BIZNIQUE / LINEAR ALGEBRA / 39
Build a matrix from simpler pieces
Build a matrix from simpler pieces
Keep one, two or three components and inspect what the reconstruction loses.
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THE IDEA
Read the example.
Read the example.
SVD represents a matrix as a sum of rank-one components. Keeping the largest singular components gives a best rank-r approximation in Frobenius norm. The example exposes every entry of a tiny matrix and its reconstruction.
Aᵣ = Σᵢ₌₁ʳ σᵢuᵢvᵢᵀ
Use it well
Use it well
A compact factorisation is not automatically an interpretable or useful model. The approximation criterion must fit the intended use, and different factorisations impose different assumptions.
TRY IT YOURSELF
A quick check.
A quick check.
If the discarded singular values are 2 and 1, what is the Frobenius reconstruction error?
Check your reasoning
Check your reasoning
√(2² + 1²) = √5, approximately 2.236.
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