Understanding Open-Set Recognition by Jacobian Norm and Inter-Class Separation

TL;DR

This work investigates open-set recognition (OSR) by linking the separation between known and unknown classes to differences in the Jacobian norm $\lVert \frac{\partial \boldsymbol{f}(\boldsymbol{x})}{\partial \boldsymbol{x}} \rVert_F$ of the representation function. It shows that intra-class learning reduces the Jacobian norm on known classes, while inter-class learning increases it for unknown samples, creating a discriminative gap that enables OSR without exposure to unknowns. Building on this insight, the authors introduce a marginal one-vs-rest (m-OvR) loss to promote strong inter-class separation and pair it with auxiliary techniques (data augmentation, weight decay, self-supervision) to maximize the Jacobian-norm disparity. They also propose using a sample-wise loss at inference to detect unknowns by leveraging the Jacobian-norm gap and proximity to class prototypes. Across standard OSR benchmarks, their approach yields improved unknown-class detection (AUC/Macro-F1) and competitive closed-set accuracy, validating the theory and highlighting inter-class learning as a key driver for OSR.

Abstract

The findings on open-set recognition (OSR) show that models trained on classification datasets are capable of detecting unknown classes not encountered during the training process. Specifically, after training, the learned representations of known classes dissociate from the representations of the unknown class, facilitating OSR. In this paper, we investigate this emergent phenomenon by examining the relationship between the Jacobian norm of representations and the inter/intra-class learning dynamics. We provide a theoretical analysis, demonstrating that intra-class learning reduces the Jacobian norm for known class samples, while inter-class learning increases the Jacobian norm for unknown samples, even in the absence of direct exposure to any unknown sample. Overall, the discrepancy in the Jacobian norm between the known and unknown classes enables OSR. Based on this insight, which highlights the pivotal role of inter-class learning, we devise a marginal one-vs-rest (m-OvR) loss function that promotes strong inter-class separation. To further improve OSR performance, we integrate the m-OvR loss with additional strategies that maximize the Jacobian norm disparity. We present comprehensive experimental results that support our theoretical observations and demonstrate the efficacy of our proposed OSR approach.

Figures (16)

• Figure 1: During the closed-set metric learning, the model learns only over the known classes $C_k$, but the learning also changes the representation of unknown class. We ask why. We discover that the intra-class learning diminishes the Jacobian norm of known class representations, while the inter-class learning increases the Jacobian norm of the unknown. The resulting disparity in Jacobian norm separates the unknown from the known.
• Figure 2: The summary of our theory on how a model becomes aware of the unknown by the closed-set metric learning over the known classes.
• Figure 3: The distribution of Jacobian norms of representations before and after training. Although the model is trained only on the known class data, the model learns to increase the Jacobian norm of unknown class representation, while lowering the Jacobian norm of the known class representation.
• Figure 4: Given known class samples $\boldsymbol{x}_0 \in C_i$ and $\boldsymbol{x}_1 \in C_2$ from two different classes $C_i$ and $C_j$, we linearly interpolate between $\boldsymbol{x}_0$ and $\boldsymbol{x}_1$ by $\boldsymbol{x}_t := (1-t)\boldsymbol{x}_0 + t\boldsymbol{x}_1$. Then, we measure the Jacobian norm of the representation $\boldsymbol{f}(\boldsymbol{x}_t)$. When $t \approx 0.5$, the interpolated sample $x_t$ passes through the open set, where unknown class samples arise.
• Figure 5: Several metrics are measured while a discriminative model (ours) is trained. (a) The discriminative quality of known class representations is measured in DBI. (b) The averaged inter-class distances between known classes. (c) The Jacobian norm difference between the known and unknown classes. (d) The degree of separation between known and unknown class representations. All metrics are improved as the discriminative model learns.

Theorems & Definitions (14)

• Proposition 1
• Theorem 2
• Corollary 3
• Corollary 4
• Corollary 5
• Proposition 6
• Proposition 7
• Proposition 8
• proof : Proof of Proposition \ref{['prop:collapse']}
• proof : Proof of Theorem \ref{['thm:grad_norm']}