{"title": "Tensor Subspace Analysis", "book": "Advances in Neural Information Processing Systems", "page_first": 499, "page_last": 506, "abstract": null, "full_text": "Tensor Subspace Analysis\n\nXiaofei He1\n\nDeng Cai2\n\nPartha Niyogi1\n\n1 Department of Computer Science, University of Chicago\n\n{xiaofei, niyogi}@cs.uchicago.edu\n\n2 Department of Computer Science, University of Illinois at Urbana-Champaign\n\ndengcai2@uiuc.edu\n\nAbstract\n\nPrevious work has demonstrated that the image variations of many ob-\njects (human faces in particular) under variable lighting can be effec-\ntively modeled by low dimensional linear spaces. The typical linear sub-\nspace learning algorithms include Principal Component Analysis (PCA),\nLinear Discriminant Analysis (LDA), and Locality Preserving Projec-\ntion (LPP). All of these methods consider an n1 \u00d7 n2 image as a high\ndimensional vector in Rn1\u00d7n2, while an image represented in the plane\nis intrinsically a matrix. In this paper, we propose a new algorithm called\nTensor Subspace Analysis (TSA). TSA considers an image as the sec-\nond order tensor in Rn1 \u2297 Rn2, where Rn1 and Rn2 are two vector\nspaces. The relationship between the column vectors of the image ma-\ntrix and that between the row vectors can be naturally characterized by\nTSA. TSA detects the intrinsic local geometrical structure of the tensor\nspace by learning a lower dimensional tensor subspace. We compare our\nproposed approach with PCA, LDA and LPP methods on two standard\ndatabases. Experimental results demonstrate that TSA achieves better\nrecognition rate, while being much more ef\ufb01cient.\n\n1\n\nIntroduction\n\nThere is currently a great deal of interest in appearance-based approaches to face recogni-\ntion [1], [5], [8]. When using appearance-based approaches, we usually represent an image\nof size n1 \u00d7 n2 pixels by a vector in Rn1\u00d7n2. Throughout this paper, we denote by face\nspace the set of all the face images. The face space is generally a low dimensional mani-\nfold embedded in the ambient space [6], [7], [10]. The typical linear algorithms for learning\nsuch a face manifold for recognition include Principal Component Analysis (PCA), Linear\nDiscriminant Analysis (LDA) and Locality Preserving Projection (LPP) [4].\n\nMost of previous works on statistical image analysis represent an image by a vector in\nhigh-dimensional space. However, an image is intrinsically a matrix, or the second or-\nder tensor. The relationship between the rows vectors of the matrix and that between the\ncolumn vectors might be important for \ufb01nding a projection, especially when the number\nof training samples is small. Recently, multilinear algebra, the algebra of higher-order\ntensors, was applied for analyzing the multifactor structure of image ensembles [9], [11],\n[12]. Vasilescu and Terzopoulos have proposed a novel face representation algorithm called\nTensorface [9]. Tensorface represents the set of face images by a higher-order tensor and\n\n\fextends Singular Value Decomposition (SVD) to higher-order tensor data. In this way, the\nmultiple factors related to expression, illumination and pose can be separated from different\ndimensions of the tensor.