We sincerely thank reviewer for pointing out these important issues

Ans. For W1:

To address concern on statistical significance, we provide standard deviation and mean statistics for GMPQ-TE in Tables 4, 7, and 8. The “Mean ± Std” rows offer evidence of performance stability across diverse models. In terms of model diversity, our experiments include non-convolutional architectures such as Vision Transformers (ViT and Swin), which differ fundamentally from CNNs in design. While current evaluation focuses on vision tasks, we acknowledge limitation and consider extension to non-vision domains, including language and graph-based tasks, as an important future direction. We plan to explore these settings in follow-up work to further validate generality of our approach.

Table 4: Results for image classification on ResNet-18 (PTQ)

Table 7: Results for image classification across various ViT and Swin transformer models (PTQ).

*Table 8: Results for image classification on ResNet-18, ResNet-50 and MobileNet-V2 (QAT). *

Ans. For W2:

Compared to prior sensitivity metrics such as orthogonality (OMPQ) and attribution rank distance (GMPQ), TE offers a unified advantage in both efficiency and transferability. OMPQ relies on inter-layer orthogonality statistics, which are inherently tied to the structural assumptions of convolutional networks and do not generalize well to non-convolutional architectures such as Vision Transformers. GMPQ, on the other hand, measures attribution rank similarity, requiring multiple rounds of gradient-based saliency computations and access to labeled data, making it computationally expensive and difficult to scale. In contrast, our method leverages intrinsic topological complexity of intermediate feature maps, requiring only a single forward pass and no access to gradients or labels. This model-agnostic formulation enables TE to operate efficiently while generalizing across architectures and calibration datasets, as demonstrated by its strong performance on both CNNs and Transformers.

The experiments demonstrate superiority of TE in both efficiency and performance compared to OMPQ and GMPQ. In addition, we provide a theoretical analysis highlighting natural suitability of TE for mixed-precision quantization (see Reviewer pnyk, Ans. For W1).

Ans. For W3:

Table shows that quantized models achieve low inference latency and high throughput across both GPU and edge platforms. On RTX 3090, MobileNet-V2 runs at over 800 FPS with only 1.2 ms latency. Even on Jetson Nano, it maintains real-time performance under a 10W power budget. These results confirm that GMPQ-TE enables efficient and hardware-friendly deployment without sacrificing accuracy.

The applicability of TE is validated on a diverse range of architectures, including lightweight models such as MobileNet-V2, non-convolutional structures like ViT and Swin, and CNNs of varying depth (e.g., ResNet18 vs. ResNet50). This demonstrates that TE is not limited to large or deep convolutional models. Fundamentally, TE is architecture-agnostic: it is derived directly from structural properties of intermediate feature maps, and does not rely on architectural assumptions such as depth, convolutional inductive bias, or model width. This enables it to generalize well even to shallow or narrow models. Nonetheless, we acknowledge value of further exploring TE on ultra-compact or task-specialized architectures.

Regarding the question part, the answer can be found above.

We sincerely thank reviewer for pointing out these important issues

Ans. For W1-2

Thanks, we add experiments compared with EMQ, SDQ, and MetaMix. The experimental results for the object detection task and the new visual model (ViT and its variants) can be found in Appendix B.

MobileNet-V2 (PTQ)

ResNet-18 (PTQ)

MobileNet-V2 (QAT)

ResNet-18 (QAT)

Overall, compared with the above methods, GMPQ-TE achieves comparable or better accuracy in PTQ and QAT scenarios with fewer Params.

Ans. For Q1:

For different models, the relationship between bit width assignment and TE is as follows:

ResNet-18:

Params: 5.3

Bit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [5.9, 5.8, 5.7, 5.6, 5.4, 5.2, 4.9, 4.8, 4.7, 3.9, 3.8, 3.5, 3.3, 3.1, 2.9, 2.8, 2.6, 4.5]

Params: 3.8

Bit Assignment: [8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3]

TE Values: [5.9, 5.2, 5.0, 4.8, 4.6, 3.8, 3.6, 3.5, 3.3, 3.2, 3.1, 2.9, 2.8, 2.6, 2.5, 2.4, 2.3, 2.2]

Params: 3.5

Bit Assignment: [6, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2]

