Efficient and Generalizable Mixed-Precision Quantization via Topological Entropy
Authors: Nan Li
Yonghui Su
Lianbo Ma
Venue: NeurIPS 2025 poster ()
Keywords: Mixed-Precision Quantization; Topological Entropy; Generalizable
Primary area: deep_learning
TL;DR: Topological Entropy for MPQ

Abstract: Network quantization effectively reduces both memory footprints and inference time of deep neural networks, enabling their deployment on resource-constrained devices. To fully utilize the multiple bit-width arithmetic operations of the hardware, mixed-precision quantization (MPQ) is developed to assign different bit-widths to each layer. However, the quantization policy obtained by existing MPQ methods struggles to achieve the objectives of efficiency and generalization simultaneously. In this paper, we propose an efficient and generalizable MPQ based on topological entropy (TE) (GMPQ-TE). Specifically, TE, derived from \textit{topological data analysis}, effectively measures the quantization sensitivity of each layer by using the minibatch of data with the same label. Furthermore, we observe that TE remains consistent across various datasets and shows a strong correlation with both quantized model accuracy and bit-width. Thus, MPQ is formulated as a single-pass linear programming problem, obtaining a generalizable quantization policy in a few seconds (11s on MobileNet-V2). Extensive experiments show that the quantization policy obtained on CIFAR-10 can generalize to ImageNet and PASCAL VOC. GMPQ-TE achieves a competitive accuracy-complexity trade-off compared to state-of-the-art MPQ methods.

Efficient and Generalizable Mixed-Precision Quantization via Topological Entropy
Nan Li
School of Artificial Intelligence, Institute of Big Data Science and Industry Shanxi University College of Software, Northeastern University linan10$sxu, edu. cn
Yonghui Su
Lianbo Ma*
College of Software, Northeastern University 23714000stu. neu. edu.
College of Software, Northeastern University malb@swc. neu. edu.
Abstract
Network quantization effectively reduces both memory footprints and inference time of deep neural networks, enabling their deployment on resource-constraing devices. To fully utilize the multiple bit- wit-width arithmetic operations of the hardware, precision quantization (MPQ) is developed to assign different bit-widths to each layer. However, the quantization policy obtained by existing MPQ methods struggles to achieve the objectives of efficiency and generalization simultaneously. In this paper, we propose an efficient and generalizable MPQ based on topological entropy (TE) (GMPQ-TE). Specifically, TE, derived from topological data anal- ysss, effectively measures the quantization sensitivity of each layer by using the minibatch of data with the same label. Furthermore, consistent across various datasets and shows strong correlation with both quan- tized model accuracy and bit-width. Thus, MPQ is formulated linear programming problem, obtaining a generalizable quantization policy in a few seconds Is MobileNet- V2). Extensive experiments show that the quantization policy obtained on CIFAR-10 can generalize to ImageNet and PASCAL VOC. GMPQ-TE achieves competitive accuracy-complexity trade-off compared state-of-the-art MPQ methods.
observe that TE remains
single-pass
Introduction
Deep neural networks have gained increasing attention in image classification Sandler et al. 2018 He [2016), semantic segmentation Strudel al. [2021] [Li et al. [2022]. object detection Wang et al. 201 19], Zou et al. [2023), and other vision tasks Xu et al. [022], Li et al. (024)) However, due their extremely high complexity, it impractical directly deploy mobile devices with limited battery capacity and computational resources Wang et al. [202I] Therefore, there is need for the model compression method based on the given hardware configurations. Recently, various compression methods have been developed to reduce the model complexity, such pruning He et al.
2017 knowledge distillation Gou et al. (021], quantization Wang al. [202T] Koryakovskiy et al. and compact architecture design Sandler et al. [2018). Jiang et al. [2023]. Among these methods, quantization aims at mapping weight or activation lower bit-width for compression and
2023
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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acceleration Wang et al. 2021]. To fully utilize arithmetic operations with variable bit-width in hardware platforms, mixed preccsson quantization (MPQ) is presented to configure bit-width for each layer, achieving better trade-off between complexity and accuracy Ma et al. 2023
However, conventional MPQ methods
Wang et al. [2019]]. Cai and Vasconce-
ImogeNet
Conventional MPQ
ImaggeNe Not
los [2020] are subject the limitations of achieving both efficiency and generaliza- tion simultaneously. This is because that I): It is commonly an iterative search in which candidate quantization policies in each gen- eration are required to train and evaluate on the given datasets Chen et al. 2019],
Iterative
D
2
Finetune
0
Bit
Search
2
CIFARR-10
GMPQ-TE
ImargeNee
Single-pass
pass
2 Bit
Finetune
Search
Bit.....**
Gemenalze
Same Labee
Sun et al.] [2022a]; 2) Existing approaches generally depend on the given datasets to search for the optimal quantization policy, which cannot be generalized across various datasets due to differences in the distribu- tion of various datasets Cai and Vasconce.
Figure 1: Conventional MPQ methods require the con- sistency of datasets between bit-width search and model deployment, while GMPQ-TE obtains the optimal quan- tization policy by using the minibatch of data with the same label and generalizes it to other datasets.
los [2020). Wang al. 2024]. The quanti-
zation policy needs to be re-searched on the rare or large-scale datasets Deng et al. (2009] posing significant computational and generalization challenges.
Accordingly, we design an efficient and generalizable MPQ via topological entropy (TE) (GMPQ-TE). Different from the ex- isting approaches that require iteration and cannot generalize to various datasets, GMPQ-TE only requires single-pass search process on minibatch of data with the same label to compute the TE and ob- tain the quantization policy that can gen- eralized across various datasets (see Figure D.. Specifically, TE measures the stabil- ity of the representation (i.e., the quanti- zation sensitivity) of a layer or network model concerning common features in a minibatch of data with the same label. Thus, TE possesses superior cross-dataset consistency, which ensures that the obtained quantization policy can be generalized other datasets. Furthermore, we observe that: 1) Model TE negatively correlates with quantization performance across different networks. 2) TE of layer positively correlated with bit-width. Based on this, we minimize the model TE as an objective function and construct a linear programming problem, where a larger bit-width is assigned to the layer with larger TE under specific hardware constraints. Thus, the optimal quantization policy can be obtained by solving the linear programming problem. Our contributions can be listed as follows:
10
20
40
10K
2KK
30 -K
8KK
12K
HMQ
12K Fewar
GMPQ
IK x Fewer
R-GMPQ
OMPQ
Non-gansraizzable
Ours
Generalizable
Figure 2: Comparison of the search cost used to obtain the optimal quantization policy on MobileNet- V2 be- tween GMPQ-TE and other MPQ methods.
We first attempt to use TE for measuring the quantization sensitivity of network model or layer by using a minibatch of data with the same label. The quantization policy solved by the TE can efficiently be generalized across various datasets. Based two positive correlations, a single-pass linear programming is designed for MPQ. Such linear programming is solved in a few seconds.
We provide comprehensive theoretical analysis of the performance degradation boundary and resolution- and -ndeependnn of TE. Furthermore, we theoretically prove that is feasible to integrate TE into quantization-awart training.
We conduct extensive experimental results on image classification and object detection, which show that the quantization policy generalized from CIFAR-I0 to ImageNet and PASCAL VOC achieves competitive accuracy-complextty trade-off compared with the state-of-the-- MPQ methods. Additional real-world deployment results on diverse hard- ware platforms could further verify the advantages of GMPQ-TE.

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2 Background Knowledge
Mixed- Quantization. Prior MPQ methods are a search-based approach limited by expensive quantization policy evaluations and iterative search process Wang et al. (2019a]. Cai and Vasconcelos 2020]. For example, HAQ Wang al. (2019b) performs 600 quantization policy evaluations. Most recently, GMPQ Wang et al. [2021] and its extended version R-GMPQ Wang et al.] [2024) design the generalizable framework via the attribution rank preservation, which focuses on the contribution of each input component to network output. The experimental results prove that it possesses good consistency across various datasets. Thus, the quantization policy searched by GMPQ and R-GMPQ can be generalized across multiple datasets. However, it still requires massive computational overhead (2.8 GPU hours).
To reduce the searching cost, several works develop some "critics" to judge the quantization sensi-
tivity of the layer, such Hessian eigenvalue Dong al. [011)] orthogonality Ma et al. (2023). quantization entropy Sun et al. [2022b), first-order information Chauhan et al. [2023]. and layer-wise
importance Tang al. [2022]. These metrics often require random minibatch of data measure the
sensitivity of the layer. Among them, OMPQ Ma et al. 20023 and LIMPQ Tang et al. [002 define
MPQ as single-pass linear programming problem, which solved in less than a second. Especially OMPQ requires only 64 images from ImageNet to calculate the optimal quantization policy for ResNet-18/50. However, due the significant differences in the distribution and resolution of ran- domly selected data across different datasets, the quantization policy fails to generalize effectively to other datasets. Additionally, this type of linear programming approach, except for OMPQ. operates in the post-training quantization (PTQ) and does not integrate with quantization-awee aware training (QAT).
In contrast, GMPQ-TE addresses both search efficiency and generalization. This is because TE focuses on the ability of a model or layer to preserve common features of data with the same label rather than their ability represent random data, eliminating the effects of the obvious difference in the distribution of random data from different datasets. Also, TE can be integrated into QAT and proven thcoretically(see App. C.I). The theoretical analysis (see App. C.2) demonstrate the resolution- and habel-ndeependnt nature of TE. Specifically, data with different labels from different datasets do not affect the evaluation of quantization sensitivity. Moreover, the single-pass linear programming problem, built upon the property of TE, is solved in under a few seconds.
Topological Data Analysis. It aims to explore relationships between neural network and input data Ghrist 2008) based on the key insight "data has shape" Carlsson 2014). Typically, natural images with the same label share common features and the locations of these features spatially correlated globally. Effective functions highlight these features with high activation values, preserving the global spatial pattern. Conversely, ineffective functions fail represent these features effectively, resulting in chaotic and vague representations. Based on this observation, recent works have attempted to use topological data analysis to explain the effect of network Rieck et al. [2019), Gabrielsson and Carlsson [019]] Guss and Salakhutdinov] (2018) Hofer et al. [2017] or basic operators (c.g.. ReLu Naitzat et al. [2020] function Zhao and Zhang (022 on performance. For example, Rieck al. Gabrielsson and Carlsson [019] design neural persistence based on topological complexity of
network structure, which can provide criteria for early stopping. Guss al. Guss and Salakhutdinoy
[218 empirically provide the correlation between neural network expressiveness and the topological complexity of dataset. Zhao al. Zhhao and Zhang [2022) construct feature entropy to evaluate the effectiveness of functions via topological data analysis. Compared to the above studies, TE based on the observation from work Zhao and Zhang [2022] can finely evaluate functions from the channel level. Furthermore, we pioneer to discover its relationship with quantized model accuracy and confirm its validity in MPQ from the view of TE.
Key Concept. Here, we mainly introduce the concepts from graph theory used in constructing TE.
Clique. It refers to complete subgraph from a graph, where clique is subset of nodes in the graph and every pair of nodes in this subset is mutually connected Carlsson 2014].
Clique Filtration. Given a graph, filtration is constructed by gradually adding cliques (complete subgraphs) during the filtration process. Each clique corresponds to a threshold at which it forms in the graph, and additional cliques are added as the threshold changes. This filtration process helps analyze the evolving topological structures the graph as the threshold varies Guss and Salakhutdinov 20188
3

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Insensitive Layer
Low Topologiral
32 Bit
Entropy
Bit
6 Bit
32 Bit
Constraint
h Sensitive Layer
kct
5 Bit
32 Bit
DGi Topological
Entropy
Bit
Constraint
Feasible
Bit
Reglon
htt pa
MLPp
he
(a) Topological Analysis
(b) I Topological
(C) Linear
Entropy
Programming
Figure 3: The overall framework of the proposed GMPQ-TE method. (a) Analyze a minibatch of data with the same label and obtain Betti time. (b) Calculate TE based on Betti time distribution. (c) Linear programming problem constructed by TE to derive optimal generalizable quantization policy.
Persistent Homology. It is used to capture how the data evolves in terms of its structure at various scales. Specifically, it starts by tracking simple "starting points" in the data and gradually adds components (e.g. cliques), observing how these topological features "form" and "disappear" ZZao and Zhang 2022]. By calculating the homology of the data, persistent homology identifies topological features (such connected components, loops, voids, etc.) and provides insights into their birth and death across different scales. This is especially useful in understanding complex data structures.
Betti Curve. It is significant criterion of persistent homology. It provides visualization of the "persistence" of topological features in data across different dimensions Zhao and Zhang (2022). The Betti curve depicts the evolution of these values (for example, how Betti numbers change through filtration) and helps us understand how topological features evolve across different scales of the data, where Betti numbers describe the number of topological features in each dimension (e.g., the 1-st Betti number is equal to the number of circles the graph).
Birth Time. For each topological feature (e.g., circle), the birth time refers to the time or scale at which the feature first forms. It marks the initiation of the feature's 'ss existence in the data, It crucial for capturing the fundamental structure of the data Bochner [1948]
Methodology
3.1 Topological Analysis
As shown in Figure 3 given the input data with the same label X' E Rexkxw, a neural network with L-layers can be considered as an input-to-output mapping function consisting of functions. The output feature maps of each function are O Re xA which can be unfolded along the channel dimension obtain feature maps (F= (f(1), f(2), f()"))) Each feature map f(i) 18 xw be viewed grid structure (+(+) R
Intuitively, the spatial patterns within the elements reveal specific regular relationships among their coordinate indices. It is natural to capture such relationship using graph structure and tackle through topological analysis.
Graph Construction. We construct the edge-weighter graph G = (V, E) for the grid structure (1)) where V and are the vertex set and the edge set, respectively. The weighted adjacency matrix W of G is defined as W E Rh xw" WW = (1()) location of an element in W and TT()) Note that each element W is the weight of edge in G, which conveys the intensity of the corresponding point in the (1) Based the typical implementation of
j,K where (j, k) denotes the coordinate index of the
We provide construction method across different architectures in App.

