On Uniformly Scaling Flows: A Density-Aligned Approach to Deep One-Class Classification

TL;DR

This work proves how training a USF via maximum-likelihood reduces to a Deep SVDD objective with a unique regularization that inherently prevents representational collapse, and advocates using USFs as a drop-in replacement for non-USFs in modern anomaly detection architectures.

Abstract

Unsupervised anomaly detection is often framed around two widely studied paradigms. Deep one-class classification, exemplified by Deep SVDD, learns compact latent representations of normality, while density estimators realized by normalizing flows directly model the likelihood of nominal data. In this work, we show that uniformly scaling flows (USFs), normalizing flows with a constant Jacobian determinant, precisely connect these approaches. Specifically, we prove how training a USF via maximum-likelihood reduces to a Deep SVDD objective with a unique regularization that inherently prevents representational collapse. This theoretical bridge implies that USFs inherit both the density faithfulness of flows and the distance-based reasoning of one-class methods. We further demonstrate that USFs induce a tighter alignment between negative log-likelihood and latent norm than either Deep SVDD or non-USFs, and how recent hybrid approaches combining one-class objectives with VAEs can be naturally extended to USFs. Consequently, we advocate using USFs as a drop-in replacement for non-USFs in modern anomaly detection architectures. Empirically, this substitution yields consistent performance gains and substantially improved training stability across multiple benchmarks and model backbones for both image-level and pixel-level detection. These results unify two major anomaly detection paradigms, advancing both theoretical understanding and practical performance.

Figures (8)

• Figure 1: Schematic comparison of Deep SVDD and Normalizing Flows. The left panel shows the input data space: a nonlinearly structured 2D “moons” dataset with outliers. Top-right depicts Deep SVDD’s latent space: a learned mapping $\omega(\cdot, W)$ concentrates normal samples near a center $c$ inside a hypersphere, while anomalies lie outside; the non-uniform distribution emphasizes representation bias. Bottom-right shows the Normalizing Flows latent space: an invertible mapping $\phi(\cdot, W)$ moves data towards a standard Gaussian (illustrated by 1$\sigma$ and 2$\sigma$ contours) and relegates outliers to low-density regions, with $\phi^{-1}(\cdot, W)$ indicating the reverse mapping.
• Figure 2: 2D Gaussian mixture experiment visualization. (a): True data distribution (b): Latent space of Deep SVDD (c): Latent space of non-USF (d): Latent space of USF. Point color encodes the true data density. Dashed lines show countours of the base distribution ($\sigma$--$3\sigma$) (flows) or the decision threshold (Deep SVDD) based on the 99th distance percentile. The USF shows direct density alignment between data and latent spaces, while Deep SVDD and non-USF show noticable discrepancies.
• Figure 3: Visualization of the transformation’s volume change in different areas of the data distribution for USFs and non-USFs. (a,c): Sample and contours of the true data distribution. The determinant of the respective transformation Jacobian is color coded. (b,d): Sample and ideal contours of centered data latents of the (b) USF and the (d) non-USF. The dashed contours show the contour lines of the flows base distribution (centered, $\sigma$ -- $3\sigma$). The color of each sample encodes the absolute value of the determinant of the transformation Jacobian. The USF enforces a constant Jacobian determinant, ensuring uniform scaling, whereas the non-USF adapts local densities through variable volume changes to correct alignment discrepancies.
• Figure 4: Scatter plots of the true log-likelihoods against the latent norms for the GM experiments (2, 8, 32, 128 dimensions). (a) -- (d): Deep SVDD (e) -- (h): Non-USF (i) -- (l) USF. USFs maintain a monotonic relationship between latent norm and data density. Non-USFs follow this behavior at first, but the relationship collapses at 128 dimensions. Deep SVDD, by contrast, transitions from a seemingly random latent structure to distinct homogeneous clusters as dimensionality increases.
• Figure 5: Scatter plots of the true log-likelihoods against the estimated log-likelihoods for the GM experiments (2, 8, 32, 128 dimensions). (a) -- (d): Non-USF (e) -- (h): USF. Across increasing dimensionality, the estimated likelihood quality remains largely comparable between both flow models.

Theorems & Definitions (3)

• Definition 1
• Proposition 1
• proof