TE Values: [5.1, 3.9, 3.8, 3.6, 3.5, 3.2, 3.1, 3.0, 2.9, 2.8, 2.7, 2.6, 2.5, 2.3, 2.2, 2.1, 2.0, 2.0]

ResNet-50:

Params: 11.2

Bit Assignment: [8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [7.0, 6.8, 6.7, 6.6, 6.5, 6.4, 6.3, 6.2, 6.1, 6.0, 5.9, 5.8, 5.7, 5.6, 5.6, 5.5, 5.4, 5.3, 5.3, 5.2, 5.2, 5.1, 5.1, 5.0, 5.0, 4.9, 4.9, 4.8, 4.8, 4.7, 4.6, 4.4, 4.3, 4.2, 4.2, 4.1, 4.1, 4.0, 4.0, 3.9, 3.9, 3.8, 3.8, 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 6.1]

Params: 9.4

Bit Assignment: [8, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5]

TE Values: [6.8, 6.1, 6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.4, 5.3, 5.2, 5.1, 5.0, 4.9, 4.9, 4.8, 4.8, 4.7, 4.6, 4.6, 4.5, 4.4, 4.3, 4.2, 4.2, 4.1, 4.1, 4.0, 4.0, 3.9, 3.9, 3.8, 3.8, 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 3.4, 3.3, 3.3, 3.2, 3.2, 3.1, 3.1, 3.2, 3.3, 3.0, 3.1]

Params: 8.1

Bit Assignment: [6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [6.8, 6.4, 6.3, 6.2, 6.1, 6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.4, 5.3, 5.2, 5.1, 5.0, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.9, 3.8, 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 3.3, 3.3, 3.2, 3.2, 3.1, 3.1, 3.1, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 6.6]

MobileNet-V2:

Params: 1.4

Bit Assignment: [8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.4, 5.3, 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.8, 3.7, 3.6, 3.5, 3.5, 3.4, 3.3, 3.2, 3.1, 3.0, 2.9, 2.8, 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0, 2.0, 2.0, 1.9, 1.8, 1.8, 1.7, 1.6, 1.6, 1.5, 1.4, 5.2]

Params: 1.3

Bit Assignment: [8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.4, 5.3, 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.8, 3.7, 3.6, 3.5, 3.5, 3.4, 3.3, 3.2, 3.1, 3.0, 2.9, 2.8, 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0, 1.9, 1.8, 1.7, 1.6, 1.6, 1.5, 1.5, 1.5, 1.5, 1.5, 5.0]

Params: 1.1

Bit Assignment: [6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4]

TE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.9, 3.8, 3.7, 3.6, 3.5, 3.4, 3.3, 3.2, 3.1, 3.0, 2.9, 2.8, 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0, 1.9, 1.8, 1.7, 1.6, 1.5, 1.5, 5.4, 1.5, 1.4, 1.5, 1.5, 1.3, 1.5]

Ans. For Q2:

Compared to Hessian-based methods such as HAWQ, our proposed TE-based strategy significantly reduces the search overhead. GMPQ-TE achieves a 2947$\times$ reduction in search time on ResNet-18 and a 5268$\times$ reduction on ResNet-50.

Ans. For Q3:

Thanks. We selected CIFAR-10 as the calibration dataset for TE calculation due to its lightweight size and availability, which enables efficient entropy computation during the search phase. Importantly, the proposed topological entropy is theoretically proven to be both resolution- and label-independent (see Appendix A.2). This guarantees that the relative ordering of layer-wise entropy remains consistent across datasets.

To empirically validate this, we compute TE using CIFAR-10 and apply the resulting quantization policy directly to ImageNet and PASCAL VOC. The bit assignment remains nearly identical across datasets (see Sec. 4.2), and the quantized models exhibit strong generalization in terms of accuracy and compression (Tables 1-2). These results confirm that TE-based quantization policy is robust to calibration dataset selection.

