09 · Matrix Factorizations
Part IV, Block 5 of Tensors for Machine Learning. Section 01 drew the map of factorizations. This section walks it, on real data, and answers the two questions a practitioner actually has: which one do I reach for on this data, and what does it cost me?
The organizing idea is that all six are constrained optimizations, and the constraint is what gives each factorization its shape. QR minimizes ‖y − Xβ‖ subject to an orthonormal Q. The truncated SVD minimizes ‖A − B‖_F subject to rank(B) ≤ k, and Eckart–Young–Mirsky proves nothing else does better. NMF minimizes the same quantity subject to W, H ≥ 0 — worse on error by construction, and chosen anyway, because the components come out as parts you can name.
This is also where eigendecomposition finally gets named, an hour before section 10 leans on the same machinery.
Study here, then run the real thing: notebook 09 on Colab · the handbook · workshop home
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