Deconditional Downscaling with Gaussian Processes

1 Introduction . Spatial observations often operate at limited resolution due to practical constraints . For example , remote sensing atmosphere products [ 2 , 3 , 4 , 5 ] provide measurement of atmospheric properties such as cloud top temperatures and optical thickness , but only at a low resolution . Devising methods to refine low-resolution ( LR ) variables for local-scale analysis thus plays a crucial part in our understanding of the anthropogenic impact on climate When high-resolution ( HR ) observations of different covariates are available , details can be instilled into the LR field for refinement . This task is referred to as statistical downscaling or spatial disaggregation and models LR observations as the aggregation of an unobserved underlying HR field . For example , multispectral optical satellite imagery [ 6 , 7 ] typically comes at higher resolution than atmospheric products and can be used to refine the latter . Statistical downscaling has been studied in various forms , notably giving it a probabilistic treatment [ 8 , 9 , 10 , 11 , 12 , 13 ] , in which Gaussian processes ( GP ) [ 14 ] are typically used in conjunction with a sparse variational formulation [ 15 ] to recover the underlying unobserved HR field . Our approach follows this line of research where we do not observe data from the underlying HR groundtruth field . On the other hand , deep neural network ( DNN ) based approaches [ 16 , 17 , 18 ] study this problem from a different setting , where they often assume that both HR and LR matched observations are available for training . Then , their approaches follow a standard supervised learning setting in learning a mapping between different resolutions . ∗Indicates equal contribution †Department of Statistics , Oxford , UK , OX1 3LB . < siu.chau @ stats.ox.ac.uk , shahine.bouabid @ stats.ox.ac.uk , dino.sejdinovic @ stats.ox.ac.uk > 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . However , both lines of existing methods require access to bags of HR covariates that are paired with aggregated targets , which in practice might be infeasible . For example , the multitude of satellites in orbit not only collect snapshots of the atmosphere at different resolutions , but also from different places and at different times , such that these observations are not jointly observed . To overcome this limitation , we propose to consider a more flexible mediated statistical downscaling setup that only requires indirect matching between LR and HR covariates through a mediating field . We assume that this additional field can be easily observed , and matched separately with both HR covariates and aggregate LR targets . We then use this third-party field to mediate learning and downscale our unmatched data . In our motivating application , climate simulations [ 19 , 20 , 21 ] based on physical science can serve as a mediating field since they provide a comprehensive spatiotemporal coverage of meteorological variables that can be matched to both LR and HR covariates . Formally , let bx = { x ( 1 ) , . . . , x ( n ) } ⊂ X be a general notation for bags of HR covariates , f : X → R the field of interest we wish to recover and z̃ the LR aggregate observations from the field f . We suppose that bx and z̃ are unmatched , but that there exists mediating covariates y , ỹ ∈ Y , such that ( bx , y ) are jointly observed and likewise for ( ỹ , z̃ ) as illustrated in Figure 1 . We assume the following aggregation observation model z̃ = EX [ f ( X ) |Y = ỹ ] + ε with some noise ε . Our goal in mediated statistical downscaling is then to estimate f given ( bx , y ) and ( ỹ , z̃ ) , which corresponds to the deconditioning problem introduced in [ 1 ] . Motivated by applications in likelihood-free inference and tasktransfer regression , Hsu and Ramos [ 1 ] first studied the deconditioning problem through the lens of reproducing kernel Hilbert space ( RKHS ) and introduced the framework of Deconditional Mean Embeddings ( DME ) as its solution . In this work , we first propose a Bayesian formulation of deconditioning that results into a much simpler and elegant way to arrive to the DME-based estimator of Hsu and Ramos [ 1 ] , using the conditional expectations of f . Motivated by our mediated statistical downscaling problem , we then extend deconditioning to a multi-resolution setup and bridge the gap between DMEs and existing probabilistic statistical downscaling methods [ 9 ] . By placing a GP prior on the sought field f , we obtain a posterior distribution of the downscaled field as a principled Bayesian solution to the downscaling task on indirectly matched data . For scalability , we provide a tractable variational inference approximation and an alternative approximation to the conditional mean operator ( CMO ) [ 22 ] to speed up computation for large multi-resolution datasets . From a theoretical stand point , we further develop the framework of DMEs by establishing it as a two-staged