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Published byΑμάρανθος Παπάζογλου Modified over 5 years ago
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Maximum likelihood estimation of intrinsic dimension
Authors: Elizaveta Levina & Peter J. Bickel presented by: Ligen Wang
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Plan Problem Some popular methods MLE approach Statistical behaviors
Evaluation Conclusions
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Problem Facts: Why discover this low-D structure? Our target:
Many real-life high-D data are not truly high-dimensional Can be effectively summarized in a space of much lower dimension Why discover this low-D structure? Help to improve performance in classification and other applications Our target: How much is this lower dimension exactly, i.e., the intrinsic dimension Importance of this lower dimension: If our estimation is too low, features are collapsed onto the same dimension If too high, the projection becomes noisy and unstable
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Some popular methods PCA LLE ISOMAP Etc.
Decides the dimension by users by how much covariance they want to preserve LLE User provides the manifold dimension ISOMAP Provides error curves that can be ‘eyeballed’ to estimate dimension Etc.
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MLE approach
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A little math
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Statistical behaviors
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Evaluation – 1: on manifold
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Evaluation – 2: near manifold
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Evaluation – 3: real-world data
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Conclusions MLE produces good results on a range of simulated (both non-noisy and noisy) and read datasets Outperforms two other methods Suffers from a negative bias for high dimensions Reason: approximation is based on observations falling in a small sphere, which requires very large sample size when the dimension is high Good news: in reality, the intrinsic dimensions are low for most interesting applications
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