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the sublevel set (see Eq. we can construct the undirected subgraphs ( = (V(1), E("),, where v(e) V and g(r) C E. Its weighted adjacency matrix is denoted as w(w).
W I( j.b 2 w())
(1)
where I() is indicator function. w(() is the v-th value in the descend ordering of elements of W W j.k is larger than or equal to w((), then w requires to construct subgraphs based on the weights of the edges. If the weights of the edges are not
j.k equals 1, and conversely, it equals O. Graph filtration
symmetric, it may lead some unreasonable graph structures, such as wrong edge connections or
To ensure that w is symmetric, we make the
unidirectional connections Zhao and Zhang 2022 following adjustment.
W max Wlo), (W(r)))
By utilizing Eq. 2 we obtain the graph filtration G() C ((2) C... G. In this sublevel set filtration (i.e.. 9:), the process commences with the vertex set. The edge weights are then ranked from the maximum Wmax to the minimum Wmii- and the threshold parameters are systematically decreased from Wmax to Wmin- each step, the corresponding edges are incorporated to form the threshold subgraph g(v))
Clique Filtration. To further explore the structural information among the elements in the threshold subgraphs, we use topological invariants to capture high-level abstraction of structural information.
Here, each subgraph ((v) is converted to clique complex K,, based on persistent homology method
Horak et al. 2009]. this way, we can get clique complex filtration (K1 C K22 C C K)) corresponding to graph filtration.
The clique complex filtration describes the evolution of structural information in graph 9 along with the decreasing threshold parameter (v).
k-th Betti number be regarded as the number of k-dimensional circle or hole in complex. Due to the fact that grid structure U is matrix of h² x w", the complex contains higher-order structures (e.g., circle). Thus, each element (K,,) in clique complex filtration quantified by 1-st Betti number 31 (K.,)) based on persistent homology theory Hatcher (2002) which can characterize the number of circle structural information in complex K...
Betti Curve and Birth Time.
K (KC..)
Intuitively, many meaningful patterns in (66 would lead the circle structure in the clique complex filtration. The number of circle is typically used as an important quantitative index for expressing patterns. Hence, the 1-st Betti number (K,)), 1-st Betti curves ((U(),, v) for U1),, Here, we employ birth time GUU)) interpret and extract core characterization of 1-st Betti curves 3(U()), v).
{1.-.. hr} could be arranged into so called
b((())) inf (80 v) + 0)
Birth time refers to the moment when circle structure begins to appear (i.c., (Betti number & 0)) in the clique complex filtration. signifies essential changes in filtration, indicating that regularized spatial patterns of notable components begin emerge within (6), Conversely, when no spatial pattern is detected in the components of (+(), ((U(() v) remains 0, indicating the absence of birth time. This situation typically arises when the function fails to effectively represent the images (i.e.,
the majority the values in (i) are equal to 0). We provide an example about Betti curve and and birth time in App. B
3.2 Topological Entropy
one layer quantization-insesiitti the output feature maps can still perceive common features about set of images with the same label at lower bit-width. This is, the birth time obtained from
5

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each realization of U should be relatively close. In other words, the performance of a quantization- insensitive layer for a certain label should remain stable across all images with the same label. This is the key idea behind assessing the performance of layer.
Birth Time Distribution. Sampling experiments. Thus, birth time is random variable (denoted b(m, ).. We construct the probability space (S2, Σ, P) for b(m, U where the elements in sample space Ω are the unit (+) resulted from the images with same label, Σ could be set as common discrete o-fieeld and probability measure P is uniformly distributed on f2 since each image has equal chance to be selected the input of network model. This probability space satisfies the following probability distribution,
of images with the same label be regarded statistical
# Puro (a)) P(b(m, U(0) x)) 2j=1
I(b(U(()) = x) #(22)
The degree concentration of Prr (3r) gives a direct view of the expressiveness of U for the images with the same label. Specifically, if the distribution presents close to degenerate- style, means that the underlying common features of the images with the same label could be stably perceived by U On the contrary, the distribution presents close uniform-like style when features are perceived almost blindly, indicating that U+) is invalid for the images with the same label. In summary, the degree of concentration of Pire (x) is supposed to be quantification sensitivity indicator of Utt))
Considering function with e* output features, use the weighted entropy to further measure the quantification sensitivity of output features of the function (called TE Ht).
H Σ
Hi
(6)
where
|2
H4 = - D Pows log (r)) 0, = Hi
(7)
Σi=1 Hi
where H4 and w, are the TE and weight corresponding to (+(), respectively. A lower H4 means that (1() exhibits quantization-insesiine Thus, a lower weight 0, should be assigned for a lower H4.
3.3 Linear Programming for MPQ
TE directly indicates the quantification sensitivity of the layers (i.e., functions) in the network without taking into account the different dataset distributions, guiding the configuration of generalizable quantization policy. Generally, sensitive layers should be assigned a larger bit-width to enhance their representational capability Liu al. (2018). Thus, assign larger bit-width to layers with higher TE to maximize representational capability.
We perform extensive experiments that present sufficient and reliable evidence for such an assertion. Specifically, we first sample various quantization policies for ResNet-18 and Faster R- C-CN.. We then perform tieettnnng to optimize model performance. Simultaneously. the overall TE of the sampled models is calculated separately. Interestingly, as shown in Figure we find that model TE and performance are positively correlated with the sum of TE of each layer. Based on this finding, we set minimizing TE as our objective function and formulate a linear programming problem to derive final quantization policy.

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Recket
Rechet 50
550 5 V0B 36
faster CMN & Restier
Accuracy150
ASCUTWOYTM
IAAP/
Figure 4: Relationship between TE and accuracy/mAP for different quantization policies on ResNet- 18, ResNet-50, SDD & VGG-16 and Faster R-CNN & ResNet-18.
Objective
min
H(0, B0
1=1
(8)
Constraint
M(
8 EN. HO > HD) = B B0) ST,
where (B(0) represents the model size of the i-th layer under B(0) bit-width quantization, while T denotes the target model size. B refers to the optimal bit-width configuration, where all ( are integers (i.e., B(1) E N+), larger TE corresponds to a larger bit-width configuration (i.e., H(0 H() = B > B Maximizing the objective function involves assigning a larger bit-width to more unstable layers, thereby implicitly enhancing the representational capacity of the model. Notably, Eq. [ can be solved in only a few seconds on single CPU. Furthermore, we theoretically prove performance degradation bound for GMPQ-TE (see App. C.3).
Remark: The objective of MPQ to allocate bit-widths (B; E N+) each layer (i E (I....,4) LJ) under a resource budget T, to minimize total performance degradation min(B,) AB,)), s.t. E (B,) T, where (L,( denotes the task loss increase due to quantization of layer i, and M(B, is the corresponding resource cost. We define TE H; from the birth-time distribution P Prico (x) of same-label inputs at layer i. high H; implies structural inconsistency and greater sensitivity to quantization. Quantization introduces perturbations to the weights. For layer the quantized weight is given by W. W; + SW, |@W; S E.
Assuming a Lipschitz continuous forward operator f()), the pre- and post-quantization feature maps are (+) (W, ((0) f(0 (W, X). By Lipschitz continuity, the output de- viation is bounded by ||0/(00 O < L. IW, - Will < L.s. This perturbation in feature space leads to structural changes the persistence diagrams, with bottleneck distance bounded by (D(((,)) (()())) s || - [. Given that entropy is computed over the birth-time histogram derived from the persistence diagram,
infer the E difference in their birth-time dis- tributions P Pooll 11 S a, I Will S age. Shannon entropy is Lipschitz continuous with respect to & distance, yielding the following bound (H) - -H (B) < C;- I - Pirco |/1 < Crase. We now relate entropy variation task loss degradation. The increase in loss can estimated by second-order Taylor expansion AL; S &
_ kx 6W,11.. We further observe that the Li distance between the birth-time distributions before and after quantization can upper bounded in terms of TE. Specifically, since Shannon entropy is Lipschitz-continuuu with respect its input distribution, we have & Pl Powe l1) s (IH H!!²)). This confirms that TE is an upper-bound surrogate of quantization induced performance degradation. Based on this insight, we propose a TE-guided MPQ objective minçB.) // H, B; s.t. Dt. ((,) < T' To ensure alignment with sensitivity ranking, enforce H1 H1 + B, Bj. This ordering remains consis- tent during training due to entropy drift stability, shown by HH!") H < Ag log M, L;
sign( H, - H,) sign( sign(H!()) H,("),, where sign H,) denotes the relative ordering between layer sensitivities. This ensures that layers identified as more sensitive in the initial phase continue to
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receive higher precision during optimization. Thus, TE is suitable for mixed- seeeed-precisoo quantization with sufficient theoretical justification.
4
Experiments
We first introduce the datasets and the experimental settings. Then, we demonstrate why TE possesses good cross-dataset properties through ablation experiments. Finally, we compare GMPQ-TF to state-of-the-art MPQ approaches for image classification and object detection.
4.1 Datasets and Implementation Details
Following studies Wang et al. [2021] and Wang et al. 2024 configuration, for im-
age classification, we use (eeng al. 2009] evaluate the quantized net-
5.0
batch 51ze-512
4.5
$220-188
butch sizemod
works, where ResNet-18, ResNet-50 He
8.0
bwithh sinwm32
al. 2016], MobileNet-V2 V2 [Sandler et
3.5
[2018]. ViT Dosoviiskky et al.
[2020] and
Swin transformer Liu et al.
2021 are
3.0
treated as the baseline architectures. In
2.5
terms of object detection, PASCAL VOC
2.0
Everingham et al. (2015) is employed to validate the effectiveness of GMPQ-TE,
condt
30
Figure 5: Relationship between different batch sizes and TE on ResNet-18.
where VGG-16 Karen 2014] with SSD
framework Liu et al. 2016 and ResNet-18
with Faster R-CNN Ren et al. [2016] are
used as the baseline architectures. Regarding evaluation metrics, common criteria include model storage cost (Params.), computational cost (BOPs), BOPs compression ratio (Comp), and search cost (s). Top- 1/5 accuracy (%) and mean average precision (mAP) (%) Everingham et al. (2015) specifically used for image classification and object detection, respectively. By setting different constraints T. we can obtain the quantized networks with different aecuracy-complexin trade-offs. In addition, follow study Wang al. [2021] fine-tune the quantized networks that are found on different tasks. All experiments are perform on an NVIDIA Geforce GTX 3090Ti.
4.2 Ablation Study
Normally, more accurate measures of quantization sensitivity can be achieved by sampling as many images with similar labels as possible. However, excessive data sampling may lead to sig-
1) Batch size:
5.0
Cbsss-aiplane
abttmbutooobble Class-bird
4.5
Classudig
8.0
nificant computational overhead. We set
different batch sizes (i.e., 32, 64, 128, 256,
3.5
and 512) on CIFAR- 10 to verify the effect of batch size on quantification sensitivity, where ResNet-18 18 as baseline model is used to test the TE of each layer. batch of images with the label "cat" is selected as input for ResNet-18. Figure 5 illustrates the TE of each layer under different batch sizes. Overall, the trend of TE variation is consistent across batch sizes. Furthermore, batch size increases. Since the differences in TE for batch sizes 128, 256, and 512 are insignificant, consider the computational efficiency and choose 128 in all experiments.
3.0
2.5
Figure 6: Relationship between different labels from the same dataset and TE on ResNet-18.
observe that the TE of each layer stabilizes the
2) Labels from the different datasets:
We compute TE derived from the labels of different datasets to validate its cross-dataset property. Specifically, we first use ResNet-18 as the baseline architecture and compute TE for each layer obtained from the same labels ("dog" and "cat") of the three datasets (CIFAR-10, ImageNet, PASCAL VOC), shown in Figure 7 (a). We observe that images with the
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5.0
Cass-dde (CE1AB110)
dag (imageket)
50-
Cassecat (inageNet
Class.ddo dog IGFAR 10)
Peature entmerr
K.
4.5
Cisdmmat (CHARR 18)
Casamua (MAGAL vod
Cisdemaat (mageheet)
Cles--ii INSSCA, VOcI
Trature antnerr
6.0
Clsseccor IPASCAL Voci
Clasurcher INDC vok)
Classaapok (magehet)
Obssellig (CIFAR 18)
3.5
1.0
25
2.5
(a) Same label images
(b) Different label images
Figure 7: Relationship between labels from different datasets and TE on ResNet-18.
same labels on different datasets do not affect the calculation of TE. We then use different labels to further check the consistency of TE across datasets, shown in Figure (b). Similarly, TE is consistent across different labels in different datasets. The above phenomenon indicates that TE has good cross dasssed property. That is, the quantization strategy obtained through TE has good generalization ability.
3) Labels from the same dataset: To exclude the effect of labels on TE, we select "airplane", "automobile", "bird", and "dog" in CIFAR-10 for analysis, where ResNet-18 serves as the benchmark architecture. The experimental results are shown in Figure [ We can observe that TE of each layer does not change dramatically with the change of labels. This proves that the TE is independent the labels. Therefore, there is no need to consider the effect of labels when using TE for quantitative sensitivity analysis of each layer. This reflects the robustness of the proposed TE concerning labels.
4.3 Comparison with State-of-the-art Methods
We compare the proposed method with the state- of-the-art quantization approaches, including
Table 1: Results for image classification on MobileNet-V V2 (PTQ).
OMPQ Ma et al.) [023), GMPQ |Wang et al.)
[2021] R- R-GMPQ Wang et al.] [2024]. DJPQ Wang et al. 2020), EdMIDS Cai and Vasconce- et al. [2019) Li
Params.
BOPs
Comp.
Top-1
Top
13,4
337.9
71,9
90.3
los [2020].
RQ
2.7
11.9
28.4
68
ct al. ALQ Qu 2000)) HMQ Habi al.
GMPQI
1,4
10.4
32.6
71.5
90.1
6.1K
R-GMPQ1
54,1
90.2
[220 HAQ Wang et al. 2019a). BP-NAS Yu
GMPO-TE
HAQ
1.4
8.3
41
69.5
88.8
18.3K
ct al. 2020]. RQ Louizos et al. [2020], EMQ Dong al. 2003), and DQ Uhlich et al. [2019].
DIPO
1.9
7.9
43
69.3
43.9K
GMPQ
1.2
7.4
45.8
70.4
89.4
9.3K
R-GMPO
46.7
20.9
89.7
10.4K
There are two types from which these experi-
GMPO-TE
46.1
mental data derived: duplication of data from the original literature and reproduction based on the open-source code or the quantized architec- tural information. i is implemented by ourselves using open source code. is implemented based on the quantized architectural information pro-
EMQ
1.7
3.2
64.4
70.9
120.8K
DQ
1.7
4.9
68.7
69.7
77.7K
EMQ
1.5
70.75
few seconds
GMPO
4.8
69.7
69.5
89.
10K
R-GMPQ
OMPQ
1.5
12.3
27.3
GMPQ-TE
1.3
4.5
74.8
69.8
89.7
vided by the authors due to lack of open source code. The best and second-best are color coded. The performances of the full- full-precision baseline models are also provided for better comparison. We use CIFAR-10 to find the suitable quantization policy for all baseline architectures. Then, the baseline architectures with suitable quantization policies are deployed on ImageNet or PASCAL VOC. In addition, the results for ResNet-13
18, 18, ResNet-50, VIT, Swin transformer and ResNet-18 18 with Faster R-CNN under PTQ can be found in App. D.I Results under QAT for ResNet-18, ResNet-50, MobileNet-Vi V2, VIT, Swin transformer
be found in App. D.2
Results on ImageNet: Table D record the experimental results on ImageNet with MobileNet-V2 as the baseline model, respectively. Compared to the generalizable MPQ approaches (i.e., GMPQ and R- GMPQ). GMPQ-TE achieves good post-quantization accuracy over multiple baseline architectures with less cost. Compared to OMPQ, GMPQ-TE gets better quantization performance although GMPQ-TE uses 2s more on MobileNet-V2.

