the data. Structured Principal Component Analysis (SPCA)

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1 Strutured Prinipal Component Analysis Kristin M. Branson and Sameer Agarwal Department of Computer Siene and Engineering University of California, San Diego La Jolla, CA Abstrat Many tasks involving high-dimensional data, suh as fae reognition, suffer from the urse of dimensionality: the number of training samples required to aurately learn a lassifier inreases exponentially with the dimensionality of the data. Strutured Prinipal Component Analysis (SPCA) redues the dimensionality of the data while preserving its disriminative power. The algorithm finds lusters of similar features, where the similarity between features is measured using the lass-onditional Chi-squared distane between the distributions of the features. As features in a luster are similar and thus redundant, an entire luster an be represented by a small number of prinipal omponents extrated from eah luster. We test the algorithm on two fae reognition databases, the Ekman and Friesen Pitures of Faial Affet Database and the Yale Fae Database, with enouraging results. 1. Introdution Many tasks in mahine learning and omputer vision require learning a lassifier from a small number of highdimensional training samples. These tasks are partiularly diffiult beause the potential omplexity of a lassifier inreases exponentially with the dimensionality of the data. For example, onsider the task of image lassifiation. The goal is to learn a simple lassifier, say a pereptron, that will aurately lassify novel images. The number of parameters to learn is more than the number of pixels in the image, and the number of samples required to aurately and onfidently learn a pereptron is many more than the number of parameters. A standard dataset has over, pixels per image and training images, thus an aurate pereptron annot be learned from this data. Fortunately, real-world data sets ontain large amounts of redundany, thus the data an be represented by small(er) sets of features. The pixels in an image are redundant, as images are highly strutured. JPEG ompression and image subsampling exploit redundany to redue dimensionality while retaining most of the image struture. Two properties make dimensionality redution for lassifiation tasks effiient. First, lassifiation tasks are restrited to small domains. For example, in fae reognition, all data samples are images of faes. As all faes share the same struture and deviations between different fae images are small, a few features an represent any fae. Seond, dimensionality redution with the ultimate goal of learning a lassifier need only preserve the properties of the data relevant to the lassifiation task. For example, if the lassifiation task is fae expression reognition, only the features relevant to desribing the posed expression must be preserved, not the features desribing the person s identity. Reent researh has shown that features useful for identity reognition are orthogonal to those useful for expression reognition [4]. Inluding features enoding identity inreases noise and omplexity that will obfusate lassifier learning algorithms. Most dimensionality redution tehniques do not take advantage of this seond property. These unsupervised algorithms instead ignore the lass labels of the data and find the features that best represent all the properties of the data. In this paper, we present a new supervised algorithm for dimensionality redution, Strutured Prinipal Component Analysis (SPCA), that preserves the lass-onditional struture of the data. If two features are similar within every lass, then given one feature, the seond feature does not add muh information useful for lassifying the data. This means the dimensionality an be redued by replaing lusters of features that are similar within every lass with a small number of features. SPCA strutures the features of the data into groups with high within-lass similarity, then, for eah luster, performs Prinipal Component Analysis (PCA) on the data projeted on the features in that luster. SPCA thus finds a linear projetion of the data that preserves lass disriminability.. Related Work SPCA was oneived with the faults of two lassial algorithms in mind, Prinipal Component Analysis and Fisher s Linear Disriminant Analysis. In this setion, we disuss these two algorithms and their flaws. In addition, we disuss Fator Analysis, whih shares some ideas with SPCA. 1

