DIRT

DIRT(
    negloglik: Callable[[Tensor], Tensor],
    neglogpri: Callable[[Tensor], Tensor],
    preconditioner: Preconditioner,
    bases: Basis1D | List[Basis1D],
    bridge: Bridge | None = None,
    tt_options: TTOptions | None = None,
    dirt_options: DIRTOptions | None = None,
    prev_approx: Dict[int, SIRT] | None = None,
)

Deep (squared) inverse Rosenblatt transport.

Parameters

negloglik : Callable[[Tensor], Tensor]: A function that receives an \(n \times d\) matrix of samples and returns an \(n\)-dimensional vector containing the negative log-likelihood function evaluated at each sample.
neglogpri : Callable[[Tensor], Tensor]: A function that receives an \(n \times d\) matrix of samples and returns an \(n\)-dimensional vector containing the negative log-prior density evaluated at each sample.
bases : Basis1D | List[Basis1D]: A list of sets of basis functions for each dimension, or a single set of basis functions (to be used in all dimensions), used to construct the functional tensor trains at each iteration.
bridge : Bridge | None = None: An object used to generate the intermediate densities to approximate at each stage of the DIRT construction.
tt_options : TTOptions | None = None: Options for constructing the FTT approximation to the square root of the ratio function (i.e., the pullback of the current bridging density under the existing composition of mappings) at each iteration.
dirt_options : DIRTOptions | None = None: Options for constructing the DIRT approximation to the target density.
prev_approx : Dict[int, SIRT] | None = None: A dictionary containing a set of SIRTs generated as part of the construction of a previous DIRT object.

References

Cui, T and Dolgov, S (2022). Deep composition of tensor-trains using squared inverse Rosenblatt transports. Foundations of Computational Mathematics 22, 1863–1922.

Methods

Name	Description
eval_potential	Evaluates the potential function.
eval_potential_cond	Evaluates the conditional potential function.
eval_rt	Evaluates the deep Rosenblatt transport.
eval_irt	Evaluates the deep inverse Rosenblatt transport.
eval_cirt	Evaluates the conditional inverse Rosenblatt transport.
eval_irt_pullback	Evaluates the pullback of a density function.
eval_cirt_pullback	Evaluates the pullback of a conditional density function.
random	Generates a set of random samples.
sobol	Generates a set of QMC samples.

eval_potential

DIRT.eval_potential(
    xs: Tensor,
    subset: str | None = None,
    n_layers: int | None = None,
)

Evaluates the potential function.

Returns the joint potential function, or the marginal potential function for the first \(k\) variables or the last \(k\) variables, corresponding to the pullback of the reference measure under a given number of layers of the DIRT.

Parameters

xs : Tensor: An \(n \times k\) matrix containing a set of samples from the approximation domain.
subset : str | None = None: If the samples contain a subset of the variables, (i.e., \(k < d\)), whether they correspond to the first \(k\) variables (subset='first') or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the current DIRT construction to use when computing the potential. If not specified, all layers will be used when computing the potential.

Returns

neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential function of the target density evaluated at each element in xs.

eval_potential_cond

DIRT.eval_potential_cond(
    ys: Tensor,
    xs: Tensor,
    subset: str = 'first',
    n_layers: int | None = None,
)

Evaluates the conditional potential function.

Returns the conditional potential function evaluated at a set of samples in the approximation domain.

Parameters

ys : Tensor: An \(n \times k\) matrix containing samples from the approximation domain.
xs : Tensor: An \(n \times (d-k)\) matrix containing samples from the approximation domain.
subset : str = 'first': Whether ys corresponds to the first \(k\) variables (subset='first') of the approximation, or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the deep inverse Rosenblatt transport to push the samples forward under. If not specified, the samples will be pushed forward through all the layers.

Returns

neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential function of the approximation to the conditional density of \(X \textbar Y\) evaluated at each sample in xs.

eval_rt

DIRT.eval_rt(xs: Tensor, subset: str | None = None, n_layers: int | None = None)

Evaluates the deep Rosenblatt transport.

Parameters

xs : Tensor: An \(n \times k\) matrix of samples from the approximation domain.
subset : str | None = None: If the samples contain a subset of the variables, (i.e., \(k < d\)), whether they correspond to the first \(k\) variables (subset='first') or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the deep inverse Rosenblatt transport to push the samples forward under. If not specified, the samples will be pushed forward through all the layers.

Returns

rs : Tensor: An \(n \times k\) matrix containing the composition of mappings evaluated at each value of xs.
neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential function of the pullback of the reference density under the current composition of mappings evaluated at each sample in xs.

eval_irt

DIRT.eval_irt(
    rs: Tensor,
    subset: str | None = None,
    n_layers: int | None = None,
)

Evaluates the deep inverse Rosenblatt transport.

Parameters

rs : Tensor: An \(n \times k\) matrix containing samples from the reference domain.
subset : str | None = None: If the samples contain a subset of the variables, (i.e., \(k < d\)), whether they correspond to the first \(k\) variables (subset='first') or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the deep inverse Rosenblatt transport to pull the samples back under. If not specified, the samples will be pulled back through all the layers.

