For the Given Tensor: A ∈ R

1×2×3

A(:, :, 1) = 

1 0

, A(:, :, 2) = 

1 −1

, A(:, :, 3) = 

0 1

Compute the HOSVD and find:

• Core tensor S,

• Factor matrices U

(1)

, U

(2), and U

(3)

.

The Higher-Order Singular Value Decomposition (HOSVD) is a generalization of the singular value decomposition to tensors. For a tensor \( A \in \mathbb{R}^{1 \times 2 \times 3} \), HOSVD decomposes \( A \) into:

\[
A = S \times_1 U^{(1)} \times_2 U^{(2)} \times_3 U^{(3)},
\]

where:
- \( S \) is the core tensor,
- \( U^{(1)}, U^{(2)}, U^{(3)} \) are the factor matrices for the respective modes.

Given the tensor \( A \):

\[
A(:, :, 1) = 
\begin{bmatrix}
1 & 0
\end{bmatrix}, 
A(:, :, 2) = 
\begin{bmatrix}
1 & -1
\end{bmatrix}, 
A(:, :, 3) = 
\begin{bmatrix}
0 & 1
\end{bmatrix}.
\]

### Steps to Compute HOSVD:

1. **Unfold the Tensor**: For each mode (1, 2, and 3), the tensor is unfolded into a matrix.
   - Mode-1 unfolding: \( A_{(1)} \in \mathbb{R}^{1 \times 6} \),
   - Mode-2 unfolding: \( A_{(2)} \in \mathbb{R}^{2 \times 3} \),
   - Mode-3 unfolding: \( A_{(3)} \in \mathbb{R}^{3 \times 2} \).

2. **Compute SVD for Each Unfolding**:
   - Perform SVD on \( A_{(1)}, A_{(2)}, A_{(3)} \) to extract \( U^{(1)}, U^{(2)}, U^{(3)} \).

3. **Compute the Core Tensor \( S \)**:
   - The core tensor is computed as:
     \[
     S = A \times_1 U^{(1)T} \times_2 U^{(2)T} \times_3 U^{(3)T}.
     \]

Let us compute this explicitly. numpy as np
from numpy.linalg import svd

# Define the tensor A
A = np.zeros((1, 2, 3))
A[:, :, 0] = [[1, 0]]
A[:, :, 1] = [[1, -1]]
A[:, :, 2] = [[0, 1]]

# Unfold the tensor along each mode
A1 = A.reshape(1, -1)  # Mode-1 unfolding
A2 = A.transpose(1, 0, 2).reshape(2, -1)  # Mode-2 unfolding
A3 = A.transpose(2, 1, 0).reshape(3, -1)  # Mode-3 unfolding

# Perform SVD on each unfolding
U1, S1, Vh1 = svd(A1, full_matrices=True)
U2, S2, Vh2 = svd(A2, full_matrices=True)
U3, S3, Vh3 = svd(A3, full_matrices=True)

# Compute the core tensor S
S_core = np.tensordot(U1.T, A, axes=(1, 0))  # mode-1
S_core = np.tensordot(S_core, U2.T, axes=(1, 0))  # mode-2
S_core = np.tensordot(S_core, U3.T, axes=(1, 0))  # mode-3

# Display results
U1, U2, U3, S_core### HOSVD Results:

1. **Factor Matrices**:
   - \( U^{(1)} \):
     \[
     U^{(1)} = \begin{bmatrix} -1 \end{bmatrix}
     \]
   - \( U^{(2)} \):
     \[
     U^{(2)} = \begin{bmatrix} -0.7071 & 0.7071 \\ 0.7071 & 0.7071 \end{bmatrix}
     \]
   - \( U^{(3)} \):
     \[
     U^{(3)} = \begin{bmatrix} -0.4082 & 0.7071 & -0.5774 \\ -0.8165 & 0 & 0.5774 \\ 0.4082 & 0.7071 & 0.5774 \end{bmatrix}
     \]

2. **Core Tensor \( S \)**:
   The core tensor \( S \) is:
   \[
   S = \begin{bmatrix}
   \begin{array}{ccc}
   \begin{bmatrix} 1.1196 & -0.9856 & 0.8804 \\ 0.6969 & 0.1691 & -0.6969 \end{bmatrix}
   \end{array}
   \end{bmatrix}
   \]

If you'd like a detailed breakdown of any part or further clarifications, let me know! Here are some related questions to consider:

---

1. How is the unfolding of tensors performed in general?
2. Why is HOSVD useful in applications like compression or feature extraction?
3. What is the relationship between the core tensor \( S \) and the original tensor \( A \)?
4. How do factor matrices in HOSVD differ from principal components in PCA?
5. What are the computational challenges in performing HOSVD for higher-dimensional tensors?

**Tip:** When implementing HOSVD, always verify the orthogonality of the factor matrices to ensure accuracy.

For the Given Tensor: A ∈ R 1×2×3 A(:, :, 1) = 1 0 , A(:, :, 2) = 1 −1 , A(:, :, 3) = 0 1 Compute the HOSVD and find: • Core tensor S, • Factor matrices U(1), U(2), and U(3).

Learn how to perform Higher-Order Singular Value Decomposition (HOSVD) on a 1x2x3 tensor. This detailed solution includes the decomposition of the tensor into the core tensor S and the factor matrices U(1), U(2), and U(3). Suitable for advanced undergraduate and graduate students in linear algebra and tensor decomposition.

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