---

Results on PASCAL VOC: As shown in Ta-
Table 2: Results for object detection on SSD VGG-16. (PTQ)
ble 2 GMPQ-TE achieves the best trade-off
between mAP and compression ratio SSD VGG-16. For example, GMPQ-TE obtains 71.1% % mAP with only 32.6 Mb parameters and 758 BOPs, outperforming R-GMPQ in both ac-
Methods
Params.
Comp.
Cost.
pull-precision
105.5
27787.7
72.4
HAQ
42.7
847.2
32.8
70.9
225K
HAQ-C
42.9
819.7
33.9
67.6
18.3K
curacy and computational efficiency under the same parameter budget. Notably, it also sur- passes EdMIPS by 2.2% mAP while using sig- nificantly fewer parameters (32.6 Mb vs. 33.5
EdMIPS
33.5
958.2
29
69.4
5,4K
GMPO
36.6
70.5
5.7K
R-GMPQ
32.6
761.3
36.5
70.8
6.4K
GMPO-TE
32.6
758.4
36.6
71.1
HAQ
35.5
430.15
64.6
69.1
244.4K
HAQ-C
36.3
445.3
62.4
66.4
24.4K
Mb) and computational cost (27 Meanwhile, the Faster R-CNN & ResNet-18
5.4K).
EdMIPS
29.4
68.7
108.7K
GMPQI
24.7
413.5
67.2
69.2
6.4K
model compressed by GMPQ-TE yields 0.7% mAP improvement over GMPQ, while reducing the cost by approximately 250 times, demon- strating its superior efficiency-accuracy balance.
R-GMPQ1
70.3
GMPQ-TE
24.7
405.4
68.6
70.6
23
4.4 Hardware Efficiency
Table 3 compares the hard- ware performance of dif- ferent quantization methods on two FPGA platforms (XC7Z020 and XC7Z045) using ResNet-18 on Ima- geNet. For each FPGA board, the utilization per- centages of look-up table (LUT) and digital signal processing (DSP) blocks are reported for each quantization method. The LUTT utilization, in partieular, reflects the effectiveness of the quantization method in accelerating inference. Among all methods, GMPQ-TE consistently achieves the best overall performance. On XC7Z020, it attains the highest LUT utilization (53%) and throughput (84 GOP/s), while reducing latency to 42.3 ms, significantly lower than GMPQ (57.3 ms), EdMIPS (61.4 ms), and R-GMPQ (52.1 ms). On the more powerful XC7Z045, GMPQ-TV EMPQ-TE further improves throughput to 413 GOP/s and lowers latency to 9.4 ms, outperforming G-MP (398 GOP/s, 10.4 ms) and GMPQ (367 GOP/s, 12.3 ms). These results demonstrate that GMPQ-TE TE offers superior hardware efficiency and inference speed compared to existing methods across different FPGA platforms.
Table 3: Comparison of different quantization methods on two FPGA
boards for ResNet-18 on ImageNet.
Resulis FPGA XC72020
Resulis
FPGA XC72INS
Methods
Utiiization
Tarougliput
Latency
Unlization
Throghhpu (GOP/s)
Laiency
LUT
DSP
(GOP/s)
(mS)
LUT
DSP
(ms)
GMPQ EAMIPS
52%
367
39%
100%
61
61.4
41%
100%
316
15.9
R-GMPQ
GMPQ-TE
53%
100%
84
42.3
62%
100%
413
9.4
5 Conclusion
In this paper, we propose GMPQ-TE for handling the efficiency and generalization. By utilizing the properties of TE, GMPQ-TE can be constructed as single-pass linear programming. The quantization policy obtained on CIFAR- 10 is well generalized to ImageNet and PASCAL VOC. The proposed GMPQ-TE outperforms peer MPQ in terms of both accuracy and complexity. Further, the ablation studies verify the effectiveness of the TE. GMPQ-TE, offers promising direction for MPQ. In the future, we will explore new "topological data analysis" for MPQ on more complex tasks.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China under Grant (24.0747,,7 in part by the Fundamental Research Funds for the Central Universities (No.N2417003).
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19

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Please refer to our LLM policy fattps://neurd/: httpp/////////ooeeeeeee///////// for what should or should not be described.
20

---

A Feature Map Construction Across Different Architectures
Case 1 - Convolution-basel architectures. For convolutional networks, the output feature map U(+) " is naturally defined on regular 2D grid. This grid already encodes the true Euclidean neighborhood structure assumed by our clique filtration process. Therefore, we can directly flatten the n h² w² vertices to form E I "x* without any padding.
Case - Transformer-bassd architectures. Vision Transformers produce a sequence of tokens, where N is often not perfect square. Tokens be rearranged into p q grid according their spatial positions in the original image, but typically # and some positions remain empty. A direct flattening of the 1D token sequence into N x N adjacency matrix would impose arbitrary linear ordering on the vertices. This ordering does not correspond to the true Fuclidean neighborhood relationships in the image plane, and would distort the clique filtration process by creating spurious edges between spatially unrelated tokens. As result, the birth-death times of topological features would be inconsistent with those obtained from genuine 2D grids.
To address this, embed the p X token grid into the smallest enclosing square of size [VN] x [VN], and fill the empty positions with zero-valued tokens. This construction has two benefits:
1. It ensures that adjacency is always defined on a square 2D lattice, identical in form the convolutional case, enabling fair cross gross-architectury comparison.
2. By choosing the minimal square size, we limit the number of padded vertices, thereby minimizing any dilution of the adjacency structure.
From topological perspective, the padded region forms contractible subcomplex with trivial higher-order homology. In the filtration, zero-valued vertices outside the original token set do not create new non-trive trivill cycles. Therefore, persistent homology and the resulting topological entropy is invariant under this padding step.
After padding, the feature map has size hi x w = [VN] x [VN], and is flattened into an adjacency matrix of shape (h' w²) x (h' w²). Each matrix entry Aue encodes the weighted connection between vertices u and under this consistent 2D spatial embedding.
B Betti Curves and Birth Times
We illustrate the concept of Betti curves and birth times using a simple clique filtration process over four nodes labeled 1-4. Let the filtration sequence be denoted as (7(1), (7(1),-(2) (())), where each step incrementally adds edges to build higher-order cliques:
7 (L); add edge (1, 2)
22) add edge (3,4)
continue adding edges, no cycle yet
( form a 1-cycle (1 2 - 4 3 3 1)
7 (5); add edge (2, 3), creating additional cliques
( complete graph with all edges
To capture the emergence of topological features, define birth-time indicator function 3(i, v, Ui.) which equals if the i-th feature appears for the first time at filtration step v, and otherwise. A typical curve looks like = [0, 1, 2, 3, 4, 5, 6]
B [0. 0, 0, 0. 1, 0, 0)
This indicates that a 1 I-dimensional loop born at e 4, corresponding to the closure of cycle. Notably, although edges are added at v 5 and = 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.
21

---

C Theoretical Analysis
C.1 Integration of Quantiattoon-wware Training
Similarly the study Ma al. (2023), we integrate the TE into QAT and theoretically show that it has very little effect on the final performance. The details are as follows:
Let
W = (W10) W E E.
(9)
where (0)) is the floating parameters (i.e., weights) obtained by GMPQ-TE. Here is the number of quantized layers and N = Σi=1 n, with nı = dim W, For every layer fixes a bit-width b, E Z,o and clipping radius a; > 0. Defining quantization step Ag. = 2 2- a; and symmetric quantizer Qa, (w) clip (arnddw/An.... Aq.i iteration updates the floating copy via
( W T7, VC(W()),
ost<
(10)
The forward path employs the quantized tensor (Qb, (W("),.. Qb (W. A single SGD step cannot move any weight across two adjacent quantizer levels. Hence,
Sozykin et al.
[002]]
W
w:
Aqi
(11)
For every layer and every iteration summing Eq.
II
over all steps yields global drift
e ||!! w/ 1o
Ag.i
(V)).
(12)
2
Because 2 A one bin width, one has Qo, Q (w."))
For convolution BN- ReLU block, perturbing the weight tensor by E (1) L (t) change in the output feature map, i.e.
in lo produces at most
(U(W())) (.(W (1) S. T
I 2.5.
(13)
Here Li is the Loc loc operator norm the block. The bottleneck distance between two persistence diagrams McCleary and Patel (2018) is upper-bounded by the Loo distance of their generating functions. Thus, dn(D(UU ((U!")) )) s L bottleneck displacement of at most Lee (t)
implies that each sample's birth time moves by no more than one bin. Consequently, ||
Popor 11 < 2L, Because Shannon entropy Lin [191 satisfies
2_ log S log M, on the
interval [1/M,, 1), finally obtain
|H - H < 2L,(log M,) < L:(log M,) Ag.i
(14)
with Mj the number of histogram bins and Ag,i the quantization step of layer i.
Having bounded each single-layer entropy variation in Eq. [4 aggregate those bounds and compare them with the initial separation among layers. To control all layers simultaneously we introduce worst-case entropy drift 6max == max; L, Llog M:) Aq,i and record the minimum initial
gap == min, (H()) ( > 0. Whenever the inequality δmax < holds, each layer can move by most half the gap hence no pair of layers can swap their order. Applying Eq. 14 every pair,
we obtain
H 11 (H1 (0) + 28max # 0,
(15)
22

---

Thus,
Vi レ J, t. sign ( 1 -) sign ( - :
(16)
That the TE ranking established 0 preserved for the entire QAT trajectory. Because the mixed-pree mixed-precissoon linear programming (LP) enforces H; > H = by > bj. and Eq. 16 keeps this
ordering intact, the integer-feasible polytope == {b E Z H > H b, > by is identical
for every iteration Therefore the bit-width vector 3 that was optimal at 0 stays both feasible and optimal throughout QAT.
At step & the LP objective can be decomposed
L EH
(b)
Σ H
(1) b, (b)+ -H, by.
(17)
Because |H! bounded:
11
8max and by v Umax the perturbation of the objective value uniformly
V ( (0) (b) < max Umax L,
Ab B.
(18)
Classical sensitivity theory Popsel 1963] for integer LP states that the optimal basis cannot change if every coefficient variation is smaller than the minimum reduced-cost gap (2 7/2 in our case). Due to 6max < 7/2, the unique optimum 3* of (0()) therefore remains the optimum for all subsequent iterations.
B" argmin (b)) bEB
Vt.
(19)
The forward path always employs the unchanged bit-width vector (see Eq. 19 The half-interval drift property guarantees that no weight ever crosses a quantizer boundary. Therefore, the integer weights themselves remain identical throughout training. The inference graph at step is the same as at t = 0, and the classification performance cannot vary: (t) (c) Vt.
C.2 Resolution- and Label-indeeendent
Convolution-assed architectures
Let dataset X (xx1) and x (2:2), where X RRixx: and X Blxxw2 Pra (1)( and Pte (x) are the distribution of pixel values from X and X respectively. PFx (x) and PFg (z) are distribution of output feature maps from X and B respectively. A single convolutional layer with kernel K acts on an image patch x via
T(x) : Zu,o Σ Ks, Fy Tu-s,
(20)
8,
where the receptive field S = {(8, 1) Kat * 0} is local. Given any input distribution Pr, the induced feature distribution is Pr(z) = fō(z (x dx. The TE of Pp is HrE(Pr) =
EM (k( log PF(k), where (k(k) the birth-time histogram and M the bin count. Because T depends only pixels inside S, we may rewrite feature distribution as
Pz(z) f0(z ((x)) dxs,
(21)
where Xg is the sub-tensor on S. Applying Eq. to to the two datasets yields
Pr (z) = f 8(2 6(2 ((x)) (xs) dxs, Pra (z) = f 0 T(xs)) (x5ss) dxg.
(22)
Images of the same class share identical local distribution Osada et al. 2002
23

