Towards a Geometric Understanding of Tensor Learning via the t-Product

We sincerely thank the reviewer for the thoughtful and constructive comments, and for recognizing several key strengths of our work, including the generality across discrete and continuous settings, the development of t-manifold theory for learning, and the modeling value of the Bidirectional Tensor Representation (BTR). We respond to each concern in detail below.

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Weakness 1: Critical assumptions such as transform-domain smoothness and the existence of a unit element

We appreciate this important observation. As our work represents the first attempt to develop a differential-geometric framework grounded in the algebraic structure of the t-product, we have adopted assumptions that enable conceptual clarity, mathematical consistency, and compatibility with existing t-SVD-based models.

• Transform-domain smoothness is imposed on frequency components, rather than on raw tensors. This aligns with the slice-wise nature of t-SVD, and reflects well-established spectral regularities in real-world data such as images, videos, and signals. This assumption facilitates the definition of local charts and geometric operations consistent with transform-based algebra. We further elaborate on this point in Remarks 2 and 4 of the appendix.
• Existence of a unit element is standard in discrete settings (e.g., DFT/DCT), where the unit corresponds to the inverse transform of an all-ones vector. In continuous settings, the unit becomes a distribution (e.g., Dirac delta), which falls outside smooth function spaces but can be addressed via standard extensions—such as working in distributional rings or applying unitization techniques from C*-algebra. These extensions are discussed in Footnote 2.

We fully acknowledge that these assumptions may not hold universally. Nonetheless, they allow us to define smoothness, curvature, and differential operators in a tractable and algebraically meaningful way. Future work will aim to relax these assumptions, e.g., by exploring weak derivatives, piecewise smooth transforms, or multiscale settings.

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Weakness 2: Denoising experiments structurally align with BTR—what about generalization?

We agree that the denoising task naturally benefits from the bidirectional structure of BTR. However, our intention is not to present BTR as a universal model, but rather to use it as an illustrative instantiation of how ideas from t-product geometry—particularly dual t-module coupling—can inform the design of structured tensor models.

To demonstrate generality beyond the denoising scenario, we also present in Appendix F.1:

• Image clustering tasks using the same BTR formulation, showing its utility in unsupervised settings;
• Poisson tensor completion, a challenging inverse problem involving sparse, signal-dependent noise. This task is not structurally aligned with denoising, and highlights the adaptability of the framework to non-Gaussian data.

These examples cover a range of regimes including unsupervised learning, inverse recovery, and robustness under complex noise distributions. While our experiments are scoped to controlled settings, they collectively show that the proposed geometric perspective is not limited to a narrow application domain.

Moreover, as emphasized in our conclusion, this work primarily aims to establish a theoretical foundation for t-product-based geometry. Practical extensions—such as more advanced models, scalable solvers, and applications in large-scale systems—are part of a broader research trajectory, several directions of which are outlined in Appendix F.2.

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Weakness 3: The paper does not experimentally explore assumption violations or failure modes

We fully agree that understanding the limitations of the framework under assumption violations is important. While the scope of this paper focuses on theoretical development, we have taken a first step by conducting synthetic t-manifold fitting experiments to empirically validate convergence under controlled noise conditions.

In these experiments, we generate noisy samples from ground-truth t-manifolds (t-circle and t-torus), and apply our proposed fitting procedure. As shown below, both average and maximum pointwise recovery errors decrease consistently with increasing sample size, thereby supporting our theory:

While these toy-scale studies do not exhaustively explore failure scenarios, they provide encouraging evidence that the estimation procedure is robust under moderate perturbations. Extending these results to stress-test the framework under sharp discontinuities, corrupted transforms, or ill-conditioned structures is an exciting direction for future work.

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Question 1: What types of problems could break the assumptions?

The framework may become inadequate under the following conditions:

• Highly irregular or non-smooth signals, where frequency-domain regularity does not hold;
• Use of non-orthogonal, non-invertible, or data-driven transforms, which may not preserve t-product algebra;
• Severe data degradation, such as extreme sparsity, heteroscedasticity, or unstructured noise, which challenges the stability of spectral low-rank models.

Robust extensions may require multiscale analysis, adaptive basis learning, or regularization techniques beyond low-rank priors.