2 Finally, we emphasize the dissimilarity of SPCA and mixture algorithms like Mixtures of Gaussians. The lassial unsupervised dimensionality redution algorithm is PCA. PCA selets the orthonormal features among the linear ombinations of the original features that maximize the variane of the projeted data. While PCA is optimal in terms of its riterion, it is generally not optimal for lassifiation tasks as PCA ignores the lass labels. In the ase of faial expression reognition, maximizing the variane of the projeted data is not ideal, for fae images vary more over identity than expression. Most features found will not be useful in expression lassifiation. Fisher s Linear Disriminant Analysis (LDA), on the other hand, uses a supervised riterion to hoose a set of orthonormal features from all linear transformations of the original features. The features are seleted to minimize the variane of the data within eah lass while maximizing the variane of the means of eah lass of data. The standard tradeoff between these two goals is to maximize the quotient: the variane of the means of eah lass divided by the summed variane within eah lass. Note that LDA only depends on the mean and variane of the data. However, these two statistis are suffiient to desribe the data only if the data is normally distributed. If this assumption does not hold, then it is not lear that LDA is optimizing the right riterion. For example, in many domains the distribution of a feature ann be bimodal within a lass. Another problem with LDA is that it an selet at most 1 features, where is the number of lasses. In most ases, this is not enough to generalize to novel data samples. The theme of Fator Analysis (FA) is similar to that of SPCA: if features are highly orrelated, they an be represented by a few features in the diretions of their orrelation. FA represents eah D-dimensional data sample x i by a d-dimensional fator z i suh that z i probabilistially haraterizes as muh of the orrelation between eah dimension of x i as possible. Therefore, given z i, the dimensions of eah x i are independent. There are two main differenes between SPCA and FA that make SPCA a more powerful method. Most importantly, FA is not a supervised algorithm. SPCA is supervised beause it uses the lassonditional proximity of the features distributions to measure similarity, as opposed to the unsupervised orrelation between features used by FA. This results diretly from the deep embedding of the similarity measure of FA in the algorithm, whereas in SPCA the similarity measure is expliit and flexible. Beause it is unsupervised, FA finds features that are not useful for the ultimate lassifiation task. Seond, FA represents similar features by features in the diretion of maximum ovariane, whereas SPCA represents similar features by features in the diretion of maximum variane. The maximum variane is a more robust measure, sine if the variane of even one feature of a group in a ertain diretion is large, this diretion is most probably important for representing the data. Thus SPCA does not rely on the grouping of features being exat. When one thinks of lustering, it is diffiult not to think of lustering the data samples. However, Fator Analysis and SPCA group together features, not data samples. Mixture methods like Mixtures of Gaussians and Mixtures of Prinipal Component Analyzers find soft lusterings of the data samples, not the dimensions. 3. SPCA Algorithm Desription Figure 1: Illustration of the SPCA algorithm. On the left are four training data samples, two in eah lass. Eah box represents a feature of the data. Color enodes feature values. SPCA groups together features that have similar distributions within lasses. The lusters are shown on the left. PCA is performed on the data projeted on eah luster of features separately. SPCA finds features that preserve the lass-onditional struture of the data. It lusters the features of the data into groups that have similar lass-onditional distributions. No one luster neessarily has more disriminative power than any other luster. The hypothesis is that eah luster of features an be represented by just a few features, and that the olletion of these few features from eah luster will preserve the disriminability of the data. SPCA is an algorithmi framework beause the pairwise similarity measure, the lustering algorithm used to group similar variables, and the method used to hoose representative features from eah luster an all be varied. In this setion, we desribe the instantiation of the SPCA framework we implemented. First, the pairwise distane between eah pair of features is measured by the lass-onditional Chisquared distane between the distributions of the features. Seond, the Normalized Cut riterion is used to luster the features. Finally, a few features are hosen to represent eah luster of features by PCA. 3.1 Feature Similarity Measure A supervised measure of the similarity between two features, u and v, is the lass-onditional distane between the distributions of the features. This is the weighted sum of the within-lass distane between the distributions, d f u f v P 1 where is the number of lasses, d is an unsupervised funtion of distane between distributions, and P is the