In addition, we demonstrate calibration dataset dependency from an experimental perspective. Specifically, using ResNet-18 as the base model, we compute the topological entropy on both CIFAR-10 and ImageNet as calibration datasets. Based on the measured entropy values, the final quantization policies are derived accordingly. The results are summarized as follows:

CIFAR-10:

Bit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [5.9, 5.8, 5.7, 5.6, 5.4, 5.2, 4.9, 4.8, 4.7, 3.9, 3.8, 3.5, 3.3, 3.1, 2.9, 2.8, 2.6, 4.5]

ImageNet:

Bit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 5]

TE Values: [19.0, 18.7, 18.4, 18.1, 17.8, 16.0, 15.6, 15.2, 14.8, 12.5, 12.3, 12.0, 11.7, 11.5, 11.2, 10.9, 10.6, 13.5]

Although TE values differ numerically when using different calibration datasets (e.g., CIFAR-10 vs. ImageNet), the derived quantization policies remain nearly identical, as reflected in the consistent bit assignments across both cases.

Ans. For Q4

We provide a theoretical justification in Appendix A.2 showing that TE is invariant to input resolution, under the assumption of consistent local class-conditional distributions. Specifically, for any two input resolutions, the patch-level distribution remains stable, and the attention or convolutional operators act locally and shift-equivariantly. This leads to equivalent feature distributions.

In practice, as shown in Ans. For Q3, we observe that the absolute values of TE may scale with resolution due to increased sample support, but the layer-wise ranking of TE values remains consistent. This relative ranking is sufficient for solving the linear programming that determines the quantization bit-width configuration.

Therefore, resolution changes do not compromise the effectiveness, generalizability, or consistency of the quantization policy derived from TE. This property enables GMPQ-TE to produce robust policies even when the calibration dataset differs in resolution from the deployment domain.

Ans. For Q5

No, GMPQ-TE does not apply knowledge distillation from the full-precision model during QAT. Instead, GMPQ-TE solely relies on the topological entropy derived from same-label calibration data to guide the quantization policy. The QAT phase in our framework fine-tunes the quantized model without using soft targets or logits from a full-precision teacher model.

We appreciate your further questions.

Ans for W1-2-(1)

Thank you for your valuable comment and for pointing out the comparison with MataMix and SDQ on MobileNet-V2 under QAT. Overall, GMPQ-TE forms a Pareto dominance relationship with the comparison algorithm. We acknowledge that GMPQ-TE yields a slightly lower Top-1 accuracy (70.42%) and higher BOPs (7.1) compared to SDQ (4.89, 72%). However, in terms of search costs (GPU hours), GMPQ-TE is significantly lower than MataMix (7.3 ＜ 13.4). SDQ does not provide specific search costs. In addition, in terms of Params, GMPQ-TE is also less than MataMix. GMPQ-TE is designed with a distinct objective: to derive a generalizable and hardware-friendly quantization policy that can be transferred across datasets, rather than being specifically optimized for a single dataset.

Ans for W1-2-(1)

Thank you for your thoughtful question. We appreciate the observation and agree that the result appears counter-intuitive. However, this observation is possible. Specifically, this phenomenon may be attributable to the inherent differences in the objectives and optimization dynamics of PTQ and QAT under the GMPQ-TE framework. In our design, the bit-width configuration is obtained prior to training via a linear programming problem based on topological entropy. For the PTQ setting, the bit-width assignment tends to concentrate precision on entropy-sensitive layers under tight complexity constraints, resulting in an efficient allocation that preserves pre-trained representational capacity. In contrast, the QAT configuration is derived under a relaxed constraint, allowing slightly more generous bit-width assignments. However, the ensuing fine-tuning process introduces training noise and quantization-aware perturbations. This can occasionally degrade performance in highly compact architectures, where model capacity is already constrained and even minor representational shifts may impair accuracy.

Ans for Q1

Thanks for your constructive suggestion. Due to the rebuttal policy, we are unable to include additional figures or external links in this response. However, we will incorporate such visualizations in the final version of the paper. Specifically, we plan to annotate the TE plots with the corresponding bit-width assignments for all baseline models, which will help readers better understand how TE guides layer-wise precision configuration.

To further clarify the relationship between TE and bit-width assignment, we summarize two key empirical observations. First, layers with higher TE values are consistently assigned higher bit-widths, reflecting greater quantization sensitivity. Second, within each bit-width group, the TE values vary within a bounded range, indicating that the assignment is based on relative ordering and entropy grouping rather than strict thresholds.

