公式（原创实用版）目录1.概述2.halcon convexity 计算公式的推导3.halcon convexity 在实际应用中的例子4.总结正文1.概述Halcon convexity 是一种用于描述数据集中凸包的数学概念。

在计算机视觉和机器学习领域，凸包是一种重要的数据结构，它可以用来描述一组点集的最小凸包。

Halcon convexity 提供了一种有效的计算凸包的方法，它通过计算数据集中所有点到某一点集的凸包的并集，从而得到整个数据集的凸包。

2.halcon convexity 计算公式的推导Halcon convexity 的计算公式可以由以下步骤推导得到：假设我们有一个由 n 个点组成的数据集 S，我们首先选择一个点 x 作为参考点，然后计算数据集中所有点到参考点的凸包，记为 C(x)。

然后我们再选择一个点 y，计算数据集中所有点到参考点 y 的凸包，记为C(y)。

最后，我们计算 C(x) 和 C(y) 的并集，得到整个数据集的凸包 C。

我们可以用以下公式来计算 C：C = (x) ∪ (y)其中，(x) 表示数据集中所有点到参考点 x 的凸包，(y) 表示数据集中所有点到参考点 y 的凸包。

3.halcon convexity 在实际应用中的例子Halcon convexity 在实际应用中有很多例子，例如在计算机视觉中，它可以用来检测图像中的凸包，从而得到目标物体的边界。