\n\nIn this paper, we propose a new algorithm for image (human faces in particular) represen-\ntation based on the considerations of multilinear algebra and differential geometry. We call\nit Tensor Subspace Analysis (TSA). For an image of size n1 \u00d7 n2, it is represented as the\nsecond order tensor (or, matrix) in the tensor space Rn1 \u2297 Rn2. On the other hand, the face\nspace is generally a submanifold embedded in Rn1 \u2297 Rn2. Given some images sampled\nfrom the face manifold, we can build an adjacency graph to model the local geometrical\nstructure of the manifold. TSA \ufb01nds a projection that respects this graph structure. The\nobtained tensor subspace provides an optimal linear approximation to the face manifold\nin the sense of local isometry. Vasilescu shows how to extend SVD(PCA) to higher order\ntensor data. We extend Laplacian based idea to tensor data.\n\nIt is worthwhile to highlight several aspects of the proposed approach here:\n\n1. While traditional linear dimensionality reduction algorithms like PCA, LDA and\nLPP \ufb01nd a map from Rn to Rl (l < n), TSA \ufb01nds a map from Rn1 \u2297 Rn2 to\nRl1 \u2297 Rl2 (l1 < n1, l2 < n2). This leads to structured dimensionality reduction.\n2. TSA can be performed in either supervised, unsupervised, or semi-supervised\nmanner. When label information is available, it can be easily incorporated into\nthe graph structure. Also, by preserving neighborhood structure, TSA is less sen-\nsitive to noise and outliers.\n\n3. The computation of TSA is very simple. It can be obtained by solving two eigen-\nvector problems. The matrices in the eigen-problems are of size n1\u00d7n1 or n2\u00d7n2,\nwhich are much smaller than the matrices of size n \u00d7 n (n = n1 \u00d7 n2) in PCA,\nLDA and LPP. Therefore, TSA is much more computationally ef\ufb01cient in time\nand storage. There are few parameters that are independently estimated, so per-\nformance in small data sets is very good.\n\n4. TSA explicitly takes into account the manifold structure of the image space. The\n\nlocal geometrical structure is modeled by an adjacency graph.\n\n5. This paper is primarily focused on the second order tensors (or, matrices). How-\never, the algorithm and analysis presented here can also be applied to higher order\ntensors.\n\n2 Tensor Subspace Analysis\n\nIn this section, we introduce a new algorithm called Tensor Subspace Analysis for learning a\ntensor subspace which respects the geometrical and discriminative structures of the original\ndata space.\n\n2.1 Laplacian based Dimensionality Reduction\n\nProblems of dimensionality reduction has been considered. One general approach is based\non graph Laplacian [2]. The objective function of Laplacian eigenmap is as follows:\n\n(f (xi) \u2212 f (xj))2 Sij\n\nmin\n\nf Xij\n\nwhere S is a similarity matrix. These optimal functions are nonlinear but may be expensive\nto compute.\n\nA class of algorithms may be optimized by restricting problem to more tractable families\nof functions. One natural approach restricts to linear function giving rise to LPP [4]. In this\n\n\fpaper we will consider a more structured subset of linear functions that arise out of tensor\nanalysis. This provided greater computational bene\ufb01ts.\n\n2.2 The Linear Dimensionality Reduction Problem in Tensor Space\n\nThe generic problem of linear dimensionality reduction in the second order tensor space\nis the following. Given a set of data points X1, \u00b7 \u00b7 \u00b7 , Xm in Rn1 \u2297 Rn2, \ufb01nd two trans-\nformation matrices U of size n1 \u00d7 l1 and V of size n2 \u00d7 l2 that maps these m points to a\nset of points Y1, \u00b7 \u00b7 \u00b7 , Ym \u2208 Rl1 \u2297 Rl2 (l1 < n1, l2 < n2), such that Yi \u201crepresents\u201d Xi,\nwhere Yi = U T XiV . Our method is of particular applicability in the special case where\nX1, \u00b7 \u00b7 \u00b7 , Xm \u2208 M and M is a nonlinear submanifold embedded in Rn1 \u2297 Rn2.