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Question 2: What are the scalability limits and bottlenecks?

The primary bottleneck lies in the use of t-SVD and tensor nuclear norm computations, which require multiple SVDs over transformed tensor slices and scale cubically in spatial dimensions. To enhance scalability, several directions are promising:

• Approximate or truncated t-SVD using randomized or low-rank approximations;
• Sparse or structured transform representations that reduce per-slice complexity;
• Neural surrogates or geometry-aware models that emulate t-product operations without full decomposition.

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Question 3: What are other applications where this framework may be useful?

We appreciate the reviewer’s interest in the broader applicability of the proposed framework. As summarized in Appendix F.2, the t-product geometry introduced in this work provides a general modeling foundation that may inform multiple lines of research involving structured tensor data. While the present paper focuses on the theoretical formulation, we outline below several areas where the proposed tools could be potentially useful:

• Representation learning: The use of t-modules enables spectral-aware learning without flattening, which may benefit structured data modalities such as hyperspectral imaging, EEG, or multi-view video.
• Low-rank regularization: Bi-module structures support regularization along multiple modes, which may offer improved robustness to direction-dependent noise or missingness, as demonstrated in our Poisson tensor completion experiments.
• Optimization over structured manifolds: The framework introduces transform-consistent gradients and charts, which may be incorporated into algorithms for constrained tensor optimization, analogous in spirit to Riemannian methods.
• Graph-based tensor learning: The t-Laplacian supports frequency-adaptive graph construction over tensor data, which may improve local structure modeling in spatiotemporal or multimodal tasks.
• Generative modeling: By leveraging low-dimensional t-manifolds, the framework may provide a means of encoding geometric priors in generative models, potentially improving coherence in the transform domain.
• Spectral manifold learning: Transform-slice geometry allows defining neighborhoods and distances directly in the frequency domain, offering an alternative perspective to traditional Euclidean manifold learning for tensor data.

While these directions remain exploratory at this stage, they follow naturally from the geometric and algebraic constructions introduced in this work. We hope that the framework may serve as a foundation for future advances in both theoretical modeling and practical algorithm design.

We sincerely thank the reviewer for the thoughtful and constructive feedback. We appreciate the recognition of our theoretical development and your helpful suggestions for clarifying motivation, assumptions, and empirical interpretation. Below, we address each of the raised concerns in detail.

Weakness 1: Motivation is ethereal; practical implications unclear; no theoretical verification of impact on model performance

We appreciate the reviewer’s comment and fully agree that grounding the motivation in concrete modeling considerations is essential. Our work is driven by the increasing use of transform-based tensor models—particularly those based on the t-SVD—in applications such as video modeling, multi-modal learning, and graph analysis. These models operate in the transform domain, yet lack a unified geometric framework that:

• Formalizes the algebraic role of t-scalars (e.g., tubes or time-domain functions) in structured representations;
• Bridges discrete (e.g., frame sequences) and continuous (e.g., temporal signals) tensor data under a common formulation;
• Enables transform-consistent differential tools (metrics, gradients, Laplacians) for principled analysis and regularization.

To address these gaps, we introduce the framework of t-product geometry, which offers a principled foundation for modeling and learning in tensor spaces governed by transform-based algebra. By grounding geometric reasoning in the structure of the t-product, the framework supports consistent reasoning across spectral modes and provides a theoretical basis for learning over such spaces.

While this paper focuses on establishing the conceptual and theoretical underpinnings, we also explore initial modeling implications through the Bidirectional Tensor Representation (BTR), which draws directly on the t-product structure. In particular, BTR couples orthogonal t-modules to encode bidirectional regularities, departing from traditional flat models. This design reflects the potential of geometric constraints to guide modeling, and BTR achieves competitive results in image clustering and video denoising. We emphasize, however, that these examples serve to illustrate how the theory may inform future model design, rather than exhaustively validate the framework.

Furthermore, to provide preliminary empirical support for the theoretical results beyond conceptual modeling, we conducted controlled synthetic experiments for t-manifold fitting. Specifically, we sampled data from frequency-domain-defined ground-truth structures (t-circle and t-torus), added Gaussian noise, and applied our proposed fitting procedure. The recovery error consistently decreased with sample size, aligning with the convergence guarantees established in our theoretical results:

These results provide preliminary empirical evidence for the viability of our t-manifold estimation procedure and its consistency with the theoretical learning framework.

