Slow feature analysis

Slow Feature Analysis is an unsupervised learning algorithm that aims to learn invariant or at least slowly changing features from a vector signal. It is based on the major axis transformation .

Problem Description

If an input signal is given, an input / output function is sought for which varies as little as possible and is not constant. ${\displaystyle x(t)}$ ${\displaystyle g(x)}$ ${\displaystyle y(t)=g(x(t))}$ ${\displaystyle g}$

Formally one writes:

A -dimensional input signal is given with . Find a -dimensional input / output function that generates the -dimensional output with for each . The following constraints must be met for all of them : ${\displaystyle n}$ ${\displaystyle x(t)=[x_{1}(t),\dots ,x_{n}(t)]}$ ${\displaystyle t\in [t_{0},t_{1}]}$ ${\displaystyle m}$ ${\displaystyle g(x)=[g_{1}(x),\dots ,g_{m}(x)]}$ ${\displaystyle x(t)}$ ${\displaystyle m}$ ${\displaystyle y(t)=[y_{1}(t),y_{2}(t),\dots ,y_{m}(t)]}$ ${\displaystyle y_{i}(t)=g_{i}(x(t))}$ ${\displaystyle i\in \{1,\dots ,m\}}$ ${\displaystyle i\in \{1,\dots ,m\}}$

${\displaystyle {\begin{aligned}\Delta _{i}=\Delta (y_{i}):&=\langle {\dot {y}}_{i}^{2}\rangle &{\text{ ist minimal}}\\\langle y_{i}\rangle &=0&{\text{(Mittelwert)}}\\\langle y_{i}^{2}\rangle &=1&{\text{(Varianz)}}\\\forall i'<i:\langle y_{i'}y_{i}\rangle &=0&{\text{(Dekorrelation)}}\end{aligned}}}$

where the derivative is denoted by and an average over time is: ${\displaystyle {\dot {y}}}$ ${\displaystyle t}$ ${\displaystyle \langle \cdot \rangle }$

${\displaystyle \langle f\rangle :={\frac {1}{t_{1}-t_{0}}}\int \limits _{t_{0}}^{t_{1}}f(t)\mathrm {d} t}$

Web links

• Laurenz Wiskott et al .: Slow feature analysis. In: Scholarpedia. 6, No. 4, 2011, 5282.

Individual evidence

1. ↑ Laurenz Wiskott, Terrence J. Sejnowski: Slow Feature Analysis: Unsupervised Learning of Invariances . In: Neural Computation . tape 14 , no. 4 , 2002, p. 715-770 , doi : 10.1162 / 089976602317318938 .