---

PIA (xx))) PIn (xs) =: Ps(xs).
(23)
Substituting Eq. 23 into Eq. 22 gives PFA (z) = PFn (z) - - (xs)) dxs, and Hre( Prx) = HTE Pro) )- Thus, TE does not depend on the global resolution of the input image.
Let C' be the class label. The convolution acts information bottleneck (IB):
[[F; C) & I(ICO),
(24)
where -(:))) denotes mutual information. Pixels inside the receptive field have identical class- conditioned distributions:
Ps|C=e (xs) = Ps|C=es (xs);
Vel, C2.
(25)
Hence the class-conditional feature distributions coincide:
Pri (z) = PF(Cmeoaz))
(26)
Their topological entropies are equal,
HTE HTE(Pr(==)) HTE(PrIC=o)
(27)
Defining the Kullback-Leibler divergence DKL (PriC=a || Pr(C=()), Pinsker's inequality Fedotov ct al. 20033
implies
PF|C=s Pric=ea 11 22DLL.
(28)
Eq. 26 gives DKL = 0, Thus
|FPCCe - PFIC=ool11 0.
(29)
Using the entropy Lipschitz bound Polyanskiy and Wu 2016 [H()) H(g) s (log M) 4|l yields
HTE PFIc=e.) - HTE (ene(PFFCae PF(C=cx) =
(30)
The IB objective at a hidden layer is mingru [T(F:T) 3 (FF(FF)))) C)}, where / is trade-off parameter. The optimum satisfies
I(F; const,
I(F;C) 0,
(31)
Thus, TE is proportional to the IB cost:
HTR (Pr) (rp)
00 I(;1),
A> 0.
(32)
00
Minimising TE is therefore IB-consistent. Collecting Eqs.
32
we
have
that
HTE is independent to input resolution and class label, i.e., HTm p Transformer-bssed architectures
HTT
p(2)
Let the input an attention block be X Wxx where N is the number of tokens and is the channel dimension. An attention operation can be expressed as:
QKT
Attn(X)
bias
V
(33)
VTT
Q XW K K K,
XW
(34)
where a is the row-wise softmax, and bias may include fixed masks or positional encodings. Multi- head attention concatenation of several such heads followed by linear projection. Let the block output be F Attn(X) e [B Nxd
24

---

Assumption A1 (Class-nvaariance within attention domains). For any attention receptive domain (which may be the entire token set for global attention, window for local attention, or sparsified neighborhood), the multiset of tokens Xs has class-conditional distribution that is invariant across labels:
Px = Pxslo2 == Ps- VC1, 02
(35)
The attention mapping Ag Xs + Zs is a composition of linear maps, scaling, softmax weight- ing, and aggregation, all of which are measurable and Lipschitz bounded domains. The class- conditional output distribution is:
Prya (=) / f x== Ao(Xs)) A=xss())) Pxsler (Xs) dXs
(36)
Substituting Assumption A1 into the above yields:
Pree (2) (=))2(=) / 82= Ao Ps(Xs)
(37)
which is directly analogous to Eq. 27 Appendix for convolutional mappings.
Let U be the scalar field derived from F (e.g., channel aggregation or reshaped into 2D grid). For attention architectures where N is a perfect square, we embed the p token layout into the minimal square [VN] [VN] and zero-pad empty positions. The padded region is contractible (H2 = 0, Vk 2 1) and does not introduce spurious topological features.
From U, construct weighted adjacency matrix and perform sublevel set filtration to obtain the birth-time histogram Pu (k)). The topological entropy is:
M
HTE(PU) = Σ (k) log (k(k).
(38)
Because PEy. = Prey. the induced Pola=zz is also identical across labels. By the Lipschitz
continuity of entropy with respect to total variation distance:
[HTE(PP)Ee)) - (1j(()))))) s O(M) || ( U/e Pellall.
(39)
and here |l p(l) 111 O, we have exact equality:
HTe(Putel = HTE( (Pree)) V2,, C2.
(40)
Under Assumption A1, the equality of class-conditional token distributions through any measurable attention mapping (global, local, multi-head, with or without positional encodings) guarantees that the resulting birth-time histograms are label-independent. Padding in attention architectures serves only to unify the 2D adjacency structure and does not affect the birth-time distribution topology.
C.3 Performance Degradation Boundary
We provide theoretical performance degradation. The details follows:
Let
( (W(0)
( E R
(41)
where (0) is the floating parameters (i.c., weights) obtained by GMPQ-TE. After finetuning phase,
(t+11) w n VZ(W()).
Ost<T,
(42)
with the mini-batch cross-entropy Seidenschwarz et al. 2021
((W) = 1 Σ
(2,
25

---

Because a straight-through estimator Yin et al. (2019) never pushes weight farther than half quantization interval, the cumulative drift satisfies
& = W w 11
S
A q3
Isis..
(43)
2
For a same-label batch Qg we denote Pv, (k) = [etal Eoen, (W,(,((,x))) = kb. and recall the
TE H.WW (W) Σx Pv. (k) log Pv. (k). A standard Lipschitz estimate for convolution-NN-RR
blocks Jordan and Dimakis 2020 gives [[ - (w)))l I L, where L, S 2.5 is an operator norm.
By the notteneck-distanne stability of persistent homology He et al. 20244. we obtain
((U s L,
(44)
Thus, every sample's birth time moves by at most one histogram bin, i.e. || 1/011, S 22, Ei. Since - x log æ| < log M, on [1/M,, 1],
H H < 2L, (log M,) E;
(45)
where M2 the number of bins. Elm (W) b,. Using inequality (Eq. 45) and the trivial bound b, S bmaxx we obtain
We defines the mixed--reeissin objective as ((W)
f - (0) < 2 bmax Limx (log Mmax) L mmxx 6max == maxx Eg
(46)
Writing H, = VM w: L (( and T; := V ((S)), where & E [W Whi lies on the line segment connecting the initial and finetuned weights. A per-layer Taylor series yields
Chn (()) HAW,HAW, + EET,
(47)
where AW, == Wfinn - W( H; is layer-wise Hessian of L at ( Dong et al. [2019]. T; is third-order derivative tensor evaluated at &, is global apeetral-norm bound. R is global third-order bound max; ||Ti|loo.
Assuming H.lllop $ X, |T I R (both empirically S Chauhan et al. 2023 ), and |AW; < ne? ||AW S n²=,, we obtain
C _
2 M 2 max + 1R m L 3 max
(48)
Let Tiin > 0 denote the smallest soft-max margin on the training set. A standard PAC-Bayes argument McAllester [1998 implies
1 Tmin
An
Aoc
| L(0)/.
(49)
Combining Eqs. 48 with
49
obtain
A L
Rin R L
Acc fin Acc
2
(50)
Emax
2Tmin
67min
By Eq. 43
the global drift Smax <
A A.min 2 (bmin) a. Thus, the r.h.s. of Eq. 49 is 0 26min ).
Even for bmin = the term is < 10 for deeper compression it decays exponentially. Under the half-interval drift condition Eq.
43 the topological-entrogn objective satisfies T finn T10)
O(2 O22 Amm)) while performance is perturbed by at most Acc - ! = 0 20min ), where
Dynn min; b, the smallest bit-width in the network. Thus, GMPQ-TE cannot cause any practically measurable performance degradation.
26

---

D Experimental Results
D.1 Results for PTQ
Table Results for image classification on
Table 4: Results for image classification on ResNet- (PTQ).
ResNet-50 (PTQ).
Methods
Params.
BOPs
Comp.
Top-1
Top-5
Cost.
Fue-pseccsion
3952.6
12274
2017
97x
HAWQ
13.1
61.3
64.3
15.3
92.4
131.7K
(veng)
26.4
65.3
66.
84.1
HAQ
12.2
50.3
78.6
75.5
92.4
243.7K
GMIN-TE
143
2011
243
R-GMPQ1
GMPOI
12.4
53
74.6
76.1
92.7
92.9
7,9K
9K
Mus 53
21.00.511
60.77.0.44 0.34
GMPQ-TE
93
11:30
003
BENNAS
11.3
33.2
119
15.7
92.8
12%.1K
156
1807
91.1
13K
GMPO1
30.7
128.6
75.2
92.1
9.7K
Mosa 5M
1547
118.33 0.31
16-1 11
R-GMPQI
GMPO-TE
7,9
29.5
133.9
76.1
92.7
23
2310
59.52
EDMIPSS
13.9
15.6
254.2
72.1
90.6
75.4K
601
GMPO
15.7
252.2
73.6
91.2
12.2K
GVPO-TT
1019
R-GMPQI
Moce
15
OMPO
18.7
15.6
253.3
74,28
91.6
GMPQ-TE
10.2
15.5
251.9
74.3
91.8
20
The performance of different quantization methods on ResNet-18 and ResNet--50 is summarized Tables annd Among all quantization methods, GMPQ-TE consistently outperforms its peers across multiple metrics including accuracy, compression, and cost. GMPQ-TE achieves Top-1 accuracy of 70.3% and Top-5 accuracy of 89.1%, while maintaining the lowest cost at 19 s. This is a significant improvement over R- R-GMPQ, which achieves Top-1 of 69.4% and Top-5 of 88.9% but has a higher cost 2.5K S. GMPQ-TE also demonstrates superior compression (67.6% compared to other methods, confirming its high efficiency in balancing model size and performance. GMPQ-TE further extends its advantage in ResNet-50 with a Top-1 accuracy of 80.6% and Top-5 accuracy of 85.1%, outperforming R- R-GMPQ (75.9% for Top-1, 82.8% for Top-5) by significant margin. The compression of GMPQ-TE is also outstanding at 76.8%, which ensures efficient resource utilization. Additionally, GMPQ-TE reduces the cost significantly compared to other methods, such BP-NAS, which achieves similar Top-1 accuracy of 76.3% but at much higher cost of 128.1K
For object detection tasks (see Table 6 GMPQ-TE continues to perform strongly in terms of both accuracy (mAP) and cost. GMPQ-T TE on Faster R-CNN ResNet- 18 achieves an mAP of 74.2, surpassing R-GMPQ (73.9) and GMPQ (73.5), with noticeable reduction in cost to 25 s. compared to R-GMPQ (2. s) and GMPQ (140K s). This result highlights GMPQ-TE's ability to maintain high mAP while significantly reducing computational cost, making it suitable for real-time applications where both accuracy and resource constraints are critical.
The results across various VIT and Swin Transformer models show that GMPQ-TE provides competi- tive performance even transformer-based architectures, as shown in Table For VIT-S and ViT-B, GMPQ-TE achieves Top-1 accuracy of 80.6% and 85.1 1% respectively, outperforming PTQ4ViT and
Table Results for object detection on Faster R-CNN & ResNet-18 (PTQ).
Methods
Params.
BOPs
Comp.
MAP
Cost.
Faster
R-CNN ResNet-18
Full-precision
47.4
22534.8
74.5
HAQ
8.3
324.5
65.8
73.5
140K
HAQ-C
8.5
337.9
66.7
70.7
14.7K
EdMIPS
9.3
361.7
62.3
72.3
59.7K
GMPQi
6.4
337.9
66.7
73.9
1.8K
R-GMPQt
7.2
324.7
69.4
74.3
2.1K
GMPQ-TE
6.4
325.8
69.3
74.2
HAQ
303.7
74.2
73.2
126.7K
HAQ-C
7.6
310.4
72.6
70.4
18.7K
EdMIPS
18.7
348.8
71.1
71.8
65.1K
GMPQi
6.2
286.3
78.7
73.2
1.8K
R-GMPQ
6.8
284.5
79.2
73.6
2.1K
GMPQ-TE
6.2
283.1
79.6
73.9
22
27

