Measure transport via polynomial density surrogates

Figure 1.  Exemplary sets on which a function $ f $ satisfying Assumption 2.3 is allowed to be zero (left) and is not allowed to be zero (right)

Figure 2.  For a multimodal distribution $ \pi $ on $ [0, 1] $, its density $ { }{f_{{\pi}}} $ will vanish or be very small in certain parts of the domain. By consequence, the inverse transport $ S = T^{-1} $ (corresponding to the antiderivative of $ { }{f_{{\pi}}} $) will be (almost) constant in those areas, and the transport $ T = S^{-1} $ can exhibit discontinuities. Even if $ { }{f_{{\pi}}} $ and $ S $ are $ C^\infty $ functions that are well approximated by polynomials, the same is in general not true for $ T $

Figure 3.  Test density $ { }{\hat{f}_ {{\pi}}} $ as in (41)

Figure 4.  Uniform grid, uniform density $ f_\mu $ and uniformly i.i.d. samples $ ( {\boldsymbol x}_k)_{k = 0}^{800} $ with $ {\boldsymbol x}_k \sim \mu $

Figure 5.  Grid, density and samples from Figure 4 as transformed by a transport $ T_1 $ constructed from a surrogate of (41) in $ \mathbb{P}_{\Lambda_1} $ with $ \Lambda_1 = \{{ {{\mathit{\boldsymbol{\nu}}}} \in {\mathbb N}^{{2}}}\, :\, {\| { {{\mathit{\boldsymbol{\nu}}}}} \|_{1}\le 15}\} $

Figure 6.  Grid, density and samples from Figure 4 as transformed by a transport $ T_2 $ constructed from a surrogate of (41) in $ \mathbb{P}_{\Lambda_2} $ with $ \Lambda_2 = \{{ {{\mathit{\boldsymbol{\nu}}}} \in {\mathbb N}^{{2}}}\, :\, {\| { {{\mathit{\boldsymbol{\nu}}}}} \|_{1}\le 65}\} $

Figure 7.  Monte-Carlo approximation of the Hellinger distance between the target measure $ \pi $ with $ f_\pi $ as in (41) and $ T_\sharp \mu $ in dependence of the cardinality of the surrogate ansatz space $ \mathbb{P}_\Lambda $. Further shown are the WLS error, which differs only by a constant from $ {\rm{dist}}_{{\rm{H}}}(\pi, T_\sharp\mu) $, and the predicted exponential convergence rate (with the unknown constant $ \beta $ fitted to $ 0.12 $)

Figure 8.  Test density $ { }{f_{{\pi}}} $ composed of three isolated hat functions with square support. The density is zero in the white regions of the domain

Figure 9.  Density surrogate in $ \mathbb{P}_{\Lambda} $ for $ \Lambda = \{{ {{\boldsymbol{\nu}}} \in {\mathbb N}^2}\, :\, {\| { {{\boldsymbol{\nu}}}} \|_{1}\le 25}\} $ along with the image of a regular grid and uniform distributed samples under the corresponding transport $ T $

Figure 10.  Monte-Carlo approximation of the Hellinger distance between the target measure $ \pi $ with density $ f_\pi $ as in Figure 8 and $ T_\sharp \mu $ in dependence of the cardinality of the density surrogate ansatz space $ \mathbb{P}_\Lambda $, as well as the WLS error

Figure 11.  Hat functions $ (\varphi_{\ell, i})_{i = 0}^{2^{\ell-1}-1} $ for $ \ell \in \{0, \dots, 3\} $

Figure 12.  A random signal function $ s $

Figure 13.  Convoluted signal $ c(y) $ and noisy measurements $ (m_j)_{j = 1}^{10} $

Figure 14.  Monte-Carlo approximation of the Hellinger distance between the target measure $ \pi $ with density $ f_\pi $ as in (45) and $ T_\sharp \mu $ in dependence of the cardinality of the density surrogate ansatz space $ \mathbb{P}_\Lambda $ for different parameter dimensions $ d\in {\mathbb N} $

Figure 15.  CPU runtime $ t $ (in seconds) of computing a single evaluation of $ T $ (with the bisection parameter $ r $ in Algorithm 8 set to $ r = 30 $) in dependence of the cardinality of the density surrogate ansatz space $ \mathbb{P}_\Lambda $ for different parameter dimensions $ d\in {\mathbb N} $. Each datapoint is the average of $ 3 $ evaluations