Geometry-Aware Classification Layer (GACL)

• The paper introduces GACL, a Softmax-based layer that couples feature norm and angular recognizability to assess sketch quality without explicit annotations.
• It employs dual-task optimization and soft binning to simultaneously learn recognition and quality estimation, mitigating early training bias.
• Empirical results show that GACL aligns with human quality judgments (78.13% agreement) while improving sketch recognition and generation performance.

Geometry-Aware Classification Layer (GACL) is a Softmax-based geometry-aware classification layer introduced for annotation-free human sketch quality assessment. Its defining move is to make the feature magnitude $q(x)=\|f(x)\|_2$ semantically meaningful, so that recognizability learning and quality estimation become a dual task under a cross-entropy classification objective. In this formulation, better sketches are treated as easier to recognize, closer to their class center, and farther from the decision boundary, allowing the learned feature norm to function as a usable quality score without explicit quality annotations (Yang et al., 28 Jul 2025).

1. Problem setting and conceptual scope

GACL was proposed for human free-hand sketch quality assessment, a setting in which convenient human quality labels are absent and quality is described as highly subjective and holistic rather than tied to simple distortions such as noise or blur. The central objective is to learn a metric $q(x)$ that ranks sketches from bad to good without quality supervision. The motivating observation is that quality is already implicit in the geometry of a recognition model: a well-drawn sketch should be more recognizable, more tightly associated with its class, and farther from the decision boundary. Standard Softmax classification, however, is described as suppressing this signal because it is trained only to separate classes and often normalizes away norm information (Yang et al., 28 Jul 2025).

Within that framing, GACL is not merely a post hoc scoring rule. It is a classification layer designed to preserve and exploit feature magnitude instead of discarding it. The resulting model is a quality-aware recognizer: the learned feature norm becomes a quantitative quality metric, while the angular geometry of class separation remains the mechanism for recognition. The paper further presents GACL as agnostic to network architecture changes and to the underlying sketch representation, provided that the network produces a feature vector $f(x)$ whose magnitude can be used.

A common misconception is to treat GACL as a generic label for any geometry-aware classifier. In the source paper, the term is much narrower. It refers specifically to a Softmax-style classification layer that modifies the target logit so that feature magnitude carries quality information. Related geometry-aware methods exist in other domains, but they are not introduced under the same module name and do not necessarily use feature norm as the quality variable.

2. Formal definition and classification objective

The formal core of GACL is a modification of normalized Softmax. Instead of using only the target-class angular term, GACL replaces the target logit $s\cos\theta_{i,y_i}$ with a learned compound function $A(q_i,\theta_{y_i})$, where the instance score is defined by

$q_i \equiv q(x_i)=\|f(x_i)\|_2.$

The GACL objective is written as

$$L_{GACL}(q_i,\theta_{y_i}) = -\log\frac{e^{A(q_i,\theta_{y_i})}}{e^{A(q_i,\theta_{y_i})}+R},$$

with

$R=\sum_{j=1, j\neq i}^C e^{s\cos\theta_{i,y_j}}.$

In this construction, the target logit depends jointly on quality or magnitude and on recognizability or angular distance. The non-target logits remain in the standard Softmax-style form through $R$. The paper’s explicit interpretation is that feature magnitude is not a nuisance scale but the learned quality score itself (Yang et al., 28 Jul 2025).

This design distinguishes GACL from ordinary classification heads. A standard classifier may consume a feature vector after global pooling and map it to logits without assigning semantic meaning to feature norm. GACL instead makes norm part of the supervised objective. It is also distinct from a prototype-distance classifier or a geodesic-margin classifier: the paper does not define a dedicated class-prototype layer or a manifold-distance decision rule. Its mechanism is a modified Softmax target logit whose geometry couples angular recognizability with instance-dependent magnitude.

The source also analyzes vanilla normalized Softmax and argues that, under simplifying assumptions, optimization can become dominated by feature magnitude in a way that harms angular discriminability. GACL is motivated as a way to control that interaction rather than allowing magnitude to emerge as an uncontrolled by-product.

3. Geometric interpretation and theoretical characterization

The paper formalizes GACL as a dual-task optimization problem. Recognizability learning is carried by the angular term $\theta$, while magnitude learning is carried by $q(x)$0. The layer is designed so that these two quantities co-evolve rather than one destroying the other. This is the basis for the paper’s claim of a geometric interpretation: better quality implies easier recognition (Yang et al., 28 Jul 2025).

