Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank

Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.

W1: In this figure, we tested the ability of multiple low-rank tensor norms (including TNN, ETR, LTSpN and LTR) to approximate the true rank of three-dimensional tensors. Specifically, we constructed a series of three-dimensional tensors with a fixed true rank of 3, while varying singular values that progressively increase from 0 to 9. The horizontal axis represents the singular values, while the vertical axis represents the approximation value of the rank function. Such setup allows us to evaluate how well each method approximates the true rank under different singular value distributions.

We will revise the figure to include a detailed caption explaining the experimental setup. We hope these changes address the reviewer’s concerns.

W2: To clarify, our learning process follows a transductive learning paradigm, where the model leverages both labeled and unlabeled data during training but only uses the labels from the training set for supervision. The matrix $\bf S$ is a filtering matrix designed to ensure that the optimization process only utilizes the label information from the training set, while excluding any label information from the test set. Specifically, $\bf S$ is defined as a diagonal matrix:

$\mathbf{S}_{ii}=\begin{cases}1&\text{if the }i\text{-th sample belongs to the training set,}\\\\0&\text{otherwise (samples belongs to the test set).}\end{cases}$

Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.

W1: We agree that comparing the proposed Laplace Tensor Rank (LTR) with the widely used Tensor Nuclear Norm (TNN) is essential to highlight the advantages of our method. In fact, we have already performed a comparison of the LTR function with the TNN function and the results are shown in Fig. 3. According to the figure, LTR provides a tighter approximation to the true rank function compared to TNN, especially for larger singular values. Theoretically, LTR’s nonconvex formulation more aggressively suppresses small (noise-corrupted) singular values while preserving larger (signal-carrying) ones, leading to a more accurate low-rank representation. This property is not fully captured by TNN, which tends to over-penalize larger singular values due to its convex nature.

However, we fully agree with the reviewer that comparing LTR with TNN in the experiments would better highlight its effectiveness. Thus, we compared TMvML with its variant TMvML-TNN, where the LTR was replaced by TNN to capture the low-rank tensor structure, and we report the results for the numerical experiments:

Experimental results demonstrate that TMvML consistently outperforms TMvML-TNN across all datasets. This compelling evidence proves that our proposed LTR is more effective than traditional TNN in modeling the complex high-order correlations in multi-view multi-label learning tasks, particularly in preserving discriminative singular values while suppressing noise-corrupted ones.

W2: We agree that ablation study to validate the rotation operation in the tensor classifier construction would provide concrete evidence of the rotation’s contribution to capturing label consistency and view correlations. We compared TMvML with a variant that removes the rotation operation (denoted as TMvML-NoRot), and the results are summarized below:

The results show that TMvML consistently outperforms TMvML-NoRot across all datasets. This significant performance gap highlights the critical role of rotation in extraction of both cross-view consistent correlations and multi-label semantic relationships simultaneously.

W3: Thank you for pointing this out. We will enlarge the font of the coordinate axes to improve readability and ensure better clarity in the visualization.

Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.

W1(C1): Existing tensor-based methods have primarily been applied to multi-view clustering tasks for mining higher-order feature correlations, while some matrix-based methods employ low-rank constraints to capture label semantic relevance. To the best of our knowledge, our proposed TMvML represents the first attempt to utilize tensor structures for MVML tasks, designed to simultaneously model both multi-view high-order correlations and multi-label co-occurrence patterns. The tensor formulation provides a natural and effective framework for capturing the intrinsic multi-dimensional relationships in MVML data that conventional matrix-based approaches cannot fully characterize.

Although direct comparisons with other tensor methods are unavailable, we validate the effectiveness of our proposed Laplace Tensor Rank (LTR) by comparing TMvML with its variant TMvML-TNN, where we replace LTR with the traditional Tensor Nuclear Norm (TNN) for low-rank tensor structure approximation. Experimental results demonstrate that TMvML consistently outperforms TMvML-TNN across all datasets. This compelling evidence proves that our proposed LTR is more effective than traditional TNN in modeling the complex high-order correlations in multi-view multi-label learning tasks.

W2(C2): The introduction of the additional exponential term e^δ in the modified Laplace function, $f_{\mathrm{LTR}}(x)=1-\exp\left(-\frac{e^\delta x}{\delta}\right)$, provides several key advantages over the original Laplace function, $f_{\mathrm{Laplace}}(x)=1-\exp\left(-\frac{x}{\delta}\right)$.

