Normalizing Flow Maps¶

← Normalizing Flows | Conditional Flow Matching →

Normalizing Flow Maps (NFM; Ruscio et al., 2026) learn a two-time invertible pushforward \(f_{s,t}\) along a prescribed probability path. The endpoint map \(f_{0,1}\) is a normalizing flow with an exact change-of-variables likelihood, while intermediate time pairs provide few-step transport.

The torchregress tabular implementation uses time-conditioned affine coupling layers. For an admissible pair \(s<t\),

The forward endpoint transports Gaussian noise to targets: \(y=f_{0,1}(z\mid c)\) with \(z\sim\mathcal N(0,I)\). Sampling uses this forward map; likelihood evaluation uses its inverse and inverse log Jacobian. Each time argument can independently be a scalar or one value per batch row.

Training combines pointwise flow-map matching,

with the endpoint NLL. This is a lean tabular instantiation, not the image-scale autoregressive NFM of the paper.

arch="tarflow" enables a more ambitious tokenized variant: explicit context and time tokens followed by one token per target coordinate, with a causal multi-layer Transformer and exact sequential inversion. tarflow_layers controls the number of transformer blocks per map layer and tarflow_heads the attention heads. This is a tabular refinement, not the image-scale autoregressive architecture.

from torchregress.models import NormalizingFlowMap

nfm = NormalizingFlowMap(n_features=2, context_dim=64, n_layers=4)
loss = nfm.training_loss(context, target)
logp = nfm.log_prob(context, target)
samples = nfm.sample(context, n_samples=128)   # [B, S, 2]

When to use it¶

NFM is a research direction for combining exact likelihoods with flow-map few-step transport. For routine tabular regression with calibrated NLL, NSF remains the safer default; test NFM when few-step/consistency behavior matters.

References¶