3 probability of lass. Beause f u, f v, and P are unknown, they must be estimated from the training data. The distributions are estimated as the histogrammed data; h u i is the number of samples of lass suh that the value of feature u falls within the interval i. The distane between the distributions is estimated as the distane between the histogrammed data, d h u h v. The lass probability is estimated by the Maximum Likelihood Estimate, P n n, where n is the total number of samples and n is the number of samples of lass. Any distane metri may be used for the unsupervised distane d. We hose the Chi-squared distane metri: d h u h v k i i h u i h v i h u i h v i where k is the number of intervals into whih the data is divided. The Chi-squared distane was hosen beause it is the standard, historially used metri to ompare histogrammed data. The lass-onditional pairwise distane between features u and v is therefore 1 k i i h u i h v i h u i h v i P 3. Clustering Using Normalized Cut SPCA uses the Normalized Cut algorithm to luster the features so that features in the same luster are similar, while features in different lusters are dissimilar []. Thus, SPCA lusters the features so that intra-luster affinity is maximized while inter-luster affinity is minimized, where affinity is group similarity. The similarity between features u and v is inversely proportional to the distane between them, d u v : W u v e d u v σ (σ is a onstant that desribes what distanes are onsidered far). The inter-luster affinity between lusters S 1 and S is: Aff S 1 S W u v u S 1 v S Similarly, the intra-luster affinity of luster S is: Aff S S W u v u v S The riterion funtion minimized by Normalized Cut is: NCut S 1 S Aff S 1 S Aff Aff S 1 S S 1 S 1 S Aff S S 1 S This quantity inreases with inter-luster affinity and dereases with intra-luster affinity. The membership vetor, y, that indiates whih luster eah feature should be in would ideally be disrete valued, with a single value for eah lass. Finding the optimal disrete-valued membership vetor is an NP-hard problem. However, y an be approximated by solving a generalized eigenvetor problem, W y λdy. The pairwise affinity matrix, W, is a N x N matrix, where N is the original number of features in the data. Eah element of the affinity matrix is the pairwise similarity W u v between two features, u and v. The degree matrix, D, is a diagonal matrix in whih eah diagonal element represents the total similarity of a feature to all other features. That is, D u u N v 1 W u v [11]. The vetor y is thresholded to determine whih features are members of the same luster. The above formulation an be extended to a k- partitioning of the graph by using additional eigenvetors [9]. We do so by staking the nd to the k th eigenvetors olumnwise, normalizing the rows of the resulting matrix, and performing k-means lustering on them. Given that our data is high-dimensional, solving the eigenvetor problem is a omputationally intensive task. However, our high dimensional data is highly redundant, i.e. there are a large number of features in our data that are similar to eah other, implying that a number of rows of our weight matrix W are similar to eah other. Having made this observation, we approximate the eigenvetor deomposition by solving the problem for a random sample from the data and extrapolating the resulting eigenvetors to the full dataset. This is known as the Nyström approximation. The original eigenvetor problem has omplexity O D 3 in the dimensionality of the data. Using the Nyström approximation we an ompute the eigenvalue deomposition in O s 3 D, where s is the number of samples used. Empirial evidene shows that for data with a lear lustering struture, a fairly small number of samples an be used to approximate the eigenvetors to a small error [7]. 3.3 Representation of Eah Cluster SPCA lusters the features of the data into groups that, beause of their high affinity, an be represented by a small number of omponents to redue dimensionality. As illustrated in Figure 1, if features u 1 u m are grouped into one luster, then a few features are hosen based on the data samples X 1 x 1 u 1 x 1 u T m through X n x n u 1 x n u T m, where x 1 x n are the training data samples. This is repeated for eah luster. The onise representation losest to the atual data in eah luster is the top prinipal omponents of the data. PCA hooses the omponents that minimize the sumsquared distane between the projeted data and the original data. These omponents are the eigenvetors of the sample ovariane matrix, i X i µ X i µ T in order of the orresponding eigenvalue. Thus, there are two parameters in SPCA: the number of lusters and the number of features extrated from eah luster. As in PCA, the mean of the 3

