Discriminative linear transforms for feature normalization and speaker adaptation

discriminator()函数discriminator()函数是一种用于人工智能和机器学习领域的函数，主要用于区分不同类型的数据和信息。

该函数通常用于将数据分为两个类别，分别为真和假，以判断所输入的信息是否是真实有效的。

在机器学习中，discriminator()函数被广泛应用于对数据集中的样本进行分类。

例如，当我们在训练生成对抗网络（GAN）时，该函数将用于对生成的图像和真实的图像进行区分。

在这种情况下，discriminator()函数是判别器，因为它必须区分出生成的图像是否与真实图像相似。

在深度学习领域中，discriminator()函数还可以用于对图像、文本、语音等不同类型的数据进行分类。

在这个过程中，数据集中的样本被随机分配到不同的类别中，该函数会自动学习如何将这些数据分类，以便在未来的预测和决策中更加精确和可信。

除了上述应用之外，discriminator()函数还可以用于识别垃圾邮件、恶意软件和诈骗行为等方面。

通过将数据集中的样本分为真实和虚假，该函数能够识别出那些不真实的信息，从而帮助用户防止不必要的骗局和欺诈行为。

在编写discriminator()函数时，需要注意以下几点：1. 建立适当的训练数据集，以确保该函数可以有效地分类各种不同类型的数据。

2. 使用适当的算法和技术来训练并优化该函数，以便使其能够准确地识别不同类别的数据。

3. 通过不断改进和调整函数参数，确保其在不同场景和应用中的性能和效果都能够得到充分的体现和发挥。

总之，discriminator()函数是一种非常有用的函数，可以用于对各种数据和信息进行分类和区分。

通过不断改进和优化该函数的性能和效果，我们可以更好地应对各种数据分类和识别的挑战，从而大大提高机器学习和人工智能技术的应用效果和价值。

离散傅里叶变换的算术傅里叶变换算法张宪超1，武继刚1，蒋增荣2，陈国良1（1.中国科技大学计算机科学与技术系，合肥230027；2.国防科技大学系统工程与数学系，长沙410073）摘要：离散傅里叶变换（DFT）在数字信号处理等许多领域中起着重要作用.本文采用一种新的傅里叶分析技术—算术傅里叶变换（AFT）来计算DFT.这种算法的乘法计算量仅为0（N）；算法的计算过程简单，公式一致，克服了任意长度DFT传统快速算法（FFT）程序复杂、子进程多等缺点；算法易于并行，尤其适合VLSI设计；对于含较大素因子，特别是素数长度的DFT，其速度比传统的FFT方法快；算法为任意长度DFT的快速计算开辟了新的思路和途径.关键词：离散傅里叶变换（DFT）；算术傅里叶变换（AFT）；快速傅里叶变换（FFT）中图分类号：TN917文献标识码：A文章编号：0372-2112（2000）05-0105-03An Algorithm for Computing DFT Using Arithmetic Fourier TransformZHANG Xian-chao1，WU Ji-gang1，JIANG Zeng-rong2，CHEN Guo-iiang1（1.Dept.of CompUter Science&Technology，Unio.of Science&Technology of China，Hefei230027，China；2.Dept.of System Engineering&Mathematics，National Unio.of Defense Technology，Changsha410073，China）Abstract：The Discrete Fourier Transform（DFT）piays an important roie in digitai signai processing and many other fieids.In this paper，a new Fourier anaiysis technigue caiied the arithmetic Fourier transform（AFT）is used to compute DFT.This aigorithm needs oniy0（N）muitipiications.The process of the aigorithm is simpie and it has a unified formuia，which overcomes the disadvantage of the traditionai fast method that has a compieX program containing too many subroutines.The aigorithm can be easiiy performed in paraiiei，especiaiiy suitabie for VLSI designing.For a DFT at a iength that contains big prime factors，especiaiiy for a DFT at a prime iength，it is faster than the traditionai FFT method.The aigorithm opens up a new approach for the fast computation of DFT.Key words：discrete Fourier transform（DFT）；arithmetic Fourier transform（AFT）；fast Fourier transform（FFT）!引言离散傅里叶变换（DFT）在数字信号处理等许多领域中起着重要作用.但DFT的计算量很大（N点DFT需0（N2）乘法和加法）.因此，DFT的快速计算问题非常重要.1965年，Cooiey 和Tukey开创了快速傅里叶变换（FFT）方法，使N点DFT的计算量从0（N2）降到0（N iog N），开辟了DFT的快速计算时代.但FFT的计算仍较复杂，且对不同长度的DFT其计算公式不一致，致使任意长DFT的FFT程序非常复杂，包含大量子进程.1988年，Tufts和Sadasiv［1］提出了一种用莫比乌斯反演公式（Mibius inversion formuia）计算连续函数的傅立叶系数的方法并命名为算术傅立叶变换（AFT）.AFT有许多良好的性质：其乘法量仅为0（N）；算法简单，并行性好，尤其适合VLSI设计.因此很快得到广泛关注，并在数字图像处理等领域得到应用.AFT已成为继FFT后一种新的重要的傅立叶分析技术［2~5］.根据DFT和连续函数的傅立叶系数的关系，可以用AFT 计算DFT.这种方法保持了AFT的良好性质，且具有公式一致性.大量实验表明，同直接计算相比，AFT方法可以将DFT的计算时间减少90%，对含较大素因子，特别是其长度本身为素数的DFT，它的速度比传统的FFT快.从而它为DFT快速计算开辟了新的途径."算术傅立叶变换本文采用文［3］中的算法.设A（t）为周期为T的函数，它的傅立叶级数只含有限项，即：A（t）=a0+!Nn=1a n cos2!f0t+!Nn=1b n sin2!f0t（1）其中：f0=1/T，a0=1T"TA（t）dt.令：B（2n，!）=12n!2n-1m=0（-1）m A（m2nT+!T），-1<!<1（2）则傅立叶系数a n和b n可以由下列公式计算：a n=!［N/n］l=1，3，5，…U（l）B（2nl，0）b n=!［N/n］l=1，3，5，…U（l）（-1）（l-1）/2B（2n，14nl），n=1，…，N（3）第5期2000年5月电子学报ACTA ELECTR0NICA SINICAVoi.28No.5May2000其中：!（l ）=I ，（-I ）r ，0{，l =I l =p I p 2…p r 3p 使p 2\l为莫比乌斯（M bioLS ）函数.这就是AFT ，其计算量为：加法：N 2+［N /2］+［N /3］+…+I -2N ；乘法：2N.AFT 需要函数大量的不均匀样本点，而在实际应用中，若计算函数前N 个傅立叶系数，根据奈奎斯特（NygLiSt ）抽样定律，只需在函数的一个周期内均匀抽取2N 个样本点.这时可以用零次插值解决样本不一致问题.文献［2、3］已作了详细的分析，本文不再重复.3DFT 的AFT 算法3.1DFT 的定义及性质定义1设X I 为一长度为N 的序列，它的DFT 定义为：Y I =Z N-II =0X I w II ，I =0，I ，…，N -I ；w =e -i 2!