As an example, we report below the TE values and corresponding bit-width assignments for ResNet-18, where the total parameter count is 5.3M. This configuration is generated by solving the linear programming formulation under a predefined hardware constraint.

Bit-width Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6]

TE Values: [5.9, 5.8, 5.7, 5.6, 5.4, 5.2, 4.9, 4.8, 4.7, 3.9, 3.8, 3.5, 3.3, 3.1, 2.9, 2.8, 2.6, 4.5]

We observe that all layers with TE values above 5.4 are assigned 8 bits, and layers with TE in the range [2.6, 3.9] are mostly assigned 4 bits. This demonstrates the monotonic relationship between entropy and precision, while also highlighting the smooth, non-binary nature of the bit-width transition.

We sincerely thank reviewer for pointing out these important issues

Ans. For W1:

TE does not assume strict topological similarity. It leverages local statistical consistency of features. Crucially, if shared semantic structures are heavily occluded, distinguishing object becomes fundamentally difficult under any model.

The objective of MPQ is to allocate bit-widths $( B_i \in \mathbb{N}^+ )$ to each layer $( i \in \{1, \dots, L\} )$ under a resource budget $T$, to minimize total performance degradation:

$\min_{\{B_i\}} \sum_{i=1}^L \Delta \mathcal{L}_i(B_i)$

$\text{s.t.}$ $\sum_{i=1}^L M(B_i) \leq T$

where $( \Delta \mathcal{L}_i(B_i) )$ denotes task loss increase due to quantization of layer $i$, and $M(B_i )$ is corresponding resource cost.

We define topological entropy $H_i$ from birth-time distribution $P_{U^{(i)}}(x)$ of same-label inputs at layer $i$. A high $H_i$ implies structural inconsistency and greater sensitivity to quantization. Quantization introduces perturbations to the weights. For layer $i$, let the quantized weight be:

$\hat{W}_i = W_i + \delta W_i, \quad \| \delta W_i \| \leq \varepsilon$

Assuming a Lipschitz continuous forward operator $f^{(i)}$, the pre- and post-quantization feature maps become:

$U^{(i)} = f^{(i)}(W_i, X), \quad \hat{U}^{(i)} = f^{(i)}(\hat{W}_i, X)$

By Lipschitz continuity, the output deviation is bounded by:

$\| U^{(i)} - \hat{U}^{(i)} \| \leq L_i \cdot \| W_i - \hat{W}_i \| \leq L_i \varepsilon$

This perturbation in feature space leads to structural changes in the persistence diagrams, with bottleneck distance bounded by:

$d_B(D(U^{(i)}), D(\hat{U}^{(i)})) \leq \| U^{(i)} - \hat{U}^{(i)} \|$

Given that the entropy is computed over the birth-time histogram derived from the persistence diagram, we can infer the $\ell_1$ difference in their birth-time distributions:

$\| P_{U^{(i)}} - P_{\hat{U}^{(i)}} \|_1 \leq \alpha_i \cdot \| W_i - \hat{W}_i \| \leq \alpha_i \varepsilon$

Shannon entropy is Lipschitz continuous with respect to $\ell_1$ distance, yielding the following bound:

$| H_i^{(0)} - H_i^{(B)} | \leq C_i \cdot \| P_{U^{(i)}} - P_{\hat{U}^{(i)}} \|_1 \leq C_i \alpha_i \varepsilon$

We now relate entropy variation to task loss degradation. The increase in loss can be estimated by a second-order Taylor expansion:

$\Delta \mathcal{L}_i \leq \frac{1}{2} \lambda_i \| \delta W_i \|^2 + \frac{1}{6} \kappa_i \| \delta W_i \|^3$

We further observe that the $L_1$ distance between the birth-time distributions before and after quantization can be upper bounded in terms of topological entropy. Specifically, since Shannon entropy is Lipschitz-continuous with respect to its input distribution, we have:

$\Phi\left( \| P_{U^{(i)}} - P_{\hat{U}^{(i)}} \|_1 \right) \leq \Phi\left( | H_i^{(0)} - H_i^{(B)} | \right)$

This confirms that topological entropy is an upper-bound surrogate of quantization-induced performance degradation. Based on this insight, we propose a TE-guided MPQ objective:

