AdaptiveTransportMaps: ATM

What is the ATM algorithm?

Package contains methods and algorithms for learning monotone triangular transport maps given access to samples from a target density or evaluations of the target. The Adaptive Transport Map algorithm is a greedy approach for learning parameteric map approximation using a sparse feature expansion.

More information on the methodology for this sofware can be found in this preprint. The authors are Ricardo Baptista (MIT), Youssef Marzouk (MIT), and Olivier Zahm (INRIA). The software is currently being maintained by the Uncertainty Quantification Group at MIT.

Software development team

Ricardo Baptista (MIT), Paul-Baptiste Rubio (MIT)

E-mails: [email protected], [email protected]

Installation

The ATM algorithm is implemented in MATLAB and does not require additional packages. Check installation by typing runtests in MATLAB from inside the testing folder.

Example

Let the target density be the product of two conditional Gaussians . Our goal is to approximate the density given 1000 i.i.d. samples in .

First, we standardize the samples with a linear transport map as

    G = GaussianPullbackDensity(2, true);
    G = G.optimize(X);
    Xnorm = G.evaluate(X);

Next, we define the pullback density of a monotone triangular map represented using expansions of Hermite polynomials through a two-dimensional standard Gaussian density as

    ref = IndependentProductDitribution({Normal(), Normal()});
    basis = HermiteProbabilistPolyWithLinearization();
    S = identity_map(1:2, basis);
    PB = PullbackDensity(S, ref);

To optimize the map we use a greedy algorithm that enriches the basis functions in the map starting from an identity function. We run this procedure for 5 iterations for each component. In practice, we recommend using a cross-validation procedure to select the optimal number of terms in each expansion.

    [PB, ~] = PB.greedy_optimize(Xnorm, [], [5,5], 'max_terms');

The final map CM is given by the composition of G and PB as

    CM = ComposedPullbackDensity({G, PB}, ref);

To plot the approximate density, we evaluate CM.log_pdf on a grid of points. The true and approximate densities are given below.

We can also generate 100 samples from the approximate density by inverting the map at samples from the standard Gaussian reference using

    Xsamples = G.S.inverse(PB.S.inverse(randn(1000,2)));

We can evaluate the approximate conditional density using the second component of the map. We can evaluate the pullback density of the Gaussian reference through the second component at x as CM.log_pdf(x,2).