\n\n2.3 Optimal Linear Embeddings\n\nAs we described previously, the face space is probably a nonlinear submanifold embedded\nin the tensor space. One hopes then to estimate geometrical and topological properties of\nthe submanifold from random points (\u201cscattered data\u201d) lying on this unknown submanifold.\nIn this section, we consider the particular question of \ufb01nding a linear subspace approxima-\ntion to the submanifold in the sense of local isometry. Our method is fundamentally based\non LPP [4].\nGiven m data points X = {X1, \u00b7 \u00b7 \u00b7 , Xm} sampled from the face submanifold M \u2208 Rn1 \u2297\nRn1, one can build a nearest neighbor graph G to model the local geometrical structure of\nM. Let S be the weight matrix of G. A possible de\ufb01nition of S is as follows:\n\nkXi\u2212Xj k2\n\nt\n\n,\n\ne\u2212\n\n0,\n\nif Xi is among the k nearest\nneighbors of Xj, or Xj is among\nthe k nearest neighbors of Xi;\notherwise.\n\n(1)\n\nSij =\uf8f1\uf8f4\uf8f4\uf8f2\n\uf8f4\uf8f4\uf8f3\n\nwhere t is a suitable constant. The function exp(\u2212kXi \u2212 Xjk2/t) is the so called heat\nkernel which is intimately related to the manifold structure. k \u00b7 k is the Frobenius norm of\nij. When the label information is available, it can be easily\n\nmatrix, i.e. kAk =qPiPj a2\n\nincorporated into the graph as follows:\n\nSij =( e\u2212\n\n0,\n\nkXi\u2212Xj k2\n\nt\n\n,\n\nif Xi and Xj share the same label;\notherwise.\n\n(2)\n\nLet U and V be the transformation matrices. A reasonable transformation respecting the\ngraph structure can be obtained by solving the following objective functions:\n\nkU T XiV \u2212 U T XjV k2Sij\n\n(3)\n\nmin\n\nU,V Xij\n\nThe objective function incurs a heavy penalty if neighboring points Xi and Xj are mapped\nfar apart. Therefore, minimizing it is an attempt to ensure that if Xi and Xj are \u201cclose\u201d\nthen U T XiV and U T XjV are \u201cclose\u201d as well. Let Yi = U T XiV . Let D be a diagonal\n\nmatrix, Dii =Pj Sij. Since kAk2 = tr(AAT ), we see that:\n\nkU T XiV \u2212 U T XjV k2Sij =\n\n1\n\n1\n\n1\n\n=\n\n2Xij\n2Xij\n= tr(cid:16)Xi\n\ntr(cid:0)YiY T\n\nDiiYiY T\n\ni + YjY T\n\nj \u2212 YiY T\n\nj \u2212 YjY T\n\ni \u2212Xij\n\nSijYiY T\n\nj (cid:17)\n\n2Xij\n\ntr(cid:0)(Yi \u2212 Yj)(Yi \u2212 Yj)T(cid:1) Sij\ni (cid:1) Sij\n\n\fDiiU T XiV V T X T\n\nSijU T XiV V T X T\n\nDiiXiV V T X T\n\nSijXiV V T X T\n\n= tr(cid:16)Xi\n= tr(cid:16)U T(cid:0)Xi\n= tr(cid:0)U T (DV \u2212 SV ) U(cid:1)\nwhere DV = Pi DiiXiV V T X T\n\ntr(AT A), so we also have\n\n.\n\ni U \u2212Xij\ni \u2212Xij\n\nkU T XiV \u2212 U T XjV k2Sij\n\nj U(cid:17)\nj (cid:1)U(cid:17)\n\ni and SV = Pij SijXiV V T X T\n\nj . Similarly, kAk2 =\n\n1\n\n1\n\n1\n\n=\n\n=\n\ni Yi + Y T\n\n2Xij\n2Xij\n2Xij\n= tr(cid:16)Xi\n= tr(cid:16)V T(cid:0)Xi\n= tr(cid:0)V T (DU \u2212 SU ) V(cid:1)\n\ni Yi \u2212Xij\n\nDiiX T\n\nDiiY T\n\n.\n\ntr(cid:0)(Yi \u2212 Yj)T (Yi \u2212 Yj)(cid:1) Sij\ntr(cid:0)Y T\n\nj Yj \u2212 Y T\n\ni Yj \u2212 Y T\n\nj Yi(cid:1) Sij\n\nSijY T\n\ni Yj(cid:17)\ni U U T Xi \u2212Xij\n\nX T\n\ni U U T Xj(cid:1)V(cid:17)\n\nwhere DU = Pi DiiX T\nsimultaneously minimize tr(cid:0)U T (DV \u2212 SV ) U(cid:1) and tr(cid:0)V T (DU \u2212 SU ) V(cid:1).\n\ni U U T Xi and SU = Pij SijX T\n\nIn addition to preserving the graph structure, we also aim at maximizing the global variance\non the manifold. Recall that the variance of a random variable x can be written as follows:\n\ni U U T Xj. Therefore, we should\n\nvar(x) =ZM\n\n(x \u2212 \u00b5)2dP (x), \u00b5 =ZM\n\nxdP (x)\n\nwhere M is the data manifold, \u00b5 is the expected value of x and dP is the probability\nmeasure on the manifold. By spectral graph theory [3], dP can be discretely estimated by\n\nthe diagonal matrix D(Dii = Pj Sij) on the sample points. Let Y = U T XV denote\n\nthe random variable in the tensor subspace and suppose the data points have a zero mean.