Front. Appl. Math. Stat. Frontiers in Applied Mathematics and Statistics Front. Appl. Math. Stat. 2297-4687 Frontiers Media S.A. 10.3389/fams.2025.1629658 Applied Mathematics and Statistics Correction Corrigendum: Expectation-maximization alternating least squares for tensor network logistic regression Yamauchi Naoya 1 Hontani Hidekata 1 Yokota Tatsuya 1 2 * 1Department of Computer Science, Nagoya Institute of Technology, Aichi, Japan 2RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

Edited and reviewed by: Yannan Chen, South China Normal University, China

 *Correspondence: Tatsuya Yokota t.yokota@nitech.ac.jp 01 07 2025 2025 11 1629658 16 05 2025 03 06 2025 Copyright © 2025 Yamauchi, Hontani and Yokota. 2025 Yamauchi, Hontani and Yokota

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 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 expectation-maximization (EM) majorization-minimization (MM) alternating least squares (ALS) tensor networks tensor train logistic regression Pólya-Gamma (PG) augmentation section-at-acceptance Optimization

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:

β^m=(Zm⊤ΩtZm)-1Zm⊤(κ⊘ωt), β^m(ϵ)=(Zm⊤ΩtZm+ϵI)-1Zm⊤(κ⊘ωt), β^m=(Zm⊤ΩtZm)-1Zm⊤(κ⊘ωt), β^m(ϵ)=(Zm⊤ΩtZm+ϵI)-1Zm⊤(κ⊘ω).

The corrected Equations appear below:

β^m=(Zm⊤ΩtZm)-1Zm⊤κ, β^m(ϵ)=(Zm⊤ΩtZm+ϵI)-1Zm⊤κ, β^m=(Zm⊤ΩtZm)-1Zm⊤κ, β^m(ϵ)=(Zm⊤ΩtZm+ϵI)-1Zm⊤κ. 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

g(Am,v|θ(t))=||Ωt12(κ⊘ωt)-Ωt12(L(m)⊙Φ(m)⊙R(m))⊤vec(Am)       +vΩt121||22,(57)

where we put Ωt=diag(ω^1(t),ω^2(t),...,ω^N(t))∈ℝN×N, ωt=[ω^1(t),ω^2(t),...,ω^N(t)]⊤∈ℝN×N, κ=[κ1,κ2,...,κN]⊤∈ℝN, and ⊘ stands for entry-wise division. Let us put βm=[vec(Am)⊤,v]⊤ and Zm=[(L(m)⊙Φ(m)⊙R(m))⊤,1], the cost function is rewritten as

||Ωt12(κ⊘ωt)-Ωt12(L(m)⊙Φ(m)⊙R(m))⊤vec(Am)+vΩt121||22       =||Ωt12(κ⊘ωt)-Ωt12Zmβm||22       =||Ωt12(Zmβm-κ⊘ωt)||22       =||Ωt12(Zmβm-Ωt-1κ)||22       =(Zmβm-Ωt-1κ)⊤Ωt(Zmβm-Ωt-1κ)       =βm⊤Zm⊤ΩtZmβm-2βm⊤Zm⊤κ+κ⊤Ωt-1κ.

Note that κ⊘ωt=Ωt-1κ. Optimality condition is given by

∂g∂βm|βm=βm^=2Zm⊤ΩtZmβ^m-2Zm⊤κ=0.

Finally, the minimizer is given in closed form as

β^m=(Zm⊤ΩtZm)-1Zm⊤κ.

The original article has been updated.

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