在机器学习中，它可以用来对数据集进行预处理，从而提高算法的性能。

4.总结Halcon convexity 是一种重要的数学概念，它可以用来描述数据集中的凸包。

通过计算数据集中所有点到某一点集的凸包的并集，我们可以得到整个数据集的凸包。

convnext分类一、了解ConvNext分类的背景和意义随着深度学习技术的不断发展，自然语言处理（NLP）领域取得了显著的成果。

ConvNext分类作为深度学习的一种方法，逐渐引起了研究者的广泛关注。

它通过卷积神经网络（CNN）和循环神经网络（RNN）的结合，对文本进行特征提取和分类，具有较高的准确率和实用性。

二、ConvNext分类的基本原理和方法ConvNext分类的核心思想是通过卷积神经网络捕捉文本的局部特征，再通过循环神经网络对捕捉到的特征进行全局建模。

具体来说，它包括以下几个步骤：1.数据预处理：将文本转换为向量形式，如词嵌入或字符嵌入。

2.卷积层：利用卷积神经网络从输入向量中提取局部特征。

3.池化层：对卷积层得到的特征进行降维和压缩。

4.循环神经网络：将池化后的特征作为输入，进行全局序列建模。

5.全连接层：将循环神经网络的输出进行非线性变换，得到分类结果。

三、ConvNext分类在自然语言处理领域的应用ConvNext分类在自然语言处理领域具有广泛的应用，如情感分析、文本分类、命名实体识别等。

它相较于传统的机器学习方法，如朴素贝叶斯、支持向量机等，具有更高的准确率和较强的鲁棒性。

四、ConvNext分类的优缺点分析优点：1.能够有效地捕捉文本的局部特征，提高分类准确性。

2.适应性强，适用于多种自然语言处理任务。

3.参数共享，减少过拟合风险。

缺点：1.训练过程中，参数更新速度较快，可能导致梯度消失或梯度爆炸。

2.模型解释性较差，难以进行直观的解释。

五、我国在ConvNext分类领域的研究进展近年来，我国在ConvNext分类领域取得了丰硕的研究成果。

研究者们在基础理论和应用研究方面进行了深入探讨，如提出了基于注意力机制的ConvNext分类方法、多任务学习的ConvNext分类模型等。

同时，我国还在积极开展ConvNext分类在各领域的应用，如金融、医疗、教育等。

六、ConvNext分类的发展趋势和展望随着深度学习技术的不断进步，ConvNext分类在自然语言处理领域有望取得更大的突破。

深度神经网络函数近似能力证明深度神经网络在近年来取得了令人瞩目的成果，成为了机器学习和人工智能领域的热门研究方向之一。

其中一个重要的问题是深度神经网络的函数近似能力究竟如何，即它们能否有效地拟合任意复杂的函数。

在神经网络的早期发展阶段，只有浅层网络被广泛研究和应用。

然而，随着深度学习的兴起，深层神经网络展现出了强大的表达能力和泛化能力。

但是，对于深层数目和节点数目较大的神经网络，其函数拟合能力的证明仍然是一个具有挑战性的问题。

要证明深度神经网络的函数近似能力，一个常用的方法是利用万能逼近定理。

万能逼近定理表明，一个拥有足够多节点的神经网络可以以任意精度近似任意连续函数。

具体来说，给定一个定义在有限空间上的连续函数，只要神经网络具有足够多的节点，它就能以任意精度逼近这个函数。

然而，万能逼近定理并不能直接用于深度神经网络。

因为在深层网络中，节点数目呈指数级增长，从而导致计算复杂度的急剧上升。

为了解决这个问题，研究者们提出了一系列的改进算法和网络结构，如卷积神经网络和循环神经网络，以提高深层神经网络的函数拟合能力，并降低计算复杂度。

另一个证明深度神经网络函数近似能力的方法是通过实际的实验和应用。

近年来，深度神经网络在许多领域中取得了显著的应用成果，包括图像识别、自然语言处理、语音识别等。

这些成功案例表明，深度神经网络具有强大的函数近似能力，能够有效地解决复杂的现实问题。

尽管深度神经网络在实践中证明了其函数近似能力，但在理论上的证明仍然是一个开放性问题。

理论证明通常需要严格的数学推导和复杂的分析。

研究者们正在努力寻找新的方法和技巧，以解决这个问题。

其中一些研究方向包括对深度神经网络的结构和参数优化进行分析，对网络的鲁棒性和可解释性进行研究等。

深度神经网络的函数近似能力证明不仅对于了解神经网络的内在机制和性能特点具有重要意义，也对于指导深度学习的实践具有指导意义。

通过深入研究深度神经网络的函数近似能力，我们可以更好地理解神经网络的工作原理，并提出更有效的算法和模型来解决实际问题。

Department of Computer Science,University of British ColumbiaTechnical Report TR-2008-01,January2008(revised May2008)PROBING THE PARETO FRONTIER FORBASIS PURSUIT SOLUTIONSEWOUT V AN DEN BERG AND MICHAEL P.FRIEDLANDER∗Abstract.The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem.Basis pursuit denoise(BPDN)fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-offbetween the least-squares fit and the one-norm of the solution.We prove that this curve is convex and continuously differentiable over all points of interest,and show that it gives an explicit relationship to two other optimization problems closely related to BPDN.We describe a root-finding algorithm forfinding arbitrary points on this curve;the algorithm is suitable for problems that are large scale and for those that are in the complex domain.At each iteration,a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint.Only matrix-vector operations are required.The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.Key words.basis pursuit,convex program,duality,root-finding,Newton’s method,projected gradient,one-norm regularization,sparse solutionsAMS subject classifications.49M29,65K05,90C25,90C061.Basis pursuit denoise.The basis pursuit problem aims tofind a sparse solution of the underdetermined system of equations Ax=b,where A is an m-by-n matrix and b is an m-vector.Typically,m n,and the problem is ill-posed.The approach advocated by Chen et al.[15]is to solve the convex optimization problemx 1subject to Ax=b.(BP)minimizexIn the presence of noisy or imperfect data,however,it is undesirable to exactlyfit the linear system.Instead,the constraint in(BP)is relaxed to obtain the basis pursuit denoise(BPDN)problemx 1subject to Ax−b 2≤σ,(BPσ)minimizexwhere the positive parameterσis an estimate of the noise level in the data.The case σ=0corresponds to a solution of(BP)—i.e.,a basis pursuit solution.There is now a significant body of work that addresses the conditions under which a solution of this problem yields a sparse approximation to a solution of the underdetermined system;see Cand`e s,Romberg,and Tao[11],Donoho[24],and Tropp[48],and references therein.The sparse approximation problem is of vital importance to many applications in signal processing and statistics.Some important applications include image reconstruction,such as MRI[36,37]and seismic[31,32] images,and model selection in regression[26].In many of these applications,the data sets are large,and the matrix A is available only as an operator.In compressed sensing[10–12,23],for example,the matrices are often fast operators such as Fourier or wavelet transforms.It is therefore crucial to develop algorithms for the sparse reconstruction problem that scale well and work effectively in a matrix-free context.