\nThus, the weighted variance can be estimated as follows:\n\nvar(Y ) = Xi\n\nkYik2Dii =Xi\n\ntr(Y T\n\ni Yi)Dii =Xi\n\ntr(V T X T\n\ni U U T XiV )Dii\n\n= tr V T Xi\n\nDiiX T\n\ni U U T Xi! V! = tr(cid:0)V T DU V(cid:1)\n\nSimilarly, kYik2 = tr(YiY T\n\ni ), so we also have:\n\nvar(Y ) =Xi\n\ntr(YiY T\n\ni )Dii = tr U T Xi\n\nFinally, we get the following optimization problems:\n\nDiiXiV V T X T\n\ni ! U! = tr(cid:0)U T DV U(cid:1)\n\nmin\nU,V\n\ntr(cid:0)U T (DV \u2212 SV ) U(cid:1)\n\ntr (U T DV U )\n\n(4)\n\n\fmin\nU,V\n\n(5)\n\ntr(cid:0)V T (DU \u2212 SU ) V(cid:1)\n\ntr (V T DU V )\n\nThe above two minimization problems (4) and (5) depends on each other, and hence can not\nbe solved independently. In the following subsection, we describe a simple computational\nmethod to solve these two optimization problems.\n\n2.4 Computation\n\nIn this subsection, we discuss how to solve the optimization problems (4) and (5). It is easy\nto see that the optimal U should be the generalized eigenvectors of (DV \u2212 SV , DV ) and the\noptimal V should be the generalized eigenvectors of (DU \u2212 SU , DU ). However, it is dif\ufb01-\ncult to compute the optimal U and V simultaneously since the matrices DV , SV , DU , SU\nare not \ufb01xed. In this paper, we compute U and V iteratively as follows. We \ufb01rst \ufb01x U, then\nV can be computed by solving the following generalized eigenvector problem:\n\n(DU \u2212 SU )v = \u03bbDU v\n\n(6)\n\nOnce V is obtained, U can be updated by solving the following generalized eigenvector\nproblem:\n\n(DV \u2212 SV )u = \u03bbDV u\n\n(7)\nThus, the optimal U and V can be obtained by iteratively computing the generalized eigen-\nvectors of (6) and (7). In our experiments, U is initially set to the identity matrix. It is easy\nto show that the matrices DU , DV , DU \u2212 SU , and DV \u2212 SV are all symmetric and positive\nsemi-de\ufb01nite.\n\n3 Experimental Results\n\nIn this section, several experiments are carried out to show the ef\ufb01ciency and effectiveness\nof our proposed algorithm for face recognition. We compare our algorithm with the Eigen-\nface (PCA) [8], Fisherface (LDA) [1], and Laplacianface (LPP) [5] methods, three of the\nmost popular linear methods for face recognition.\n\nTwo face databases were used. The \ufb01rst one is the PIE (Pose, Illumination, and Experience)\ndatabase from CMU, and the second one is the ORL database.\nIn all the experiments,\npreprocessing to locate the faces was applied. Original images were normalized (in scale\nand orientation) such that the two eyes were aligned at the same position. Then, the facial\nareas were cropped into the \ufb01nal images for matching. The size of each cropped image in all\nthe experiments is 32\u00d732 pixels, with 256 gray levels per pixel. No further preprocessing is\ndone. For the Eigenface, Fisherface, and Laplacianface methods, the image is represented\nas a 1024-dimensional vector, while in our algorithm the image is represented as a (32 \u00d7\n32)-dimensional matrix, or the second order tensor. The nearest neighbor classi\ufb01er is used\nfor classi\ufb01cation for its simplicity.