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# Learning The Parts Of Objects By Non-Negative Matrix Factorization

NATURE, no. 6755 (1999): 788-791

WOS NATURE

摘要

Is perception of the whole based on perception of its parts? There is psychological(1) and physiological(2,3) evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations(4,5). But little is known about how brains or computers might learn the parts of objects....更多

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简介

- The authors demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text.
- The authors have applied non-negative matrix factorization (NMF), together with principal components analysis (PCA) and vector quantization (VQ), to a database of facial images.
- As shown in Fig. 1, all three methods learn to represent a face as a linear combination of basis images, but with qualitatively different results.

重点内容

- principal components analysis constrains the columns of W to be orthonormal and the rows of H to be orthogonal to each other. This relaxes the unary constraint of vector quantization, allowing a distributed representation in which each face is approximated by a linear combination of all the basis images, or eigenfaces[6]
- Unlike the unary constraint of vector quantization, these non-negativity constraints permit the combination of multiple basis images to represent a face
- Application of vector quantization, principal components analysis or negative matrix factorization involves finding the approximate factorization of this matrix V Ϸ WH into a feature set W and hidden variables H, in the same way as was done for faces
- negative matrix factorization was performed with the iterative algorithm described in Fig. 2, starting with random initial conditions for W and H
- vector quantization was done via the k-means algorithm, starting from random initial conditions for W and H

结果

- The NMF basis is radically different: its images are localized features that correspond better with intuitive notions of the parts of faces.
- An encoding consists of the coefficients by which a face is represented with a linear combination of basis images.
- This unary representation forces VQ to learn basis images that are prototypical faces.
- This relaxes the unary constraint of VQ, allowing a distributed representation in which each face is approximated by a linear combination of all the basis images, or eigenfaces[6].
- Unlike the unary constraint of VQ, these non-negativity constraints permit the combination of multiple basis images to represent a face.
- The non-negativity constraints are compatible with the intuitive notion of combining parts to form a whole, which is how NMF learns a parts-based representation.
- The exact form of the objective function is not as crucial as the non-negativity constraints for the success of NMF in learning parts.
- It is helpful to visualize the dependencies between image pixels and encoding variables in the form of the network shown in Fig. 3.
- Application of VQ, PCA or NMF involves finding the approximate factorization of this matrix V Ϸ WH into a feature set W and hidden variables H, in the same way as was done for faces.
- Learning parts for these complex cases is likely to require fully hierarchical models with multiple levels of hidden variables, instead of the single level in NMF.
- Non-negativity constraints may help such models to learn parts-based representations[13], the authors do not claim that they are sufficient in themselves.

结论

- This results in a basis that is non-global; in this representation all the basis images are used in cancelling combinations to represent an individual face, and the encodings are not sparse.
- The NMF representation contains both a basis and encoding that are naturally sparse, in that many of the components are exactly equal to zero.
- The authors propose that the one-sided constraints on neural activity and synaptic strengths in the brain may be important for developing sparsely distributed, parts-based representations for perception.

基金

- We acknowledge the support of Bell Laboratories and MIT

引用论文

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