/N（4）性质1用记号X I 、=、Y I 表示序列Y I 为序列X I 的DFT ，G I 、=、H I ，则：pX I +gG I 、=、pY I +gH I （5）因此，一个复序列的DFT 可以用两个实序列的DFT 计算.故本文只讨论实序列DFT 的计算问题.性质2设X I 为一实序列，X I 、=、Y I ，则：Re Y I =Re Y N -I ，Im Y I =-Im Y N -I （Re Y I 和Im Y I 分别代表Y I 的实部和虚部）（6）因此，对N 点实序列DFT ，只需计算：Re Y I 和Im Y I （I =0，…，「N /2）.3.2DFT 的AFT 算法离散序列的DFT 和连续函数的傅立叶系数有着密切的联系.事实上，若序列X I 是一段区间［0，T ］上的函数A （t ）经过离散化后得到的，再设A （t ）的傅立叶级数只含前N /2项，即：A （t ）=a 0+Z「N /2-II =Ia I coS2!f 0t +Z「N /2-II =I6I Sin2!f 0t（7）则DFT Y I 和傅立叶系数的关系为：Re Y I =「N /2a I /2Im Y I =「N /26I /{2，I =0，…，「N /2（8）式（7）中函数代表的是一种截频信号.对一般函数，式（8）中的“=”要改为“匀”［7］.因此，序列X I 的DFT 可以通过函数A （t ）的傅里叶系数计算.对于一般给定序列X I ，注意到在任意一个区间上，经过离散后能得到序列X I 的函数有无穷多个.对所有这些插值函数，公式（8）都近似地满足（仅式（7）中的函数精确地满足式（8））［7］.AFT 的零次插值实现实质上就是用这些插值函数中的零次插值函数代替原来的函数进行计算的.而从AFT 的零次插值实现方法可知，用AFT 计算傅里叶系数，实际上参与计算的只是函数经离散化后得到的序列，而不必知道函数本身.因此，我们可以任取一个区间，在这个区间上，把序列X算（8）中的“傅里叶系数”，再通过式（8），就可以计算出序列的DFT .算法描述如下（采用［0，I ］区间）：for I =I to 「N /2for m =0to 2I -IB （2I ，0）：=B （2I ，0）+（-I ）mX［Nm /2I +0.5］B （2I ，I /4I ）：=B （2I ，I /4I ）+（-I ）mX［Nm /2I +N /4I +0.5］endforB （2I ，0）：=B （2I ，0）/2I B （2I ，I /4I ）：=B （2I ，I /4I ）/2I endforfor =0to N -I a 0：=a 0+X （ ）/N for I =I to 「N /2for I =I to［「N /2/I ］by 2a I ：=a I +!（I ）B （2II ，0）6I ：=6I +!（I ）（-I ）（K -I ）/2B （2II ，I /4II ）endforRe Y I ：=「N /2a I /2Re Y N -I ：=Re Y I Im Y I ：=「N /2a I /2Im Y N -I ：=-Im Y I endfor endfor图IDFT 的AFT 算法程序AFT 方法的误差主要是由零次插值引起的，大量实验表明，同FFT 相比，其误差是可以接受的（部分实验结果见附录）.4算法的性能4.1算法的程序DFT 的AFT 算法具有公式一致性，且公式简单，因此算法的程序也很简单（图I ）.图2DFT 的AFT 算法进程示意为便于比较，不妨看一下FFT 的流程.图3FFT 算法进程示意可以看出，FFT 的程序中包含大量子进程，且这些子程序都较复杂.其中素数长度DFT 的FFT 算法程序尤其复杂.因此，任意长DFT 的FFT 算法其程序是非常复杂的.4.2算法的计算效率AFT 方法把DFT 的乘法计算量从0（N 2）降到0（N ），它2电子学报2000年计算时间减少90%.当DFT的长度!为2的幂时，FFT比AFT 方法快"对一般长度的DFT，当!含较大素因子时，AFT方法比FFT快；当!的因子都较小时，AFT方法不如FFT快.当DFT长度!本身为一较大素数时，AFT方法比FFT快"附录中给出部分实验结果以便比较"特别指出，对素数长度DFT，FFT的计算过程非常复杂，很难在实际中应用.而AFT方法算法简单，提供了较好的素数长度DFT快速算法"表1是两种算法计算效率较详细的比较"表1长度52191197114832417FFT效率67.30%68.03%72.50%71.23%76.22% AFT方法效率91.39#91.78#91.63#91.81#91.83# 4.3算法的并行性AFT具有良好的并行性，尤其适合VLSI设计，已有许多VLSI设计方案被提出，并在数字图像处理等领域得到应用.DFT的AFT算法继承了AFT优点，同样具有良好的并行性"5结论和展望本文采用算术傅里叶变换（AFT）计算DFT.这种方法把AFT的各种优点引入DFT的计算中来，开辟了DFT快速计算的新途径.把AFT方法同FFT结合起来，还可以进一步提高DFT的计算速度"参考文献［1］ D.W.Tufts and G.Sadasiv.The arithmetic Fourier transform.IEEE ASSP Mag，Jan.1988：13~17［2］I.S.Reed，D.W.Tufts，Xiao Yu，T.K.Troung，M.T.Shih and X.Yin.Fourier anaiysis and signai processing by use of Mobius inversion for-muiar.IEEE Trans.Acoust.Speech Speech Processing，Mar，1990，38（3）：458~470［3］I.S.Reed，Ming Tang Shih，T.K.Truong，R.Hendon and D.W.Tufts.A VLSI architecture for simpiified arithmetic fourier transform aigo-rithm.IEEE Trans.Signai Processing，May，1993，40（5）：1122~1132［4］H.Park and V.K.Prasanna.Moduiar VLSI architectures for computing the arithmetic fourier transform.IEEE.Signai Processing，June，1993，41（6）：2236~2246［5］Lovine.F.P，Tantaratanas.Some aiternate reaiizations of the arithmetic Fourier transform.Conference on Signai，system and Computers，1993，（Cat，93，CH3312-6）：310~314［6］蒋增荣，曾泳泓，余品能.快速算法.长沙：国防科技大学出版社，1993［7］E.0.布赖姆.快速傅立叶变换.上海：上海科学技术出版社，1976附录：较详细的实验结果（机型：586微机，主频：166MHz单位：秒）2的幂长度长度AFT方法基-2FFT直接算法2560.005160.002400.115120.018600.004400.4410240.075800.01100 1.81素数长度长度AFT方法FFT直接算法5210.03790.14390.449710.13400.4400 1.6014830.3103 1.0904 3.7924170.8206 2.389910.75任意长度长度因子分解AFT方法FFT直接算法13462!6370.270.44 3.1429862!1483 1.26 2.1414.8235793!1193 1.81 1.9222.1646374637 3.0821.4237.4755742!3!929 4.45 2.4752.2964364!1609 5.94 3.5772.6278933!3!8778.96 1.92105.49最大相对误差长度AFT方法FFT1024实部 2.1939>10-2 2.3328>10-2虚部 2.1938>10-29.9342>10-2 2048实部 4.2212>10-3 1.1967>10-2虚部 6.1257>10-3 4.9385>10-2 4096实部 2.3697>10-3 6.0592>10-3虚部 2.0422>10-3 2.4615>10-3张宪超1971年生"1994年、1998年分别获国防科技大学学士、硕士学位"现在中国科技大学攻读博士学位"主要研究方向为信号处理的快速、并行计算等"武继刚1963年生"烟台大学副教授，现在中国科技大学攻读博士学位"主要研究方向为算法设计和分析等"3第5期张宪超：离散傅里叶变换的算术傅里叶变换算法。