$\min_{\{B_i\}} \sum_{i=1}^{L} H_i \cdot B_i \quad \text{s.t.} \quad \sum_{i=1}^{L} M(B_i) \leq T$

To ensure alignment with sensitivity ranking, we enforce:

$H_i > H_j \Rightarrow B_i > B_j$

This ordering remains consistent during training due to entropy drift stability, as shown by:

$|H_i^{(t)} - H_i^{(0)}| \leq \Delta_{q,i} \log M_i \cdot L_i \quad \Rightarrow \quad \operatorname{sign}(H_i - H_j) = \operatorname{sign}(H_i^{(t)} - H_j^{(t)})$

where $\operatorname{sign}(H_i - H_j)$ denotes the relative ordering between layer sensitivities. This ensures that layers identified as more sensitive in the initial phase continue to receive higher precision during optimization.

Ans. For W2:

Proof of label independence for transformer models can be found in Ans. For W1-ii from Reviewer ErFk.

Resolution-Independence of Topological Entropy under Attention (Sketch Proof)

Let input $X \in \mathbb{R}^{H \times W \times C}$ be reshaped as $X \in \mathbb{R}^{N \times C}$ with $N = H \times W$. A attention layer computes:

$Z = \mathrm{softmax}\left( \frac{QK^\top}{\sqrt{d}} \right)V,\quad Q= XW_Q,\quad K = XW_K,\quad V = XW_V$

Each output token is:

$z_i = \sum_{j} \alpha_{ij} v_j,\quad \alpha_{ij} = \frac{\exp(q_i^\top k_j / \sqrt{d})}{\sum_{j'} \exp(q_i^\top k_{j'} / \sqrt{d})}$

Assume two inputs $X^{(1)}$ and $X^{(2)}$ come from different resolutions but the same class. If the local patch statistics are resolution-consistent:

$P(x_S^{(1)}) = P(x_S^{(2)}) \Rightarrow P(z_i^{(1)}) = P(z_i^{(2)}),\quad \forall i$

After reshaping $Z$ into 2D feature maps $U^{(i)}$, we construct graph $G$ from $U^{(i)}$ (per Eq. (1)-(3) in the main paper), apply clique filtration, and compute the Betti birth-time $b(U^{(i)})$. Let $P_b^{(1)}, P_b^{(2)}$ denote the distributions of birth time under different resolutions.

Since attention is a permutation-equivariant global operator that preserves relative structure, we obtain:

$P_b^{(1)} = P_b^{(2)} \Rightarrow H_{\mathrm{TE}}^{(1)} = H_{\mathrm{TE}}^{(2)}$

Thus, TE is invariant to resolution under generic mechanisms. This is a sketch proof and full derivation will be included upon acceptance.

Ans. For W3:

We explain betti curve, and birth time during the process of clique filtration through text. We consider a simple filtration sequence over four nodes labeled 1-4, denoted as (\tau^{(1)}, \tau^{(2)}, \dots, \tau^{(6)}\), where each step incrementally adds edges to build more complex clique structures:

$\tau^{(1)}$: add edge (1, 2)

$\tau^{(2)}$: add edge (3, 4)

$\tau^{(3)}$: continue adding edges, no cycle yet

$\tau^{(4)}$: form a 1-cycle: (1-2-4-3-1)

$\tau^{(5)}$: add edge (2, 3), forming more cliques

$\tau^{(6)}$: complete graph with all edges

To capture the emergence of topological features, we define a birth-time indicator function $\beta(i, v, U_k)$, which equals 1 if the $i$-th feature appears for the first time at filtration step $v$, and 0 otherwise. A typical curve looks like:

$v = [0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6] \quad \Rightarrow \quad \beta = [0,\ 0,\ 0,\ 0,\ 1,\ 0,\ 0]$

This indicates that a 1-dimensional loop is born at $v$ = 4, corresponding to the closure of a cycle. Notably, although edges are added at $v=5$ and $v=6$, they do not give rise to new independent topological features. Instead, they reinforce existing structures or create higher-order cliques, thus not being counted as new births.