---

Table 7: Results for image classification across various VIT and Swin transformer models (PTQ).
Maihoodh
VIT-5
VT--.
DaT-
Dx1-6
precision
Quan
APQ-VIT
666
79
70.49
77:76
Quantt
60
79.63
TSPTQ-VT
6/5
79.45
$2.29
65.18
20.82
77.18
80.61
8062
$2,60
8Q-5-OPTTm
6/5
79.08
$2.70
83.53
71.03
78
81
80.67
82,62
LIP QVIT
MP
80.59
$3.87
71,05
79.05
81.44
82.56
34.72
Rep
MP
30.43
$3.02
70.76
78
81
82.79
$4.57
GMPQ-TE
MP
M0.61
$3.90
85.51
71.15
79.11
X1
81180
82.88
$4.78
Mean Sid
8IL48
0.114
80.31
0.12
86.15 0.07
71.11 0.14
70.00 =.0.
8006 =: 0.06
8000 = 0.30
82 12.55 0.005
84.12 0.05
PIOAViT
4/4
42.57
30.69
78,
56 96
54 06
64
73.48
76.09
74.00
APO VIT
4/6
47.95
41.41
47.94
43.55
07.48
77.15
76.48
TSPTQ VI
4/4
52.56
50.10
77.64
48.36
45 06
45
72.48
76.30
73.25
S0-5+OPF-m
4/4
55.83
61.
80.17
55.62
68.45
76.14
73.82
77.20
76.51
PSM)-VIT
4/4
37.19
41.52
57.58
63.61
67.95
72.86
76.44
IRP-QVIT
MP
70.81
755
61.24
73:45
78
81.37
50.77
RepQ-ViT
MP
65.05
68
57.45
69.03
75.61
79.45
78.53
GMPQ-TE
MP
70,90
75:41
80.13
61.05
72.52
78:19
73390
$1.33
30.81
Men Std
70.62 1.010
75.24
0.00
20.04 0.16
60.86
1..86.10.11
72.0
1800
15.00 3
3
80.43 1:0.7
AiQ in terms of accuracy and compression. For example, GMPQ-TE achieves Top-1 accuracy of 80.6% on ViT-S, significantly higher than PTQ4ViT (78.3%). Additionally, GMPQ-TE maintains a favorable compression ratio compared other methods, ensuring efficient model deployment without sacrificing accuracy. On Swin-T, GMPQ-TE maintains a Top-1 accuracy of 80.7%, significantly outperforming traditional methods like PTQ4ViT and APQ-ViT. This confirms that GMPQ-TE's efficacy extends to transformer-based models as well, which are increasingly being used for a variety of tasks.
These generalizable results can be attributed to the TE, which can prevent inconsistencies in quantiza- tion sensitivity across datasets due to distribution differences among the datasets. In addition, the observations on the correlation between TE and model performance as well as bit-widths help in the construction of linear programming problem, accelerating the solution of generalizable quantization proxy.
D.2
Results for QAT
Table 8: Results for image classification on ResNet- 18, ResNet-50 and MobileNet- V2 (QAT).
Methods
Params.
Comp.
BOPs
Top-1 Acc.
ResNet- 18
QDrop
5.41
8,23
29.0
69.76
HMQAT
4.31
10.34
20.9
69.63
QuanDCL
4.49
9.92
21.8
69.51
MataMix
>35
72
SDQ
5.2
15.7
69.1
EMQ
6.69
71
72.28
GMPQ-TE
4.21
117
20.3
69.8
Mean + Std
4.21
115 + 2.31
20.1 + 1.66
69.63 + 0.10
ResNet-50
QDrop
13.14
7.4
61.7
75.45
EPTQ
13.14
7.4
123.5
75.45
HMQAT
9.45
10.29
51.5
75.48
GMPQ-TE
7.9
131.4
25.1
76.32
Mean + Std
7.9
128.1 + 4.25
24.7 + 1.47
76.26 = 0.01
MobileNet-V2
MataMix
>8
73
SDQ
1.8
4.89
72
QDrop
5.41
8.23
29.0
69.76
HMQAT
1.22
10.98
7.71
70.81
EPTQ
1.26
10.63
31.68
70.39
GMPQ-TE
47.2
7.1
70.42
Mean + Std
1.1
48.2 1.37
7.1 + 2.94
70.32 + 0.43
28

---

Table 9: Results for image classification across various ViT and Swin transformer models (QAT).
Methods
DeñT-T
DeñT-S
DeiT-B
Swin-T
Swin-S
Full-precision
72.21
79.85
81.80
81.20
83.23
LSQ
54.45
68.00
70.30
70.40
72.40
Q-VIT
50.37
72.10
74.20
74.70
76.90
OFQ
64.33
75.72
78.52
Mix-LSQ
64.19
73.88
76.58
75.13
79,49
Mix-OFQ 67.87
76.39
78.26
78.71
81.23
GMPQ-TE 67.90
76.51
77.94
78.73
81.33
Table 10: Experimental results for ResNet-18 on average model bit-
Method
Params.
BOPs
Comp.
Top-1
Top-5
ResNet- 18-1
5.4
30.1
60.6
69.14
88.73
GMPQ-TE--
5.3
27.4
67.6
70.40
89.40
ResNet- 18-2
3.7
23.4
107
68.20
88.13
GMPQ-TE-2
3.8
15.7
118
69.40
89.40
ResNet- 18-3
3.5
7.4
254.3
66.10
87.40
GMPQ-TE-3
3.5
7.1
260.9
68.30
88.90
As shown in Table the results for ResNet-18, ResNet-50, and MobileNet-V2 V2 show that GMPQ-TE outperforms the other quantization methods terms of both accuracy and efficiency across all three architectures. GMPQ-TE TE achieves Top-1 accuracy of 69.8%, slightly outperforming Qrrop (69.76%) and HMQAT (69.63% (9....).. Importantly, GMPQ TE also exhibits the best compression (117) and BOPs (20.3), indicating its superior efficiency in terms of model size and computational cost. Other methods like QuanDCL and HMQAT show drop in performance and higher computational overhead compared to GMPQ-TE. The pattern of superiority continues with GMPQ-TE, achieving Top-1 accuracy of 76.32%, outperforming HMQAT (75.45%) and EPTQ (75.48%). GMPQ-TE also demonstrates better compression (131.4) and BOPs (25.1), showcasing its effective trade-off between accuracy and computational efficiency. For MobileNet- V2, GMPQ-TE -E achieves Top-1 accuracy of 70.42%, slightly outperforming QDrop and HMQAT (both 69. 69.76% with significantly better compression value (47.2) and lower BOPs (7.1). This demonstrates that GMPQ-TE can maintain competitive accuracy while dramatically reducing the model's size and computation.
The performance of GMPQ-TE on various ViT and Swin Transformer models further demonstrates its effectiveness in QAT, as shown in Table 9 For VIT-T and VIT-B, GMPQ-TE achieves Top-1 accuracy 67.9% and 77.9 77.9%, respectively. These results are competitive with or surpass other methods like Mix-LSQ and Mix-OFQ. For instance, Mix-OFQ achieves 77. 77.94% on ViT-B, while GMPQ-TE provides better overall performance with improved computational efficiency. GMPQ-TE continues perform strongly with Top-1 accuracy of 78.7% on Swin-T and 81.33% on Swin-S, which outperforms Mix-LSQ and LSQ in both accuracy and compression. This highlights GMPQ-TE' robust performance across variety of transformer architectures, demonstrating that it is well-suited for handling more complex and varied models.
The proposed GMPQ-TE method is effective in QAT, providing superior performance in terms of accuracy, compression, and computational efficiency across multiple models, including ResNet, MobileNet, and Transformer architectures. These results confirm that GMPQ-TE is promising solution for efficient model deployment, and its effectiveness in QAT is evident.
D.3 Results for Average Model
Tables 10 and present the experimental results for the average model bit-width on common archi- tectures, namely ResNet-18 18 and MobileNet-V. V2. The results show that GMPQ-TE effectively reduces the average bit-width and BOPs while maintaining even improving accuracy. For example, in the ResNet- 18-2 setting, GMPQ- TE lowers the BOPs from 23.4 15.7 and simultaneously increases computational efficiency from 107 to 118, accompanied by +1.2% gain in Top-1 accuracy, Similar trends are observed in MobileNet-V2 where GMPQ-TE achieves lower bit-width configurations with consistent accuracy improvements, such as +2.6% Top-1 gain in the MobileNet- V2-2 setting, These

---

Table 11: Experimental results for MobileNet- V2 on average model bit-width.
Method
Params.
BOPs
Comp.
Top-1
Top-5
MobileNet-Var V2-1
1.14
8.7
82.6
71.04
88.70
GMPQ-TE: TE-1
1.10
4.5
74.8
71.20
89.90
MobileNet-Ve V2-2
1.24
7.1
55.7
67.20
87.31
GMPQ-TE-2
1.30
7.3
46.1
69.80
89.70
MobileNer- V2-3
1.56
49.2
70.24
89.73
GMPQ-TE-3
1.40
10.1
33.4
71.80
90.20
Bi (CIPAR 10)
TE (CIPAR-
Ri is (InageNet) (InageNet)
Bir Assigrm igrmont
TE Values
Figure 8: Comparison of layer-wise bit assignment and TE values for ResNet-18 18 under different calibration datasets.
findings confirm that GMPQ-TE provides a favorable trade-off between quantization bit-width and predictive performance.
D.4 Bit Assignment and TE Values
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 guantization policies are derived accordingly (see Figure. 8) For different models, the relationships between bit width assignment are shown in Figures
D.5 Real-Worid Hardware Deployment
Table 12 shows that quantized models achieve low inference latency and high throughput across both GPU and edge platforms. On RTX 3090, MobileNet-V V2 runs at over 800 FPS with only 1.2 ms latency. Even on Jetson Nano, it maintains real-time performance under 1OW power budget. These results confirm that GMPQ-TE enables efficient and serdware-ffiendle deployment without sacrificing accuracy.
The applicability of TE is validated 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 convolutional models. Fundamentally, TE is architecture-agnosiic structural properties of intermediate feature maps, and does not rely architectural assumptions such depth, convolutional inductive bias, or model width. This enables it to generalize well even shallow or narrow models. Nonetheless, we acknowledge the value of further exploring TE on ultra-compact or task-specialized architectures.
ResNet50). This demonstrates that TE is not limited large or deep
derived directly from the
30

---

7.5
Bic Assigroont
81 Toneecct
TF Values
6.5
0.0
4.8
10
20
50
Bit Assi
7.0
Vallues
B11 Mest WesTammmn
Bagges
5.0
5.0
4.6
3.5
Parens: 9.4
Assignaent
7.0
1811 0000 207M00
6.0
5.0
50
Parmms:
11.2
Figure Bit assignment and TE values for ResNet-50.
BIt Ass farmontnnt TE. Yo lues
Jasi graient Kalsos
Bit Asafamont Yaluns
Dac Iestimaann ma
16 9alues
11:
2.6
16
10
18
10
12
14
20
Figure 10: Bit assignment and TE values for 18.
Table 12: Quantized models achieve low inference latency and high throughput across both GPU and edge platforms. On RTX 3090, MobileNet- V2 runs over 800 FPS with only 1.2 ms latency. Even on Jetson Nano, it maintains real-time performance under 10W power budget. These results confirm that GMPQ-TE enables efficient and mordware-friendee deployment without sacrificing accuracy.
Model
Platform
Inference Latency (ms)
Throughput (FPS)
Avg. INTS Kemel Time (ms)
Power Consumption
ResNet-18
NVIDIA RTX 3090
~1.1
4470
~115
Res-81118
Jetson Xavier NX
1332
-75
13.9
<-18
MobileNea-V2
MobileNet-V2 V2
Jesson Nano (10W)
~245
-38
25.3
~8.5
31

---

6.5
Bit Assigmenn
6.0
TE Values
0.5
5.0
4.5
4.0
0.0
2.5
1.5
20
40
60
Parans:
1.4
1.3
BL Assigment
7.0
TE Values
6.3
6.0
0.5
5.0
4.5
3.5
1.0
10
20
30
40
50
60
Parans:
1,3
Bitt Assi ininnnt
Bic Ass geneent
6.3
6.5
Parans:
1.
Figure 11: Bit assignment and TE values for MobileNet- V2.
32


