cs.LGMar 27, 2026

Silhouette Loss: Differentiable Global Structure Learning for Deep Representations

Authors: Matheus Vinícius Todescato, Joel Luís Carbonera

Organizations: Institute of Informatics UFRGS Porto Alegre, Brazil

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

Learning discriminative representations is a central goal of supervised deep learning. While cross-entropy (CE) remains the dominant objective for classification, it does not explicitly enforce desirable geometric properties in the embedding space, such as intra-class compactness and inter-class separation. Existing metric learning approaches, including supervised contrastive learning (SupCon) and proxy-based methods, address this limitation by operating on pairwise or proxy-based relationships, but often increase computational cost and complexity. In this work, we introduce Soft Silhouette Loss, a novel differentiable objective inspired by the classical silhouette coefficient from clustering analysis. Unlike pairwise objectives, our formulation evaluates each sample against all classes in the batch, providing a batch-level notion of global structure. The proposed loss directly encourages samples to be closer to their own class than to competing classes, while remaining lightweight. Soft Silhouette Loss can be seamlessly combined with cross-entropy, and is also complementary to supervised contrastive learning. We propose a hybrid objective that integrates them, jointly optimizing local pairwise consistency and global cluster structure. Extensive experiments on seven diverse datasets demonstrate that: (i) augmenting CE with Soft Silhouette Loss consistently improves over CE and other metric learning baselines; (ii) the hybrid formulation outperforms SupCon alone; and (iii) the combined method achieves the best performance, improving average top-1 accuracy from 36.71% (CE) and 37.85% (SupCon2) to 39.08%, while incurring substantially lower computational overhead. These results suggest that classical clustering principles can be reinterpreted as differentiable objectives for deep learning, enabling efficient optimization of both local and global structure in representation spaces.

Figures & tables

 CECE+SILSupConSupCon2CE+SIL+SupCon2ProxyNCACenter
DatasetTop1Top5Top1Top5Top1Top5Top1Top5Top1Top5Top1Top5Top1Top5
CIFAR-100.83910.99380.83930.99420.84370.99240.85050.99390.85140.99410.8446–0.84380.9934
CIFAR-1000.52830.80570.53220.80680.50420.79810.53780.81440.53860.80680.5305–0.50700.7977
Caltech-1010.44040.63470.45950.63830.38570.60070.44170.64040.46920.66310.4416–0.38320.5709
FGVC-Aircraft0.17760.48210.17040.43410.11100.37380.21090.52090.20430.47850.1947–0.12360.3966
Oxford Flowers0.19600.47210.26560.54560.17910.46540.21060.50250.27400.56110.2210–0.20180.4789
Table 1: Top-1 and top-5 classification accuracy across the datasets.
Figure 1: Training loss and validation accuracy curves across training epochs for the FGVC-Aircraft dataset.
Figure 2: Training loss and validation accuracy curves across training epochs for the Caltech-101 dataset.
Figure 3: Training loss and validation accuracy curves across training epochs for the Caltech-256 dataset.
Figure 4: Training loss and validation accuracy curves across training epochs for the Stanford Cars dataset
Figure 5: Training loss and validation accuracy curves across training epochs for the CIFAR-10 dataset.
Figure 6: Training loss and validation accuracy curves across training epochs for the CIFAR-100 dataset.
Figure 7: Training loss and validation accuracy curves across training epochs for the Oxford Flowers dataset.
More in this topic

Representation Learning

Representation learning develops data transformations that expose useful structure for prediction, transfer, generation, and control.

See all 799 papers

Explore similar work

CardsList
Apr 2, 2026

DDCL: Deep Dual Competitive Learning: A Differentiable End-to-End Framework for Unsupervised Prototype-Based Representation Learning

A persistent structural weakness in deep clustering is the disconnect between feature learning and cluster assignment. Most architectures invoke an external clustering step, typically k-means, to produce pseudo-labels that guide training, preventing the backbone from directly optimising for cluster quality. This paper introduces Deep Dual Competitive Learning (DDCL), the first fully differentiable end-to-end framework for unsupervised prototype-based representation learning. The core contribution is architectural: the external k-means is replaced by an internal Dual Competitive Layer (DCL) that generates prototypes as native differentiable outputs of the network. This single inversion makes the complete pipeline, from backbone feature extraction through prototype generation to soft cluster assignment, trainable by backpropagation through a single unified loss, with no Lloyd iterations, no pseudo-label discretisation, and no external clustering step. To ground the framework theoretically, the paper derives an exact algebraic decomposition of the soft quantisation loss into a simplex-constrained reconstruction error and a non-negative weighted prototype variance term. This identity reveals a self-regulating mechanism built into the loss geometry: the gradient of the variance term acts as an implicit separation force that resists prototype collapse without any auxiliary objective, and leads to a global Lyapunov stability theorem for the reduced frozen-encoder system. Six blocks of controlled experiments validate each structural prediction. The decomposition identity holds with zero violations across more than one hundred thousand training epochs; the negative feedback cycle is confirmed with Pearson -0.98; with a jointly trained backbone, DDCL outperforms its non-differentiable ablation by 65% in clustering accuracy and DeepCluster end-to-end by 122%.

Giansalvo Cirrincione

Lab. LTI, Université de Picardie Jules Verne, Amiens, France

ClusteringDeep ClusteringUnsupervised Representation LearningRepresentation Learning
AskPDF
Jul 14, 2026

Contrastive-Collapsed Loss for Flexible and Geometrically Optimal Embeddings and Faster Convergence

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Blanca Cano-Camarero, Ángela Fernández-Pascual, José R. Dorronsoro

1Departamento de Ingeniería Informática, Universidad Autónoma de Madrid, Calle Francisco Tomás y Valiente, 11, 28049, Madrid, España

Geometric Representation LearningNeural CollapseSupervised Contrastive Learning
AskPDF
May 13, 2026

A Unified Geometric Framework for Weighted Contrastive Learning

Contrastive learning (CL) aims to preserve relational structure between samples by learning representations that reflect a similarity graph. Yet, the geometry of the resulting embeddings remains poorly understood. Here we show that weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, where the weighting scheme specifies the target geometry to be realized by the representation. This viewpoint yields exact characterizations of the optimal embeddings for several supervised and weakly supervised objectives. In supervised classification, both SupCon and Soft SupCon (a dense relaxation of it where pairs from distinct classes have small non-zero similarity) collapse samples within each class to a single prototype. However, while balanced SupCon recovers the classical regular simplex geometry, class imbalance breaks this symmetry: SupCon induces non-uniform inter-class similarities depending on class sizes, whereas Soft SupCon preserves a regular simplex geometry regardless of class imbalance. In continuous-label settings, our framework reveals a different failure mode: y-Aware CL generally cannot attain its entropic optimum unless the labels lie on a hypersphere, exposing a mismatch between Euclidean label weights and spherical latent similarity. By contrast, geometrically consistent choices such as Euclidean-Euclidean weighting or X-CLR admit unique optimal embeddings. Our results show that the choice of weighting scheme determines whether contrastive learning is geometrically realizable, degenerate, or inconsistent, providing a principled framework for designing contrastive objectives.

Raphael Vock, Edouard Duchesnay, Benoit Dufumier

GAIA Lab, NeuroSpin, CEA, CNRS · Université Paris-Saclay, Gif-sur-Yvette, France

Contrastive LearningNeural Representation GeometrySupervised Contrastive Learning
AskPDF