\n\nIn short, the recognition process has three steps. First, we calculate the face subspace from\nthe training set of face images; then the new face image to be identi\ufb01ed is projected into\nd-dimensional subspace (PCA, LDA, and LPP) or (d \u00d7 d)-dimensional tensor subspace\n(TSA); \ufb01nally, the new face image is identi\ufb01ed by nearest neighbor classi\ufb01er. In our TSA\nalgorithm, the number of iterations is taken to be 3.\n\n3.1 Experiments on PIE Database\n\nThe CMU PIE face database contains 68 subjects with 41,368 face images as a whole. The\nface images were captured by 13 synchronized cameras and 21 \ufb02ashes, under varying pose,\nillumination and expression. We choose the \ufb01ve near frontal poses (C05, C07, C09, C27,\n\n\f \n\n70\n\n)\n\n%\n\n(\n \ne\nt\na\nr\n \nr\no\nr\nr\n\nE\n\n60\n\n50\n\n40\n\n30\n\n0\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n \n\n60\n\n50\n\n)\n\n%\n\n(\n \n\ne\n\nt\n\na\nr\n \nr\no\nr\nr\n\n40\n\nE\n\n30\n\n20\n\n400\n\n0\n\n300\n\n200\n\n100\nDims d (d\u00b7 d for TSA)\n(a) 5 Train\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n \n\n50\n\n40\n\n)\n\n%\n\n(\n \n\ne\n\nt\n\na\nr\n \nr\no\nr\nr\n\nE\n\n30\n\n20\n\n10\n\n \n\n35\n\n30\n\n25\n\n20\n\n)\n\n%\n\n(\n \n\ne\n\nt\n\na\nr\n \nr\no\nr\nr\n\nE\n\n15\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n600\n\n0\n\n200\n\n400\nDims d (d\u00b7 d for TSA)\n(b) 10 Train\n\n800\n\n400\n\n600\n\n200\nDims d (d\u00b7 d for TSA)\n(c) 20 Train\n\n10\n\n5\n\n0\n\n1000\n\n800\n\n400\n\n600\n\n200\nDims d (d\u00b7 d for TSA)\n(d) 30 Train\n\n1000\n\nFigure 1: Error rate vs. dimensionality reduction on PIE database\n\nTable 1: Performance comparison on PIE database\n10 Train\n\n5 Train\n\nMethod\n\nBaseline\nEigenfaces\nFisherfaces\n\nLaplacianfaces\n\nTSA\n\nMethod\nBaseline\nEigenfaces\nFisherfaces\n\nLaplacianfaces\n\nTSA\n\nerror\ndim\n69.9% 1024\n69.9% 338\n31.5%\n67\n30.8%\n67\n27.9% 112\n\n20 Train\n\nerror\ndim\n38.2% 1024\n38.1% 889\n15.4%\n67\n14.1% 146\n9.64% 132\n\ntime(s)\n\n-\n\n0.907\n\n1.843\n\n2.375\n0.594\n\ntime(s)\n\n-\n\n14.328\n\n35.828\n\n39.172\n7.125\n\nerror\ndim\n55.7% 1024\n55.7% 654\n22.4%\n67\n21.1% 134\n16.9% 132\n\n30 Train\n\nerror\ndim\n27.9% 1024\n27.9% 990\n7.77%\n67\n7.13% 131\n6.88% 122\n\ntime(s)\n\n-\n\n5.297\n\n9.609\n\n11.516\n2.063\n\ntime(s)\n\n-\n\n15.453\n\n38.406\n\n47.610\n15.688\n\nC29) and use all the images under different illuminations and expressions, thus we get 170\nimages for each individual. For each individual, l(= 5, 10, 20, 30) images are randomly\nselected for training and the rest are used for testing.\n\nThe training set is utilized to learn the subspace representation of the face manifold by using\nEigenface, Fisherface, Laplacianface and our algorithm. The testing images are projected\ninto the face subspace in which recognition is then performed. For each given l, we average\nthe results over 20 random splits. It would be important to note that the Laplacianface\nalgorithm and our algorithm share the same graph structure as de\ufb01ned in Eqn. (2).\n\nFigure 1 shows the plots of error rate versus dimensionality reduction for the Eigenface,\nFisherface, Laplacianface, TSA and baseline methods. For the baseline method, the recog-\nnition is simply performed in the original 1024-dimensional image space without any di-\nmensionality reduction. Note that, the upper bound of the dimensionality of Fisherface is\nc \u2212 1 where c is the number of individuals. For our TSA algorithm, we only show its per-\nformance in the (d \u00d7 d)-dimensional tensor subspace, say, 1, 4, 9, etc. As can be seen, the\nperformance of the Eigenface, Fisherface, Laplacianface, and TSA algorithms varies with\nthe number of dimensions. We show the best results obtained by them in Table 1 and the\ncorresponding face subspaces are called optimal face subspace for each method.