Weakness 2: Assumptions may not always hold, limiting practical support

We appreciate the reviewer’s thoughtful observation. As this work represents the first attempt to formalize a differential geometric framework over the t-product algebra, we adopt several simplifying assumptions aimed at ensuring conceptual clarity, mathematical consistency, and alignment with existing practices in transform-based tensor modeling.

• Transform-domain smoothness is imposed on the frequency components of tensor data rather than on raw entries. This mirrors the slice-wise structure of t-SVD and reflects the spectral regularity widely observed in image, video, and signal processing tasks. It enables geometry to be defined in a transform-consistent manner. Further discussion is provided in Appendix Remarks 2 and 4.
• Existence of a unit element is guaranteed under common discrete transforms (e.g., DFT, DCT), where the unit corresponds to the inverse image of an all-ones vector. In the continuous case (e.g., Fourier transform), the unit becomes the Dirac delta distribution, which lies outside the smooth function space. As noted in Footnote 2, this can be addressed by extending the scalar ring to include distributions or by adjoining a formal identity via standard techniques from C*-algebra.
• Algebraic interpretation of t-scalars as abstract units (rather than physically observed signals) allows the construction of coherent tensor modules. This perspective supports the extension of differential operators in a consistent way, particularly under the sheaf-theoretic framework employed in our geometric formulation.

We acknowledge that these assumptions may not hold in every practical setting. However, they allow for a unified, algebraically grounded definition of smoothness, locality, and geometry that is compatible with the t-product. They are consistent with the structures underlying many successful t-SVD-based models. Relaxing these assumptions—for instance, to allow piecewise smooth transforms, multiscale dictionaries, or weakly-defined differential operators—is an important and promising direction for future research.

Regarding the concern on “limiting practical support,” we emphasize that this work is theory-driven: its primary contribution lies in establishing a principled geometric foundation for tensor learning with the t-product. Rather than aiming to exhaustively address diverse application scenarios, our objective is to develop a tractable and expressive formalism that can serve as a basis for future extensions. As illustrated in the BTR case study, even partial realizations of the proposed geometry can already inform effective modeling strategies. We believe that a rigorous theoretical foundation is a necessary step toward the systematic advancement of practical tensor geometric methods.

Weakness 3: Experimental validation is relatively flimsy

We appreciate this concern and would like to clarify the intended role of the empirical section. This work is theory-centered, aiming to introduce a geometric foundation for transform-based tensor learning. Accordingly, our experiments are not designed as exhaustive performance evaluations, but as targeted illustrations of how the proposed framework can inform model design in diverse settings. We highlight three concrete directions:

• Image clustering and video denoising via BTR: The Bidirectional Tensor Representation (BTR) model draws on the structural insights of t-product geometry. By coupling two orthogonal t-modules, it encodes directional low-rank constraints that depart from flat tensor models. This provides a concrete instantiation of the idea that curvature can be introduced through algebraic interactions in the t-product framework.
• Poisson tensor completion: This task extends BTR to inverse problems under non-Gaussian observation models. The fact that the same geometric regularization principle adapts naturally to photon-limited settings underscores the flexibility of the approach, especially in low-sample regimes that challenge standard Euclidean or matrix-based assumptions.
• t-manifold fitting (new): As noted in our response to Weakness 1, we have additionally conducted small-scale synthetic experiments to assess whether t-manifolds can be reliably recovered from noisy data. These experiments are intentionally limited in scope and not intended as comprehensive validation. However, the observed trends—decreasing recovery error with increasing sample size—are consistent with the convergence behavior predicted by our theory. We view these results as exploratory, serving primarily to illustrate the feasibility of the theoretical framework under controlled settings rather than to establish practical robustness.

Together, these case studies demonstrate the modeling implications of the proposed geometry. While we do not claim exhaustive experimental validation, the presented results show that the framework is not only theoretically well-structured but also practically adaptable to a range of modeling scenarios.

Question 1: Why is unifying discrete and continuous tensor modeling essential, given existing work such as [R1]?

