Four decompositions, four bargains

Method	Main idea	Storage	Best fit
CP	Sum of rank-1 components	R(I + J + K)	Individually interpretable components; uniqueness can matter
Tucker / HOSVD	One low-dimensional subspace per mode, plus a core	R₁R₂R₃ + IR₁ + JR₂ + KR₃	Different modes need different ranks
Tensor Train (TT)	Chain of small cores	≈ O(N · I · r²)	Very high-order tensors
t-SVD	FFT along mode 3, matrix SVDs, inverse FFT	Depends on retained tubal rank	Order-3 tensors with a meaningful third mode

Each keeps a different structure. CP looks for individual components, Tucker allows a different rank per mode, Tensor Train stops a high-order core growing exponentially, and t-SVD preserves third-mode structure through the FFT.