f.linear用法-回复线性回归是一种常用的机器学习算法，用于预测连续型变量的值。

它基于对自变量和因变量之间的线性关系进行建模，并以此建立预测模型。

在实际应用中，我们经常使用f.linear函数来进行线性回归。

本文将介绍f.linear 函数的用法，并通过一步一步的解释，帮助读者理解如何使用f.linear进行线性回归分析。

1. 简介f.linear是PyTorch框架中的一个函数，用于执行线性回归任务。

它接受多个输入张量，并对它们进行线性操作。

在该函数中，每个输入张量都会被视为一个样本，而每个样本都包含一个或多个特征。

例如，如果我们有100个样本，每个样本有3个特征，我们可以将这100个样本表示为一个形状为(100, 3)的张量。

2. f.linear函数的参数f.linear函数有三个主要的参数: input、weight和bias。

其中，input是输入的张量，weight是线性层的权重张量，而bias是线性层的偏置张量。

下面我们会介绍每个参数的作用。

- input：输入张量，该张量的形状应为(batch_size, n_features)。

其中，batch_size表示样本数量，n_features表示每个样本的特征数量。

- weight：权重张量，该张量的形状应为(output_features,input_features)，其中output_features表示线性层的输出特征数量，input_features表示线性层的输入特征数量。