∗Department of Computer Science,University of British Columbia,Vancouver V6T1Z4,B.C., Canada({ewout78,mpf}@cs.ubc.ca).Friedlander is corresponding author.This work was supported by the NSERC Collaborative Research and Development Grant334810-05.May30,200812 E.van den BERG and M.P.FRIEDLANDERWe present an algorithm,suitable for large-scale applications,that is capable of finding solutions of (BP σ)for any value of σ≥0.Our approach is based on recasting (BP σ)as a problem of finding the root of a single-variable nonlinear equation.At each iteration of our algorithm,an estimate of that variable is used to define a convex optimization problem whose solution yields derivative information that can be used by a Newton-based root finding algorithm.1.1.One-norm regularization.The convex optimization problem (BP σ)is only one possible statement of the one-norm regularized least-squares problem.In fact,the BPDN label is typically applied to the penalized least-squares problem (QP λ)minimize xAx −b 22+λ x 1,which is the problem statement proposed by Chen et al.[14,15].A third formulation,(LS τ)minimize x Ax −b 2subject to x 1≤τ,has an explicit one-norm constraint and is often called the Lasso problem [46].The parameter λis related to the Lagrange multiplier of the constraint in (LS τ)and to the reciprocal of the multiplier of the constraint in (BP σ).Thus,for appropriate parameter choices of σ,λ,and τ,the solutions of (BP σ),(QP λ),and (LS τ)coincide,and these problems are in some sense equivalent.However,except for special cases—such as A orthogonal—the parameters that make these problems equivalent cannot be known a priori.The formulation (QP λ)is often preferred because of its close connection to convex quadratic programming,for which many algorithms and software are available;some examples include iteratively reweighted least squares [7,section 4.5]and gradient projection [27].For the case where an estimate of the noise-level σis known,Chen et al.[15,section 5.2]argue that the choice λ=σ√2log n has important optimality properties.However,this argument hinges on the orthogonality of A .We focus on the situation where σis approximately known—such as when we can estimate the noise levels inherent in an underlying system or in the measurements taken.In this case it is preferable to solve (BP σ),and here this is our primary goal.An important consequence of our approach is that it can also be used to efficiently solve the closely related problems (BP )and (LS τ).Our algorithm also applies to all three problems in the complex domain,which can arise in signal processing applications.1.2.Approach.At the heart of our approach is the ability to efficiently solve a sequence of (LS τ)problems using a spectral projected-gradient (SPG)algorithm [5,6,18].As with (QP λ),this problem is parameterized by a scalar;the crucial difference,however,is that the dual solution of (LS τ)yields vital information on how to update τso that the next solution of (LS τ)is much closer to the solution of (BP σ).Let x τdenote the optimal solution of (LS τ).The single-parameter functionφ(τ)= r τ 2with r τ:=b −Ax τ(1.1)gives the optimal value of (LS τ)for each τ≥0.As we describe in section 2,its derivative is given by −λτ,where λτ≥0is the unique dual solution of (LS τ).Importantly,this dual solution can easily be obtained as a by-product of the minimization of (LS τ);thisPROBING THE PARETO FRONTIER FOR BASIS PURSUIT SOLUTIONS3 is discussed in section2.1.Our approach is then based on applying Newton’s method tofind a root of the nonlinear equationφ(τ)=σ,(1.2)which defines a sequence of regularization parametersτk→τσ,where xτσis a solutionof(BPσ).In other words,τσis the parameter that causes(LSτ)and(BPσ)to share the same solution.There are four distinct components to this paper.Thefirst two are related to the root-finding algorithm for(BPσ).The third is an efficient algorithm for solving (LSτ)—and hence for evaluating the functionφand its derivativeφ .The fourth gives the results of a series of numerical experiments.Pareto curve(section2).The Pareto curve defines the optimal trade-offbetween the two-norm of the residual r and the one-norm of the solution x.The problems (BPσ)and(LSτ)are two distinct characterizations of the same curve.Our approach uses the functionφto parameterize the Pareto curve byτ.We show that for all points of interest,φ—and hence the Pareto curve—is continuously differentiable.We are also able to give an explicit expression for its derivative.This surprising result permits us to use a Newton-based root-finding algorithm tofind roots of the nonlinear equation (1.2),which correspond to points on the Pareto curve.Thus we canfind solutions of (BPσ)for anyσ≥0.Rootfinding(section3).Each iteration of the root-finding algorithm for(1.2) requires the evaluation ofφandφ at someτ,and hence the minimization of(LSτ). This is a potentially expensive subproblem,and the effectiveness of our approach hinges on the ability to solve this subproblem only