- General Terms and tools
- Hebbian Learning
- Source Separation
- Infomax ICA
- Second Order Source Separation
- Stochastic Optimization
- k-means Clustering
- Pairwise Clustering
- Self-Organising Maps
- Locally Linear Embedding
- Estimation Theory
- Density Estimation
- Kernel Density Estimation
- Parametric Density Estimation
- Mixture Models - Estimation Models
- Density Estimation
Independent Component Analysis (ICA)
ICA allows the reconstruction of mixed signals. This could for example be multiple speakers on one audio track.
- Needs some prior knowledge
- The number of sources to be recovered is a parameter to the algorithm. It is possible to choose a higher number of sources and afterwards remove sources that are only noise.
- sources must have non-Gaussian distributions
- source amplitudes cannot be recovered
number of observations
number of dimensions
Mixing Matrix (must be invertible for ICA to work)
Different methods leverage different prior knowledge
- Vanishing cross-correlation functions (QDIAG, FFDIAG) (more or less invented by the institute)
- non-Gaussianity (fastICA)
- Statistical Independence
- See above
- Probability distribution of th target space
- our cost function
- is the vector with the results of our approximation function
- is a freely chosen sigmoid cdf function that depends on .
- training error of a specific datapoint.
Infomax ICA uses empirical risk minimization(ERM) to do gradient ascent.
We want to maximize which relies on the real distribution of the data. Since this distribution is unkown we use the proxy error function . uses a sum over all the data points instead of an integral over the real distribution. Otherwise the two are identical.
The Infomax cost function Infomax is the measure of how much mutual information two random variables have. It can also be expressed as the KL divergence(is equivalent?). We try to minimize the mutual information, because we believe that our sources are independent This is why we need to maximize the cost function
Possible exam task Independent source signals work better if they match .
Find a suitable
Gradient ascent for w
- Batch Learning
- Online Learning Taylor approximation used because inverse is hard to find.
The natural gradient is faster than normal gradient descent.
While normal gradient changes the parameters by a fixed rate, the natural gradient changes the outcome distribution by a constant distance. This distance in distributions is measured by the KL Divergence.1
Step size is normalized at each step
Second Order (Blind)Source Separation
SOBSS allows the separation of sources after temporal shift or some other form of noise. sources are not iid but assumed to be time correlated / sequential.
Principle (As ambiguous as necessary because I just transcribed this in the lecture) Try different so that matches in height.
- time shift
Cost function minimization depends on the algorithm used on top.
For additional robustness remove from the set of possible solutions. This avoids some trivial solutions.
- Whitened data
- Kurtosis is known (prior knowledge)
- sensitive to outliers
is orthogonal matrix.
Find the maximum negentropy based on contrast function.
We can show that
Optimize for curtosis
- Batch learning
- Online Learning
- Initialize with random vector of unit length
- <0: sub-gaussian. (looks like rectangle) uniform
- 0: Gaussian normal
- >0: super-gaussian(looks like triangle) laplace
Kevin Frans 2016, A[sic!] intuitive explanation of natural gradient descent ↩