• The modified function offers enhanced flexibility by allowing dynamic adjustment of the growth rate and magnitude through $e^δ$. When $δ$ is large, $e^δ$ amplifies $x$, making the function grow faster for small singular values. When δ is small, the effect of $e^δ$ is reduced, and the function behaves similarly to the original Laplace function. This adaptability makes the modified function more versatile in handling different data distributions.

• The modified function exhibits faster convergence for large values of $x$ due to the exponential scaling $e^δ$, improving optimization efficiency in tasks involving large-scale data or high-dimensional tensors.

Thank you again for this valuable feedback. Please let us know if there is any additional information we can provide to assist with your evaluation.

W3(Q1): We agree that theoretical guarantees for convergence are critical for ensuring the reliability and robustness of the optimization framework. We have added formal theoretical proofs ensuring convergence to a stationary point and reported the convergence theorem and its detailed proof in our rebuttal to Reviewer miAu.

W1: The convergence of TMvML is guaranteed through the validation presented in Theorem 1, with comprehensive and rigorous proof below.

Theorem 1: Let $\\{\mathcal{P}_k = ({\bf Z}_k^{v}, {\bf E}_k^{v}, {\bf A}_k^{v}, {\bf W}_k^{v}, {\bf B}_k^{v}, \mathcal{C}\_k, \mathcal{G}\_k)\\}\_{k=0}^{\infty}$ be the sequence generated by Algorithm 1, then the sequence $\{\mathcal{P}_k\}$ meets the following two principles:

1). $\\{\mathcal{P}_k\\}$ is bounded;

2). Any accumulation point of $\\{\mathcal{P}_k\\}$ is a KKT point of Algorithm 1.

To prove Theorem 1, we first introduce two lemmas.

Lemma 1 [1]: Let $\mathcal{H}$ be a real Hilbert space with inner product $\langle \cdot, \cdot \rangle$, norm $\\|\cdot\\|$, and dual norm $\\|\cdot\\|^{dual}$. For $y \in \partial \\|x\\|$, we have $\\|y\\|^{dual} = 1$ if $x \neq 0$ and $\\|y\\|^{dual} \leq 1$ if $x = 0$.

Lemma 2 [2]: Let $F: \mathbb{R}^{m \times n} \to \mathbb{R}$ be defined as $F(\mathbf{X}) = f(\sigma(\mathbf{X}))$, where ${\bf X} = {\bf U} \mathrm{Diag}(\sigma({\bf X})) {\bf V}^T$ is the SVD of ${\bf X}$, $r = \min(m,n)$, and $f: \mathbb{R}^r\to\mathbb{R}$ is differentiable and absolutely symmetric at $\sigma(\mathbf{X})$. Then,

$\frac{\partial F(\mathbf{X})}{\partial\mathbf{X}}=\mathbf{U}\mathrm{Diag}(\partial f(\sigma(\mathbf{X}))) \mathbf{V}^T.$

where $\partial f(\sigma(\mathbf{X})) = (\frac{\partial f(\sigma_1(x))}{\partial \mathbf{X}}, \dots, \frac{\partial f(\sigma_r(x))}{\partial\mathbf{X}}).$

Proof of the first part: On the $k+1$ iteration, from the updating rule of $\mathbf{E}_{k+1}$, the first-order optimal condition should be satisfied.

0=\alpha\partial\left\\|{\bf E}_{k+1}^v\right\\|\_{2,1}+\mu_k({\bf E}\_{k+1}^v-({\bf X}^v-{\bf{Z}\_{k+1}}^v{\bf A}^v+{\bf B}\_k^v/\mu\_k))=\alpha\partial\left\\|{\bf E}\_{k+1}^v\right\\|\_{2,1}-{\bf B}\_{k+1}^v,

Thus, we have

\frac{1}{\alpha}[\mathbf{B}\_{k+1}^{v}]\_{i,j}=\partial\left\\|\left[\mathbf{E}\_{k+1}^{v}\right]\_{:,j}\right\\|\_{2},

The $\ell\_{2}$ norm is self-dual, so based on the Lemma 1, we have \left\\|\frac{1}{\alpha}[\mathbf{B}\_{k+1}^{v}]\_{:,j}\right\\|\_{2}\geq1. So the sequence $\\{\mathbf{B}\_{k+1}^{v}\\}$ is bounded.