- bias：偏置张量，该张量的形状应为(output_features,)。

当使用偏置时，输出特征将会加上这个偏置值。

3. 使用f.linear进行线性回归接下来我们将通过一个例子来解释如何使用f.linear进行线性回归。

我们假设有一个数据集，其中包含了房屋的面积和价格两个特征。

我们想要通过建立一个线性回归模型来预测房屋的价格。

首先，我们需要准备我们的数据集。

18IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 1, JANUARY 2008MPCA: Multilinear Principal Component Analysis of Tensor ObjectsHaiping Lu, Student Member, IEEE, Konstantinos N. (Kostas) Plataniotis, Senior Member, IEEE, and Anastasios N. Venetsanopoulos, Fellow, IEEEAbstract—This paper introduces a multilinear principal component analysis (MPCA) framework for tensor object feature extraction. Objects of interest in many computer vision and pattern recognition applications, such as 2-D/3-D images and video sequences are naturally described as tensors or multilinear arrays. The proposed framework performs feature extraction by determining a multilinear projection that captures most of the original tensorial input variation. The solution is iterative in nature and it proceeds by decomposing the original problem to a series of multiple projection subproblems. As part of this work, methods for subspace dimensionality determination are proposed and analyzed. It is shown that the MPCA framework discussed in this work supplants existing heterogeneous solutions such as the classical principal component analysis (PCA) and its 2-D variant (2-D PCA). Finally, a tensor object recognition system is proposed with the introduction of a discriminative tensor feature selection mechanism and a novel classification strategy, and applied to the problem of gait recognition. Results presented here indicate MPCA’s utility as a feature extraction tool. It is shown that even without a fully optimized design, an MPCA-based gait recognition module achieves highly competitive performance and compares favorably to the state-of-the-art gait recognizers. Index Terms—Dimensionality reduction, feature extraction, gait recognition, multilinear principal component analysis (MPCA), tensor objects.I. INTRODUCTION HE term tensor object is used here to denote a multidimensional object, the elements of which are to be addressed by more than two indices [1]. The number of indices used in the description defines the order of the tensor object and each index defines one of the so-called “modes.” Many image and video data are naturally tensor objects. For example, color images are 3-D (third-order tensor) objects with column, row, and color modes [2]. Gait silhouette sequences, the input to most if not all gait recognition algorithms [3]–[7], as well as other grayscale video sequences can be viewed as third-order tensorsManuscript received May 14, 2006; revised October 31, 2006 and January 2, 2007; accepted March 1, 2007. This work was supported in part by the Ontario Centres of Excellence through the Communications and Information Technology Ontario Partnership Program and the Bell University Labs, University of Toronto, Toronto, ON, Canada. This paper was presented in part at the Biometrics Consortium/IEEE 2006 Biometrics Symposium, Baltimore, MD, September 19–21, 2006. H. Lu and K. N. Plataniotis are with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: kostas@). A. N. Venetsanopoulos was with the The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada. He is now with the Ryerson University, Toronto, ON M5B 2K3, Canada. Digital Object Identifier 10.1109/TNN.2007.901277Twith column, row, and time modes. Naturally, color video sequences are fourth-order tensors with the addition of a color mode. In the most active area of biometrics research, namely, that of face recognition, 3-D face detection and recognition using 3-D information with column, row, and depth modes, in other words, a third-order tensor, has emerged as an important research direction [8]–[10]. Moreover, the research problem of matching still probe images to surveillance video sequences can be viewed as a pattern recognition problem in a third-order tensorial setting [11]. Beyond biometrics signal analysis, many other computer vision and pattern recognition tasks can be also viewed as problems in a multilinear domain. Such tasks include 3-D object recognition tasks [12] in machine vision, medical image analysis, and content-based retrieval, space-time analysis of video sequences for gesture recognition [13] and activity recognition [14] in human-computer interaction (HCI), and space-time super resolution [15] for digital cameras with limited spatial and temporal resolution. The wide range of applications explains the authors’ belief that a comprehensive study of a specialized feature extraction problem, such as multilinear feature extraction, is worthwhile. A typical tensor object in pattern recognition or machine vision applications is commonly specified in a high-dimensional tensor space. Recognition methods operating directly on this space suffer from the so-called curse of dimensionality [16]: Handling high-dimensional samples is computationally expensive and many classifiers perform poorly in high-dimensional spaces given a small number of training samples. However, since the entries of a tensor object are often highly correlated with surrounding entries, it is reasonable to assume that the tensor objects encountered in most applications of interest are highly constrained and thus the tensors are confined to a subspace, a manifold of intrinsically low dimension [16], [17]. Feature extraction or dimensionality reduction is thus an attempt to transform a high-dimensional data set into a low-dimensional equivalent representation while retaining most of the information regarding the underlying structure or the actual physical phenomenon [18]. Principal component analysis (PCA) is a well-known unsupervised linear technique for dimensionality reduction. The central idea behind PCA is to reduce the dimensionality of a data set consisting of a larger number of interrelated variables, while retaining as much as possible the variation present in the original data set [19]. This is achieved by transforming to a new set of variables, the so-called principal components (PCs), which are uncorrelated, and ordered so that the first few retain most of the original data variation. Naive application of PCA to tensor objects requires their reshaping into vectors with1045-9227/$25.00 © 2007 IEEELU et al.: MPCA: MULTILINEAR PRINCIPAL COMPONENT ANALYSIS OF TENSOR OBJECTS19high dimensionality (vectorization), which obviously results in high processing cost in terms of increased computational and memory demands. For example, vectorizing a typical gait silhouette sequence of size (120 80 20) results in a vector with dimensionality (192 000 1), the singular value decomposition (SVD) or eigendecomposition processing of which may be beyond the computing processing capabilities of many computing devices. Beyond implementation issues, it is well understood that reshaping breaks the natural structure and correlation in the original data, removing redundancies and/or higher order dependencies present in the original data set and losing potentially more compact or useful representations that can be obtained in the original form [20]. Vectorization as PCA preprocessing ignores the fact that tensor objects are naturally multidimensional objects, e.g., gait sequences are 3-D objects, instead of 1-D objects. Therefore, a dimensionality reduction algorithm operating directly on a tensor object rather than its vectorized version is desirable. Recently, dimensionality reduction solutions representing images as matrices (second-order tensors) rather than vectors (first-order tensors) have been introduced. A 2-D PCA algorithm is proposed in [21], where the image covariance matrix is constructed using image matrices as inputs. However, a linear transformation is applied only to the right-hand side of the input image matrices. As a result, image data is projected in one mode only, resulting in poor dimensionality reduction. The less restrictive 2-D PCA algorithm introduced in [20] takes into account the spatial correlation of the image pixels within a localized