approximately.We present rate-of-convergence results for the case whereφandφ are known only approximately.This is in contrast to the usual inexact-Newton analysis[22],which assumes thatφis known exactly.We also give an effective stopping rule for determining the required accuracy of each minimization of(LSτ).Projected gradient for Lasso(section4).We describe an SPG algorithm for solving (LSτ).Each iteration of this method requires an orthogonal projection of an n-vector onto the convex set x 1≤τ.In section4.2we give an algorithm for this projection with a worst-case complexity of O(n log n).In many important applications,A is a Fourier-type operator,and matrix-vector products with A and A T can be obtained with O(n log n)cost.The projection cost is typically much smaller than the worst-case, and the dominant cost in our algorithm consists of the matrix-vector products,as it does in other algorithms for basis pursuit denoise.We also show how the projection algorithm can easily be extended to project complex vectors,which allows us to extend the SPG algorithm to problems in the complex domain.Implementation and numerical experiments(sections5and6).To demonstrate the effectiveness of our approach,we run our algorithm on a set of benchmark problems and compare it to other state-of-the-art solvers.In sections6.1and6.2we report numerical results on a series of(BPσ)and(BP)problems,which are normally considered as distinct problems.In section6.3we report numerical results on a series of(LSτ)problems for various values ofτ,and compare against the equivalent(QPλ)formulations.In section6.4we show how to capitalize on the smoothness of the Pareto curve to obtain quick and approximate solutions to(BPσ).1.3.Assumption.We make the following blanket assumption throughout:Assumption1.1.The vector b∈range(A),and b=0.This assumption is only needed in order to simplify the discussion,and it implies that(BPσ)is feasible for all4 E.van den BERG and M.P.FRIEDLANDERσ≥0.In many applications,such as compressed sensing[10–12,23],A has full row rank,and therefore this assumption is satisfied automatically.1.4.Related work.Homotopy approaches.A number of approaches have been suggested for solving (BPσ),many of which are based on repeatedly solving(QPλ)for various values ofλ. Notable examples of this approach are Homotopy[41,42]and Lars[26],which solve (QPλ)for essentially all values ofλ.In this way,they eventuallyfind the value ofλthat recovers a solution of(BPσ).These active-set continuation approaches begin withλ= A T b ∞(for which the corresponding solution is xλ=0),and gradually reduceλin stages that predictably change the sparsity pattern in xλ.The remarkable efficiencyof these continuation methods follows from their ability to systematically update the resulting sequence of solutions.(See Donoho and Tsaig[25]and Friedlander and Saunders[28]for discussions of the performance of these methods.)The computational bottleneck for these methods is the accurate solution at each iteration of a least-squares subproblem that involves a subset of the columns of A.In some applications(such as the seismic image reconstruction problem[32])the size of this subset can become large, and thus the least-squares subproblems can become prohibitively expensive.Moreover, even if the correct valueλσis known a priori,the method must necessarily begin with λ= A T b ∞and traverse all critical values ofλdown toλσ.Basis pursuit denoise as a cone program.The problem(BPσ)withσ>0can be considered as a special case of a generic second-order cone program[8,Ch.5]. Interior-point(IP)algorithms for general conic programs can be very effective if the matrices are available explicitly.Examples of general-purpose software for cone programs include SeDuMi[45]and MOSEK[39].The software package 1-magic[9] contains an IP implementation specially adapted to(BPσ).In general,the efficiency of IP implementations relies ultimately on their ability to efficiently solve certain linear systems that involve highly ill-conditioned matrices.Basis pursuit as a linear program.The special caseσ=0corresponding to(BP) can be reformulated and solved as a linear program.Again,IP methods are known to be effective for general linear programs,but many IP implementations for general linear programming,such as CPLEX[16]and MOSEK,require explicit matrices.The solver PDCO[44],available within the SparseLab package,is capable of using A as an operator only,but it often requires many matrix-vector multiplications to converge, and as we report in section6.2,it is not generally competitive with other approaches. We are not aware of simplex-type implementations that do not require A explicitly.Sampling the Pareto curve.A common approach for obtaining approximate solu-tions to(BPσ)is to sample various points on the Pareto curve;this is often accomplished by solving(QPλ)for a decreasing sequence of values ofλ.As noted by Das and Den-nis[19],and more recently by Leyffer[35],a uniform distribution of weightsλcan lead to an uneven sampling of the Pareto curve.In contrast,by instead parameterizing the Pareto curve byσorτ(via the problem(BPσ)or(LSτ)),it is possible to obtain a more uniform sample of the Pareto curve;see section6.4.Projected gradient.Our application of the SPG algorithm to solve(LSτ)follows Birgin et al.