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1 2 3 4 5 6 7 8 9 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 1 rank k of the approximation ‖A − Aₖ‖ measured, 2-norm σₖ₊₁, from theory measured, Frobenius the first two agree to 4.3·10⁻⁹ worst |‖A−Aₖ‖₂ − σₖ₊₁| / σₖ₊₁ 4.3·10⁻⁹ worst Frobenius discrepancy 4.3·10⁻⁹ κ = 10⁹; 30 random rank-3 matrices, none closer the error is σₖ₊₁
Svd
2
The best approximation there is
1 rung · spectra
1 2 3 4 5 6 7 10¹ 10² 10³ 10⁴ 10⁵ number of indices numbers entries: 6^d stored: 4n(d − 1) exponential against linear entries at d = 6 4.7·10⁴ numbers stored 120 ratio 389 slope against d 24 ‖T − T_tt‖ ⁄ ‖T‖ 1.4·10⁻¹⁵ one line is n^d the other is a constant per index
Tensor train
1
The format that does not notice the dimension
1 rung · tensor
0 0.25 0.5 0.75 1 10⁻¹ 1 10¹ share of the noise placed in the matrix least-squares error ÷ total least-squares error equally accurate total least squares ahead ordinary least squares ahead the model, not the method advantage, all noise in b 0.28 advantage, all noise in A 2.5 seeds at each share 40 the same total noise at every point and only where it sits changes
Total least squares
1
When the matrix is wrong too
1 rung · leastsquares
10⁻¹ 1 10¹ 10² 10³ 10⁻¹¹ 10⁻¹⁰ 10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ frequency ω |H(iω)| one function, two routes states 24 relative, ω ≤ 100 10⁻¹⁴ against the terms 3.9·10⁻¹⁵ ‖H‖∞ 0.09 24 states one function of one variable
Transfer function
1
A model that is a rational function
1 rung · reduction
6 × 6 × 6, rank 3 · 9 ≥ 8 1.0000 2 × 2 × 2, rank 3 · 6 < 8 0.0207 6 × 6 matrix, rank 3 · no condition 0.1089 worst agreement between two runs about the factors, 0 to 1 every run fits to 3·10⁻¹³ every run fits to 1.2·10⁻¹¹ every run factorises to 2.4·10⁻¹⁵ and none agrees with another the only thing that improves tensor, Kruskal holds 1 tensor, Kruskal fails 0.021 matrix 0.11 worst residual 1.2·10⁻¹¹ three successful fits one recoverable answer
Uniqueness
1
A factorisation that is unique for once
1 rung · tensor
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