Ans. For W4:

We carefully correct errors or unclear notations, as follows:

L162: given multiple instances $\mathcal{X}$ $\in$ $\mathbb{R}^{b \times c\times h\times w}$ ($b$ is the number of instances) with same label in a minibatch.

L166: $U^{(i)} \in$ $\mathbb{R}^{h^{\ast}\times w^{\ast}}$

L204: The notation for birth time $b$ is revised to $b_t$.

Eq. (5): $inf$ (infimum) means the smallest filtration index $v$ at which the Betti number becomes non-zero (i.e., the earliest moment a topological feature appears).

Eq. (6): The variable $m$ denotes the index of input images sampled from the same class. We define a standard discrete $\sigma$-field $\Sigma$ over the finite sample space $\Omega$ to construct a valid probability space $(\Omega, \Sigma, P)$.

L248: The“positively” is revised to “negatively”

Eq. (9): The objective function is revised to maximize.

Ans. For Q1

The feature map $U \in \mathbb{R}^{h \times w}$ is regarded as a scalar field defined over a two-dimensional spatial domain. Each spatial location $(j, k)$ corresponds to a vertex in graph, forming vertex set $V = \{ v_{jk} \}$ with cardinality $|V| = h \cdot w$. The edges in the graph are defined based on spatial adjacency, rather than similarity of feature values. This construction preserves the inherent spatial structure of feature map. As a result, adjacency matrix $A$ is a square matrix of size $(h^{\ast} \cdot w^{\ast}) \times (h^* \cdot w^*)$. The notation $h^{\ast} \times w^{\ast}$ used in Eq.~(1) is intended to describe 2D spatial shape of scalar field, rather than shape of adjacency matrix itself.

Ans. For Q2

We appreciate the reviewer’s concern. To clarify, Eq.~(1) does not define an adjacency matrix in the graph-theoretic sense. Instead, it represents a scalar field $f: \mathbb{Z}^2 \rightarrow \mathbb{R}$, where $U_{jk}$ denotes the activation value at spatial coordinate $(j,k)$. This scalar field is used as a filtration function in persistent homology, consistent with standard topological data analysis practices [1-2].

Separately, we define a graph $G = (V, E)$ over the spatial domain, where each vertex corresponds to a pixel, and edges are defined using spatial adjacency. This graph yields a binary, symmetric adjacency matrix $A \in \mathbb{R}^{(h^{\ast} \cdot w^{\ast}) \times (h^{\ast} \cdot w^{\ast})}$, independent of pixel values. The graph captures spatial topology, which enables the construction of a simplicial complex via clique expansion. Persistent homology is then applied to the evolving structure as nodes are activated via thresholding over the scalar field [3].

Using feature map activations as a scalar field is theoretically justified and practically effective. These activations encode rich semantic and spatial information at each layer, and serve as a natural filtration function that reflects the network’s learned representations. Importantly, they are not used to construct adjacency matrix or define similarity between nodes, but only to control progressive inclusion of vertices during filtration [4].

To summarize, Eq. (1) does not define an adjacency matrix. Rather, it provides a scalar field used in filtration, while the true graph topology is defined separately via spatial adjacency.

Reference

[1] Topological data analysis with applications

[2] On time-series topological data analysis: New data and opportunities

[3] Applications of topological data analysis in oncology

[4] A short survey of topological data analysis in time series and systems analysis

We sincerely thank reviewer for pointing out these important issues.

Ans. For W1

1. Geometric Interpretation of Feature Maps. Each feature map $U^{(i)} \in \mathbb{R}^{h \times w}$ is treated as a 2D grid where activations indicate semantic saliency. We construct an edge-weighted graph $G = (V, E)$ with adjacency $W_{j,k} = U^{(i)}_{j,k}$, enabling topological analysis. This allows us to assess semantic consistency across same-label samples, crucial for quantization robustness.

