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

Naoya Yamauchi¹Hidekata Hontani¹Tatsuya Yokota^1,2*

• ¹Nagoya Institute of Technology, Nagoya, Japan
• ²RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

4 There were errors in the formula in the paper. We had originally derived the correct equations (shown in 5 Section of Derivation of Corrections) and had implemented the computer program correctly. However, 6 mistakes occurred during the paper writing stage, and we had not noticed it until now.We report that the error was caused by a simple typo. We confirmed that only the mathematical formulas 8 (58), ( 59), ( 73) and (74) in the paper were incorrect, and that they were correctly implemented in 9 the computer programs used for the experiments. Therefore, this revision does not change any of the 10 experimental results or the conclusions of the original paper.In the published article, there was an error. Some formulas have typos.Corrections have been made to Equations ( 58), ( 59), ( 73) and (74) in Section 4. These previously stated:14 βm = (Z ⊤ m Ω t Z m ) -1 Z ⊤ m (κ ⊘ ω t ),(58)β(ϵ) m = (Z ⊤ m Ω t Z m + ϵI) -1 Z ⊤ m (κ ⊘ ω t ), (59) βm = (Z ⊤ m Ω t Z m ) -1 Z ⊤ m (κ ⊘ ω t ),(73)β(ϵ) m = (Z ⊤ m Ω t Z m + ϵI) -1 Z ⊤ m (κ ⊘ ω). (74) βm = (Z ⊤ m Ω t Z m ) -1 Z ⊤ m κ,(58)β(ϵ) m = (Z ⊤ m Ω t Z m + ϵI) -1 Z ⊤ m κ, (59) 20 βm = (Z ⊤ m Ω t Z m ) -1 Z ⊤ m κ, (73) 21 β(ϵ) m = (Z ⊤ m Ω t Z m + ϵI) -1 Z ⊤ m κ. (74)The authors apologize for these errors and state that these do not change the scientific conclusions of the 22 article in any way. The original article has been updated.Here we only show the derivation of equation ( 58). The remaining three corrections can be derived in a 25 similar manner.First, equation ( 58) is the weighted least squares solution forg(A m , v|θ (t) ) = ∥Ω 1 2 t (κ ⊘ ω t ) -Ω 1 2 t (L (m) ⊙ Φ (m) ⊙ R (m) ) ⊤ vec(A m ) + vΩ 1 2 t 1∥ 2 2 ,(57)where we put Ω t = diag(ω (t) 1 , ω(t) 2 , ..., ω(t) N ) ∈ R N ×N , ω t = [ω (t) 1 , ω(t) 2 , ..., ω(t) N ] ⊤ ∈ R N , κ = [κ 1 , κ 2 , ..., κ N ] ⊤ ∈ R N ,Z m = [(L (m) ⊙ Φ (m) ⊙ R (m) ) ⊤ , 1], the cost function is rewritten as 30 ∥Ω 1 2 t (κ ⊘ ω t ) -Ω 1 2 t (L (m) ⊙ Φ (m) ⊙ R (m) ) ⊤ vec(A m ) + vΩ 1 2 t 1∥ 2 2 = ∥Ω 1 2 t (κ ⊘ ω t ) -Ω 1 2 t Z m β m ∥ 2 2 = ∥Ω 1 2 t (Z m β m -κ ⊘ ω t )∥ 2 2 = ∥Ω 1 2 t (Z m β m -Ω -1 t κ)∥ 2 2 = (Z m β m -Ω -1 t κ) ⊤ Ω t (Z m β m -Ω -1 t κ) = β ⊤ m Z ⊤ m Ω t Z m β m -2β ⊤ m Z ⊤ m κ + κ ⊤ Ω -1 t κ.Note that κ ⊘ ω t = Ω -1 t κ. Optimality condition is given by ∂g ∂β m βm= βm = 2Z ⊤ m Ω t Z m βm -2Z ⊤ m κ = 0.(1)