Three constraints are imposed on $q(x)$1. The geometry constraint requires quality to increase when a sample is geometrically closer to the class center:

$q(x)$2

The co-optimisation constraint requires updates to $q(x)$3 not to destroy progress in $q(x)$4. With one gradient-descent step,

$q(x)$5

$q(x)$6

the requirement is that the $q(x)$7-update remain non-destructive, yielding the condition

$q(x)$8

The optimality constraint requires a valid optimum for $q(x)$9 in a bounded interval $f(x)$0. Assuming convexity in $f(x)$1, the gradient must change sign at the interval endpoints:

$f(x)$2

and because

$f(x)$3

the corresponding condition becomes

$f(x)$4

The paper’s proof sketch further interprets $f(x)$5 as the distance to the class decision boundary. Starting from a hard classification form, introducing an instance-adaptive margin $f(x)$6, and then using $f(x)$7 and $f(x)$8 approximations, the source arrives at a soft form that matches GACL when instantiated as

$f(x)$9

Under that instantiation, $s\cos\theta_{i,y_i}$0 plays the role of the adaptive margin $s\cos\theta_{i,y_i}$1. The resulting interpretation is explicit: $s\cos\theta_{i,y_i}$2 measures closeness to the class center, $s\cos\theta_{i,y_i}$3 measures distance from the decision boundary, and higher quality corresponds to smaller angular deviation, larger boundary margin, and larger feature magnitude.

4. Training regime, annotation-free learning, and soft binning

A defining property of GACL is that it is trained without human quality annotations. Only ordinary category labels are required. The quality signal is described as emerging implicitly from the recognition task through the geometry of the learned feature space. In that sense, GACL treats feature magnitude and recognizability learning as a dual task and optimizes them under a single cross-entropy classification loss (Yang et al., 28 Jul 2025).

The paper also identifies a training-stage bias. If GACL starts assigning fine-grained quality scores too early, it may conflate semantic subclusters with actual quality. The given example is that the model may prefer “frontal bunnies” over “side-view bunnies” simply because the former are easier to classify during early training. To mitigate this, the authors introduce soft binning, which discretizes the continuous score into ordered bins and temporarily suppresses fine-grained discrimination within the same bin until class boundaries are better formed.

The soft-binning mechanism is defined by

$s\cos\theta_{i,y_i}$4

where $s\cos\theta_{i,y_i}$5 are fixed bin boundaries. During early training, $s\cos\theta_{i,y_i}$6 replaces $s\cos\theta_{i,y_i}$7 in the GACL loss. The source describes this as remaining differentiable while preventing premature, biased score learning.

The architectural scope is deliberately broad. GACL is presented as plug-compatible with vector sketches encoded by BiLSTM, raster sketches encoded by CNN or ConvNeXt, point-cloud or sequence-like sketch encoders with Transformers, and even natural image classifiers. The key requirement is the existence of a feature vector whose magnitude can be used as $s\cos\theta_{i,y_i}$8. This suggests a broader interpretation of GACL as a general-purpose data re-weighting or scoring layer, although the explicit formulation in the paper remains tied to the modified Softmax geometry.

5. Empirical evidence and application domains

The main empirical validation is a large human study with 160,000 pairwise trials. Across 8 sketch categories, the paper reports that GACL’s quality ranking agrees with human judgment 78.13% of the time. For the Cosine variant, soft binning improves human agreement from 72.35% to 78.13%. The paper also reports human preference over the soft-binned scoring of 81.32% in one evaluation. These results are presented as evidence that the GACL-induced feature magnitude aligns with human quality perception (Yang et al., 28 Jul 2025).

In sketch recognition, GACL is reported to improve recognition accuracy over standard sketch classifiers. The best reported recognition result among the paper’s variants is 82.52% top-1 for Ours-Cos, compared with 81.51% for SketchAA. The geometric explanation offered by the paper is that better sketches become both easier to recognize and farther from the decision boundary, allowing a recognition model to function simultaneously as a quality assessor.

The paper further uses $s\cos\theta_{i,y_i}$9 as a critic for quality-guided sketch generation. In latent-space optimization, it defines

$A(q_i,\theta_{y_i})$0

This procedure is used to move generated sketches toward better-looking outputs while preserving identity. The paper also adapts the idea to ControlNet-based sketch-to-image generation, where a quality-enhanced sketch is reported to improve structural consistency and visual fidelity.