[5]closely for minimization of general nonlinear functions over arbitrary convex sets.The method they propose combines projected-gradient search directions with the spectral step length that was introduced by Barzilai and Borwein[1].A nonmonotone linesearch is used to accept or reject steps.The key ingredient of Birgin et al.’s algorithm is the projection of the gradient direction onto a convex set,which in our case is defined by the constraint in(LSτ).In their recent report,FigueiredoPROBING THE PARETO FRONTIER FOR BASIS PURSUIT SOLUTIONS5Fig.2.1:A typical Pareto curve(solid line)showing two iterations of Newton’s method.The first iteration is available at no cost.et al.[27]describe the remarkable efficiency of an SPG method specialized to(QPλ). Their approach builds on the earlier report by Dai and Fletcher[18]on the efficiency of a specialized SPG method for general bound-constrained quadratic programs(QP s).2.The Pareto curve.The functionφdefined by(1.1)yields the optimal value of the constrained problem(LSτ)for each value of the regularization parameterτ.Its graph traces the optimal trade-offbetween the one-norm of the solution x and the two-norm of the residual r,which defines the Pareto curve.Figure2.1shows the graph ofφfor a typical problem.The Newton-based root-finding procedure that we propose for locating specific points on the Pareto curve—e.g.,finding roots of(1.2)—relies on several important properties of the functionφ.As we show in this section,φis a convex and differentiable function ofτ.The differentiability ofφis perhaps unintuitive,given that the one-norm constraint in(LSτ)is not differentiable.To deal with the nonsmoothness of the one-norm constraint,we appeal to Lagrange duality theory.This approach yields significant insight into the properties of the trade-offcurve.We discuss the most important properties below.2.1.The dual subproblem.The dual of the Lasso problem(LSτ)plays a prominent role in understanding the Pareto curve.In order to derive the dual of(LSτ), wefirst recast(LSτ)as the equivalent problemr 2subject to Ax+r=b, x 1≤τ.(2.1) minimizer,xThe dual of this convex problem is given bymaximizeL(y,λ)subject toλ≥0,(2.2)y,λwhere{ r 2−y T(Ax+r−b)+λ( x 1−τ)}L(y,λ)=infx,ris the Lagrange dual function,and the m-vector y and scalarλare the Lagrange multipliers(e.g.,dual variables)corresponding to the constraints in(2.1).We use the6 E.van den BERG and M.P.FRIEDLANDERseparability of the infimum in r and x to rearrange terms and arrive at the equivalent statementL(y,λ)=b T y−τλ−supr {y T r− r 2}−supx{y T Ax−λ x 1}.We recognize the suprema above as the conjugate functions of r 2and ofλ x 1, respectively.For an arbitrary norm · with dual norm · ∗,the conjugate function of f(x)=α x for anyα≥0is given byf∗(y):=supx {y T x−α x }=0if y ∗≤α,∞otherwise;(2.3)see Boyd and Vandenberghe[8,section3.3.1].With this expression of the conjugate function,it follows that(2.2)remains bounded if and only if the dual variables y and λsatisfy the constraints y 2≤1and A T y ∞≤λ.The dual of(2.1),and hence of (LSτ),is then given bymaximizey,λb T y−τλsubject to y 2≤1, A T y ∞≤λ;(2.4)the nonnegativity constraint onλis implicitly enforced by the second constraint.Importantly,the dual variables y andλcan easily be computed from the optimal primal solutions.To derive y,first note that from(2.3),supr{y T r− r 2}=0if y 2≤1.(2.5)Therefore,y=r/ r 2,and we can without loss of generality take y 2=1in(2.4). To deriveλ,note that as long asτ>0,λmust be at its lower bound,as implied by the constraint A T y ∞≤λ.Hence,we takeλ= A T y ∞.(If r=0orτ=0,the choice of y orλ,respectively,is arbitrary.)The dual variable y can then be eliminated, and we arrive at the following necessary and sufficient optimality conditions for the primal-dual solution(rτ,xτ,λτ)of(2.1):Axτ+rτ=b, xτ 1≤τ(primal feasibility);(2.6a)A T rτ ∞≤λτ rτ 2(dual feasibility);(2.6b)λτ( xτ 1−τ)=0(complementarity).(2.6c)2.2.Convexity and differentiability of the Pareto curve.LetτBPbe the optimal objective value of the problem(BP).This corresponds to the smallest value ofτsuch that(LSτ)has a zero objective value.As we show below,φis nonincreasing,and thereforeτBPis thefirst point at which the graph ofφtouches the horizontal axis.Our assumption that0=b∈range(A)implies that(BP)is feasible,and thatτBP>0. Therefore,at the endpoints of the interval of interest,φ(0)= b 2>0andφ(τBP)=0.