Reviews:
[{"review_id": "F4gV8zIguj", "reviewer": "Reviewer_ErFk", "summary": "The authors analyze topological factors affecting the quantization sensitivity of neural networks from a geometric perspective. Topological entropy, defined as inter-channel information based on Birth Time distribution, is developed as a metric for quantization sensitivity. The proposed topological entropy is theoretically demonstrated to possess resolution- and label-independent properties. Furthermore, the authors formulate an LP problem to solve for the optimal precision distribution in GMPQ with the topological entropy metric. A theoretical performance degradation bound for this quantization scheme is established and theoretical analysis on the applicability to QAT is provided. Experimental results validate the label independence of topological entropy, and report quantization results with GMPQ-TE across standard datasets, demonstrating competitive performance.", "questions": "1. What topological structure does non-square weighted adjacency matrix represent? How can Betti numbers be properly defined on such a matrix?\n\n2. Is there mathematical guarantee that the transformation in Equation (3) preserves the validity of topological entropy measurement? \n\n3. Is there substantive evidence demonstrating that applying clique filtration to a Graph whose adjacency matrix is exactly feature matrix effectively extracts 'expressing patterns' as asserted? Could more intuitive validation be provided?\n\n4. Can the theoretical properties of your proposed topological entropy be formally proven under Attention mechanisms?\n\n5. To what extent are the consistent trends in Figures 5-7 directly attributable to feature map size variations rather than feature map patterns? Does the Linear Programming algorithm account for this scaling effect? \n\n6. Could the authors provide results on common architectures such as ResNet-18 or MobileNet-V2?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. This paper establishes a novel framework for mixed-precision quantization, theoretically guaranteed to be label-independent thus achieve competitive performance in GMPQ tasks.\n\n2. The GMPQ-TE method maintains strong model performance while significantly reducing quantified computational costs.\n\nWeaknesses:\n\n1. The geometric interpretation of the feature map space and the motivation for the proposed formulation of topological entropy are ambiguous. Furthermore, the significance of applying filtration operations to the feature map space lacks theoretical support. \n\ni) The weighted adjacency matrix $\\mathbf W$ might be non-square, which could contradict with fundamental principles in graph theory. It remain unclear what topological structure is represented and whether Betti numbers can be properly defined on such a matrix.\n\nii) The transformation in Equation (3) could discard up to half of the feature matrix's information. There is no mathematical guarantee that this operation preserves the validity of topological entropy measurement. Additionally, Equation (3) appears to be an assignment operation rather than a mathematical formulation.\n\niii) Substantive evidence is necessary to demonstrate that applying clique filtration to a Graph whose adjacency matrix is exactly feature matrix effectively extracts 'expressing patterns' as asserted. \n\n2. Theoretical or empirical explanations are not sufficient to substantiate the experimental findings. To my best knowledge, theoretical justification for the label-independence of topological entropy relies on properties specific to convolutional kernels. It remains a problem whether the theoretical properties of the proposed topological entropy can be formally proven under Attention mechanisms, despite experimental results demonstrate that the quantization scheme remains effective for Transformer models.\n\n3. Some of experimental validation is unconvincing and undermines the reliability of the conclusions. \n\ni) Based on my understanding, the value range of Birth Time appears intrinsically scaled with graph size. This implies that Birth Time distributions naturally exhibit greater dispersion on larger feature maps, causing topological entropy values to systematically increase with feature map size. It is hard to determine the consistent trends in Figures 5-7 is due to feature map size variations rather than feature map patterns.\n\nii) it is not specified whether the Linear Programming algorithm accounts for this scaling effect. If not, the quantization process risks degenerating into mere feature map size sorting.\n\niii) Experimental results for comparing quantization performance with average model bit-width on common architectures such as ResNet-18 or MobileNet-V2 are missing.\n\n4. Neither relevant repository links nor access instructions appear to be provided in the manuscript.", "was_revised": true, "final_justification": "I would like to thank the authors for providing detailed explanation on their work. After rebuttal, I agree that this work introduces an interesting idea of using topological entropy for mixed precision quantization and achieves impressive experimental performance. To this end, I raise my rating.", "rebuttal": "**We sincerely thank reviewer for pointing out these important issues.**\n\n**Ans. For W1** \n\n*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.\n\n*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:\n\n$b(U^{(i)}) = \\inf \\{v \\mid \\beta_1(K^{(v)}) \\neq 0 \\}$\n\nwhere $\\beta_1(K^{(v)})$ is first Betti number at filtration level $v$. Across a batch of $M$ same-label samples, we define distribution:\n\n$P_{U^{(i)}}(x) = \\frac{1}{M} \\sum_{m=1}^M \\mathbb{I}(b(U^{(i)}_m) = x)$\n\nand compute Shannon entropy:\n\n$H_i = - \\sum_x P_{U^{(i)}}(x) \\log P_{U^{(i)}}(x)$\n\nLow $H_i$ 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.\n\n*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,\n\n$ f(j,k) = U^{(i)}_{j,k}, \\quad \\text{with } (j,k) \\in \\mathcal{M} $\n\nGiven this scalar function, sublevel set at threshold $\\tau$ is defined as:\n\n$\\mathcal{M}_\\tau = f^{-1}([-\\infty, \\tau]) = { (j,k) \\in \\mathcal{M} \\mid f(j,k) \\le \\tau }$\n\nAs $\\tau$ 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 $\\tau$. \n\nThis process defines a filtration $\\mathcal{M}_\\tau$, a fundamental object in persistent homology. \n\nTo compute homology, we construct simplicial complexes over $\\mathcal{M}_\\tau$. \n\nIn 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:\n\n$H_k(K_{\\tau_1}) \\to H_k(K_{\\tau_2}) \\to \\cdots \\to H_k(K_{\\tau_L})$\n\nPersistent 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**).\n\n**Ans. For W1-i** \n\nYes, 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:\n\n$f_{j,k} = \\frac{1}{d} \\sum_{c=1}^{d} U_{(j,k), c}, \\quad \\text{for } (j,k) \\in \\{1, \\dots, h\\} \\times \\{1, \\dots, w\\}$\n\nIf $N$ is not a perfect square, we embed grid into a minimal square domain:\n\n$n = \\left\\lceil \\sqrt{N} \\right\\rceil$\n\nWe 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.\n\nWe 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:\n\n$\\bar{X}_\\tau = \\{ x \\in \\bar{X} \\mid \\tilde{f}(x) \\ge \\tau \\}$\n\nSince $\\tilde{f}(x) = 0$ 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 $\\tau > 0$.\n\nEven 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 $, and:\n\n$H_k(C) = 0, \\quad \\text{for all } k \\ge 1$\n\nThus, the inclusion map $X_\\tau \\hookrightarrow \\bar{X}_\\tau$ induces an isomorphism:\n\n$H_k(X_\\tau) \\cong H_k(\\bar{X}_\\tau), \\quad \\forall k \\ge 1$\n\nWe 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.\n\n**Ans. For W1-ii** \n\nWe 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.\n\n**Ans. For W1-iii**\n\nWe 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.\n\n**Ans. For W2** \n\nLet $\\tilde{f} : \\mathcal{M} \\rightarrow \\mathbb{R}$ be the scalar field from intermediate activations. Attention mechanisms produce structured outputs over $\\mathcal{M} \\subset \\mathbb{Z}^2$. Define:\n\n$\\mathcal{M}_\\tau := \\{ x \\in \\mathcal{M} \\mid \\tilde{f}(x) \\ge \\tau \\}$\n\nFrom this we build $K_\\tau$ and compute birth times $b_i$, defining:\n\n$\\mathrm{TE} := -\\sum_x P(x) \\log P(x), \\quad P(x) = \\frac{1}{M} \\sum_{i=1}^M \\delta(b_i - x)$\n\nAll steps in TE (projection, padding, filtration, homology, entropy) are label-free. For input $(x,y)$:\n\n$\\mathrm{TE}(x, y) = \\Phi(\\tilde{f}(x)), \\quad \\frac{\\partial \\mathrm{TE}}{\\partial y} = 0$\n\nTherefore, TE is invariant to label supervision and is a purely geometric, architecture-agnostic descriptor.\n\n**Ans. For W3-i** \n\nWe 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.\n\n**Ans. For W3-ii** \n\nTE 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.\n\nThe 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.\n\nIn practice, some small feature maps still receive higher bit-widths, showing that the LP solution captures structural sensitivity beyond size. \n\n**Ans. For W3-iii** \n\nResults for average bit-width:\n\n### ResNet-18\n\n| Method       | Params | BOPs | Comp | Top-1 | Top-5 |\n|--------------|--------|------|------|--------|--------|\n| ResNet-18-1  | 5.4    | 30.1 | 60.6 | 69.14  | 88.73  |\n| GMPQ-TE-1    | 5.3    | 27.4 | 67.6 | 70.4   | 89.4   |\n| ResNet-18-2  | 3.7    | 23.4 | 107  | 68.2   | 88.13  |\n| GMPQ-TE-2    | 3.8    | 15.7 | 118  | 69.4   | 89.4   |\n| ResNet-18-3  | 3.5    | 7.4  | 254  | 66.1   | 87.4   |\n| GMPQ-TE-3    | 3.5    | 7.1  | 261  | 68.3   | 88.9   |\n\n### MobileNet-V2\n\n| Method         | Params | BOPs | Comp | Top-1 | Top-5 |\n|----------------|--------|------|------|--------|--------|\n| MobileNet-V2-1 | 1.14   | 8.7  | 82.6 | 71.04  | 88.7   |\n| GMPQ-TE-1      | 1.1    | 4.5  | 74.8 | 71.2   | 89.9   |\n| MobileNet-V2-2 | 1.24   | 7.1  | 55.7 | 67.2   | 87.31  |\n| GMPQ-TE-2      | 1.3    | 7.3  | 46.1 | 69.8   | 89.7   |\n| MobileNet-V2-3 | 1.56   | 9.4  | 49.2 | 70.24  | 89.73  |\n| GMPQ-TE-3      | 1.4    |10.1  | 33.4 | 71.8   | 90.2   |\n\n**Ans. For W4:**\n\nWe provide source code related to the paper in the supplementary materials.\n\n**Regarding the question part, the answer can be found above.**\n\n**References**\n\n[1]Persistence images: A stable vector representation of persistent homology\n\n[2]On time-series topological data analysis: New data and opportunities"}, {"review_id": "fFWeMMzO5a", "reviewer": "Reviewer_pnyk", "summary": "This paper presents a bit-width assignment scheme based on topological entropy for MPQ. It efficiently measures the sensitivity of each layer without iterative optimization, given a minibatch of data containing the same label. The sensitivity is estimated with the entropy of birth time distribution, which generalizes well across different datasets. Experimental results show the effectiveness of the proposed method.", "questions": "My major concern is the clarity of the paper. I will raise my ratings if the authors clarify my concerns. In addition to the weaknesses, I have a few questions as follows:\n\nIt is hard to understand why graph in Eq.~(1) has an adjacency matrix of $h^* \\times w^*$, if the number of vertices is equal to the number of pixels in the feature map. I think the authors need to clarify how the set of vertices $V$ is formulated, in addition to the shape of adjacency matrices.\n\nIn addition, the adjacency matrix in Eq.~(1) does not seem to represent similarity between pixels; I think it is just the pixel value, according to the notation. Is there any reason why the adjacency matrix is set to the pixel value of intermediate feature maps?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "[Strengths]\n\nThe authors propose a sensitivity metric for MPQ that does not require iterative optimization. The bit-widths for each layer are assigned within seconds, based on the solutions of a linear programming problem (Eq.~(9)). It also generalizes well to other datasets.\n\nExperimental results clearly demonstrate the effectiveness of the proposed criteria for MPQ. The paper also includes results on various tasks (e.g., classification, detection) and transformer-based models (e.g., ViT, Swin), demonstrating the generalizability of the proposed component. It also includes results on various quantization scenarios (e.g., QAT, PTQ).\n\n[Weaknesses]\n\n1. I find it difficult to see a clear justification for why topological entropy serves as a reliable proxy for estimating layer sensitivity, despite the explanation provided in L225–230 and the empirical results (Fig. 6). The authors argue that common features across images can be stably captured when the entropy of birth time distributions exhibits a degenerate-like behavior. However, this claim relies on the assumption that images within the same class contain topologically similar objects. In practice, real-world images often contain objects that are truncated, occluded, or vary significantly in appearance even for the images with the same label, potentially invalidating this assumption. Furthermore, the justification remains largely empirical—the authors demonstrate that their sensitivity metric generalizes across datasets, but no theoretical grounding is provided to explain why topological entropy is inherently suited for this purpose.\n\n2. The proof in Appendix A.2 assumes convolution layers with fixed receptive fields; however, it remains unclear whether the properties of topological entropy can be extended to transformer-based models, which do not contain convolutional layers.\n\n3. I think a figure visualizing betti curve, and birth time during the process of clique filtration would strengthen the paper’s presentation, especially for readers not familiar with graph theory and topological data analysis.\n\n4. This paper contains a few errors and unclear notations as follows:\n\nL162: If there are multiple instances in a minibatch, the dimension should be $b \\times c \\times h \\times w$, where $b$ is the number of instances.\n\nL166: A missing notation $\\in$.\n\nL204: The notation for birth time needs to be revised.\n\nEq. (5): The notation $inf$ needs to be explained.\n\nEq. (6): The notation $m$ needs to be explained. In addition, the authors state that $\\sum$ could be set as a common discrete $\\sigma$-field, but there are no explanations about $\\sigma$-field.\n\nL248: According to the graph in Fig. 4, “positively” should be replaced with “negatively”.\n\nEq. (9): According to the explanation from L254, the objective function needs to be maximized, rather than minimized.", "was_revised": true, "final_justification": "The authors have clarified the relationship between the task loss and topological entropy, and have further demonstrated the applicability of topological entropy to transformer-based architectures. They have also provided a clear formulation of the adjacency matrix in Eq.~(1). To this end, I am updating my rating from \"borderline reject\" to \"borderline accept\".", "rebuttal": "**We sincerely thank reviewer for pointing out these important issues**\n\n**Ans. For W1:** \n\nTE 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. \n\nThe 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:\n\n$\\min_{\\{B_i\\}} \\sum_{i=1}^L \\Delta \\mathcal{L}_i(B_i)$\n\n$\\text{s.t.}$ $\\sum_{i=1}^L M(B_i) \\leq T$\n\nwhere $( \\Delta \\mathcal{L}_i(B_i) )$ denotes task loss increase due to quantization of layer $ i $, and $ M(B_i )$ is corresponding resource cost. \n\nWe 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:\n\n$\\hat{W}_i = W_i + \\delta W_i, \\quad \\| \\delta W_i \\| \\leq \\varepsilon$\n\nAssuming a Lipschitz continuous forward operator $f^{(i)}$, the pre- and post-quantization feature maps become:\n\n$U^{(i)} = f^{(i)}(W_i, X), \\quad \\hat{U}^{(i)} = f^{(i)}(\\hat{W}_i, X)$\n\nBy Lipschitz continuity, the output deviation is bounded by:\n\n$\\| U^{(i)} - \\hat{U}^{(i)} \\| \\leq L_i \\cdot \\| W_i - \\hat{W}_i \\| \\leq L_i \\varepsilon$\n\nThis perturbation in feature space leads to structural changes in the persistence diagrams, with bottleneck distance bounded by:\n\n$d_B(D(U^{(i)}), D(\\hat{U}^{(i)})) \\leq \\| U^{(i)} - \\hat{U}^{(i)} \\| $\n\nGiven 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:\n\n$\\| P_{U^{(i)}} - P_{\\hat{U}^{(i)}} \\|_1 \\leq \\alpha_i \\cdot \\| W_i - \\hat{W}_i \\| \\leq \\alpha_i \\varepsilon$\n\nShannon entropy is Lipschitz continuous with respect to $\\ell_1$ distance, yielding the following bound:\n\n$| H_i^{(0)} - H_i^{(B)} | \\leq C_i \\cdot \\| P_{U^{(i)}} - P_{\\hat{U}^{(i)}} \\|_1 \\leq C_i \\alpha_i \\varepsilon$\n\nWe now relate entropy variation to task loss degradation. The increase in loss can be estimated by a second-order Taylor expansion:\n\n$\\Delta \\mathcal{L}_i \\leq \\frac{1}{2} \\lambda_i \\| \\delta W_i \\|^2 + \\frac{1}{6} \\kappa_i \\| \\delta W_i \\|^3$\n\nWe 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:\n\n$\\Phi\\left( \\| P_{U^{(i)}} - P_{\\hat{U}^{(i)}} \\|_1 \\right) \\leq \\Phi\\left( | H_i^{(0)} - H_i^{(B)} | \\right)$\n\nThis 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:\n\n$\\min_{\\{B_i\\}} \\sum_{i=1}^{L} H_i \\cdot B_i \\quad \\text{s.t.} \\quad \\sum_{i=1}^{L} M(B_i) \\leq T$\n\nTo ensure alignment with sensitivity ranking, we enforce:\n\n$H_i > H_j \\Rightarrow B_i > B_j$\n\nThis ordering remains consistent during training due to entropy drift stability, as shown by:\n\n$|H_i^{(t)} - H_i^{(0)}| \\leq \\Delta_{q,i} \\log M_i \\cdot L_i \\quad \\Rightarrow \\quad \\operatorname{sign}(H_i - H_j) = \\operatorname{sign}(H_i^{(t)} - H_j^{(t)})$\n\nwhere $\\operatorname{sign}(H_i - H_j)$ 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.\n\n**Ans. For W2:** \n\nProof of label independence for transformer models can be found in **Ans. For W1-ii from Reviewer ErFk**. \n\n*Resolution-Independence of Topological Entropy under Attention (Sketch Proof)*\n\nLet 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:\n\n$Z = \\mathrm{softmax}\\left( \\frac{QK^\\top}{\\sqrt{d}} \\right)V,\\quad Q= XW_Q,\\quad K = XW_K,\\quad V = XW_V$\n\nEach output token is:\n\n$z_i = \\sum_{j} \\alpha_{ij} v_j,\\quad \\alpha_{ij} = \\frac{\\exp(q_i^\\top k_j / \\sqrt{d})}{\\sum_{j'} \\exp(q_i^\\top k_{j'} / \\sqrt{d})}$\n\nAssume two inputs $X^{(1)}$ and $X^{(2)}$ come from different resolutions but the same class. If the local patch statistics are resolution-consistent:\n\n$P(x_S^{(1)}) = P(x_S^{(2)}) \\Rightarrow P(z_i^{(1)}) = P(z_i^{(2)}),\\quad \\forall i$\n\nAfter 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.\n\nSince attention is a permutation-equivariant global operator that preserves relative structure, we obtain:\n\n$P_b^{(1)} = P_b^{(2)} \\Rightarrow H_{\\mathrm{TE}}^{(1)} = H_{\\mathrm{TE}}^{(2)}$\n\nThus, TE is invariant to resolution under generic mechanisms. *This is a sketch proof and full derivation will be included upon acceptance.*\n\n**Ans. For W3:** \n\nWe 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:\n\n$\\tau^{(1)}$: add edge (1, 2)\n\n$\\tau^{(2)}$: add edge (3, 4)\n\n$\\tau^{(3)}$: continue adding edges, no cycle yet\n\n$\\tau^{(4)}$: form a 1-cycle: (1-2-4-3-1)\n\n$\\tau^{(5)}$: add edge (2, 3), forming more cliques\n\n$\\tau^{(6)}$: complete graph with all edges\n\nTo 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:\n\n$v = [0,\\ 1,\\ 2,\\ 3,\\ 4,\\ 5,\\ 6] \\quad \\Rightarrow \\quad \\beta = [0,\\ 0,\\ 0,\\ 0,\\ 1,\\ 0,\\ 0]$\n\nThis 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. \n\n**Ans. For W4**: \n\nWe carefully correct errors or unclear notations, as follows:\n\nL162: 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.\n\nL166: $U^{(i)} \\in$ $\\mathbb{R}^{h^{\\ast}\\times w^{\\ast}}$\n\nL204: The notation for birth time $b$ is revised to $b_t$.\n\nEq. (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).\n\nEq. (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 $(\\Omega, \\Sigma, P)$.