4 data samples in eah luster is not represented in SPCA. 4 Experiments The SPCA algorithm was ompared to PCA and LDA on three sets of data. The first set is a syntheti set designed to demonstrate the weaknesses of PCA and LDA. The seond set is the Ekman and Friesen POFA database, with the task of expression reognition. The third set is the Yale Fae database, with the task of identity reognition. SPCA ahieves % auray on the syntheti data, ompared to SPCA and LDA whih did no better than hane. SPCA also outperforms PCA and LDA on the POFA database. SPCA outperforms PCA on the Yale database and has similar performane to LDA. 4.1 Syntheti Data PCA and LDA both have weaknesses that limit their effetiveness. If a feature has high variane but is unorrelated with the lass labels, PCA will highly represent this feature beause of its variane, negleting features with smaller variane but more orrelated with the lassifiation of the data. On the other hand, LDA assumes the lass-onditional distribution of the data over eah feature is normal. Suppose this assumption is false, for instane if a feature s data for one lass is bimodally distributed and for another lass is normally distributed. This is the ase for the pixels in the smiles (whih may or may not show teeth) of happy faes versus pixels in the mouths of sad faes. As LDA hooses the omponents that separate the lass means as muh as possible, it will hoose to offset the means of the bimodal and normal distributions. This ould result in one of the modes of the bimodal distribution being projeted to nearly the same value as the mean of the normal distribution. Class-Conditional Distributions of Features f x 1 f x g x 1 g x h x Figure : Distributions of the features of the synthesized data. With these limitations in mind, we synthesized training and test samples, all i.i.d. Eah sample has features with three possible distributions, f x, g x, and h x. Only the features with distribution f or g are useful in lassifiation. These distributions are shown in Figure. f x 1 and g x are bimodal distributions, with modes 5 and a standard deviation of 1. f x and g x 1 are normal distribution with mean and standard deviation 1. h x is uniformally distributed between and 1. features have distribution f, features have distribution g, and 8 features have distribution h. The optimal dimensionality redution tehnique for this data set would ignore all 8 features of distribution h and use any of the features of distribution f or g. Figure 3(a-d) shows the projetion of the data on the features hosen by SPCA, LDA, and PCA. When grouping the data into three lusters, SPCA put all but two of the features with distribution f in one luster, all but one of the features with distribution g in the seond luster, and all the rest of the features in the third luster. Thus the first and seond prinipal omponents generated by SPCA are useful in disriminating the data, while the third is not. SPCA performs equally well when only two lusters of the features are found. As hypothesized, LDA was not able to separate the test data. It was able to find a projetion to separate the training data, but this projetion relied heavily on the features of distribution h whih are not orrelated with the lassifiation. Thus, when generalizing to the test data, LDA fails. PCA was distrated by the 8 features of distribution h that were not orrelated with the lassifiation, and thus was unable to separate the training and test data. In fat, SPCA performs well while the other two algorithms fail on data in whih the separation between the modes of the bimodal distribution is small. For separations greater than.1, SPCA ahieves % auray using a nearest-neighbor lassifier. No matter how small the separation, LDA and PCA are not able to separate the data, despite the distributions approahing a normal distribution, as shown in Figure 3(e). These experiments on the syntheti data set show that SPCA is robust to features that are unorrelated with lassifiation, unlike PCA. They also show that SPCA is robust to non-normal distributions of the data, unlike LDA. 4. The Ekman and Friesen POFA Database SPCA, PCA, and LDA were tested on the Ekman and Friesen Database of Pitures of Faial Affet [6]. This data set inludes 14 trained ators posing six expressions, plus neutral. There are 1 greysale images in this data set, 96 of whih are not neutral. Examples from are shown in Figure 4(a). An expression lassifier must generalize over identity and onentrate only on the expression in an image. A supervised algorithm would be able to find a more aurate and onise representation that is tailored to expression reognition, in omparison to PCA. However, PCA signifiantly outperforms LDA, by a margin of % auray. We hypothesized that this was partially due to the limited number of omponents LDA an extrat (6 1 5). Even trying 4