neighborhood. Two linear transforms are applied to both the left- and the right-hand sides of the input image matrices. Thus, projections in both modes are calculated and better dimensionality reduction results are obtained according to [22]. Similarly to the solutions introduced in [21] and [22], the so-called tensor subspace analysis algorithm of [23] represents the input image as a matrix residing in a tensor space and attempts to detect local geometrical structure in that tensor space by learning a lower dimensional tensor subspace. For the theoretically inclined reader, it should be noted that there are some recent developments in the analysis of higher order tensors. The higher order singular value decomposition (HOSVD) solution, which extends SVD to higher order tensors, was formulated in [24] and its computation leads to the calculation of (the order) different matrix SVDs of unfolded matrices. An alternating least square (ALS) algorithm for the approximation of higher order tenbest ranksors was studied in [1], where tensor data was projected into a lower dimensional tensor space iteratively. The application apof HOSVD truncation and the best rankproximation to dimensionality reduction in independent component analysis (ICA) was discussed in [25]. These multilinear algorithms have been used routinely for multiple factor analysis [26], [27], where input data such as images are still represented as vectors but with these vectors arranged into a tensor for the subsequent analysis of the multiple factors involved in image/video formation. It should be added that in [25]–[27], the tensor data under consideration is projected in the original coordinate without data centering. However, for classification/recognition applications where eigenproblem solutions are attempted,the eigendecomposition in each mode can be influenced by the mean (average) of the data set. Recently, there have been several attempts to develop multilinear subspace algorithms for tensor object feature extraction and classification. In [28], a heuristic MPCA approach based on HOSVD was proposed. The MPCA formulation in [29] targets optimal reconstruction applications (where data is not centered) with a solution built in a manner similar to that of [1]. It should be noted that the solution in [29] was focused on reconstruction not recognition and that it did not cover a number of important algorithmic issues, namely, initialization, termination, convergence, and subspace dimensionality determination. When applied to the problem of tensor object recognition, the methodology described in [29] uses all the entries in the projected tensor for recognition although the discrimination power of these entries varies considerably. There is also a recent work on multilinear discriminant analysis (MLDA) [30], [31], named discriminant analysis with tensor representation (DATER), where an iterative algorithm similar to ALS of [1] is utilized in order to maximize a tensor-based discriminant criterion. Unfortunately, this MLDA variant does not converge and it appears to be extremely sensitive to parameter settings [32]. As the number of possible subspace dimensions for tensor objects is extremely high (e.g., there are 225 280 possible subspace dimensions for the gait recognition problem considered in this work), exhaustive testing for determination of parameters is not feasible. Consequently, the algorithmic solution of [30] and [31] cannot be used to effectively determine subspace dimensionality in a comprehensive and systematic manner. Motivated by the works briefly reviewed here, this paper introduces a new MPCA formulation for tensor object dimensionality reduction and feature extraction. The proposed solution follows the classical PCA paradigm. Operating directly on the original tensorial data, the proposed MPCA is a multilinear algorithm performing dimensionality reduction in all tensor modes seeking those bases in each mode that allow projected tensors to capture most of the variation present in the original tensors. The main contributions of this paper include the following. 1) The introduction of a new MPCA framework for tensor object dimensionality reduction and feature extraction using tensor representation. The framework is introduced from the perspective of capturing the original tensors’ variation. It provides a systematic procedure to determine effective representations of tensor objects. This contrasts to previous work such as those reported in [16], [26], and [27], where vector, not tensor, representation was used, and the works reported in [20], [21], and [23], where matrix representation was utilized. It also differs from the works reported in [1], [24], and [25], where tensor data were processed as part of a reconstruction/regression solution. Furthermore, unlike previous attempts, such as the one in [29], design issues of paramount importance in practical applications, such as the initialization, termination, convergence of the algorithm, and the determination of the subspace dimensionality, are discussed in detail. 2) The definition of eigentensors and -mode eigenvalues as counterparts of the eigenvectors and eigenvalues in clas-20IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 1, JANUARY 2008sical PCA. The geometrical interpretation of these concepts is provided, enabling a deeper understanding of the main principles and facilitating the application of multilinear feature extraction. 3) The presentation of a recognition system that selects discriminative tensor features from tensor objects and uses a novel weighting method for classification. This differs from traditional vector-based object recognition systems [16] that often encounter computational and memory difficulties when dealing with tensor object inputs. It also differs from [29], where all of the projected features were used for recognition. 4) The development of a solution to the gait recognizer by representing gait sequences as tensor samples and extracting discriminative features from them. This is a more natural approach that differs from [3]–[7], where either silhouettes or heuristic features derived from silhouettes were used as features. The rest of this paper is organized as follows. Section II introduces basic multilinear algebra notations, concepts, and the notion of multilinear projection for dimensionality reduction. In Section III, the problem of MPCA is formulated and an iterative solution is presented. Initialization procedures, termination criteria, convergence, and subspace dimensionality are discussed in detail. The connection to PCA and 2-D PCA is illustrated. The computational aspects of the proposed framework are also discussed in this section. The problem of tensor object recognition is discussed in Section IV. Section V lists experiments on both synthetic data sets and true application data. Synthetic data sets are used to verify the properties of the proposed methodology while gait data sets are used to demonstrate performance on a recognition problem of particular importance. Finally, Section VI summarizes the major findings of this work. II. MULTILINEAR PROJECTION OF TENSOR OBJECTS This section briefly reviews some basic multilinear concepts used in the MPCA framework development and introduces the multilinear projection of tensor objects for the purpose of dimensionality reduction. A. Notations and Basic Multilinear Algebra Table I lists the fundamental symbols defined in this paper. The notations followed are those decreed by convention in the multilinear algebra, pattern recognition, and adaptive learning literature. Thus, in this paper, vectors are denoted by lowercase boldface letters, e.g., , matrices by uppercase boldface, e.g., , and tensors by calligraphic letters, e.g., . Their elements are denoted with indices in brackets. Indices are denoted by lowercase letters and span the range from 1 to the uppercase letter of . To indicate part of a vector/mathe index, e.g., trix/tensor, “:” denotes the full range of the corresponding index denotes indices ranging from to . Throughout