\n\nL248: The“positively” is revised to “negatively”\n\nEq. (9): The objective function is revised to maximize.\n\n\n**Ans. For Q1** \n\nThe 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. \n\n**Ans. For Q2** \n\nWe 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].  \n\nSeparately, 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].\n\nUsing 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].\n\nTo 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.\n\n**Reference**\n\n[1]  Topological data analysis with applications\n\n[2] On time-series topological data analysis: New data and opportunities\n\n[3] Applications of topological data analysis in oncology\n\n[4] A short survey of topological data analysis in time series and systems analysis"}, {"review_id": "hjt9IZKbRo", "reviewer": "Reviewer_8djE", "summary": "The paper proposes GMPQ-TE, mixed precision quantization method by leveraging topological entropy. The metric is computed using minibatches of data with constructing adjacency matrix based on spatial dimension of the intermediate featuremap of each layer. Based on the measured entropy from distribution, bit-width per each layer is decided with proposed linear programming method leading to assign lower bit-widths to lower layers..", "questions": "- Question\n1. What is the result of bit-width assigned per layer with the entropy displayed on the paper? Showing selected bit-width per budget on entropy figures will give more understanding of entropy to bit-width selection.\n2. How much benefits TE can offer compared to hessian based method like HAWQ in terms of search cost? Can we say TE is better than HAWQ in this aspect?\n3. Why did you set calibration data as CIFAR10 to get topological entropy? Are there any calibration dataset dependency for measuring entropy? To state that the proposed TE is practical, its robustness on calibration data must be demonstrated.\n4. Are there any difference on entropy measurement when using different resolution of calibration dataset? (Like measuring the difference between only using CIFAR10 as calibration dataset and Imagenet as calibration dataset) Since the proposed method is relying on the spatial information of the feature map, resolution of data can affect the result of measurement.\n5. Is distillation with full precision model applied on QAT?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strength\n1. The paper firstly proposes utilizing entropy of a distribution from graph reflecting spatial patterns on intermediate feature map of a model. Existing methods does not reflect the information from the feature map on spatial dimension for bit-width selection.\n2. The topological entropy method offers fast searching of bit-width compared to existing methods with iterative searching on PTQ and learning based searching on QAT.\n3. Clear figures and equations for describing idea proposed in paper. Figure 3 will help a lot for readers to understand the concept of the paper.\n\n- Weakness\n1. Lack of comparison with state-of-the-art models on PTQ, i.e., EMQ [1].\n2. Lack of comparison with state-of-the-art models on QAT, i.e., SDQ [2], MetaMix [3].\n3. Evaluation is conducted on a bit old models in classification task, i.e., MobileNet-v2, ResNet-18.\n\n[1] Dong et al., Emq: Evolving training-free proxies for automated mixed precision quantization\n\n[2] Huang et al., SDQ: Stochastic Differentiable Quantization with Mixed Precision\n\n[3] Kim et al., MetaMix: Meta-state Precision Searcher for Mixed-precision Activation Quantization", "was_revised": true, "final_justification": "The authors were not able to demonstrate clear benefits over existing works in QAT. The proposed method yields inferior results while requiring significantly more computation. In particular, highly optimized models such as MobileNets are inherently difficult to quantize and the authors have failed to show that their method provides any advantage in such scenarios.", "rebuttal": "**We sincerely thank reviewer for pointing out these important issues**\n\n**Ans. For W1-2** \n\nThanks, 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.\n\n### MobileNet-V2 (PTQ)\n\n| Method  | Params. | BOPs   | Top-1 | Cost     |\n|---------|---------|--------|-------|----------|\n| EMQ | 1.5  |     | 70.75    | few seconds   |\n| GMPQ-TE | 1.4    | 4.5   | 71.8| 11|\n\n### ResNet-18 (PTQ)\n\n| Method  | Params. | BOPs   | Top-1  | Cost     |\n|---------|---------|--------|--------|----------|\n| EMQ | 4 |    | 69.92    | few seconds   |\n| GMPQ-TE | 3.8    | 7.1   | 69.6   | 16 |\n\n\n### MobileNet-V2 (QAT)\n\n| Method  | Params. | BOPs   | Top-1 | \n|---------|---------|--------|-------|\n| MataMix |         | ＞8    | 73    | \n| SDQ     | 1.8     | 4.89   | 72    | \n| GMPQ-TE | 1.1     | 7.1   | 70.42    |\n\n### ResNet-18 (QAT)\n\n| Method  | Params. | BOPs   | Top-1  | \n|---------|---------|--------|--------|\n| MataMix |         | ＞35   | 72     | \n| SDQ     | 5.2     | 15.7   | 69.1   | \n| EMQ     | 6.69    | 71     | 72.28  | \n| GMPQ-TE     | 4.21     | 20.3  | 69.8   | \n\nOverall, compared with the above methods, GMPQ-TE achieves comparable or better accuracy in PTQ and QAT scenarios with fewer Params. \n\n**Ans. For Q1:** \n\nFor different models, the relationship between bit width assignment and TE is as follows:\n\n*ResNet-18:* \n\nParams: 5.3\n\nBit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE 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]\n\nParams: 3.8\n\nBit Assignment: [8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3]\n\nTE 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]\n\nParams: 3.5\n\nBit Assignment: [6, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2]\n\nTE 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]\n\n*ResNet-50:* \n\nParams: 11.2\n\nBit Assignment: [8, 6, 6, 6, 6, 6, 6, 6, 6, 6,\n 6, 6, 6, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE Values: [7.0, 6.8, 6.7, 6.6, 6.5, 6.4, 6.3, 6.2, 6.1, 6.0,\n 5.9, 5.8, 5.7, 5.6, 5.6, 5.5, 5.4, 5.3, 5.3, 5.2,\n 5.2, 5.1, 5.1, 5.0, 5.0, 4.9, 4.9, 4.8, 4.8, 4.7,\n 4.6, 4.4, 4.3, 4.2, 4.2, 4.1, 4.1, 4.0, 4.0, 3.9,\n 3.9, 3.8, 3.8, 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 6.1]\n\nParams: 9.4\n\nBit Assignment: [8, 6, 6, 6, 6, 6, 6, 5, 5, 5,\n 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 5]\n\nTE Values: [6.8, 6.1, 6.0, 5.9, 5.8, 5.7, 5.6, 5.5, 5.4, 5.3,\n 5.2, 5.1, 5.0, 4.9, 4.9, 4.8, 4.8, 4.7, 4.6, 4.6,\n 4.5, 4.4, 4.3, 4.2, 4.2, 4.1, 4.1, 4.0, 4.0, 3.9,\n 3.9, 3.8, 3.8, 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 3.4,\n 3.3, 3.3, 3.2, 3.2, 3.1, 3.1, 3.2, 3.3, 3.0, 3.1]\n\nParams: 8.1\n\nBit Assignment: [6, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 5, 5, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE Values: [6.8, 6.4, 6.3, 6.2, 6.1, 6.0, 5.9, 5.8, 5.7, 5.6,\n 5.5, 5.4, 5.3, 5.2, 5.1, 5.0, 5.0, 4.9, 4.8, 4.7,\n 4.6, 4.5, 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.9, 3.8,\n 3.7, 3.7, 3.6, 3.5, 3.5, 3.4, 3.3, 3.3, 3.2, 3.2,\n 3.1, 3.1, 3.1, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 6.6]\n\nMobileNet-V2: \n\nParams: 1.4\n\nBit Assignment: [8, 8, 8, 8, 8, 8,\n 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,\n 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 5,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5,\n 5.4, 5.3, 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5,\n 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.8, 3.7, 3.6, 3.5,\n 3.5, 3.4, 3.3, 3.2, 3.1, 3.0,\n 2.9, 2.8, 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0,\n 2.0, 2.0, 1.9, 1.8, 1.8, 1.7, 1.6, 1.6, 1.5, 1.4, 5.2]\n\nParams: 1.3\n\nBit Assignment: [8, 8, 8, 8, 8, 8,\n 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,\n 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 5,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5,\n 5.4, 5.3, 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5,\n 4.4, 4.3, 4.2, 4.1, 4.0, 3.9, 3.8, 3.7, 3.6, 3.5,\n 3.5, 3.4, 3.3, 3.2, 3.1, 3.0,\n 2.9, 2.8, 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0,\n 1.9, 1.8, 1.7, 1.6, 1.6, 1.5, 1.5, 1.5, 1.5, 1.5, 5.0]\n\nParams: 1.1\n\nBit Assignment: [6, 6, 6, 6, 6, 6,\n 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,\n 5, 5, 5, 5, 5, 5,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,\n 4, 4, 4, 4, 6, 4, 4, 4, 4, 4,  4] \n\nTE Values: [6.0, 5.9, 5.8, 5.7, 5.6, 5.5,\n 5.2, 5.1, 5.0, 4.9, 4.8, 4.7, 4.6, 4.5, 4.4, 4.3,\n 4.2, 4.1, 4.0, 3.9, 3.9, 3.8,\n 3.7, 3.6, 3.5, 3.4, 3.3, 3.2, 3.1, 3.0, 2.9, 2.8,\n 2.7, 2.6, 2.5, 2.4, 2.3, 2.2, 2.1, 2.0, 1.9, 1.8,\n 1.7, 1.6, 1.5, 1.5, 5.4, 1.5, 1.4, 1.5, 1.5, 1.3, 1.5]\n\n**Ans. For Q2:**\n\nCompared 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. \n\n**Ans. For Q3:**\n\nThanks. 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. \n\nTo 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.\n\nIn 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:\n\nCIFAR-10:\n\nBit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 6]\n\nTE 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]\n\nImageNet:\n\nBit Assignment: [8, 8, 8, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 5]\n\nTE 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]\n\nAlthough 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. \n\n**Ans. For Q4** \n\nWe 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.\n\nIn 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.\n\nTherefore, 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.\n\n\n**Ans. For Q5** \n\nNo, 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."}, {"review_id": "4SXK3DJBQj", "reviewer": "Reviewer_E1Lz", "summary": "The paper introduces a mixed-precision quantization method (GMPQ-TE) based on topological entropy for compressing deep neural networks to fit resource-constrained devices. Using topological data analysis, the method measures layer-wise quantization sensitivity with topological entropy, enabling a single-pass linear programming approach to determine the quantization policy. Experiments demonstrate that a policy trained on CIFAR-10 generalizes to ImageNet and PASCAL VOC.", "questions": "Besides the weaknesses above:\n\nThe results in Tables 4, 7, and 8 lack error bars or confidence intervals. Please provide a statistical analysis and specify the method used for computation. \n\nHardware deployment is mentioned but lacks specific performance metrics. Please provide detailed deployment results for platforms such as GPUs, TPUs, or edge devices. \n\nThe paper does not sufficiently compare topological entropy to other quantization sensitivity metrics. Please include comparative experiments or theoretical analysis to highlight its unique advantages.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\nThe application of topological entropy to mixed-precision quantization is a novel contribution, leveraging topological data analysis to assess quantization sensitivity. By analyzing the topological structures of feature maps via Betti curves and persistent homology, the method links quantization sensitivity to topological entropy.\n\nThe method’s efficiency and generalization are impactful for deploying models on resource-constrained devices (e.g., mobile platforms). Its cross-dataset generalization reduces the need for dataset-specific policy searches, and the label-independence of topological entropy enhances its applicability across diverse scenarios.\n\nGMPQ-TE employs a single-pass linear programming approach to assign bit-widths based on topological entropy, thereby avoiding the iterative search overhead associated with traditional methods. \n\nWeaknesses:\n\nExperimental results (Tables 4, 7, 8) lack statistical significance analysis, which may undermine confidence in their reliability. The evaluation is limited to vision tasks with no exploration of non-vision tasks or non-convolutional architectures.\n\nThe paper does not sufficiently compare topological entropy to other quantization sensitivity metrics, limiting the demonstration of its unique advantages.\n\nReal-world hardware deployment is mentioned, but it lacks detailed performance metrics, which restricts the evaluation of its practical applicability. The applicability of topological entropy to small networks or specific architectures remains underexplored.", "was_revised": true, "final_justification": "Thank the authors for the submission and the feedback during the rebuttal. Also, thanks for the other reviewers' comments, which all help the reviewer to finalize the evaluation (still keep positive).", "rebuttal": "**We sincerely thank reviewer for pointing out these important issues**\n\n**Ans. For W1:** \n\nTo 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.\n\n*Table 4: Results for image classification on ResNet-18 (PTQ)*\n| Methods        | Params.   | BOPs     | Comp.     | Top-1     | Top-5     | Cost  |\n|----------------|-----------|----------|-----------|-----------|-----------|--------|\n| Full-precision | 46.8      | 1853.4   | –         | 69.7      | 89.2      | –      |\n| HAWQ           | 5.8       | 34       | 54.5      | 68.5      | –         | 56K    |\n| GMPQ†          | 5.4       | 28.4     | 65.2      | 69.2      | 89.1      | 1.8K   |\n| R-GMPQ†        | 5.3       | 27.9     | 66.4      | 70.4      | 89.7      | 1.8K   |\n| OMPQ∗          | 6.7       | 75       | 24.7      | 70.1      | 89.3      | –      |\n| **GMPQ-TE**    | 5.3       | 27.4     | 67.6      | 70.3      | 89.4      | 19     |\n| → Mean ± Std   | 5.3 | 27.90±0.41 | 66.40±0.98 | 69.97±0.54 | 89.40±0.24 | 19.03±0.03 |\n| APoT           | 4.6       | 16.3     | 11.38     | 69.8      | –         | –      |\n| GMPQ†          | 3.8       | 15.7     | 118       | 69.4      | 89.4      | 2.1K   |\n| R-GMPQ†        | 3.8       | 15.6     | 118.7     | 69.4      | 89.1      | 2.5K   |\n| **GMPQ-TE**    | 3.8       | 15.7     | 118       | 69.6      | 89.4      | 18     |\n| → Mean ± Std   | 3.8 | 15.67±0.05 | 118.23±0.33 | 69.47±0.09 | 89.30±0.14 | 18±0.21|\n| ALQ            | 3.4       | 7.2      | 256       | 66.4      | –         | 138.6K |\n| EdMIPS         | 4.7       | 7.2      | 258       | 65.9      | 86.5      | 34.2K  |\n| GMPQ†          | 3.7       | 7.2      | 255.8     | 67.1      | 88        | 3.2K   |\n| R-GMPQ†        | 3.5       | 7.2      | 258.5     | 67.9      | 88.7      | 3.9K   |\n| **GMPQ-TE**    | 3.5       | 7.1      | 260.9     | 68.3      | 88.9      | 16     |\n| → Mean ± Std   | 3.5| 7.17±0.05 | 258.40±2.08 | 67.77±0.50 | 88.53±0.39 | 15.4±0.32 |\n\n*Table 7: Results for image classification across various ViT and Swin transformer models (PTQ).*\n| Methods       | W/A   | ViT-S     | ViT-B     | ViT-L     | DeiT-T    | DeiT-S    | DeiT-B    | Swin-T    | Swin-S    | Swin-B    |\n|---------------|-------|-----------|-----------|----|----|---|-----|------|------|-----------|\n| Full-precision| 32/32 | 81.39     | 84.53     | 85.84     | 72.18     | 79.85     | 81.80     | 81.37     | 83.21     | 85.27     |\n| PTQ4ViT       | 6/6   | 78.63     | 81.65     | 84.79     | 69.62     | 76.28     | 80.25     | 80.47     | 82.38     | 84.01     |\n| PD-Quant      | 6/6   | 70.84     | 75.82     | -         | -         | 78.33     | -         | -         | -         | -         |\n| APQ-ViT       | 6/6   | 79.10     | 82.21     | -         | 70.49     | 77.76     | 80.42     | -         | 82.67     | 84.18     |\n| NoisyQuant    | 6/6   | 79.65     | 82.32     | -         | -         | 77.43     | 80.70     | 80.51     | 82.86     | 84.68     |\n| TSPTQ-ViT     | 6/6   | 79.45     | 82.29     | 85.18     | 70.82     | 77.18     | 80.61     | 80.62     | 82.60     | 84.16     |\n| SQ-b+OPT-m    | 6/6   | 79.98     | 82.70     | 85.53     | 71.03     | 78.70     | 81.25     | 80.67     | 82.62     | 84.50     |\n| LRP-QViT      | MP    | 80.59     | 83.87     | -         | 71.03     | 79.93     | 81.44     | -         | 82.86     | 84.72     |\n| RepQ-ViT      | MP    | 80.43     | 83.62     | -         | 70.76     | 78.90     | 81.27     | -         | 82.79     | 84.57     |\n| **GMPQ-TE**   | MP    | 80.61     | 83.90     | 85.51     | 71.13     | 79.11     | 81.49     | 80.69     | 82.88     | 84.78     |\n| → Mean ± Std  |       | 80.48±0.14| 83.31±0.12| 85.13±0.07| 71.11±0.14| 79.06±0.15| 80.66±0.08| 80.09±0.10| 82.55±0.06| 84.12±0.05 |\n| PTQ4ViT       | 4/4   | 42.57     | 30.69     | 78.38     | 36.96     | 34.08     | 64.39     | 73.48     | 76.09     | 74.02     |\n| APQ-ViT       | 4/4   | 47.95     | 41.41     | -         | 47.94     | 43.55     | 67.48     | -         | 77.15     | 76.48     |\n| TSPTQ-ViT     | 4/4   | 52.56     | 50.10     | 77.64     | 48.36     | 45.08     | 69.45     | 72.48     | 76.30     | 73.28     |\n| SQ-b+OPT-m    | 4/4   | 55.88     | 61.84     | 80.07     | 55.62     | 68.43     | 76.14     | 73.82     | 77.20     | 76.51     |\n| PSAQ-ViT      | 4/4   | 37.19     | 41.52     | -         | 57.58     | 63.61     | 67.95     | -         | 72.86     | 76.44     |\n| LRP-QViT      | MP    | 70.81     | 75.37     | -         | 61.24     | 72.43     | 78.13     | -         | 81.37     | 80.77     |\n| RepQ-ViT      | MP    | 65.05     | 68.48     | -         | 57.43     | 69.03     | 75.61     | -         | 79.45     | 78.32     |\n| **GMPQ-TE**   | MP    | 70.90     | 75.41     | 80.13     | 61.03     | 72.52     | 78.19     | 73.90     | 81.43     | 80.84     |\n| → Mean ± Std  |       | 70.62±0.10| 75.24±0.08| 79.94±0.14| 60.86±0.11| 72.04±0.11| 78.04±0.11| 73.66±0.07| 81.22±0.10| 80.43±0.17 |\n\n\n*Table 8: Results for image classification on ResNet-18, ResNet-50 and MobileNet-V2 (QAT). *\n| Methods    | Params | Comp.  | BOPs  | Top-1 Acc. |\n|------------|--------|--------|-------|------------|\n| QDrop      | 5.41   | 8.23   | 29.0  | 69.76      |\n| HMQAT      | 4.31   | 10.34  | 20.9  | 69.63      |\n| QuanDCL    | 4.49   | 9.92   | 21.8  | 69.51      |\n| GMPQ-TE    | 4.21   | 117    | 20.3  | 69.80      |\n| → Mean ± Std |  4.21   | 115±2.31  |  20.1±1.66    | 69.63±0.10 |\n| QDrop      | 13.14  | 7.4    | 61.7  | 75.45      |\n| EPTQ       | 13.14  | 7.4    | 123.5 | 75.45      |\n| HMQAT      | 9.45   | 10.29  | 51.5  | 75.48      |\n| GMPQ-TE    | 7.9    | 131.4  | 25.1  | 76.32      |\n| → Mean ± Std |   7.9    |   128.1±4.25    |  24.7±1.47   |76.26±0.01 |\n| QDrop      | 5.41   | 8.23   | 29.0  | 69.76      |\n| HMQAT      | 1.22   | 10.98  | 7.71  | 70.81      |\n| EPTQ       | 1.26   | 10.63  | 31.68 | 70.39      |\n| GMPQ-TE    | 1.1    | 47.2   | 7.1   | 70.42      |\n| → Mean ± Std |  1.1     |   48.2±1.37    |   7.1± 2.94  |70.32±0.43|\n\n**Ans. For W2:** \n\nCompared 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.\n\nThe 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**).\n\n**Ans. For W3:** \n\nTable 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.\n\n| Model        | Platform             | Inference Latency (ms) | Throughput (FPS) | Avg. INT8 Kernel Time (ms) | Power Consumption (W) |\n|--------------|----------------------|-------------------------|------------------|-----------------------------|------------------------|\n| ResNet-18    | NVIDIA RTX 3090      | ~2.1                    | ~470             | 2.3                         | ~115                   |\n| ResNet-18    | Jetson Xavier NX     | ~13.2                   | ~75              | 13.9                        | ~18                    |\n| MobileNet-V2 | NVIDIA RTX 3090      | ~1.2                    | ~800             | 1.4                         | ~105                   |\n| MobileNet-V2 | Jetson Nano (10W)    | ~24.5                   | ~38              | 25.3                        | ~8.5                   |\n\nThe 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.\n\n**Regarding the question part, the answer can be found above.**"}]