Returns

xs : Tensor: An \(n \times k\) matrix containing the corresponding samples from the approximation domain, after applying the deep inverse Rosenblatt transport.
neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential function of the pullback of the reference density under the current composition of mappings, evaluated at each sample in xs.

eval_cirt

DIRT.eval_cirt(
    ys: Tensor,
    rs: Tensor,
    subset: str = 'first',
    n_layers: int | None = None,
)

Evaluates the conditional inverse Rosenblatt transport.

Returns the conditional inverse Rosenblatt transport evaluated at a set of samples in the approximation domain.

Parameters

ys : Tensor: A matrix containing samples from the approximation domain. The matrix should have dimensions \(1 \times k\) (if the same realisation of \(Y\) is to be used for all samples in rs) or \(n \times k\) (if a different realisation of \(Y\) is to be used for each samples in rs).
rs : Tensor: An \(n \times (d-k)\) matrix containing samples from the reference domain.
subset : str = 'first': Whether ys corresponds to the first \(k\) variables (subset='first') of the approximation, or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the DIRT object to use when evaluating the CIRT. If not specified, all layers will be used.

Returns

xs : Tensor: An \(n \times (d-k)\) matrix containing the realisations of \(X\) corresponding to the values of rs after applying the conditional inverse Rosenblatt transport.
neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential function of the approximation to the conditional density of \(X \textbar Y\) evaluated at each sample in xs.

eval_irt_pullback

DIRT.eval_irt_pullback(
    potential: Callable[[Tensor], Tensor],
    rs: Tensor,
    subset: str | None = None,
    n_layers: int | None = None,
)

Evaluates the pullback of a density function.

This function evaluates \(\mathcal{T}^{\sharp}f(r)\), where \(\mathcal{T}(\cdot)\) denotes the inverse Rosenblatt transport and \(f(\cdot)\) denotes an arbitrary density function.

Parameters

potential : Callable[[Tensor], Tensor]: A function that takes an \(n \times k\) matrix of samples from the approximation domain, and returns an \(n\)-dimensional vector containing the potential function associated with \(f(\cdot)\) evaluated at each sample.
rs : Tensor: An \(n \times k\) matrix containing a set of samples from the reference domain.
subset : str | None = None: If the samples contain a subset of the variables, (i.e., \(k < d\)), whether they correspond to the first \(k\) variables (subset='first') or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the deep inverse Rosenblatt transport to pull the samples back under. If not specified, the samples will be pulled back through all the layers.

Returns

neglogTfrs : Tensor: An \(n\)-dimensional vector containing the potential of the pullback function evaluated at each element in rs; that is, \(-\log(\mathcal{T}^{\sharp}f(r))\).
neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential of the target function evaluated at each element in rs, pushed forward under the IRT; that is, \(-\log(f(\mathcal{T}(r)))\).

eval_cirt_pullback

DIRT.eval_cirt_pullback(
    potential: Callable[[Tensor], Tensor],
    ys: Tensor,
    rs: Tensor,
    subset: str = 'first',
    n_layers: int | None = None,
)

Evaluates the pullback of a conditional density function.

This function evaluates \(\mathcal{T}^{\sharp}f(r\|y)\), where \(\mathcal{T}(\cdot)\) denotes the inverse Rosenblatt transport and \(f(\cdot\|y)\) denotes an arbitrary conditional density function.

Parameters

potential : Callable[[Tensor], Tensor]: A function that takes an \(n \times (d-k)\) matrix of samples from the approximation domain, and returns an \(n\)-dimensional vector containing the potential function associated with \(f(\cdot\\|y)\) evaluated at each sample.
ys : Tensor: A matrix containing samples from the approximation domain. The matrix should have dimensions \(1 \times k\) (if the same realisation of \(Y\) is to be used for all samples in rs) or \(n \times k\) (if a different realisation of \(Y\) is to be used for each samples in rs).
rs : Tensor: An \(n \times (d-k)\) matrix containing a set of samples from the reference domain.
subset : str = 'first': Whether ys corresponds to the first \(k\) variables (subset='first') of the approximation, or the last \(k\) variables (subset='last').
n_layers : int | None = None: The number of layers of the deep inverse Rosenblatt transport to pull the samples back under. If not specified, the samples will be pulled back through all the layers.

Returns

neglogTfrs : Tensor: An \(n\)-dimensional vector containing the potential of the pullback function evaluated at each element in rs; that is, \(-\log(\mathcal{T}^{\sharp}f(r\\|y))\).
neglogfxs : Tensor: An \(n\)-dimensional vector containing the potential of the target function evaluated at each element in rs, pushed forward under the IRT; that is, \(-\log(f(\mathcal{T}(r)\\|y))\).

random

DIRT.random(n: int)

Generates a set of random samples.

The samples are distributed according to the DIRT approximation to the target density.

Parameters

n : int: The number of samples to generate.

Returns

xs : Tensor: An \(n \times d\) matrix containing the generated samples.

sobol

DIRT.sobol(n: int)

Generates a set of QMC samples.

The samples are distributed according to the DIRT approximation to the target density.

Parameters

n : int: The number of samples to generate.

Returns

xs : Tensor: An \(n \times d\) matrix containing the generated samples.