\n\nIt is found that our method outperforms the other four methods with different numbers\nof training samples (5, 10, 20, 30) per individual. The Eigenface method performs the\nworst. It does not obtain any improvement over the baseline method. The Fisherface and\nLaplacianface methods perform comparatively to each each. The dimensions of the optimal\nsubspaces are also given in Table 1.\n\nAs we have discussed, TSA can be implemented very ef\ufb01ciently. We show the running\ntime in seconds for each method in Table 1. As can be seen, TSA is much faster than the\n\n\f \n\n55\n\n50\n\n45\n\n)\n\n%\n\n(\n \ne\nt\na\nr\n \nr\no\nr\nr\n\nE\n\n40\n\n35\n\n30\n\n25\n\n20\n\n15\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n \n\n)\n\n%\n\n(\n \ne\nt\na\nr\n \nr\no\nr\nr\n\nE\n\n50\n\n40\n\n30\n\n20\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n0\n\n10\n\n20\n\n30\n40\nDims d\n\n50\n\n60\n\n70\n\n10\n\n0\n\n10\n\n20\n\n30\n40\nDims d\n\n50\n\n60\n\n70\n\n \n\n45\n\n40\n\n35\n\n30\n\n25\n\n20\n\n)\n\n%\n\n(\n \ne\nt\na\nr\n \nr\no\nr\nr\n\nE\n\n15\n\n10\n\n5\n\n0\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n \n\n)\n\n%\n\n(\n \ne\nt\na\nr\n \nr\no\nr\nr\n\nE\n\n40\n\n30\n\n20\n\n10\n\nTSA\nLaplacianfaces (PCA+LPP)\nFisherfaces (PCA+LDA)\nEigenfaces (PCA)\nBaseline\n\n10\n\n20\n\n30\n40\nDims d\n\n50\n\n60\n\n70\n\n0\n\n0\n\n10\n\n20\n\n30\n40\nDims d\n\n50\n\n60\n\n70\n\n(a) 2 Train\n\n(b) 3 Train\n\n(c) 4 Train\n\n(d) 5 Train\n\nFigure 2: Error rate vs. dimensionality reduction on ORL database\n\nTable 2: Performance comparison on ORL database\n\nMethod\n\nBaseline\nEigenfaces\nFisherfaces\n\nLaplacianfaces\n\nTSA\n\nMethod\nBaseline\nEigenfaces\nFisherfaces\n\nLaplacianfaces\n\nTSA\n\n2 Train\n\nerror\ndim\n30.2% 1024\n79\n30.2%\n23\n25.2%\n22.2%\n39\n20.0% 102\n\n4 Train\n\nerror\ndim\n16.0% 1024\n122\n15.9%\n9.17%\n39\n8.54%\n39\n7.12% 102\n\ntime\n\n-\n\n38.13\n60.32\n62.65\n65.00\n\ntime\n\n-\n\n141.72\n212.82\n248.90\n201.40\n\n3 Train\n\nerror\ndim\n22.4% 1024\n113\n22.3%\n39\n13.1%\n12.5%\n39\n10.7% 112\n\n5 Train\n\nerror\ndim\n11.7% 1024\n182\n11.6%\n6.55%\n39\n5.45%\n40\n4.75% 102\n\ntime\n\n-\n\n85.16\n119.69\n136.25\n135.93\n\ntime\n\n-\n\n224.69\n355.63\n410.78\n302.97\n\nEigenface, Fisherface and Laplacianface methods. All the algorithms were implemented in\nMatlab 6.5 and run on a Intel P4 2.566GHz PC with 1GB memory.\n\n3.2 Experiments on ORL Database\n\nThe ORL (Olivetti Research Laboratory) face database is used in this test. It consists of a\ntotal of 400 face images, of a total of 40 people (10 samples per person). The images were\ncaptured at different times and have different variations including expressions (open or\nclosed eyes, smiling or non-smiling) and facial details (glasses or no glasses). The images\nwere taken with a tolerance for some tilting and rotation of the face up to 20 degrees. For\neach individual, l(= 2, 3, 4, 5) images are randomly selected for training and the rest are\nused for testing.\n\nThe experimental design is the same as that in the last subsection. For each given l, we\naverage the results over 20 random splits. Figure 3.2 shows the plots of error rate versus\ndimensionality reduction for the Eigenface, Fisherface, Laplacianface, TSA and baseline\nmethods. Note that, the presentation of the performance of the TSA algorithm is different\nfrom that in the last subsection. Here, for a given d, we show its performance in the (d\u00d7d)-\ndimensional tensor subspace. The reason is for better comparison, since the Eigenface and\nLaplacianface methods start to converge after 70 dimensions and there is no need to show\ntheir performance after that. The best result obtained in the optimal subspace and the\nrunning time (millisecond) of computing the eigenvectors for each method are shown in\nTable 2.