Next, according to the updating rule of $\mathcal{G}$, the first-order optimality condition holds:

$\partial \\|\mathcal{G}\_{k+1}\\|\_{LTR}=\mathcal{C}\_{k+1}$

Let $\mathcal{U} * \mathcal{S} * \mathcal{V}^{T}$ be the t-SVD of tensor $\mathcal{G}$. Based on Lemma 2 and the definition of LTR, we have:

$\\|\partial\\|\mathcal{G}\_{k+1}\\|\_{\text{LTR}}\\|\_F^2\leq\frac{e^{2\delta}\min(n\_1,n\_2)}{\delta^2 n\_3^2}$

Thus, $\\{\mathcal{C}\_{k+1}\\}$ is bounded.

Based on the iterative method constructed in the algorithm, we can deduce:

$\mathcal{L}\_{k}(\mathbf{Z}\_{k+1}^v, \mathbf{E}\_{k+1}^v, \mathbf{A}\_{k+1}^v, \mathbf{W}\_{k+1}^v, \mathcal{G}\_{k+1}, \mathbf{B}\_k^v, \mathcal{C}\_k) \leq \mathcal{L}\_{k-1} + \frac{\rho\_k + \rho\_{k-1}}{2 \rho\_{k-1}^2} \\| \mathcal{C}\_k - \mathcal{C}\_{k-1} \\|\_F^2 + \frac{\mu\_k + \mu\_{k-1}}{2 \mu\_{k-1}^2} \sum\_v \\| \mathbf{B}\_k^v - \mathbf{B}\_{k-1}^v \\|\_F^2$

Summing both sides results in: $\mathcal{L}\_k$ is bounded, and consequently, all its components are bounded, including $\\| \mathcal{G}\_{k+1}\\|\_{\text{LTR}}$. Then, the boundedness of $\\{\mathcal{G}\_{k+1}, \mathbf{Z}\_{k+1}, \mathbf{A}\_{k+1}\\}$ is easy to prove. Therefore, the sequence $\\{\mathcal{P}\_k\\}$ is bounded.

Proof of the second part: By the Weierstrass-Bolzano theorem [3], there exists at least one accumulation point of the sequence $\\{{\mathcal{P}\_k}\\}_{k=1}^{\infty}$, denoted as $\mathcal{P}\_*$. Then, we have:

$\lim_{k\to\infty}({\bf Z}\_k^v, {\bf E}\_k^v, {\bf A}\_k^v, {\bf B}\_k^v, {\bf W}\_k^v, \mathcal{C}\_k, \mathcal{G}\_k)=({\bf Z}\_*^v, {\bf E}\_*^v, {\bf A}\_*^v, {\bf B}\_*^v, {\bf W}\_*^v, \mathcal{C}\_\*,\mathcal{G}\_\*)$

From the update rules of $\mathbf{B}\_k^v$, $\mathcal{C}\_k$, with $\\{\mathbf{B}\_k^v\\}$, $\\{\mathcal{C}\_k\\}$ bounded and the fact $\lim_{k\to\infty}\mu_{k}=\infty$, we obtain:

$\lim_{k\to\infty}(\mathbf{X}^v-\mathbf{Z}\_{k+1}^v \mathbf{A}\_{k+1}^v - \mathbf{E}\_{k+1}^v) = 0 \Rightarrow \mathbf{X}^v = \mathbf{Z}\_*^v \mathbf{A}\_*^v + \mathbf{E}\_*^v$

$\lim_{k \to \infty} (\mathcal{W}\_{k+1} - \mathcal{G}\_{k+1}) = 0 \Rightarrow \mathcal{W}\_* = \mathcal{G}\_*$

Combining the first-order optimality conditions of $\mathbf{E}\_{k+1}^v$ and $\mathcal{G}\_{k+1}$, we take the limit and obtain:

$\mathbf{B}\_*^v=\alpha \partial\\|\mathbf{E}\_*^v\\|\_{2,1}, \quad \mathcal{C}\_\*=\beta\partial\\|\mathcal{G}\_\*\\|\_{LTR}$

Therefore, the accumulation point $\mathcal{P}\_*$ generated by TMvML satisfies the KKT conditions.

[1] The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv, 2010.

[2] Nonsmooth analysis of singular values. Part I: Theory. Set-Valued Analysis, 2005.

[3] Introduction to real analysis, Wiley New York, 2000.