(2.7) As the following result confirms,the function is convex and strictly decreasing overthe intervalτ∈[0,τBP].It is also continuously differentiable on the interior of this interval—this is a crucial property.PROBING THE PARETO FRONTIER FOR BASIS PURSUIT SOLUTIONS 7Theorem 2.1.(a)The function φis convex and nonincreasing.(b)For all τ∈(0,τBP ),φis continuously differentiable,φ (τ)=−λτ,and theoptimal dual variable λτ= A T y τ ∞,where y τ=r τ/ r τ 2.(c)For τ∈[0,τBP ], x τ 1=τ,and φis strictly decreasing.Proof .(a)The function φcan be restated asφ(τ)=inf x f (x,τ),(2.8)wheref (x,τ):= Ax −b 2+ψτ(x )and ψτ(x ):= 0if x 1≤τ,∞otherwise .Note that by (2.3),ψτ(x )=sup z {x T z −τ z ∞},which is the pointwise supremum of an affine function in (x,τ).Therefore it is convex in (x,τ).Together with the convexity of Ax −b 2,this implies that f is convex in (x,τ).Consider any nonnegative scalars τ1and τ2,and let x 1and x 2be the corresponding minimizers of (2.8).For any β∈[0,1],φ(βτ1+(1−β)τ2)=inf x f (x,βτ1+(1−β)τ2)≤f βx 1+(1−β)x 2,βτ1+(1−β)τ2≤βf (x 1,τ1)+(1−β)f (x 2,τ2)=βφ(τ1)+(1−β)φ(τ2).Hence,φis convex in τ.Moreover,φis nonincreasing because the feasible set enlarges as τincreases.(b)The function φis differentiable at τif and only if its subgradient at τis unique[43,Theorem 25.1].By [4,Proposition 6.5.8(a)],−λτ∈∂φ(τ).Therefore,to prove differentiability of φit is enough show that λτis unique.Note that λappears linearly in (2.4)with coefficient −τ<0,and thus λτis not optimal unless it is at its lower bound,as implied by the constraint A T y ∞≤λ.Hence,λτ= A T y τ ∞.Moreover,convexity of (LS τ)implies that its optimal value is unique,and so r τ≡b −Ax τis unique.Also, r τ >0because τ<τBP (cf.(2.7)).As discussed in connection with(2.5),we can then take y τ=r τ/ r τ 2,and so uniqueness of r τimplies uniqueness of y τ,and hence uniqueness of λτ,as required.The continuity of the gradient follows from the convexity of φ.(c)The assertion holds trivially for τ=0.For τ=τBP , x τBP 1=τBP by definition.It only remains to prove part (c)on the interior of the interval.Note that φ(τ)≡ r τ >0for all τ∈[0,τBP ).Then by part (b),λτ>0,and hence φis strictly decreasing for τ<τBP .But because x τand λτboth satisfy the complementarity condition in (2.6),it must hold that x τ 1=τ.2.3.Generic regularization.The technique used to prove Theorem 2.1does not in any way rely on the specific norms used in the objective and regularization functions,and it can be used to prove similar properties for the generic regularized fitting problemminimize x Ax −b s subject to x p ≤τ,(2.9)8 E.van den BERG and M.P.FRIEDLANDERwhere1≤(p,s)≤∞define the norms of interest,i.e., x p=(i|x i|p)1/p.Moregenerally,the constraint here may appear as Lx p,where L may be rectangular. Such a constraint defines a seminorm,and it often arises in discrete approximations of derivative operators.In particular,least-squares with Tikhonov regularization[47], which corresponds to p=s=2,is used extensively for the regularization of ill-posed problems;see Hansen[29]for a comprehensive study.In this case,the Pareto curve defined by the optimal trade-offbetween x 2and Ax−b 2is often called the L-curve because of its shape when plotted on a log-log scale[30].If we define¯p and¯s such that1/p+1/¯p=1and1/s+1/¯s=1,then the dual of the generic regularization problem is given bymaximizey,λb T y−τλsubject to y ¯s≤1, A T y ¯p≤λ.As with(2.4),the optimal dual variables are given by y=r/ r ¯p andλ= A T y ¯s. This is a generalization of the results obtained by Dax[21],who derives the dual for p and s strictly between1and∞.The corollary below,which follows from a straightfoward modification of Theorem2.1,asserts that the Pareto curve defined for any1≤(p,s)≤∞in(2.7)has the properties of convexity and differentiability.Corollary2.2.Letθ(τ):= rτ s,where rτ:=b−Axτ,and xτis the optimal solution of(2.9).(a)The functionθis convex and nonincreasing.(b)For allτ∈(0,τBP),θis continuously differentiable,θ (τ)=−λτ,and the optimal dual variableλτ= A T yτ ¯p,where yτ=rτ/ rτ ¯s.(c)Forτ∈[0,τBP], xτ p=τ,andθis strictly decreasing.3.Rootfinding.As we briefly outlined in section1.2,our algorithm generatesa sequence of regularization parametersτk→τσbased on the Newton iterationτk+1=τk+∆τk with∆τk:=σ−φ(τk)/φ (τk),(3.1)such that the corresponding solutions xτk of(LSτk)converge to xσ.For valuesofσ∈(0, b 2),Theorem2.1implies thatφis convex,strictly decreasing,and continuously differentiable.In that case it is clear thatτk→τσsuperlinearly for all initial valuesτ0∈(0,τBP)(see,e.g.,Bertsekas[3,proposition1.4.1]).The efficiency of our method,as with many Newton-type methods for large problems,ultimately relies on the ability to carry out the iteration described by(3.1) with only an approximation ofφ(τk)andφ (τk).Although the nonlinear equation(1.2) that we wish to solve involves