Corrigendum: Expectation-maximization alternating least squares for tensor network logistic regression

A Corrigendum on
Expectation-maximization alternating least squares for tensor network logistic regression

by Yamauchi, N., Hontani, H., and Yokota, T. (2025). Front. Appl. Math. Stat. 11:1593680. doi: 10.3389/fams.2025.1593680

In the original published article, there were typographical errors in mathematical formulas (Equations 58, 59, 73, and 74). The equations were derived and implemented correctly in the computer program; however, mistakes occurred during the writing of the paper. Corrections have been made to Equations 58, 59 in Section 4.2.2 EM-ALS algorithm and Equations 73, 74 in Section 4.3.2 EM-ALS for learning multi-class TN classifiers.

Equations 58, 59, 73, and 74 previously stated:

$\begin{array}{ll}{\widehat{\beta }}_m={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m)}^{-1}{\text{Z}}_m^{\top }(\kappa \oslash {\omega }_t),& (58)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m^{(ϵ)}={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m+ϵ\text{I})}^{-1}{\text{Z}}_m^{\top }(\kappa \oslash {\omega }_t),& (59)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m)}^{-1}{\text{Z}}_m^{\top }(\kappa \oslash {\omega }_t),& (73)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m^{(ϵ)}={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m+ϵ\text{I})}^{-1}{\text{Z}}_m^{\top }(\kappa \oslash \omega ).& (74)\end{array}$

The corrected Equations appear below:

$\begin{array}{ll}{\widehat{\beta }}_m={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m)}^{-1}{\text{Z}}_m^{\top }\kappa ,& (58)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m^{(ϵ)}={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m+ϵ\text{I})}^{-1}{\text{Z}}_m^{\top }\kappa ,& (59)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m)}^{-1}{\text{Z}}_m^{\top }\kappa ,& (73)\end{array}$

$\begin{array}{ll}{\widehat{\beta }}_m^{(ϵ)}={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m+ϵ\text{I})}^{-1}{\text{Z}}_m^{\top }\kappa .& (74)\end{array}$

1 Derivation of corrections

Here we only show the derivation of Equation (58). The remaining three corrections can be derived in a similar manner.

First, Equation 58 is the weighted least squares solution for

$\begin{array}{l}g(A_m,v|{\theta }^{(t)})=\left|\right|{\Omega }_t^{\frac{1}{2}}(\kappa \oslash {\omega }_t)-{\Omega }_t^{\frac{1}{2}}{({\text{L}}^{(m)}\odot {\Phi }^{(m)}\odot {\text{R}}^{(m)})}^{\top }\text{vec}(A_m)\\ +v{\Omega }_t^{\frac{1}{2}}1|{|}_2^2,(57)& (57)\end{array}$

where we put ${\Omega }_t=\text{diag}({\widehat{\omega }}_1^{(t)},{\widehat{\omega }}_2^{(t)},\mathrm{...},{\widehat{\omega }}_N^{(t)})\in {ℝ}^{N\times N}$, ${\omega }_t={[{\widehat{\omega }}_1^{(t)},{\widehat{\omega }}_2^{(t)},\mathrm{...},{\widehat{\omega }}_N^{(t)}]}^{\top }\in {ℝ}^{N\times N}$, $\kappa ={\left[{\kappa }_1,{\kappa }_2,\mathrm{...},{\kappa }_N\right]}^{\top }\in {ℝ}^N$, and ⊘ stands for entry-wise division. Let us put ${\beta }_m={\left[\mathrm{\text{vec}}{(A_m)}^{\top },v\right]}^{\top }$ and ${\text{Z}}_m=\left[{({\text{L}}^{(m)}\odot {\Phi }^{(m)}\odot {\text{R}}^{(m)})}^{\top },1\right]$, the cost function is rewritten as

$\begin{array}{l}\left|\right|{\Omega }_t^{\frac{1}{2}}(\kappa \oslash {\omega }_t)-{\Omega }_t^{\frac{1}{2}}{({\text{L}}^{(m)}\odot {\Phi }^{(m)}\odot {\text{R}}^{(m)})}^{\top }\text{vec}(A_m)+v{\Omega }_t^{\frac{1}{2}}1|{|}_2^2\\ =\left|\right|{\Omega }_t^{\frac{1}{2}}(\kappa \oslash {\omega }_t)-{\Omega }_t^{\frac{1}{2}}{\text{Z}}_m{\beta }_m|{|}_2^2\\ =\left|\right|{\Omega }_t^{\frac{1}{2}}({\text{Z}}_m{\beta }_m-\kappa \oslash {\omega }_t)|{|}_2^2\\ =\left|\right|{\Omega }_t^{\frac{1}{2}}({\text{Z}}_m{\beta }_m-{\Omega }_t^{-1}\kappa )|{|}_2^2\\ ={({\text{Z}}_m{\beta }_m-{\Omega }_t^{-1}\kappa )}^{\top }{\Omega }_t({\text{Z}}_m{\beta }_m-{\Omega }_t^{-1}\kappa )\\ ={\beta }_m^{\top }{\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m{\beta }_m-2{\beta }_m^{\top }{\text{Z}}_m^{\top }\kappa +{\kappa }^{\top }{\Omega }_t^{-1}\kappa .\end{array}$

Note that $\kappa \oslash {\omega }_t={\Omega }_t^{-1}\kappa$. Optimality condition is given by

$\begin{array}{ll}\frac{\partial g}{\partial {\beta }_m}{|}_{{\beta }_m=\widehat{{\beta }_m}}=2{\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m{\widehat{\beta }}_m-2{\text{Z}}_m^{\top }\kappa =0.& (1)\end{array}$

Finally, the minimizer is given in closed form as

$\begin{array}{ll}{\widehat{\beta }}_m={({\text{Z}}_m^{\top }{\Omega }_t{\text{Z}}_m)}^{-1}{\text{Z}}_m^{\top }\kappa .& (58)\end{array}$

The original article has been updated.

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