Feature Map Convergence Evaluation (FMCE)

• FMCE is an evaluation methodology that quantifies the convergence of intermediate feature maps in deep neural networks using metrics like CQI and FMCS.
• It independently assesses functional modules, offering insights into training maturity and interpretability without relying solely on end-to-end evaluation.
• Extensions like FMCE-Net++ integrate FMCE as an auxiliary training signal, demonstrating effective improvements in tasks such as autonomous driving perception.

Searching arXiv for papers directly related to Feature Map Convergence Evaluation and adjacent feature-map evaluation frameworks. Feature Map Convergence Evaluation (FMCE) is an evaluation methodology for quantifying the convergence of feature maps in deep neural networks at the module level, with the stated aim of assessing how “well-trained” intermediate representations are and thereby enabling independent evaluation of functional modules rather than treating the model as an end-to-end black box (Zhang et al., 2024). In the literature represented here, FMCE appears in several closely related forms: as a feature-map analysis framework for autonomous driving perception models (Zhang et al., 2024), as a closed-loop training framework that uses predicted convergence scores as auxiliary supervision (Zhu et al., 8 Aug 2025), and as an extended quality-scoring formulation for autonomous driving models via feature map-truth similarity and dual-granularity weighting (Zhang et al., 11 Aug 2025). Related work also uses feature-map representations to assess training maturity or faithfulness through similarity, attribution, or dependency spectra, which places FMCE within a broader family of representation-centric evaluation methods (Zhang et al., 2024, Zhang et al., 2021, Hu et al., 2023).

1. Origins and problem setting

FMCE was introduced to address the absence of independent evaluation for functional modules in deep neural networks, particularly in perception systems whose internal mechanisms are otherwise difficult to inspect (Zhang et al., 2024). The motivating setting is that autonomous driving perception models are composed of multiple functional modules that interact through complex relationships, yet are predominantly optimized through end-to-end training. This creates difficulties for interpretability and optimization because intermediate modules are not evaluated independently (Zhang et al., 2024).

The core proposal in "Feature Map Convergence Evaluation for Functional Module" (Zhang et al., 2024) is a feature-map-based methodology comprising three components: the Convergence Quantification Indicator (CQI), the Feature Map Convergence Score (FMCS), and the Feature Map Convergence Evaluation Network (FMCE-Net). CQI provides a quantitative indicator of whether training has approached a flat-loss regime, FMCS assigns discrete convergence levels to extracted feature maps, and FMCE-Net predicts those convergence levels from feature maps alone (Zhang et al., 2024).

Later work extends this initial diagnostic setting in two directions. "FMCE-Net++: Feature Map Convergence Evaluation and Training" (Zhu et al., 8 Aug 2025) turns FMCE into a closed-loop training signal by attaching a pretrained, frozen FMCE-Net as an auxiliary head. "Decoupled Functional Evaluation of Autonomous Driving Models via Feature Map Quality Scoring" (Zhang et al., 11 Aug 2025) generalizes the earlier feature-map-truth similarity approach into a Feature Map Quality Score (FMQS), combining model-level and feature-level evidence through a Dual-Granularity Dynamic Weighted Scoring System. This suggests that FMCE has evolved from a convergence diagnostic into a broader representation-quality supervision framework.

2. Formalization of convergence in the original FMCE framework

The original FMCE framework defines convergence through the behavior of the training loss curve (Zhang et al., 2024). The Convergence Quantification Indicator is constructed from a smoothed loss sequence, a first-order difference, and a moving average of absolute differences over a window of epochs:

$$\widetilde{L}_m = \alpha L_{m-1} + (1 - \alpha) L_m$$

$$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$

$$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$

The model is considered converged when

$$CQI_m = \Delta \widetilde{L}_{m}^{avg} \leq \mu_{CQI}$$

where $\alpha$ is typically set to $0.85$ (Zhang et al., 2024). CQI therefore operationalizes convergence as a recent-history average of smoothed loss variation rather than as a single-epoch criterion.

After identifying the training trajectory, FMCE divides convergence into $K$ convergence phases by applying a logarithmic transform to the smoothed loss and uniformly segmenting the total decrease in log-loss:

$\widetilde{L}_{m}^{log} = \log(\widetilde{L}_m)$

$$G( \widetilde{L}^{log}) = \left| \widetilde{L}_{E_K}^{log} -\widetilde{L}_{E_0}^{log} \right|$$

$$\Delta G(\widetilde{L}^{log}) = \frac{G(\widetilde{L}^{log})}{K}$$

If $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$0 denotes the feature map generated at epoch marker $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$1 for sample $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$2, then the Feature Map Convergence Score is defined as

$$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$3

with higher $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$4 denoting higher convergence (Zhang et al., 2024). The result is a labeled dataset of feature maps indexed by convergence phase rather than by semantic class.

This formulation makes FMCS a discretized representation of training maturity. A plausible implication is that FMCE treats convergence as an intrinsic property of intermediate representations, not merely of end-task outputs.

3. FMCE-Net and supervised prediction of convergence levels

FMCE-Net is the predictive component of the original framework (Zhang et al., 2024). It is described as a lightweight CNN tailored to the size of the extracted feature maps, with $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$5-$$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$6 convolutional layers, MaxPooling, ReLU, flattening, and fully connected layers that output a $$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$7-dimensional probability vector over convergence classes (Zhang et al., 2024). The training objective is standard multiclass cross-entropy over FMCS labels.

The later FMCE-Net++ paper gives an explicit probability-vector formulation. For each feature map, FMCE-Net predicts

$$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$8

and is trained with

$$\Delta \widetilde{L}_m = \widetilde{L}_m - \widetilde{L}_{m-1}$$9

where $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$0 if the $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$1-th map belongs to class $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$2, and $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$3 otherwise (Zhu et al., 8 Aug 2025). The FMCS dataset is built by freezing a backbone at $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$4 checkpoints, extracting feature maps for the full training set at each checkpoint, and using checkpoint index as the FMCS label (Zhu et al., 8 Aug 2025).

The empirical validation reported for FMCE-Net in the original paper covers MNIST, CIFAR-10, and Mini-ImageNet with ResNet-50 and ShuffleNet v2 backbones (Zhang et al., 2024). Reported prediction results are summarized in the paper as follows:

FMCS Dataset	Accuracy	Precision
MNIST-R	0.9994	0.9994
CIFAR-R	0.9409	0.9418
Imagenet-R	0.9989	0.9989

The same table also reports an average accuracy of $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$5 with standard deviation $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$6 across the settings (Zhang et al., 2024). The paper further states that Grad-CAM visualizations become more concentrated and localized on semantically relevant object regions as FMCS increases, while lower FMCS corresponds to less focused attention (Zhang et al., 2024). This is presented as qualitative evidence that the score tracks the maturation of feature extraction.

4. Closed-loop training with FMCE-Net++

"FMCE-Net++: Feature Map Convergence Evaluation and Training" (Zhu et al., 8 Aug 2025) extends FMCE from a post hoc evaluator to a training-time auxiliary supervisor. The framework attaches a pretrained and frozen FMCE-Net to a backbone so that intermediate feature maps are routed both to the primary classifier and to the FMCE auxiliary head (Zhu et al., 8 Aug 2025).

The primary task loss is standard multiclass cross-entropy:

$$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$7

The FMCS loss uses a “fully converged” pseudo-label at class $$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$8:

$$\Delta \widetilde{L}_{m}^{avg} = \frac{1}{B} \sum_{b=1}^{B} |\Delta \widetilde{L}_{m-b+1}|$$9