only a single variableτ,the evaluation ofφ(τ)involves the solution of(LSτ),which can be a large optimization problem that is expensive to solve to full accuracy.For systems of nonlinear equations in general,inexact Newton methods assume that the Newton system analogous to the equationφ (τk)∆τk=σ−φ(τk)is solved only approximately,with a residual that is a fraction of the right-hand side.A constant fraction yields a linear convergence rate,and a fraction tending to zero yields aPROBING THE PARETO FRONTIER FOR BASIS PURSUIT SOLUTIONS9 superlinear convergence rate(see,e.g.,Nocedal and Wright[40,theorem7.2]).However, the inexact-Newton analysis does not apply to the case where the right-hand side(i.e., the function itself)is known only approximately,and it is therefore not possible to know a priori the accuracy required to achieve an inexact-Newton-type convergence rate.This is the situation that we are faced with if(LSτ)is solved approximately.As we show below,with only approximate knowledge of the function valueφthis inexact version of Newton’s method still converges,although the convergence rate is sublinear. The rate can be made arbitrarily close to superlinear by increasing the accuracy with which we computeφ.3.1.Approximate primal-dual solutions.In this section we use the duality gap to derive an easily computable expression that bounds the accuracy of the computed function value ofφ.The algorithm for solving(LSτ)that we outline in section4maintains feasibility of the iterates at all iterations.Thus,an approximate solution¯xτand its corresponding residual¯rτ:=b−A¯xτsatisfy¯xτ 1≤τ,and ¯rτ 2≥ rτ 2>0,(3.2).We where the second set of inequalities holds because¯xτis suboptimal andτ<τBPcan thus construct the approximations¯yτ:=¯rτ/ ¯rτ 2and¯λτ:= A T¯yτ ∞to the dual variables that are dual feasible,i.e.,they satisfy(2.6b).The value of the dual problem(2.2)at any feasible point gives a lower bound on the optimal value rτ 2,and the value of the primal problem(2.1)at any feasible point gives an upper bound on the optimal value.Therefore,b T¯yτ−τ¯λτ≤ rτ 2≤ ¯rτ 2.(3.3) We use the duality gapδτ:= ¯rτ 2−(b T¯yτ−τ¯λτ)(3.4) to measure the quality of an approximate solution¯xτ.By(3.3),δτis necessarily nonnegative.Let¯φ(τ):= ¯rτ 2be the objective value of(LSτ)at the approximate solution¯xτ. The duality gap at¯xτprovides a bound on the difference betweenφ(τ)and¯φ(τ).If we additionally assume that A is full rank(so that its condition number is bounded), we can also useδτto provide a bound on the difference between the derivativesφ (τ)),and¯φ (τ).From(3.3)–(3.4)and from Theorem2.1(b),for allτ∈(0,τBP¯φ(τ)−φ(τ)<δand|¯φ (τ)−φ (τ)|<γδτ(3.5)τfor some positive constantγthat is independent ofτ.It follows from the definition ofφ and from standard properties of matrix norms thatγis proportional to the condition number of A.3.2.Local convergence rate.The following theorem establishes the local con-vergence rate of an inexact Newton method for(1.2)whereφandφ are known only approximately.10 E.van den BERG and M.P.FRIEDLANDER Theorem 3.1.Suppose that A has full rank,σ∈(0, b 2),and δk :=δτk →0.Then if τ0is close enough to τσ,the iteration (3.1)—with φand φ replaced by ¯φand ¯φ—generates a sequence τk →τσthat satisfies |τk +1−τσ|=γδk +ηk |τk −τσ|,(3.6)where ηk →0and γis a positive constant.Proof .Because φ(τσ)=σ∈(0, b 2),equation (2.7)implies that τσ∈(0,τBP ).By Theorem 2.1we have that φ(τ)is continuously differentiable for all τclose enough to τσ,and so by Taylor’s theorem,φ(τk )−σ= 10φ (τσ+α[τk −τσ])dα·(τk −τσ)=φ (τk )(τk −τσ)+10 φ (τσ+α[τk −τσ])−φ (τk ) ·dα(τk −τσ)=φ (τk )(τk −τσ)+ω(τk ,τσ),where the remainder ωsatisfiesω(τk ,τσ)/|τk −τσ|→0as |τk −τσ|→0.(3.7)By (3.5)and because (3.2)holds for τ=τk ,there exist positive constants γ1and γ2,independent of τk ,such that φ(τk )−σφ (τk )−¯φ(τk )−σ¯φ (τk ) ≤γ1δk and |φ (τk )|−1<γ2.Then,because ∆τk = σ−¯φ(τk ) /¯φ (τk ),|τk +1−τσ|=|τk −τσ+∆τk |= −¯φ(τk )−σ¯φ (τk )+1φ (τk ) φ(τk )−σ−ω(τk ,τσ) ≤ φ(τk )−σφ (τk )−¯φ(τk )−σ¯φ (τk ) + ω(τk ,τσ)φ (τk )=γ1δk +γ2|ω(τk ,τσ)|=γ1δk +ηk |τk −τσ|,where ηk :=γ2|ω(τk ,τσ)|/|τk −τσ|.With τk sufficiently close to τσ,(3.7)implies that ηk <1.Applythe above inequality recursively ≥1times to obtain|τk + −τσ|≤γ1 i =1(γ1) −i δk +i −1+(ηk ) |τk −τσ|,and because δk →0and ηk <1,it follows that τk + →τσas →∞.Thus τk →τσ,as required.By again applying (3.7),we have that ηk →0.Note that if (LS τ)is solved exactly at each iteration,such that δk =0,then Theorem 3.1shows that the convergence rate is superlinear,as we expect of a standard Newton iteration.In effect,the convergence rate of the algorithm depends on the rate at which δk →0.If A is rank deficient,then the constant γin (3.6)is infinite;we thus expect that ill-conditioning in A leads to slow convergence unless δk =0,i.e.,φis evaluated accurately at every iteration.。