\n\nAs can be seen, our TSA algorithm performed the best in all the cases. The Fisherface\nand Laplacianface methods performed comparatively to our method, while the Eigenface\nmethod performed poorly.\n\n\f4 Conclusions and Future Work\n\nTensor based face analysis (representation and recognition) is introduced in this paper in\norder to detect the underlying nonlinear face manifold structure in the manner of tensor\nsubspace learning. The manifold structure is approximated by the adjacency graph com-\nputed from the data points. The optimal tensor subspace respecting the graph structure is\nthen obtained by solving an optimization problem. We call this Tensor Subspace Analysis\nmethod.\n\nMost of traditional appearance based face recognition methods (i.e. Eigenface, Fisherface,\nand Laplacianface) consider an image as a vector in high dimensional space. Such repre-\nsentation ignores the spacial relationships between the pixels in the image. In our work, an\nimage is naturally represented as a matrix, or the second order tensor. Tensor representation\nmakes our algorithm much more computationally ef\ufb01cient than PCA, LDA, and LPP. Ex-\nperimental results on PIE and ORL databases demonstrate the ef\ufb01ciency and effectiveness\nof our method.\n\nTSA is linear. Therefore, if the face manifold is highly nonlinear, it may fail to discover\nthe intrinsic geometrical structure.\nIt remains unclear how to generalize our algorithm\nto nonlinear case. Also, in our algorithm, the adjacency graph is induced from the local\ngeometry and class information. Different graph structures lead to different projections. It\nremains unclear how to de\ufb01ne the optimal graph structure in the sense of discrimination.\n\nReferences\n\n[1] P.N. Belhumeur, J.P. Hepanha, and D.J. Kriegman, \u201cEigenfaces vs. \ufb01sherfaces: recognition\nusing class speci\ufb01c linear projection,\u201dIEEE. Trans. Pattern Analysis and Machine Intelligence,\nvol. 19, no. 7, pp. 711-720, July 1997.\n\n[2] M. Belkin and P. Niyogi, \u201cLaplacian Eigenmaps and Spectral Techniques for Embedding and\n\nClustering ,\u201d Advances in Neural Information Processing Systems 14, 2001.\n\n[3] Fan R. K. 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Terzopoulos, \u201cMultilinear Subspace Analysis for Image Ensembles,\u201d\n\nIEEE Conference on Computer Vision and Pattern Recognition, 2003.\n\n[10] K. Q. Weinberger and L. K. Saul, \u201cUnsupervised Learning of Image Manifolds by SemiDe\ufb01nite\nProgramming,\u201d IEEE Conference on Computer Vision and Pattern Recognition, Washington,\nDC, 2004.\n\n[11] J. Yang, D. Zhang, A. Frangi, and J. Yang, \u201cTwo-dimensional PCA: a new approach to\nappearance-based face representation and recognition,\u201dIEEE. Trans. Pattern Analysis and Ma-\nchine Intelligence, vol. 26, No. 1, 2004.\n\n[12] J. Ye, R. Janardan, Q. Li, \u201cTwo-Dimensional Linear Discriminant Analysis ,\u201d Advances in\n\nNeural Information Processing Systems 17, 2004.\n\n\f", "award": [], "sourceid": 2899, "authors": [{"given_name": "Xiaofei", "family_name": "He", "institution": null}, {"given_name": "Deng", "family_name": "Cai", "institution": null}, {"given_name": "Partha", "family_name": "Niyogi", "institution": null}]}