A second extension is sketch attribution and noisy-label analysis. The paper states that low quality scores can be used to detect malicious or nonsensical inputs. On Clothing1M-style noisy labels, it reports around 35.14% of samples flagged as wrong under one setting, and at 40% noisy labels it states that GACL can identify about 80% of erroneous labels with $A(q_i,\theta_{y_i})$1 precision. The source also describes a qualitative separation in the $A(q_i,\theta_{y_i})$2-$A(q_i,\theta_{y_i})$3 space: noisy labels tend to form a “U-shape,” whereas clean labels form an “L-shape.”

GACL is also adapted to image quality assessment by discretizing human quality ratings into bins and training a classification objective instead of regression. On six IQA benchmarks, the paper reports competitive and in several cases state-of-the-art results, including TID2013 with PLCC/SROCC $A(q_i,\theta_{y_i})$4 for GACL$A(q_i,\theta_{y_i})$5, KADID-10k with $A(q_i,\theta_{y_i})$6, and KONIQ-10k with $A(q_i,\theta_{y_i})$7. The paper’s interpretation is that noisy and subjective ratings are better handled by an adaptive confidence-like classification geometry than by naive regression.

6. Relation to other geometry-aware classification mechanisms

The name “Geometry-Aware Classification Layer” is specific to the sketch-quality paper, but the broader literature contains several related geometry-aware classification mechanisms that illuminate its position. These methods typically inject geometry into representation learning, feature transforms, or classifier inputs rather than defining the same norm-as-quality Softmax layer.

Paper	Mechanism	Relation to GACL
"Multi-scale Geometry-aware Transformer for 3D Point Cloud Classification" (Wei et al., 2023)	Multi-scale patch partitioning, sphere mapping, geometry-aware self-attention, final classification head	Geometry-aware classification architecture, but no explicit GACL module
"Geometry-Aware Spiking Graph Neural Network" (Zhang et al., 9 Aug 2025)	Manifold embeddings, curvature-aware attention, log-map + softmax classifier	Closely related geometry-aware classification mechanism, but not a standalone GACL
"A Supervised Geometry-Aware Mapping Approach for Classification of Hyperspectral Images" (Mohanty et al., 2018)	Supervised linear geometry-aware mapping before standard classifiers	Geometry-aware preprocessing and projection, not a classification layer
"Geometry-aware Deep Transform" (Huang et al., 2015)	Hybrid classification/metric deep transform preserving local geometry	Equivalent geometry-aware feature-learning mechanism, not a Softmax head called GACL

In point cloud classification, the Multi-scale Geometry-aware Transformer integrates geometry into patch construction, local feature extraction, and self-attention, then performs classification end-to-end; the provided material explicitly states that it is highly relevant and comparable to the idea behind GACL, but not a direct description of a standalone GACL module (Wei et al., 2023). In graph learning, the Geometry-Aware Spiking Graph Neural Network performs node classification from manifold embeddings through a log-map plus softmax classifier; the provided characterization describes this as an indirect precursor or closely related mechanism rather than a dedicated GACL (Zhang et al., 9 Aug 2025).

In hyperspectral imaging, the supervised geometry-aware mapping approach learns a linear projection that preserves intrinsic geometry while improving class discrimination, after which conventional classifiers such as SVM, 1NN, 3NN, 5NN, LDC, QDC, and TREE operate in the transformed space. The geometry-aware element there is the mapping, not the final classifier (Mohanty et al., 2018). In small-data deep learning, Geometry-aware Deep Transform unifies classification and metric learning by preserving within-class geometry while separating different classes; again, the geometry-aware mechanism is a learned transform rather than a modern plug-in classification head (Huang et al., 2015).

Taken together, these comparisons delimit the meaning of GACL. In the direct source, GACL is not a generic synonym for geometry-aware classification. It is a specific Softmax-based layer in which feature magnitude is intentionally preserved and optimized as a quality variable. A plausible implication is that GACL occupies a distinct niche within geometry-aware learning: it connects class geometry, instance quality, and feature norm in a single classifier objective, whereas adjacent methods more often use geometry to structure embeddings, neighborhoods, or transformations upstream of conventional classification.