Normalizing Flow

Unlike VAEs or GANs which provide only a lower bound or implicit density, Normalizing Flows compute the exact log-likelihood of any data point via the change-of-variables formula. For fraud detection, this means an anomaly score is a true probability, not a heuristic. The log-likelihood is calculated as:

• Base distribution log-probability + sum of log-determinants of Jacobians
• Enables direct comparison of anomaly scores across different transactions
• Provides mathematically rigorous thresholds for alert generation

The defining property of Normalizing Flows is bijectivity—every transformation must be perfectly reversible. This constraint ensures:

• No information loss during encoding or decoding
• The mapping from data space to latent space and back is deterministic
• The Jacobian determinant remains tractable for density evaluation
• Common architectures include RealNVP, Glow, and Masked Autoregressive Flows (MAF)

The mathematical foundation of Normalizing Flows is the change-of-variables theorem. Given a bijective transformation f: z → x, the density of x is:

• p_X(x) = p_Z(f^{-1}(x)) * |det(J_{f^{-1}}(x))|
• The Jacobian determinant accounts for volume changes during transformation
• Architectures are designed to make this determinant computationally cheap (e.g., triangular Jacobians)
• This formula enables direct sampling and density evaluation in a single framework

Normalizing Flows learn the probability density function of normal transaction behavior. Anomaly scoring proceeds as:

• Train the flow on non-fraudulent transactions only (semi-supervised novelty detection)
• Compute the log-likelihood of each incoming transaction under the learned density
• Flag transactions falling below a density threshold as anomalous
• This approach naturally captures multi-modal normal behavior (e.g., different spending patterns by merchant category)

The 'flow' in Normalizing Flow refers to the composition of simple transformations into a complex mapping. A flow is defined as:

• x = f_K ∘ f_{K-1} ∘ ... ∘ f_1(z) where each f_i is invertible
• Each transformation adds expressive power while maintaining tractability
• Common layers include affine coupling layers, actnorm, and invertible 1x1 convolutions
• The depth of the chain controls the model's capacity to capture intricate fraud patterns

Compared to GANs, Normalizing Flows offer stable and convergent training because they optimize a direct maximum likelihood objective rather than a minimax game. Benefits include:

• No mode collapse—the model is incentivized to cover all modes of the data distribution
• No discriminator to train, simplifying the architecture
• Loss curves directly reflect model quality (higher log-likelihood = better fit)
• Training stability is critical in regulated financial environments where model behavior must be predictable