3,515 research outputs found
Symbolic dynamics I. Finite dispersive billiards
Orbits in different dispersive billiard systems, e.g. the 3 disk system, are
mapped into a topological well ordered symbol plane and it is showed that
forbidden and allowed orbits are separated by a monotone pruning front. The
pruning front can be approximated by a sequence of finite symbolic dynamics
grammars.Comment: CYCLER Paper 93Jan00
Phase transition in PCA with missing data: Reduced signal-to-noise ratio, not sample size!
How does missing data affect our ability to learn signal structures? It has
been shown that learning signal structure in terms of principal components is
dependent on the ratio of sample size and dimensionality and that a critical
number of observations is needed before learning starts (Biehl and Mietzner,
1993). Here we generalize this analysis to include missing data. Probabilistic
principal component analysis is regularly used for estimating signal structures
in datasets with missing data. Our analytic result suggests that the effect of
missing data is to effectively reduce signal-to-noise ratio rather than - as
generally believed - to reduce sample size. The theory predicts a phase
transition in the learning curves and this is indeed found both in simulation
data and in real datasets.Comment: Accepted to ICML 2019. This version is the submitted pape
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