Revisiting Deep Ensemble Uncertainty for Enhanced Medical Anomaly Detection

Anomaly detection (AD) is an essential task in medical image analysis, encompassing early detection of medical diseases [1, 27] and pathological localization [24]. The primary objective of visual medical AD is to identify images containing diseases and pinpoint anomalous pixels within them. However, obtaining a sufficient number of anomalous samples that cover the vast spectrum of disease types can be challenging, as these samples often require specialized annotations [6]. Consequently, AD tasks are often formulated as one-class classification problem, wherein only normal data is utilized for model training [8].

Prevailing approaches mainly focus on reconstruction-based anomaly detection employing Autoencoders [14, 21] or Generative Adversarial Networks [24, 1]. These methods endeavor to maximize the likelihood of normal samples derived from training data. During inference, anomalies are detected based on per-pixel reconstruction error or model probability distribution. Nevertheless, these methods are limited by imprecise reconstructions or poorly calibrated likelihoods [5].

To circumvent a direct estimation of normal probability distributions, an alternative framework leveraging deep ensembles’ uncertainty has emerged. This framework comprises multiple learners that perform self-supervised tasks [6, 4] or acquires surrogate labels through a pretrained encoder [3, 23]. Typically, learners undergo randomized training with distinct weight initializations [6] or Monte-Carlo dropout [13]. The underlying hypothesis posits that diverse learners should agree on normality while disagreeing on unseen anomalies in the output space.

However, balancing the trade-off between agreement and disagreement is challenging. Randomized training may not guarantee sufficient disagreement on anomalies, as learners in ensemble learning inherently tend to adopt the simplest decision boundary [2]. This phenomenon, known as simplicity bias [25], inhibits learners’ diversity and subsequently results in minimal disagreements on anomaly outputs [19]. To address simplicity bias, previous methods attempted to induce repulsion among learners in output space [22] or weight space [9]. Nevertheless, these approaches may culminate in either underfitting of individual models [11] or neural network redundancy [18], where models possess distinct weights yet output the same [12]. Consequently, learners’ agreement on normal samples would be compromised.

In this paper, we propose a novel ensemble-based uncertainty estimation framework for medical anomaly detection called Diversified Dual-space Uncertainty Estimation (D2UE). To enhance learners’ disagreement on anomalies, we introduce a Redundancy-Aware Repulsion (RAR), which encourages learners to reconstruct training samples from more diversified feature spaces. To promote this diversification without succumbing to neural network redundancy, RAR regulates ensemble training using a similarity kernel invariant to both isotropic scaling and orthogonal transformation. During inference, disagreement on anomalies is amplified between different learners’ feature spaces (see Fig. 1(a)). Unlike output space repulsion, feature space repulsion does not result in underfitting for normal samples in output space. Consequently, normal features converge to similar reconstructions guided by reconstruction training. Moreover, to emphasize anomalous regions, we develop a Dual-Space Uncertainty (DSU) that combines uncertainties in both input and out spaces. In input space, we calculate gradients of the reconstruction error with respect to inputs, which are further combined with outputs to estimate the final uncertainty. DSU discriminates anomalies through input space disagreement even if learners exhibit minimal disagreement in output space (see Fig. 1(c)). Our primary contributions are as follows:

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  We address medical anomaly detection from an uncertainty estimation perspective. We undertake a pioneering exploration of diversity in deep ensembles’ uncertainty and propose D2UE, a novel Diversified Dual-space Uncertainty Estimation approach for medical anomaly detection.

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  We propose Redundancy-Aware Repulsion (RAR) to strike an effective balance between agreement and disagreement, thereby enhancing anomaly detection accuracy.

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  We design a Dual-Space Uncertainty (DSU) to emphasize anomalous regions, particularly when anomalies exhibit minimal disagreement in output space.

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  We conduct comprehensive experiments on five medical benchmarks with different backbones. Experimental results demonstrate the superiority of our method to state-of-the-art methods and the effectiveness of each component.

The proposed Diversified Dual-space Uncertainty Estimation (D2UE) framework consists of $N$ learners $f(\cdot)$, each possessing identical Autoencoder architectures. Anomaly detection is achieved through uncertainty estimation, where learners should agree on normal samples while disagreeing on anomalies. To this end, we propose redundancy-aware repulsion (RAR) to enhance learners’ disagreement on anomalies while maintaining agreement on normal samples. Moreover, to further emphasize anomalous regions, dual-space uncertainty (DSU) is designed to combine uncertainties in output and input spaces during inference. In the following, we will detail each component.