5 1 PC 3 1 Projetion of Test Data (SPCA) PC PC PC Training Test Projetion of Test Data (SPCA) PC 1 Projetion of Data (LDA) Projetion of Test Data (PCA) (a) (b) () (d) (e) Perent Corretly Classified Comparison of Methods for Synthesized Data SPCA LDA PCA Distane Between Modes of Bimodal Distribution Figure 3: (a) Projetion of the test data on the features hosen by SPCA, 3 lusters (b) lusters () Projetion of the training data and test data on the feature hosen by LDA. (d) Projetion of the test data on the top two Prinipal Components hosen by PCA. (e) Results of SPCA, LDA, and PCA followed by a nearest neighbor lassifier on lassifying the syntheti data, with varying distane between the modes of the bimodal distributions. (a) (b) () Figure 4: (a) Example ropped and aligned images from the POFA database (b) Example full-fae images from the Yale database () Example losely-ropped images from the Yale database. different riterion funtions whih allow LDA to produe more features does not greatly improve LDA s performane. To ompare SPCA with previous experiments in whih PCA and LDA performed well, we perform the same image preproessing. The images were aligned so that the eyes and the bottom of the top row of teeth were in the same position for all images, and ropped inside the ontours of the fae. Next, the images were subsampled and onvolved with Gabor wavelet jets of 4 Gabor filters (five sales and eight orientations), resulting in a 4,6 dimensional vetor. Finally, the outputs of the Gabor filters were z-sored (normalized so that the mean intensity value for eah pixel is zero and the standard deviation is one). After preproessing, the dimensionality of the data is redued using PCA, LDA, or SPCA. A pereptron is learned from images of 1 of the ators, training is stopped at the best performane on a held out ator, and evaluated on a novel ator [5]. SPCA only finds lusters of features with high affinity, not neessarily important to lassifiation. Clusters differ in number of features and orrelation with the lassifiation, yet the number of prinipal omponents extrated from eah is equal. Thus, eah luster is weighted equally. For the POFA data set, we added an extra layer of PCA to weight the prinipal omponents extrated by SPCA by the amount of variane of the data projeted on them. This extra layer proved neessary when using a pereptron for lassifiation, as a pereptron is greatly influened by input variables that have small variane in the training data. For example, suppose a feature has a onstant, low value for all the training data exept for one of lass. The pereptron will find this feature useful in determining lass, and ould weight its inputs to lassify an example as lass if ever the value of this feature is high. If the inonsistent value for this one training sample is merely noise, the pereptron will mistake all test examples with an inonsistent value for this variable as lass. With an extra layer of PCA added, SPCA ahieves 9.7% auray on this data set, ompared to 9% auray for PCA (using 5 prinipal omponents), and 79.3% auray ahieved by LDA. These are the optimal results obtained by SPCA, PCA and LDA. These results are impressive beause 9.7% is.3% less than the auray humans ahieve on this dataset. SPCA proved to be relatively insensitive to the number of lusters and the number of prinipal omponents extrated from eah luster. Figure 5(a) and (b) show the results of varying these parameters. While SPCA performs better with 3 lusters than lusters, the lassifiation error differene is small, %. In addition, using 3 lusters, optimal results are obtained extrating two and four prinipal omponents from eah luster, and extrating three prinipal omponents is only 1% worse in lassifiation error. 4.3 The Yale Fae Database The Yale Database [] onsists of images of 15 ators under 11 different onditions, inluding different lighting, faial expressions, and olusion effets. Identity reognition is diffiult, partiularly for PCA, beause the lassifier must generalize over all these distrations [1]. This dataset was reated with LDA in mind, thus LDA performs extremely well while PCA performs poorly in this experiment. Two experiments were performed, one in whih the images were ropped outside the fae ontour (full-fae images) and one in whih the images were ropped inside the fae ontour (losely-ropped images). Examples are shown in Figure 4. The preproessing of this dataset is the same as that of the POFA dataset. After preproessing, the dimensionality of the data is redued using PCA, LDA, or SPCA. A pereptron is trained by bakpropagation on 164 5

