= Low-Rank Approximation =

A '''low-rank approximation''' is an approximation of a matrix.

<<TableOfContents>>

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== Description ==

[[LinearAlgebra/SingularValues|Singular values]] reflect the relative contribution of certain rows and columns to a matrix. The natural extension of this interpretation is that small singular values correspond to rows and columns which don't contribute much.

Consider the example shown [[LinearAlgebra/SingularValueDecomposition#Interpretation|here]]:

{{{
julia> U, S, V = svd(A)
SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}
U factor:
5×4 Matrix{Float64}:
 -0.538764  -0.241808  -0.00729024  -0.568448
 -0.563657  -0.387293   0.366464     0.0616337
 -0.425671  -0.129738  -0.530012     0.687406
 -0.268361   0.484868  -0.589901    -0.388298
 -0.372573   0.734578   0.486575     0.223082
singular values:
4-element Vector{Float64}:
 10.354844862115025
  6.904143550221138
  2.415275359549066
  0.5257705364748064
Vt factor:
4×4 Matrix{Float64}:
 -0.639597  -0.3382    -0.575099  -0.381835
 -0.370551   0.581321  -0.350623   0.633894
  0.289698   0.667521  -0.307725  -0.613023
  0.608016  -0.319535  -0.672035   0.276737
}}}

The first singular value is substantially larger than all others. Therefore using only the rows and columns corresponding to it should arrive at an approximation that captures a large part of the information. This is called the '''rank-1 approximation'''.

{{{
julia> A_1 = U[:,1] * S[1] * V[:,1]'
5×4 Matrix{Float64}:
 3.56819  1.88676   3.20837  2.13019
 3.73306  1.97393   3.35661  2.22861
 2.81919  1.4907    2.5349   1.68304
 1.77734  0.939804  1.59811  1.06106
 2.46753  1.30476   2.21869  1.4731
}}}

This approximation can be evaluated in terms of [[Analysis/Error|RMSE]], which can be calculated through the [[LinearAlgebra/Norm|Frobenius norm]] scaled by ''(mn)^-1^''.

{{{
julia> m,n = size(A)
(5, 4)

julia> rmse = norm(A_1 - A, 2)/sqrt(m*n)
1.6397741899653446
}}}

Approximations can be made for any rank ''k'' up to the actual [[LinearAlgebra/Rank|rank]] of the SVD. As ''k'' increases, the approximation approaches '''''A'''''.



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CategoryRicottone