First, it is crucial to establish an optimization objective that explicitly encourages repulsion in learners’ feature spaces. To this end, we endow a more straightforward target to the training of a learner: to reconstruct normal samples using $\mathcal{L}_{\text{orig}}$ through more different paths supervised by all well-trained learners via $\mathcal{L}_{\text{sim}}$. By passing anomalous samples through more diversified feature spaces, learners are encouraged to exhibit more significant disagreements while maintaining consistency on normal samples during inference.

Second, it is essential to identify a suitable similarity kernel $S(\cdot)$ that eliminates neural network redundancy. To achieve this, we establish two properties for $S(\cdot)$: 1. isotropic scaling invariance: $S(P,Q)=S(\alpha P,\beta Q)$ for $\alpha,\beta\in\mathbb{R}^{+}$. 2. orthogonal transformation invariance: $S(P,Q)=S(PU,QV)$ for $U,V\in$ orthogonal transformation. Scaling invariance implies that neurons cannot deceive well-trained neurons into perceiving different features merely by re-scaling their weights during training. Similarly, orthogonal invariance prevents neurons from deceiving well-trained neurons into perceiving different features merely by spatial reordering during training. As illustrated in Fig. 3, different features may still output the same by weight re-scaling or reordering. Therefore, both isotropic and orthogonal invariance are necessary to effectively promote feature space diversity without succumbing to neural network redundancy.

In Eq. (3), $K=PP^{T}$ and $L=QQ^{T}$ ensure orthogonal invariance since $(PU)(PU)^{T}=PP^{T}$. Normalized term $\sqrt{\text{HSIC}(K,K)}\sqrt{\text{HSIC}(L,L)}$ ensures scaling invariance in $\mathcal{L}_{\text{sim}}$. Finally, the total loss $\mathcal{L}_{\text{total}}$ is formulated as follows:

where $\lambda$ controls the strength of the repulsion to the overall loss function^{†{\dagger}†{\dagger}}${\dagger}$Ablation studies of $\lambda$ and the layer of $\mathcal{L}_{\text{sim}}$ are included in supplementary materials..

where $\odot$ symbolizes the element-wise multiplication. Consequently, even if distinct learners inadvertently agree in output space, the anomalous region can still be accentuated by input space disagreement.

Datasets and Evaluation metrics. We conduct experiments on five medical datasets, encompassing various modalities such as chest X-ray, magnetic resonance imaging, and retinal fundus. Datasets include 1. RSNA: RSNA Pneumonia Detection Challenge dataset^{†{\dagger}†{\dagger}}${\dagger}$https://www.kaggle.com/c/rsna-pneumonia-detection-challenge. 2. VinDr-CXR: VinBigData Chest X-ray Abnormalities Detection dataset^{†{\dagger}†{\dagger}}${\dagger}$https://www.kaggle.com/c/vinbigdata-chest-xray-abnormalities-detection. 3. CXAD: Chest X-ray Anomaly Detection dataset [6]. 4. Brain MRI: Brain Tumor MRI dataset^{†{\dagger}†{\dagger}}${\dagger}$https://www.kaggle.com/datasets/masoudnickparvar/brain-tumor-mri-dataset. 5. LAG: Large-scale Attention-based Glaucoma dataset [17]. To ensure consistency with other studies, we adhere to the split criteria outlined in [4, 6, 7] and details can be seen in supplementary materials. We adopt the area under the ROC curve (AUC) and average precision (AP) for image-level classification.

In this paper, we presented D2UE, a Diversified Dual-space Uncertainty Estimation framework for medical anomaly detection. To effectively balance the diversity among ensemble learners and reconstruction accuracy, we introduced redundancy-aware repulsion, which compels learners to disagree on anomalies without compromising agreement on normal inputs. Further, we propose dual-space uncertainty highlighting anomalous regions during inference to enhance the model’s discrimination ability. The framework has been extensively tested on various medical benchmarks, and experimental results demonstrate the superiority of our method to state-of-the-art methods and the effectiveness of each component. In future work, we intend to explore the quantitative relationship between ensemble diversity and final performance, and reduce the time and computational cost of training model ensembles.