Project 11: Determining the Intrinsic Dimensionality of a Distribution

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Project 11: Determining the Intrinsic Dimensionality of a Distribution. Okke Formsma, Nicolas Roussis and Per Løwenborg. Outline. About the project What is intrinsic dimensionality? How can we assess the ID? PCA Neural Network Nearest Neighbour Experimental Results.
Project 11: Determining the Intrinsic Dimensionality of a DistributionOkke Formsma, Nicolas Roussisand Per LøwenborgOutline
  • About the project
  • What is intrinsic dimensionality?
  • How can we assess the ID?
  • PCA
  • Neural Network
  • Nearest Neighbour
  • Experimental Results
  • Why did we chose this Project?
  • We wanted to learn more about developing and experiment with algorithms for analyzing high-dimensional data
  • We want to see how we can implement this into a program
  • PapersN. Kambhatla, T. Leen, “Dimension Reduction by Local Principal Component Analysis” J. Bruske and G. Sommer, “Intrinsic Dimensionality Estimation with Optimally Topology Preserving Maps” P. Verveer, R. Duin, “An evaluation of intrinsic dimensionality estimators”Howdoesdimensionalityreductioninfluenceour lives?
  • Compress images, audio and video
  • Redusing noise
  • Editing
  • Reconstruction
  • This is a image goingthroughdifferentsteps in a reconstructionIntrinsic DimensionalityThe number of ‘free’ parameters needed to generate a patternEx:
  • f(x) = -x² => 1 dimensional
  • f(x,y) = -x² => 1 dimensional
  • Principal Component analysisPrincipal Component Analysis (PCA)
  • The classic technique for linear dimension reduction.
  • It is a vector space transformation which reduce multidimensional data sets to lower dimensions for analysis.
  • It is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences.
  • Advantages of PCA
  • Since patterns in data can be hard to find in data of high dimension, where the luxury of graphical representation is not available, PCA is a powerful tool for analysing data.
  • Once you have found these patterns in the data, you can compress the data, -by reducing the number of dimensions- without much loss of information.
  • ExampleProblems with PCA
  • Data might be uncorrelated, but PCA relies on second-order statistics (correlation), so sometimes it fails to find the most compact description of the data.
  • Problems with PCAFirst eigenvectorSecond eigenvectorA better solution?Local eigenvectorLocal eigenvectorsLocal eigenvectorsAnother problemIs this the principal eigenvector?Or do we need more than one?ChooseThe answer depends on your applicationLow resolutionHigh resolutionChallenges
  • How to partition the space?
  • How many partitions should we use?
  • How many dimensions should we retain?
  • How to partition the space?Vector QuantizationLloyd AlgorithmPartition the space in k setsRepeat until convergence:Calculate the centroids of each setAssociate each point with the nearest centroidLloyd AlgorithmStep 1: randomly assignSet 1Set 2Lloyd AlgorithmStep 2: Calculate centriodsSet 1Set 2Lloyd AlgorithmStep 3: Associate points with nearest centroidSet 1Set 2Lloyd AlgorithmStep 2: Calculate centroidsSet 1Set 2Lloyd AlgorithmStep 3: Associate points with nearest centroidSet 1Set 2Lloyd AlgorithmResult after 2 iterations:Set 1Set 2How many partitions should we use?Bruske & Sommer: “just try them all”For k = 1 to k ≤ dimension(set): Subdivide the space in k regions Perform PCA on each region Retain significant eigenvalues per regionWhich eigenvalues are significant?Depends on:
  • Intrinsic dimensionality
  • Curvature of the surface
  • Noise
  • Which eigenvalues are significant?Discussed in class:
  • Largest-n
  • In papers:
  • Cutoff after normalization (Bruske & Summer)
  • Statistical method (Verveer & Duin)
  • Which eigenvalues are significant?Cutoff after normalizationµx is the xth eigenvalueWith α = 5, 10 or 20.Which eigenvalues are significant?Statistical method (Verveer & Duin)Calculate the error rate on the reconstructed data if the lowest eigenvalue is droppedDecide whether this error rate is significantResults
  • One dimensional space, embedded in 256*256 = 65,536 dimensions
  • 180 images of rotatingcylinder
  • ID = 1
  • ResultsNeural Network PCABasic Computational Element - Neuron
  • Inputs/Outputs, Synaptic Weights, Activation Function
  • 3-Layer Autoassociators
  • N input, N output and M<N hidden neurons.
  • Drawbacks for this model. The optimal solution remains the PCA projection.
  • 5-Layer Autoassociators
  • Neural Network Approximators for principal surfaces using 5-layers of neurons.
  • Global, non-linear dimension reduction technique.
  • Succesfull implementation of nonlinear PCA using these networks for image and speech dimension reduction and for obtaining concise representations of color.
  • Third layer carries the dimension reduced representation, has width M<N
  • Linear functions used for representation layer.
  • The networks are trained to minimize MSE training criteria.
  • Approximators of principal surfaces.
  • Locally Linear Approach to nonlinear dimension reduction (VQPCA Algorithm)
  • Much faster than to train five-layer autoassociators and provide superior solutions.
  • This algorithm attempts to minimize the MSE (like 5-layers autoassociators) between the original data and its reconstruction from a low-dimensional representation. (reconstruction error)
  • 2 Steps in Algorithm:
  • Partition the data space by VQ (clustering).
  • Performs local PCA about each cluster center.
  • VQPCA is actually a local PCA to each cluster.We can use 2 kinds of distances measures in VQPCA:1) Euclidean Distance2) Reconstruction DistanceExample intended for a 1D local PCA:5-layer Autoassociators vs VQPCA
  • Difficulty to train 5-layer autoassociators.
  • Faster training in VQPCA algorithm. (VQ can be accelerated using tree-structured or multistage VQ)
  • 5-layer autoassociators are prone to trapping in poor local optimal.
  • VQPCA slower for encoding new data but much faster for decoding.
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