标题：convnext的语义分割算法一、概述随着深度学习技术的不断发展，语义分割作为计算机视觉领域一项重要的任务，受到了广泛关注。

在语义分割任务中，算法需要对图像中的每个像素进行分类，从而实现对图像的像素级别理解，这为许多实际应用带来了便利，如无人驾驶、医学影像分析等。

二、convnext的介绍convnext是一家专注于图像处理和计算机视觉领域的技术公司，其团队在深度学习、卷积神经网络等方面具有丰富的经验和创新能力。

在语义分割领域，convnext提出了一系列高效且性能优越的语义分割算法，为图像处理领域的发展做出了重要贡献。

三、convnext的语义分割算法原理1. 深度卷积神经网络convnext的语义分割算法基于深度卷积神经网络（DCNN），通过多层次的卷积和池化操作，从输入的图像中提取出高层次的语义信息，为后续的像素分类提供了有力支持。

2. 空洞卷积为了扩大感受野并避免信息损失，convnext的算法引入了空洞卷积机制，通过在卷积核之间引入空洞以增加采样间隔，提高了模型对远距离像素关联的感知能力，从而提高了语义分割的准确性。

3. 多尺度融合convnext的算法采用多尺度特征融合的策略，将不同尺度下提取的特征信息进行有效融合，使模型能够更好地理解图像的局部和全局结构。

4. 损失函数设计在训练过程中，convnext的算法采用了针对语义分割任务设计的多任务损失函数，同时考虑了像素级分类的准确性和语义信息的连续性，从而提高了模型的稳定性和泛化能力。

四、convnext的语义分割算法性能1. 准确性经过在多个公开数据集上的验证，convnext的语义分割算法在像素级别的分类准确率和语义信息的连续性上均取得了显著的提升，超越了之前大部分同类算法。

2. 鲁棒性convnext的算法在处理不同场景、不同光照条件下的图像时表现出了较强的鲁棒性，能够稳定地输出准确的语义分割结果，适用于多种实际应用场景。