2. Theoretical Motivation for TE. We propose TE as a quantization sensitivity metric derived from birth time distribution of topological features in filtered complexes. For a given feature map $U^{(i)}$, birth time $b(U^{(i)})$ is defined as:

$b(U^{(i)}) = \inf \{v \mid \beta_1(K^{(v)}) \neq 0 \}$

where $\beta_1(K^{(v)})$ is first Betti number at filtration level $v$. Across a batch of $M$ same-label samples, we define distribution:

$P_{U^{(i)}}(x) = \frac{1}{M} \sum_{m=1}^M \mathbb{I}(b(U^{(i)}_m) = x)$

and compute Shannon entropy:

$H_i = - \sum_x P_{U^{(i)}}(x) \log P_{U^{(i)}}(x)$

Low $H_i$ indicates consistent topological structures across samples (i.e., quantization-insensitive), while high $H_i$ reflects instability. Therefore, TE provides a theoretically grounded, label- and resolution-invariant criterion for identifying quantization-sensitive layers.

3. Theoretical Support for Filtration. Let $U^{(i)} \in \mathbb{R}^{h \times w}$ denote $i$-th output. We interpret $U^{(i)}$ as a scalar function $f: \mathcal{M} \to \mathbb{R}$, where $\mathcal{M} \subset \mathbb{R}^2$ is a discrete sampling of a compact 2D manifold (i.e., image domain). That is,

$f(j,k) = U^{(i)}_{j,k}, \quad \text{with } (j,k) \in \mathcal{M}$

Given this scalar function, sublevel set at threshold $\tau$ is defined as:

$\mathcal{M}_\tau = f^{-1}([-\infty, \tau]) = { (j,k) \in \mathcal{M} \mid f(j,k) \le \tau }$

As $\tau$ increases from $\min f$ to $\max f$, we obtain a nested sequence of subspaces. Each $\mathcal{M}_\tau$ is a topological space whose homological features change over $\tau$.

This process defines a filtration $\mathcal{M}_\tau$, a fundamental object in persistent homology.

To compute homology, we construct simplicial complexes over $\mathcal{M}_\tau$.

In discrete setting, $V_\tau = E_\tau = (x,y) \in V_\tau \times V_\tau \mid x \sim y$. Here $x \sim y$ denotes adjacency. From $G_\tau$, we build a clique complex $K_\tau$, a simplicial complex where each complete subgraph (clique) becomes a simplex. The $k$-th persistent homology over this filtration is defined as family of homology groups:

$H_k(K_{\tau_1}) \to H_k(K_{\tau_2}) \to \cdots \to H_k(K_{\tau_L})$

Persistent features (birth times) describe evolution of $k$-dimensional topological structures [1]. For 2D feature maps, $H_0$ captures components and $H_1$ captures cycles, reflecting semantic structure in $U^{(i)}$. If $f: \mathcal{M} \to \mathbb{R}$ is a Morse function, topology of $\mathcal{M}_\tau$ changes only at critical values. This extends to discrete cases, where extrema in $U^{(i)}$ drive transitions in filtration. This offers a geometric view of how features emerge. We also provide theoretical support for using TE in MPQ (see Reviewer pnyk, Ans. For W1).

Ans. For W1-i

Yes, the weighted adjacency matrix is fundamental principles in graph theory. The feature maps output by convolutional-based architectures naturally satisfy square matrices. In ViT, each layer outputs a tensor $U \in \mathbb{R}^{N \times d}$, where $N$ is number of tokens and $d$ is feature dimension. To enable topological analysis over spatial structure, we first reshape token sequence into a 2D grid if possible. We then project $d$-dimensional token vectors into scalars, e.g., by averaging:

$f_{j,k} = \frac{1}{d} \sum_{c=1}^{d} U_{(j,k), c}, \quad \text{for } (j,k) \in \{1, \dots, h\} \times \{1, \dots, w\}$

If $N$ is not a perfect square, we embed grid into a minimal square domain:

$n = \left\lceil \sqrt{N} \right\rceil$

We define $\tilde{f}$ over an extended domain $\bar{X}$, such that $\tilde{f} \equiv f$ on $X$, and $\tilde{f} \equiv 0$ elsewhere. Since support of $\tilde{f}$ lies entirely within $X$, all persistent topological features are preserved. Here, $X \subset \mathbb{R}^2$ represents set of spatial positions associated with actual tokens, and $\bar{X}$ is extended square domain. The padded matrix $\tilde{f}$ is used to define graphs and filtration in a consistent square topology.

