Foundations of Data Science

position on genome
trees
=
Phenotype; outward
manifestation, observables
Genotype:
internal code
Figure 10.6: The system of linear equations used to find the internal code for some
observable phenomenon.
sparse solution x where x is all zeros except for the two 1’s for 12345 and 678910 where
the two teams are {1,2,3,4,5} and {6,7,8,9,10}. The question is can we recover x from
b. If the matrix A had satisfied the restricted isometry condition, then we could surely
do this. Although A does not satisfy the restricted isometry condition which guarantees
recover of all sparse vectors, we can recover the sparse vector in the case where the teams
are non overlapping or almost non overlapping. If A satisfied the restricted isometry
property we would minimize ‖x‖ 1
subject to Ax = b. Instead, we minimize ‖x‖ 1
subject
to ‖Ax − b‖ ∞
≤ ε where we bound the largest error.
10.4.5 Low Rank Matrices
Suppose L is a low rank matrix that has been corrupted by noise. That is, M = L+R.
If the R is Gaussian, then principle component analysis will recover L from M. However,
if L has been corrupted by several missing entries or several entries have a large noise
added to them and they become outliers, then principle component analysis may be far
off. However, if L is low rank and R is sparse, then L can be recovered effectively from
L + R. To do this, find the L and R that minimize ‖L‖ 2 F + λ ‖R‖ 1 . Here ‖L‖2 F
is the sum
of the singular values of L. A small value of ‖L‖ 2 F
indicates a low rank matrix. Notice
that we do not need to know the rank of L or the elements that were corrupted. All we
need is that the low rank matrix L is not sparse and that the sparse matrix R is not low
rank. We leave the proof as an exercise.
An example where low rank matrices that have been corrupted might occur is aerial
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