We thank the reviewer for pointing to [R1], which presents a functional factorization approach for continuous tensor data. While [R1] contributes an important tool for signal recovery, our work addresses a different and more general question: how to endow transform-based tensor representations with a coherent geometric structure that enables reasoning across both discrete and continuous domains.

In particular, our framework differs from [R1] in three key ways:

• It defines manifolds of structured tensors by modeling local neighborhoods via t-modules, rather than relying on coordinate-dependent factorizations.
• It develops transform-consistent differential operators, including metrics, gradients, Laplacians, and geodesics, enabling spectral geometric reasoning.
• It supports theoretical learning guarantees for manifold testing, fitting, and function approximation under t-product-based structure.

The resulting framework subsumes both discrete tubes (e.g., image rows, video frames) and continuous signals (e.g., time-varying slices) within a unified algebraic-geometric language. By treating t-scalars as basic geometric entities and generalizing classical smoothness to the transform domain, we introduce a new class of non-Euclidean spaces tailored to tensor representations—beyond what is addressed in [R1].

Dear Reviewer,

Thank you for your careful evaluation and for considering our responses. We appreciate your engagement with our work and your feedback will be valuable as we refine and further develop this research.

Best regards,

The Authors

We sincerely thank the reviewer for the thoughtful and constructive review. We are grateful for the recognition of the paper’s theoretical rigor, its structured presentation, and its relevance to emerging t-product tensor models in signal and vision applications. Below, we clarify the scope, contributions, and experimental design in response to the reviewer’s concerns.

Summary of Contribution and Scope

• This work establishes a general framework for transform-consistent tensor geometry based on the t-product algebra, aiming to unify discrete and continuous tensor modeling under a common algebraic structure for tensor learning.
• Our primary contribution is the development of an algebraic-geometric foundation—drawing from sheaf-theoretic tools—that supports differential-geometric notions aligned with the algebraic properties of the t-product. This framework enables the formulation of manifold testing and fitting, learning guarantees, and geometric regularization strategies.
• While the primary focus is theoretical, we also demonstrate practical implications via a bidirectional modeling example (BTR).
• We agree that further integration with large-scale optimization remains a promising direction for future research.

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Weakness 1 & Question 3 (Experimental Validation of Theoretical Results): “The experimental section does not adequately address the proposed hypothesis testing and fitting methods for identifying low-dimensional structures from high-dimensional tensor data, nor does it explore the learning theory approach.” “Could similar experiments be conducted on simpler problems using the analyzed learning theory approach?”

We appreciate the reviewer’s concern and agree that directly validating the theoretical results is important. Our original experiments focused on showcasing the modeling utility of the BTR framework via clustering, denoising, and inverse recovery tasks, but did not directly verify the theoretical results.

To better support the theoretical components—particularly the t-manifold fitting theory—we conducted a new set of controlled synthetic experiments aimed at examining whether t-manifold structure can be recovered from noisy observations. Specifically:

• Setup: We sample data points from known low-dimensional t-manifolds (namely, a t-circle and a t-torus) defined in the transform domain. These constructions serve as illustrative examples where the underlying geometry is analytically known and can be perturbed with controlled Gaussian noise.
• Procedure: Our proposed manifold fitting algorithm is applied to the noisy observations in an unsupervised manner, without access to the ground-truth structure during estimation.
• Metric: We evaluate both the average and maximum pointwise distances between recovered and ground-truth samples, averaged over 10 independent trials.
• Results: As shown below, recovery error consistently decreases with sample size.

The observed monotonic decrease in both average and maximum recovery error with increasing sample size provides empirical evidence that our fitting procedure operates in a manner consistent with the theoretical guarantees, under moderate Gaussian noise.

We note that these experiments use simple synthetic t-manifolds as sanity checks to demonstrate viability under controlled settings. Extending to more complex or real-world manifolds and analyzing robustness to assumption violations is an important direction for future work.

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Weakness 2 & Question 1 (Relation to Prior Riemannian Optimization Approaches): (i) the lack of application-based experiments makes it difficult to assess the practical value of the framework compared to prior work such as [1], and (ii) it remains unclear whether the framework supports the same class of optimization problems and how the optimization potential of our approach compares to [1].