and this paper, the discussion is restricted to real-valued vectors, matrices, and tensors since the targeted applications, such as holistic gait recognition using binary silhouettes, involve real data only. The extension to the complex valued data sets is out of the scope of this work and it will be the focus of a forthcoming paper.TABLE I LIST OF SYMBOLSAn th-order tensor is denoted as . indices , and each It is addressed by addresses the -mode of . The -mode product of a tensor by a matrix , denoted by , is a tensor with entries . The scalar product of two tensors is defined as and the . The th Frobenius norm of is defined as “ -mode slice” of is an th-order tensor obtained by . fixing the -mode index of to be : The “ -mode vectors” of are defined as the -dimensional vectors obtained from by varying the index while keeping all the other indices fixed. A rank-1 tensor equals to the outer , which means product of vectors for that all values of indices. Unfolding along the -mode is denoted as . The column vectors are the -mode vectors of . Fig. 1 illustrates the of 1-mode (column mode) unfolding of a third-order tensor. Following standard multilinear algebra, any tensor can be expressed as the product (1) where is an orthogonal and matrix. SinceLU et al.: MPCA: MULTILINEAR PRINCIPAL COMPONENT ANALYSIS OF TENSOR OBJECTS21Fig. 1. Visual illustration of the 1-mode unfolding of a third-order tensor.Fig. 2. Visual illustration of multilinear projection: (a) projection in the 1-mode vector space and (b) 2-mode and 3-mode vectors.has orthonormal columns, [1]. A matrix representation of this decomposition can be obtained by unfolding and as(2) denotes the Kronecker product. The decomposition where can also be written as(3) i.e., any tensor can be written as a linear combination of rank-1 tensors. This decomposition is used in the following to formulate multilinear projection for dimensionality reduction. B. Tensor Subspace Projection for Dimensionality Reduction resides in the tensor (multilinear) An th-order tensor , where are the space vector (linear) spaces [23]. For typical image and video tensor objects such as 3-D face images and gait sequences, although the corresponding tensor space is of high dimensionality, tensor objects typically are embedded in a lower dimensional tensor subspace (or manifold), in analogy to the (vectorized) face image embedding problem where vector image inputs reside in a lowdimensional subspace of the original input space [33]. Thus, it is possible to find a tensor subspace that captures most of thevariation in the input tensor objects and it can be used to extract features for recognition and classification applications. To orthonormal basis vectors (prinachieve this objective, are sought for each ciple axes) of the -mode linear space is formed mode and a tensor subspace denote the matrix by these linear subspaces. Let containing the orthornormal -mode basis vectors. The prois jection of onto the tensor subspace defined as . is comThe projection of an -mode vector of by puted as the inner product between the -mode vector and the rows of . Fig. 2 provides a visual illustration of the multilinear projection. In Fig. 2(a), a third-order tensor is projected in the 1-mode vector space by a projection matrix , resulting in the projected tensor . In the 1-mode projection, each 1-mode vector of of to obtain a vector of length 5, length 10 is projected by as the differently shaded vectors indicate in Fig. 2(a). Similarly, Fig. 2(b) depicts the 2-mode and 3-mode vectors. III. MULTILINEAR PRINCIPAL COMPONENT ANALYSIS In this section, an MPCA solution to the problem of dimensionality reduction for tensor objects is introduced, researched, and analyzed. Before formally stating the objective, the following definition is needed. be a set of tensor Definition 1: Let . The total scatter of these tensamples in , where is sors is defined as . The the mean tensor calculated as -mode total scatter matrix of these samples is then defined as22IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 1, JANUARY 2008Fig. 3. Pseudocode implementation of the proposed MPCA algorithm., where is the -mode unfolded matrix of . The previous statement leads to the following formal definition of the problem to be solved. tensor objects is availA set of able for training. Each tensor object assumes values in a tensor space , is the -mode dimension of the tensor. The where MPCA objective is to define a multilinear transformation that maps the original into a tensor subspace tensor space (with , for ): , such that captures most of the variations observed in the original tensor objects, assuming that these variations are measured by the total tensor scatter. In other words, the MPCA objective is the determination of projection matrices the that maximize the total tensor scatter (4) Here, the dimensionality for each mode is assumed to be known or predetermined. Discussions on the adaptive determination of , when it is not known in advance, will be presented in Section III-F. A. MPCA Algorithm To the best of the authors’ knowledge, there is no known optimal solution which allows for the simultaneous optimiza-projection matrices. Since the projection to an tion of the th-order tensor subspace consists of projections to vector subspaces, optimization subproblems can be solved that maximizes the scatter in the -mode by finding the vector subspace. This is discussed in Theorem 1. be the soluTheorem 1: Let tion to (4). Then, given all the other projection matrices , the matrix coneigenvectors corresponding to the largest sists of the eigenvalues of the matrix(5) where(6) Proof: The proof of Theorem 1 is given in Appendix I-B. depends on , the optimization of depends on the projections in other modes and there is no closed-form solution to this maximization problem. Instead, from Theorem 1, an iterative procedure can be utilized to solve (4), along the lines of the pseudocode summarized in Fig. 3. The input tensors are centered first: . With initializations through full projection truncation (FPT), which is to be discussed in details in Section III-C, the projection matrices are computed Since the productLU et al.: MPCA: MULTILINEAR PRINCIPAL COMPONENT ANALYSIS OF TENSOR OBJECTS23Fig. 4. Visual illustration of (a) total scatter tensor, (b) 1-mode eigenvalues, (c) 2-mode eigenvalues, and (d) 3-mode eigenvalues.one by one with all the others fixed (local optimization). The local optimization procedure can be repeated, in a similar fashion as the ALS method [34], until the result converges or a maximum number of iterations is reached. Remark 1: The issue of centering has been ignored in the existing tensor processing literature. In the authors’ opinion, the main reason for the apparent lack of studies on the problem of tensor data centering is due to the fact that previously published works focused predominately on tensor approximation and reconstruction. It should be pointed out that for the approximation/reconstruction problem, centering is not essential, as the (sample) mean is the main focus of attention. However, in recognition applications where the solutions involve eigenproblems, noncentering (in other words, an average different from zero) can potentially affect the per-mode eigendecomposition and lead to a solution that captures the variation with respect to the origin rather than capturing the true variation of the data (with respect to the data center). Remark 2: The effects of the ordering of the projection matrices to be computed have been studied empirically in this work and simulation results presented in Section V indicate that altering the ordering of the projection matrix computation does not result in significant performance differences in practical situations. In the following sections, several issues pertinent to the development and implementation of the MPCA algorithm are discussed. First, in-depth understanding of the MPCA framework is provided. The properties of full projection are analyzed, and the geometric interpretation of the -mode eigenvalues is