6 Classifiation Error (%) PCs Clusters 3 Clusters (a) Classifiation Error (%) Clusters 3 Clusters (b) Classifiation Error (%) PC PC 3 PC 4 PC Clusters () Classifiation Error (%) Number of Clusters 1 PC PC 3 PC 4 PC (d) Figure 5: Comparison of parameter settings on POFA and Yale databases. (a) POFA, and 3 lusters, without an extra layer of PCA (b) POFA, and 3 lusters, with an extra layer of PCA. The numbers above eah bar are the number of prinipal omponents extrated in the extra layer of PCA () Yale, Full-fae images (d) Yale, Closely-Cropped images. of the samples and tested on a novel image. SPCA ahieves % auray on the full-fae samples, ompared to LDA whih obtains 99.4% lassifiation auray and PCA whih obtains 9% lassifiation auray. On the losely-ropped samples, LDA outperforms SPCA. LDA ahieves 97% lassifiation auray, ompared to SPCA with 94.6% auray and PCA with 76.4% auray. A omparison of the effets of the parameters for SPCA is shown in Figure 5() and (d). 5. Disussion SPCA uses a supervised measure of similarity to luster the features into groups of high intra-luster affinity and low inter-luster affinity. It extrats a small number of prinipal omponents from eah luster to represent the data. Experimentally, we have shown that the supervised measure of similarity allows SPCA to distinguish features that are orrelated with the lassifiation from those that are not. Beause of this, SPCA outperforms PCA in all experiments. We have also shown that when the assumptions made by LDA do not hold, LDA performs very poorly. In these ases, we have experimentally shown that SPCA outperforms LDA. If the assumptions made by LDA do hold, then LDA performs optimally. In addition, we hypothesize that additional experimentation with non-aligned databases will show that SPCA is more robust than PCA and LDA to small translations and rotations in the images. As stated earlier, SPCA is atually a versatile framework of algorithms. In the future, we hope to experiment with other instantiations, inluding different methods of representing the features of eah luster. Instead of seleting from the linear ombinations of the features in a luster, we ould selet diretly from the features in the luster. This would be useful in appliations in whih linear ombinations of features are meaningless. We would also like to try mutual information measures of similarity, like the Kullbak-Liebler distane. Finally, we believe that the distane measure hosen for SPCA ould be applied to LDA. In suh an algorithm, the data would be projeted onto the feature spae whih maximizes the Chi-squared distane between the distributions of the data of eah lass and minimizes the Chi-squared distane of the distributions within eah lass. Aknowledgments We would like to thank Serge Belongie, Gary Cottrell and GURU, Sanjoy Dasgupta, Virginia de Sa, Charles Elkan, and Biana Zadrozny for helpful disussion and advie. Referenes [1] Belhumeur, P. N., Hespanha, J., and Kriegman, D. J., Eigenfaes using lass speifi linear projetion, European Conferene of Computer Vision, Vol. 1, pp 45-58, [] Belhumeur, P. N. and Kriegman, D. J., The Yale Fae Database, [3] Bishop, C. M., Neural Networks for Pattern Reognition, Oxford University Press, [4] Cottrell, G. W., Branson, K. M., and Calder, A. J., Do expression and identity need separate representations?, Proeedings of the 4th Annual Conferene of the Cognitive Siene Soiety, Fairfax, Virginia, pp 83-43,. [5] Dailey, M. N., Cottrell, G. W., and Adolphs, R., A Six-Unit Network Is All You Need to Disover Happiness, Proeedings of the nd Annual Conferene of the Cognitive Siene Soiety, Mahwah, New Jersey,. [6] Ekman, P. and Friesen, W., Pitures of Faial Affet, Consulting Psyhologists, Palo Alto, [7] Fowlkes, C., Belongie, S., and Malik, J., Effiient Spatiotemporal Grouping Using the Nyström Method, Computer Vision and Pattern Reognition, Vol. 1, pp , 1. [8] Ghahramani, Z. and Hinton, G. E., The EM Algorithm for Mixtures of Fator Analyzers, Tehnial Report CRG-TR- 96-1, University of Toronto, [9] Ng, A. Y., Jordan, M. I., and Weiss, Y., On Spetral Clustering: Analysis and an Algorithm, NIPS Vol. 14,. [] Shi, J. and Malik, J., Normalized Cuts and Image Segmentation, IEEE Transations on Pattern Analysis and Mahine Intelligene, Vol. (8), pp ,. [11] Weiss, Y., Segmentation using eigenvetors: A unifying view, International Conferene on Computer Vision, Volume, pp ,

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