Author rebuttal:

Meta-review: This paper proposes GMPQ-TE, a novel and efficient method for mixed-precision quantization based on topological entropy. The core contribution is the use of topological entropy, calculated from a minibatch of same-label data, to measure layer-wise quantization sensitivity. This allows the authors to formulate the bit-width assignment as a single-pass linear programming problem, which is extremely efficient and yields a quantization policy that generalizes across different datasets (e.g., from CIFAR-10 to ImageNet). The reviewers generally agreed on the novelty and significance of this approach. Reviewers ErFk and E1Lz highlighted the innovative framework and its potential for efficient, generalizable deployment. Initial concerns were raised by Reviewer ErFk regarding the theoretical underpinnings and by Reviewer pnyk about the justification for using topological entropy as a sensitivity metric. The authors provided a detailed rebuttal that clarified the graph construction, offered theoretical proofs for its application to Transformer architectures, and added statistical analysis to their results. This successfully addressed most of the reviewers' concerns, leading Reviewer ErFk and Reviewer pnyk to raise their scores. Although some minor issues remain regarding the clarity of the formulation and QAT performance (Reviewer 8djE), the overall consensus is that the paper presents a solid and valuable contribution to the field. Therefore, I recommend accepting this paper for a poster presentation.
Decision: Accept (poster)