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173
Locality Preserving Projections
, 2002
"... Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data s ..."
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Cited by 234 (16 self)
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Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA)  a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.
Face recognition using laplacianfaces
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... Abstract—We propose an appearancebased face recognition method called the Laplacianface approach. By using Locality Preserving Projections (LPP), the face images are mapped into a face subspace for analysis. Different from Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) wh ..."
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Cited by 210 (27 self)
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Abstract—We propose an appearancebased face recognition method called the Laplacianface approach. By using Locality Preserving Projections (LPP), the face images are mapped into a face subspace for analysis. Different from Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) which effectively see only the Euclidean structure of face space, LPP finds an embedding that preserves local information, and obtains a face subspace that best detects the essential face manifold structure. The Laplacianfaces are the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the face manifold. In this way, the unwanted variations resulting from changes in lighting, facial expression, and pose may be eliminated or reduced. Theoretical analysis shows that PCA, LDA, and LPP can be obtained from different graph models. We compare the proposed Laplacianface approach with Eigenface and Fisherface methods on three different face data sets. Experimental results suggest that the proposed Laplacianface approach provides a better representation and achieves lower error rates in face recognition. Index Terms—Face recognition, principal component analysis, linear discriminant analysis, locality preserving projections, face manifold, subspace learning. 1
Unsupervised Learning of Image Manifolds by Semidefinite Programming
, 2004
"... Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition. In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be ..."
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Cited by 176 (9 self)
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Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition. In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be used to analyze high dimensional data that lies on or near a low dimensional manifold. It overcomes certain limitations of previous work in manifold learning, such as Isomap and locally linear embedding. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and solid objects.
A DataDriven Reflectance Model
 ACM TRANSACTIONS ON GRAPHICS
, 2003
"... We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space o ..."
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Cited by 151 (6 self)
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We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space of acquired BRDFs to create new BRDFs. We treat each acquired BRDF as a single highdimensional vector taken from a space of all possible BRDFs. We apply both linear (subspace) and nonlinear (manifold) dimensionality reduction tools in an effort to discover a lowerdimensional representation that characterizes our measurements. We let users define perceptually meaningful parametrization directions to navigate in the reduceddimension BRDF space. On the lowdimensional manifold, movement along these directions produces novel but valid BRDFs.
Scaling learning algorithms towards AI
, 2007
"... One longterm goal of machine learning research is to produce methods that are applicable to highly complex tasks, such as perception (vision, audition), reasoning, intelligent control, and other artificially intelligent behaviors. We argue that in order to progress toward this goal, the Machine Le ..."
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Cited by 85 (20 self)
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One longterm goal of machine learning research is to produce methods that are applicable to highly complex tasks, such as perception (vision, audition), reasoning, intelligent control, and other artificially intelligent behaviors. We argue that in order to progress toward this goal, the Machine Learning community must endeavor to discover algorithms that can learn highly complex functions, with minimal need for prior knowledge, and with minimal human intervention. We present mathematical and empirical evidence suggesting that many popular approaches to nonparametric learning, particularly kernel methods, are fundamentally limited in their ability to learn complex highdimensional functions. Our analysis focuses on two problems. First, kernel machines are shallow architectures, in which one large layer of simple template matchers is followed by a single layer of trainable coefficients. We argue that shallow architectures can be very inefficient in terms of required number of computational elements and examples. Second, we analyze a limitation of kernel machines with a local kernel, linked to the curse of dimensionality, that applies to supervised, unsupervised (manifold learning) and semisupervised kernel machines. Using empirical results on invariant image recognition tasks, kernel methods are compared with deep architectures, in which lowerlevel features or concepts are progressively combined into more abstract and higherlevel representations. We argue that deep architectures have the potential to generalize in nonlocal ways, i.e., beyond immediate neighbors, and that this is crucial in order to make progress on the kind of complex tasks required for artificial intelligence. 1 1
Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
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Cited by 81 (22 self)
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We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
Nonlinear dimensionality reduction by semidefinite programming and kernel matrix factorization
 in Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics
, 2005
"... We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximiz ..."
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Cited by 50 (5 self)
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We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximizing the variance in feature space while preserving the distances and angles between nearest neighbors. In this paper, adapting recent ideas from semisupervised learning on graphs, we show that the full kernel matrix can be very well approximated by a product of smaller matrices. Representing the kernel matrix in this way, we can reformulate the semidefinite program in terms of a much smaller submatrix of inner products between randomly chosen landmarks. The new framework leads to orderofmagnitude reductions in computation time and makes it possible to study much larger problems in manifold learning. 1
Manifold based analysis of facial expression
 Computer Vision and Pattern Recognition Workshop
"... We propose a novel approach for modeling, tracking and recognizing facial expressions. Our method works on a low dimensional expression manifold, which is obtained by Isomap embedding. In this space, facial contour features are first clustered, using a mixture model. Then, expression dynamics are le ..."
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Cited by 35 (1 self)
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We propose a novel approach for modeling, tracking and recognizing facial expressions. Our method works on a low dimensional expression manifold, which is obtained by Isomap embedding. In this space, facial contour features are first clustered, using a mixture model. Then, expression dynamics are learned for tracking and classification. We use ICondensation to track facial features in the embedded space, while recognizing facial expressions in a cooperative manner, within a common probabilistic framework. The image observation likelihood is derived from a variation of the Active Shape Model (ASM) algorithm. For each cluster in the lowdimensional space, a specific ASM model is learned, thus avoiding incorrect matching due to nonlinear image variations. Preliminary experimental results show that our probabilistic facial expression model on manifold significantly improves facial deformation tracking and expression recognition. 1.