We interpret $\tilde{f}: \bar{X} \rightarrow \mathbb{R}$ as an extended scalar field over a larger domain $\bar{X} \supset X$. For any threshold $\tau \in \mathbb{R}$, define sublevel set:

$\bar{X}_\tau = \{ x \in \bar{X} \mid \tilde{f}(x) \ge \tau \}$

Since $\tilde{f}(x) = 0$ for all $x \in \bar{X} \setminus X$, we observe that for all $\tau > 0$. Hence, all homological features computed in filtration process are unaffected by padded region as long as $\tau > 0$.

Even when $\tau \le 0$, zero-valued region may be included in sublevel set, but it does not introduce nontrivial topological features. Specifically, added region forms a contractible subcomplex $C \subset \bar{X}_\tau$, and:

$H_k(C) = 0, \quad \text{for all } k \ge 1$

Thus, the inclusion map $X_\tau \hookrightarrow \bar{X}_\tau$ induces an isomorphism:

$H_k(X_\tau) \cong H_k(\bar{X}_\tau), \quad \forall k \ge 1$

We conclude that zero-padding the feature map to form a square matrix preserves all persistent topological structures, and introduces no spurious noise to the homology computation.

Ans. For W1-ii

We clarify that this equation is not an assignment operation. Instead, it symbolically denotes a nested sequence of graphs derived from thresholded feature maps, known as a filtration in topological data analysis [2]. Each graph $G^{(t)}$ corresponds to a threshold $\tau_t$, and is constructed based on the subset of feature map locations with activation values greater than or equal to $\tau_t$, together with spatial adjacency-based edges. Equation (3) defines a multiscale filtration process over the feature map, and does not discard information. As described in Section 3.2, the complete filtration retains all meaningful activations for persistent homology. Although higher thresholds may yield graphs with fewer nodes, no information is lost.

Ans. For W1-iii

We interpret the feature matrix not as a conventional adjacency matrix, but as a scalar field over a grid. When applying clique filtration, we do not assume semantic pairwise adjacency directly. Instead, the high-valued entries define regions of activation, and their local connectivity leads to topological features which correspond to co-activated (expressing) patterns. We will clarify this interpretation and add empirical visualizations to illustrate how persistent features align with semantic structures.

Ans. For W2

Let $\tilde{f} : \mathcal{M} \rightarrow \mathbb{R}$ be the scalar field from intermediate activations. Attention mechanisms produce structured outputs over $\mathcal{M} \subset \mathbb{Z}^2$. Define:

$\mathcal{M}_\tau := \{ x \in \mathcal{M} \mid \tilde{f}(x) \ge \tau \}$

From this we build $K_\tau$ and compute birth times $b_i$, defining:

$\mathrm{TE} := -\sum_x P(x) \log P(x), \quad P(x) = \frac{1}{M} \sum_{i=1}^M \delta(b_i - x)$

All steps in TE (projection, padding, filtration, homology, entropy) are label-free. For input $(x,y)$:

$\mathrm{TE}(x, y) = \Phi(\tilde{f}(x)), \quad \frac{\partial \mathrm{TE}}{\partial y} = 0$

Therefore, TE is invariant to label supervision and is a purely geometric, architecture-agnostic descriptor.

Ans. For W3-i

We respectfully disagree that the trends in Figures 5–7 are solely due to feature map size. While larger maps may yield more persistent features, TE is computed from normalized birth time distributions and reflects the diversity of activation patterns, not absolute size.

Ans. For W3-ii

TE is computed from the normalized distribution of birth times, capturing the diversity of topological features rather than node count. This normalization mitigates the scaling effect from large feature maps.

The LP formulation does not rank layers by TE alone. Instead, it optimizes bit-widths under a global budget, balancing sensitivity (via TE) and precision cost. Consequently, layers with higher TE may not receive more bits if their contribution to accuracy is limited.

In practice, some small feature maps still receive higher bit-widths, showing that the LP solution captures structural sensitivity beyond size.

Ans. For W3-iii

Results for average bit-width:

ResNet-18

MobileNet-V2

Ans. For W4:

We provide source code related to the paper in the supplementary materials.

Regarding the question part, the answer can be found above.

References

[1]Persistence images: A stable vector representation of persistent homology

[2]On time-series topological data analysis: New data and opportunities