While [1] introduces a Riemannian manifold structure over the set of 3rd-order tensors satisfying $\mathcal{X}^\top \* \mathcal{X} = \mathcal{I}$ (termed the tensor-Stiefel manifold) and develops associated tools such as tangent spaces, Riemannian gradients, and retractions to support conjugate gradient descent (CGD), our work adopts a broader algebraic-geometric perspective rooted in the t-product algebra.

The differences can be summarized as follows:

• Modeling Scope: [1] defines a fixed manifold tailored to orthogonality-constrained tensor models. Our framework instead introduces a family of transform-consistent t-manifolds constructed from arbitrary t-modules, supporting a wider range of structural assumptions and data types (including discrete and continuous settings).

• Mathematical Tools: [1] builds on classical Riemannian geometry, using projections, retractions, and vector transports to enable manifold-constrained optimization. Our approach develops algebraic-geometric constructs—such as t-scalar modules, sheaf-induced differentials, and transform-aligned charts—that are intrinsic to the t-product structure and support learning-theoretic analysis.

• Types of Guarantees: [1] provides optimization-theoretic tools for solving constrained problems like tensor completion. Our framework focuses on statistical learning guarantees, including manifold testing, fitting, and generalization bounds, which address identifiability and sample complexity beyond algorithmic behavior.

• Optimization Potential: Although we do not develop optimization solvers, our geometric components—metrics, gradients, and bi-module regularizers—can be incorporated into Riemannian optimization pipelines.

  • When the orthogonality constraint is imposed and the metric is defined as the sum of frequency component inner products, it aligns closely with the tensor-Stiefel geometry [1].

  • Our framework is more general, supporting constraints beyond orthogonality, such as fixed tubal-rank. In tensor completion, t-manifold optimization can enforce low-rankness without orthogonality, improving flexibility while remaining compatible with [1]'s CGD.

  • Our framework also allows a $\mathbb{K}$-valued metric, enabling finer-grained spectral representation and frequency-selective positive-definiteness, which may improve stability in noisy settings, for example by prioritizing low-frequency components to mitigate noise.

  We will clarify these optimization insights in Appendix F.2 of the revision. While realizing these potentials may require non-trivial algorithmic design and is beyond the current scope of this work, our framework can, in principle, support conjugate-gradient–style methods for tasks such as tensor completion, joint diagonalization, and sparse PCA. Rather than replacing [1], it provides a unified modeling foundation to enhance the flexibility and expressiveness of transform-based optimization frameworks.

To summarize the comparison more concretely:

Preliminary Application Support

Although our current experiments do not involve explicit optimization over t-manifolds or Riemannian tools as in [1], they reflect the modeling implications of our framework. The BTR model, built on dual t-module coupling, derives from the proposed geometry and is evaluated on denoising, image clustering, and Poisson tensor completion—covering unsupervised learning, inverse recovery, and robustness to non-Gaussian noise (Appendix F.1). BTR’s dual t-module coupling suggests potential for regularizing [1]’s model in non-Gaussian settings like Poisson completion. While not optimization benchmarks, these examples show how the framework supports diverse modeling tasks. Geometric components like transform-consistent regularization and directional coupling may serve as inductive priors in future solvers.

Thus, our work complements [1]: while [1] applies Riemannian tools to a fixed manifold, we provide a general geometric foundation for broader tensor models. Though not yet integrated into solvers, our structures may inform future optimization via descent guidance or spectral regularization.

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Question 2 (Clarifying the Relationship Between Theory and Experiments)

We fully agree that this clarification is necessary. In the revision, we will state explicitly that:

• Sec. 3 presents the theoretical framework and statistical guarantees over t-manifolds;
• Sec. 4 demonstrates modeling consequences of t-product geometry, especially via the BTR structure in clustering and denoising;
• The new t-manifold fitting experiment directly validates a core theoretical result and bridges the above aspects.

While unified by t-product geometry, these sections address distinct aims: theoretical analysis versus modeling illustration. We will emphasize this distinction in the revision for clarity.

We sincerely thank the reviewer for the encouraging assessment and thoughtful feedback. We appreciate the recognition of our theoretical contributions (Theorems 1–3), empirical evaluations, and reproducibility efforts.

