Editing Mixture model (section)

===Spectral method===
Some problems in mixture model estimation can be solved using [[spectral method]]s.
In particular it becomes useful if data points ''x<sub>i</sub>'' are points in high-dimensional [[real coordinate space|real space]], and the hidden distributions are known to be [[Logarithmically concave function|log-concave]] (such as [[Gaussian distribution]] or [[Exponential distribution]]).

Spectral methods of learning mixture models are based on the use of [[Singular Value Decomposition]] of a matrix which contains data points.
The idea is to consider the top ''k'' singular vectors, where ''k'' is the number of distributions to be learned. The projection
of each data point to a [[linear subspace]] spanned by those vectors groups points originating from the same distribution
very close together, while points from different distributions stay far apart.

One distinctive feature of the spectral method is that it allows us to [[Mathematical proof|prove]] that if
distributions satisfy certain separation condition (e.g., not too close), then the estimated mixture will be very close to the true one with high probability.