Table 1.

Comparison of storage costs for approximations of an $m\times p\times n$ tensor $A$

	Basis storage		Coefficients storage	Total implicit storage
${\mathbf{U}}_k$	$kmn$	${\mathbf{C}}_k={\mathbf{S}}_k{\mathbf{V}}_k^{\top }$	$kp$	${\mathbf{A}}_k$	$k\left(mn+p\right)$
$U_{\kappa }$	$\kappa mn$	$C_{\kappa }=S_{\kappa }{\star }_{\mathrm{\text{M}}}{V}_{\kappa }^{\top }$	$\kappa pn$	$A_k$	$\kappa \left(mn+pn\right)$ + st[$\mathbf{M}$]
${\hat{U}}_{\mathit{\rho }}$	$rm$	${\hat{C}}_{\mathit{\rho }}={\hat{S}}_{\mathit{\rho }}△{\hat{V}}_{\mathit{\rho }}^{\top }$	$rp$	$A_{\mathit{\rho }}$	$r\left(m+p\right)$ + st[$\mathbf{M}$]