Notation and mathematical foundations¶

MFTT constructs monotone triangular maps from samples. This page fixes the map direction and notation shared by the single-fidelity, hierarchical, and non-hierarchical methods. The method guides develop the corresponding parameterizations and training objectives.

For broader introductions to triangular transport, see [1][2][3].

Monotone triangular maps¶

Let \(\pi\) be a target density and \(\eta\) a tractable reference density on \(\mathbb R^d\). A lower-triangular map \(\boldsymbol S:\mathbb R^d\to\mathbb R^d\) has components

\[ [\boldsymbol S(\boldsymbol x)]_k =S_k(x_1,\ldots,x_k), \qquad k=1,\ldots,d. \]

Thus, component \(k\) depends only on the first \(k\) input coordinates. MFTT uses maps that are strictly increasing in their final input,

\[ \partial_k S_k(\boldsymbol x_{\leq k}) :=\frac{\partial}{\partial x_k} S_k(x_1,\ldots,x_k)>0. \]

The Jacobian is lower triangular, so its determinant is the product of its diagonal entries:

\[ \det\nabla\boldsymbol S(\boldsymbol x) =\prod_{k=1}^d \partial_kS_k(\boldsymbol x_{\leq k})>0. \]

This structure makes density evaluation and inversion substantially simpler than for a general multivariate transformation.

Target and reference directions¶

MFTT follows the target-to-reference convention. The ideal transport satisfies

\[ \boldsymbol S_\sharp\pi=\eta. \]

Equivalently, if \(\boldsymbol X\sim\pi\), then \(\boldsymbol S(\boldsymbol X)\sim\eta\). In the library, model.evaluate(x) applies this forward direction. The inverse direction generates target samples: if \(\boldsymbol Z\sim\eta\), then \(\boldsymbol S^{-1}(\boldsymbol Z)\sim\pi\), and model.inverse(z) performs this operation.

The density induced by a map is the pullback of the reference,

\[ (\boldsymbol S^\sharp\eta)(\boldsymbol x) =\eta(\boldsymbol S(\boldsymbol x)) \left|\det\nabla\boldsymbol S(\boldsymbol x)\right|. \]

An exact transport therefore obeys

\[ \pi(\boldsymbol x) =\eta(\boldsymbol S(\boldsymbol x)) \left|\det\nabla\boldsymbol S(\boldsymbol x)\right|. \]

The library methods pullback_pdf and pullback_logpdf evaluate the learned version of this density. See shared operations for the common forward, inverse, sampling, and density interfaces.

The Knothe–Rosenblatt rearrangement¶

Under standard absolute-continuity assumptions, the monotone triangular map coupling \(\pi\) and \(\eta\) is unique up to vairable ordering. It is the Knothe–Rosenblatt (KR) rearrangement [4][5][2][6].

Triangularity reduces inversion to a sequence of scalar problems. First solve \(S_1(x_1)=z_1\) for \(x_1\), then solve \(S_2(x_1,x_2)=z_2\) for \(x_2\), and continue through component \(d\). Strict monotonicity in \(x_k\) makes each scalar solution unique when it exists. This same structure supports exact conditioning on a leading coordinate prefix [2].

Sample and fidelity conventions¶

Training samples are stored as rows. A single-fidelity dataset is

\[ \mathcal X =\{\boldsymbol x^{(j)}\}_{j=1}^{N}, \qquad \boldsymbol x^{(j)}\sim\pi, \]

and is supplied as an array with shape (N, d).

For multifidelity problems, \(\ell=0,\ldots,M\) indexes the data sources:

\[ \mathcal X_\ell =\{\boldsymbol x_\ell^{(j)}\}_{j=1}^{N_\ell}, \qquad \boldsymbol x_\ell^{(j)}\sim\pi_\ell. \]

Fidelity \(0\) is always the high-fidelity target, and fidelities \(1,\ldots,M\) are lower-fidelity sources. Multifidelity datasets are supplied as [X_0, ..., X_M]; the method guides state whether the lower-fidelity sources are ordered or treated as peers.

Reading the method guides¶

Guide	Data assumption	Learned construction
Single-fidelity maps	One sample set from one target	One KR-map approximation
Hierarchical multifidelity maps	An ordered low-to-high fidelity chain	A composition of stage maps
Non-hierarchical multifidelity maps	Unordered low-fidelity peers	One high-fidelity map informed by pretrained parent maps

The single-fidelity guide is also the common background for the component parameterization and sample-based training used inside both multifidelity methods.

Notation conventions¶

Mathematical notation	Meaning in MFTT
\(d\)	Input dimension
\(k=1,\ldots,d\)	Triangular component index
\(\boldsymbol x\)	Target-space coordinate or sample
\(\boldsymbol z\)	Reference-space coordinate or sample
\(\pi\)	Target density
\(\eta\)	Reference density; standard normal by default
\(\boldsymbol S\)	Target-to-reference triangular map
\(\boldsymbol S_\sharp\pi\)	Pushforward of \(\pi\) through \(\boldsymbol S\)
\(\boldsymbol S^\sharp\eta\)	Pullback density induced by \(\boldsymbol S\) and \(\eta\)
\(\mathcal X\) or `X`	Single-fidelity training matrix with samples as rows
\(\pi_\ell\)	Target density at fidelity \(\ell\)
\(\mathcal X_\ell\) or `X_ell[ell]`	Training matrix at fidelity \(\ell\)
\(N_\ell\)	Number of samples at fidelity \(\ell\)
\(\ell=0\)	High-fidelity source
\(\ell>0\)	Lower-fidelity source