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 – Dimensionality Estimators 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 an application

Papers N. 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”

How does dimensionality reduction influence our lives? Compress images, audio and video Redusing noise Editing Reconstruction

This is a image going through different steps in a reconstruction

Intrinsic Dimensionality The number of ‘free’ parameters needed to generate a pattern Ex: f(x) = -x²=> 1 dimensional f(x,y) = -x² => 1 dimensional

LOCAL PRINCIPAL COMPONENT ANALYSIS

Principal 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.

Example

Problems 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 PCA

First eigenvector

Second eigenvector

A better solution?

Local eigenvector

Local eigenvectors

Another problem

Is this the principal eigenvector?

Or do we need more than one?

Choose

The answer depends on your application Low resolutionHigh resolution

Challenges How to partition the space? How many partitions should we use? How many dimensions should we retain?

How to partition the space? Vector Quantization Lloyd Algorithm Partition the space in k sets Repeat until convergence: Calculate the centroids of each set Associate each point with the nearest centroid

Lloyd Algorithm Set 1 Set 2 Step 1: randomly assign

Lloyd Algorithm Set 1 Set 2 Step 2: Calculate centriods

Lloyd Algorithm Set 1 Set 2 Step 3: Associate points with nearest centroid

Lloyd Algorithm Set 1 Set 2 Step 2: Calculate centroids

Lloyd Algorithm Set 1 Set 2 Step 3: Associate points with nearest centroid

Lloyd Algorithm Set 1 Set 2 Result after 2 iterations:

How many partitions should we use? Bruske & Sommer: “just try them all” For k = 1 to k ≤ dimension(set): 1.Subdivide the space in k regions 2.Perform PCA on each region 3.Retain significant eigenvalues per region

Which eigenvalues are significant? Depends on: Intrinsic dimensionality Curvature of the (hyper-)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? (Bruske and Summer) Cutoff after normalization µ x is the xth eigenvalue With 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 dropped Decide whether this error rate is significant

Error distance for local PCA (Kambhatla and Leen) 1)Euclidean Distance 2)Reconstruction Distance

Results One dimensional space, embedded in 256*256 = 65,536 dimensions 180 images of rotating cylinder ID = 1

Results

NEURAL NETWORK PCA

Basic 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 (Local PCA 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)

5-layer Auto-associators vs. Local PCA (VQPCA) Difficulty to train 5-layer auto-associators. Faster training in VQPCA algorithm. (VQPCA can be accelerated using tree- structured or multistage VQ) 5-layer auto-associators are prone to trapping in poor local optimal. VQPCA slower for encoding new data but much faster for decoding.

5-layer Auto-associators vs. Local PCA (VQPCA) The results of 1 st paper indicate that VQPCA is not suitable for real-time applications (e.g videoconferencing) where we need very fast encoding. For decoding only (e.g image retrieval from databases), VQPCA is a good choice - accurate and fast-.

Estimate new dimensionality 2 Algorithms proposed: Local Eigenvalue Algorithm: based on the local eigenvalues of the covariance matrix in small regions in feature space The k-Nearest Neighbor Algorithm: based on the distribution of the distances from an arbitrary data vector to a number (k) of its neighbors. Both work but not always!

Comparison of Algorithms Both algorithms work with the assumption – Vectors in dataset are uniformly distributed in a small region of N-dimensional surface. Accuracy of nearest neighbor algorithm increases for k>2. In real data sets, we have noise. Nearest neighbor algorithm has less sensitivity to noise than the local Eigenvalue estimator. The NN algorithm needs fewer vectors than LEE. NN algorithm generally underestimates the intrinsic dimensionality for sets with high dimensionality. (Usually the algorithm returns a floating number dimension ) NN algorithm has problems with borders (edge effect), LLE does not suffer from this problem. NN algorithm is much faster than LLE. For very small data sets with high dimensionality the intrinsic dimensionality can not be found.

The End