introduced together with the concept of eigentensor. In the sequence, the initialization method and the construction of termination criteria are described and convergence issues are also discussed. Finally, methods for subspace dimensionality determination are proposed and the connection to PCA and 2-D PCA is discussed, followed by computational issues. B. Full Projection With respect to this analysis, the term full projection refers for to the multilinear projection for MPCA with . In this case, is an identity matrix, as it can be seen from the pertinent lemma listed in Appendix I-C. reduces to As a result, , with determined by the input tensor samples only and independent of other projection matrices. The is then obtained as the matrix comprised optimal directly without iteration, and the of the eigenvectors of in the original data is fully captured. However, total scatterthere is no dimensionality reduction through this full projection. From the properties of eigendecomposition, it can be concluded that if all eigenvalues (per mode) are distinct, the full projection matrices (corresponding eigenvectors) are also distinct and that the full projection is unique (up to sign) [35]. To interpret the geometric meanings of the -mode eigenvalues, the total scatter tensor of the full projection is introduced as an extension of the total scatter mais defined as trix [36]. Each entry of the tensor (7) and . Using the previous definition, it can be for all ), shown that for the so-called full projection ( the th -mode eigenvalue is the sum of all the entries of the th -mode slice of where(8) In this paper, the eigenvalues are all arranged in a decreasing order. Fig. 4 shows visually what the -mode eigenvalues represent. In this graph, third-order tensors, e.g., short sequences (three frames) of images with size 5 4, are projected to a tensor space of size 5 4 3 (full projection) so that a total scatter is obtained. tensor Using (3), each tensor can be written as a linear comrank-1 tensors bination of . These rank-1 tensors will be called, herecan be viewed after, eigentensors. Thus, the projected tensor as the projection onto these eigentensors, with each entry of corresponding to one eigentensor. These definitions and illustrations for MPCA help with understanding the MPCA framework in the following discussions. C. Initialization by Full Projection Truncation FPT is used to initialize the iterative solution for MPCA, columns of the full projection matrix where the first is kept to give an initial projection matrix . The correand this initialization is sponding total scatter is denoted as equivalent to the HOSVD-based solution in [28]. Although this FPT initialization is not the optimal solution to (4), it is bounded24IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 19, NO. 1, JANUARY 2008and is considered a good starting point for the iterative procedure, as will be discussed in the following. Remark 3: There are other choices of initialization such as truncated identity matrices [20], [23], [31] (named as pseudoidentity matrices) and random matrices. Simulation studies (reported in Section V) indicate that although in practical applications, the initialization step may not have a significant impact in terms of performance, it can affect the speed of convergence of the iterative solution. Since FPT results in much faster convergence, it is the one utilized throughout this work for initialization purposes. In studying the optimality, with respect to (4), of the initialization procedure, let us assume, without loss of generality, that the 1-mode eigenvectors are truncated, in other words, only the 1-mode eigenvectors are kept. In this case, Thefirst orem 2 applies. and Theorem 2: Let be the matrix of the eigenvectors of and the , respectively, and eigenvalues of . Keep only the first eigenvectors with to get , where and . Let correspond to , and the matrix of its eigenvectors and its eigenvalues be and , respectively. ThenHaving proven the nonoptimality of FPT with respect to the objective function (4), we proceed to derive the bounds for FPT in Theorem 3. denote the th -mode eigenvalue for Theorem 3: Let the -mode full projection matrix. The upper and lower bounds , the loss of variation due to the FPT (measured for by the total scatter), are derived as follows:(9) Proof: The proof is given in Appendix I-D. From (9), it can be seen that the tightness of the bounds is determined by the eigenvalues in each mode. The bounds can be observed in Fig. 4. For instance, truncation of the last eigenvector in each of the three modes results in another truncated , and thus the difference betotal scatter tensor tween and (the sum of all entries in and , respectively) is upper bounded by the total of the sums of all the entries in each truncated slice and lower bounded by the maximum sum of all the entries in each truncated slice. For FPT, the gap between the actual loss of variation and the upper bound is due to the multiple counts of the overlaps between the discarded slice in one mode and the discarded slices in the other modes of . The tightness of the bounds and depends on the , the eigenvalue characteristics (distribution) such order as the number of zero-valued eigenvalues, and the degree of . For example, for , which is the case of truncation PCA, and the FPT is the optimal solution so no results in more terms in iterations are necessary. Larger the upper bound and tends to lead to looser bound, and vice versa. In addition, if all the truncated eigenvectors correspond since , to zero-valued eigenvalues, and the FPT results in the optimal solution. D. Termination The termination criterion is to be determined in this paper using the objective function . In particular, the iterative pro, where and cedure terminates if are the resulted total scatter from the th and th iterations, respectively, and is a user-defined small number threshold (e.g., ). In other words, the iterations stop if there is little improvement in the resulted total scatter (the objective function). In addition, the maximum number of iterations allowed is set to for computational consideration. E. Convergence of the MPCA Algorithm The derivation of Theorem 1 (Appendix I-B) implies that is a nondecreasing function per iteration, the total scatter (as it either remains the same or increases) since each update in a given mode maximizes of the projection matrix , while the projection matrices in all the other modes are considered fixed. On the other hand, is upper bounded by (the variation in the original samples)For (other modes), . Furthermore, for each mode, at least for one value of . Proof: The proof is given in Appendix I-D. It can be seen from Theorem 2 that if a nonzero eigenvalue is truncated in one mode, the eigenvalues in all the other modes tend to decrease in magnitude and the corresponding eigenvectors change accordingly. Thus, the eigendecomposition needs to be recomputed in all the other modes, i.e., the projection matrices in all the other modes need to be updated. Since from Theorem 1 the computations of all the projection matrices are upinterdependent, the update of a projection matrix as well. Consequently, the dates the matrices projection matrices in all the other modes are no longer consisting of the eigenvectors of the corresponding and they need to be updated. The update con(updated) tinues until the termination criterion, which is discussed in Section III-D, is satisfied. Fig. 4 provides a visual illustration of Theorem 2. Removal of . a basis vector in one mode results in eliminating a slice of In Fig. 4, if the last nonzero (fifth) 1-mode eigenvalue is discarded [shaded in Fig. 4(b)], the corresponding (fifth) 1-mode is removed [shaded in Fig. 4(a)], resulting in a trunslice of . Discarding this slice cated total scatter tensor will affect all eigenvalues in the remaining modes, whose corresponding slices have a nonempty overlap with the discarded 1-mode slice. In Fig. 4(c) and (d), the shaded part indicates the removed 1-mode slice corresponding to the discarded eigenvalue.。

