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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
1 11 21 31 41 51 93 111.365 129.731 148.096 166.461 probes taken running estimate of the trace normal ±1 one probe, no error the exact trace 99 ±1 variance, this matrix 0 ±1 variance, rotated 57 normal variance 545 the same spectrum in a general basis costs the ±1 probe its whole advantage
Trace estimation
1
Counting what cannot be looked at
1 rung · randomised
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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