This work proposes a general geometric framework over the t-product algebra for tensor learning, aiming to unify discrete and continuous tensor modeling under a transform-consistent structure. To the best of our knowledge, it represents the first formal attempt to construct algebraic-geometric tools compatible with the t-product—drawing on sheaf-theoretic ideas from algebraic geometry to define transform-aligned differential-geometric structures, including metrics, gradients, and learning formulations, in a manner consistent with the underlying algebra. We further demonstrate its modeling potential through the bidirectional regularization strategy embodied in BTR.

Below we respond to the reviewer’s specific concerns in detail.

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Weakness 1: The authors do not provide a sufficient description for the optimization procedure used to obtain the minimizer in Eq. (3).

Response:

We appreciate the reviewer’s attention to this point. Eq. (3) plays a theoretical role in our paper: it defines an empirical risk minimization problem over a t-manifold using Tensor Neural Networks (TNNs), serving as the foundation for the generalization bound presented in Theorem 3.

Our primary aim in introducing Eq. (3) is to support theoretical analysis—not to propose a new optimization algorithm. Nonetheless, the training procedure implied by Eq. (3) follows standard practices from the TNN literature. Specifically:

• The function $f_\theta$ can be instantiated as a multi-layer TNN with t-product layers and transformed ReLU activations, as in [20].
• The empirical loss $\mathcal{L}(\theta) = \frac{1}{n} \sum_{i=1}^n \\| f_\theta(x_i) - y_i \\|_{\mathbb{K}_c}^2$ is computed in the transform domain to ensure consistency with the t-product algebra.
• The minimization can be carried out using standard optimizers (e.g., Adam) and autodiff frameworks such as PyTorch or TensorFlow, where the t-product operations are efficiently implemented.

This training paradigm is consistent with prior works on dynamic TNNs and tensor graph models (e.g., Refs. [20, 22]), and no additional implementation challenges arise in our setting. We will clarify this point in the revision to better connect the theoretical formulation with established optimization techniques.

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Weakness 2: The authors mention that the intrinsic dimension $p$ is much lower than the observed dimension $d$, but do not discuss how $p$ is estimated for the image and video datasets in Sec. 4.

Response:

We appreciate the reviewer’s insightful question regarding intrinsic dimensionality. While we mention in the text that the intrinsic dimension $p$ is much lower than the ambient dimension $d$, we do not estimate $p$ explicitly for the image or video datasets in Section 4. This is because the experiments are not intended to compute $p$ directly, but rather to explore the modeling benefits enabled by the t-product geometry—particularly its ability to represent low-dimensional structure through transform-consistent constraints.

Instead, they demonstrate how the proposed t-product geometry can inform data modeling by coupling dual t-modules in orthogonal directions, thereby introducing non-flat geometric structure into the representation. This bidirectional regularization enables the model to capture richer spatial dependencies than traditional one-sided tensor methods, reflecting the modeling flexibility and structural expressiveness offered by the t-SVD algebra—capabilities that are not naturally accessible in conventional vector-based approaches.

In our formulation, this effect is achieved through tensor nuclear norm regularization, which encourages low-dimensional structure without requiring explicit estimation of $p$.

More broadly, the reviewer’s question raises an important and novel subproblem in our framework: t-manifold dimension estimation. Unlike classical manifold learning, where intrinsic dimension estimation is typically addressed using local PCA, correlation dimension, or spectral decay, the t-product setting introduces new challenges due to its transform-consistent structure and tensor-algebraic neighborhoods.

Potential directions for addressing this include:

• Estimating local tubal ranks via patchwise t-SVD;
• Analyzing the effective rank of learned self-representation coefficients;
• Extending spectral or entropy-based estimators to structured tensor domains.

We view this as a technically meaningful extension beyond the scope of this paper, which focuses on establishing the theoretical foundations of t-product geometry. We will clarify this point in the revision and emphasize that dimensionality is not treated as a tunable or estimated parameter in our empirical section. We thank the reviewer for raising this thoughtful and forward-looking question.

Weakness 3: Typos, e.g, line 84: "t-salars"

Response:

We thank the reviewer for pointing out the typo in line 84 (“t-salars”). We will correct it to “t-scalars” in the final version, and will carefully proofread the manuscript to eliminate any remaining typographical errors.