正曲率齐性Finsler空间的分类：偶数维情形下的一种新方法
(英文)
徐熙昀;许明
【期刊名称】《首都师范大学学报（自然科学版）》
【年(卷),期】2024(45)1
【摘要】本文介绍了正曲率齐性Finsler流形的分类。

在偶数维的情形下,给出了一种新方法,证明了偶数维光滑陪集空间上有正曲率齐性Finsler度量,当且仅当其上面有正曲率齐性黎曼度量。

【总页数】7页(P124-130)
【作者】徐熙昀;许明
【作者单位】首都师范大学数学科学学院
【正文语种】中文
【中图分类】O186
【相关文献】
1.偶数维Damek-Ricci空间的曲率
2.曲率R部分为零的Finsler空间结构(英文)
3.常曲率空间中的正曲率子流形(英文)
4.三维时空中一对复主曲率类时共形齐性曲面的分类
5.三维时空中两个不同实主曲率类时共形齐性曲面的分类
因版权原因，仅展示原文概要，查看原文内容请购买。

linear discriminate analysisLinear discriminant analysis (LDA) is a statistical technique used in machine learning and pattern recognition. It is primarily used for dimensionality reduction and classification tasks. In this article, we will dive deep into the topic of LDA, step by step, to understand its working principles, assumptions, and how to apply it in practice.1. Introduction to Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA), also known as Fisher's Linear Discriminant, is a classical statistical technique that is widely used for pattern recognition and classification problems. It aims to find a linear combination of features that characterizes or separates two or more classes or groups of data points. The key idea is to maximize the between-class scatter while minimizing the within-class scatter.2. Assumptions of LDALDA makes several assumptions about the data:- The classes are linearly separable.- The features are normally distributed within each class.- The covariances of the features are equal for all classes.These assumptions are important because violating them may lead to misleading results or inaccurate classifications.3. Steps in LDALet's now discuss the step-by-step procedure for performing LDA on a given dataset:Step 1: Data preparation- Gather the dataset, ensuring that it contains labeled instances representing different classes.- Split the dataset into training and testing subsets.Step 2: Compute class means- Calculate the mean vector for each class, which represents the average value of each feature for that class.Step 3: Compute the scatter matrices- Compute the within-class scatter matrix (Sw), which measures the variation within each class.- Compute the between-class scatter matrix (Sb), which measures the variation between classes.Step 4: Solve the generalized eigenvalue problem- Compute the eigenvectors (e1, e2, ..., ed) and their corresponding eigenvalues (λ1, λ2, ..., λd) of (Sw^(-1) * Sb), where d is the number of features.Step 5: Select discriminant features- Sort the eigenvalues in descending order and choose the k largest eigenvalues.- Corresponding eigenvectors to these k largest eigenvalues will be the discriminant axes or features.Step 6: Transform the data- Project the original dataset onto the new space formed by the selected discriminant features.Step 7: Classification and evaluation- Train a classifier (e.g., logistic regression, support vector machines) on the transformed dataset.- Evaluate the performance of the classifier using appropriate metrics (e.g., accuracy, precision, recall).4. Advantages and Limitations of LDAAdvantages:- LDA reduces the dimensionality of the dataset while preserving the class discriminatory information.- It can handle multicollinearity, which is the correlation between features.- LDA assumes linear relationships between the features and the classes, making it computationally efficient.Limitations:- LDA assumes that the data is normally distributed, which may not hold true for all real-world datasets.- It may not work well with imbalanced class distributions.- LDA is a linear method, which means it may not capture complex nonlinear relationships in the data.5. Applications of LDALDA has found applications in various domains, including:- Face recognition: LDA can be used to extract discriminative features from facial images.- Document classification: LDA has been used for topic modeling to identify the underlying themes in documents.- Bioinformatics: LDA has been applied to analyze gene expressiondata and identify genes related to different classes.6. ConclusionLinear Discriminant Analysis (LDA) is a powerful statistical technique used for dimensionality reduction and classification tasks. By maximizing the between-class scatter and minimizing the within-class scatter, LDA finds the most discriminative features that separate different classes. Although it has certain assumptions and limitations, LDA has proven to be effective in a wide range of applications, making it a valuable tool in the field of machine learning and pattern recognition.。