vector-valued regressor with a natural reconstruction loss that mirrors Grünewälder et al . [ 23 ] ’ s work on conditional mean embeddings . This perspective allows us to leverage distribution regression theory from [ 24 , 25 ] and obtain convergence rate results for the deconditional mean operator ( DMO ) estimator . Under mild assumptions , we obtain conditions under which this rate is a minimax optimal in terms of statistical-computational efficiency . Our contributions are summarized as follows : • We propose a Bayesian formulation of the deconditioning problem of Hsu and Ramos [ 1 ] . We establish its posterior mean as a DME-based estimate and its posterior covariance as a gauge of the deconditioning quality . • We extend deconditioning to a multi-resolution setup in the context of the mediated statistical downscaling problem and bridge the gap with existing probabilistic statistical downscaling methods . Computationally efficient algorithms are devised . • We demonstrate that the DMO estimate minimises a two-staged vector-valued regression and derive its convergence rate under mild assumptions , with conditions for minimax optimality . • We benchmark our model against existing methods for statistical downscaling tasks in climate science , on both synthetic and real-world multi-resolution atmospheric fields data , and show improved performance . 2 Background Materials . 2.1 Notations . Let X , Y be a pair of random variables taking values in non-empty sets X and Y , respectively . Let k : X × X → R and ` : Y × Y → R be positive definite kernels . The closure of the span of their canonical feature maps kx : = k ( x , · ) and ` y : = ` ( y , · ) for x ∈ X and y ∈ Y respectively induces RKHSHk ⊆ RX andH ` ⊆ RY endowed with inner products 〈· , ·〉k and 〈· , ·〉 ` . We observe realizations bD1 = { bxj , yj } Nj=1 of bags bxj = { x ( i ) j } nj i=1 from conditional distribution PX|Y=yj , with bag-level covariates yj sampled from PY . We concatenate them into vectors x : = [ bx1 . . . bxN ] > and y : = [ y1 . . . yN ] > . For simplicity , our exposition will use the notation without bagging – i.e . where D1 = { xj , yj } Nj=1 – when the generality of our contributions will be relevant to the theory of deconditioning from Hsu and Ramos [ 1 ] , in Sections 2 , 3 and 4 . We will come back to a bagged dataset formalism in Sections 3.3 and 5 , which corresponds to our motivating application of mediated statistical downscaling . With an abuse of notation , we define feature matrices by stacking feature maps along columns as Φx : = [ kx1 . . . kxN ] and Ψy : = [ ` y1 . . . ` yN ] and we denote Gram matrices as Kxx = Φ > x Φx = [ k ( xi , xj ) ] 1≤i , j≤N and Lyy = Ψ > y Ψy = [ ` ( yi , yj ) ] 1≤i , j≤N . The notation abuse ( · ) > ( · ) is a shorthand for the elementwise RKHS inner products when it is clear from the context . Let Z denote the real-valued random variable stemming from the noisy conditional expectation of some unknown latent function f : X → R , as Z = EX [ f ( X ) |Y ] + ε . We suppose one observes another set of realizations D2 = { ỹj , z̃j } Mj=1 from PY Z , which is sampled independently from D1 . Likewise , we stack observations into vectors ỹ : = [ ỹ1 . . . ỹM ] > , z̃ : = [ z̃1 . . . z̃M ] > and define feature map Ψỹ : = [ ` ỹ1 . . . ` ỹM ] . 2.2 Conditional and Deconditional Kernel Mean Embeddings . Marginal and Joint Mean Embeddings Kernel mean embeddings of distributions provide a powerful framework for representing and manipulating distributions without specifying their parametric form [ 22 , 26 ] . The marginal mean embedding of measure PX is defined as µX : = EX [ kX ] ∈ Hk and corresponds to the Riesz representer of expectation functional f 7→ EX [ f ( X ) ] . It can hence be used to evaluate expectations EX [ f ( X ) ] = 〈f , µX〉k . If the mapping PX 7→ µX is injective , the kernel k is said to be characteristic [ 27 ] , a property satisfied for the Gaussian and Matérn kernels on Rd [ 28 ] . In practice , Monte Carlo estimator µ̂X : = 1N ∑N i=1 kxi provides an unbiased estimate of µX [ 29 ] . Extending this rationale to embeddings of joint distributions , we define CY Y : = EY [ ` Y ⊗ ` Y ] ∈ H ` ⊗ H ` and CXY : = EX , Y [ kX ⊗ ` Y ] ∈ Hk ⊗ H ` , which can be identified with the crosscovariance operators between Hilbert spaces CY Y : H ` → H ` and CXY : H ` → Hk . They correspond to the Riesz representers of bilinear forms ( g , g′ ) 7→ Cov ( g ( Y ) , g′ ( Y ) ) = 〈g , CY Y g′〉 ` and ( f , g ) 7→ Cov ( f ( X ) , g ( Y ) ) = 〈f , CXY g〉k . As above , empirical estimates are obtained as ĈY Y = 1 NΨyΨ > y = 1 N ∑N i=1 ` yi ⊗ ` yi and ĈXY = 1 NΦxΨ > y = 1 N ∑N i=1 kxi ⊗ ` yi . Again , notation abuse ( · ) ( · ) > is a shorthand for element-wise tensor products when clear from context . Conditional Mean Embeddings Similarly , one can introduce RKHS embeddings for conditional distributions referred to as Conditional Mean Embeddings ( CME ) . The CME of conditional probability measure PX|Y=y is defined as µX|Y=y : = EX [ kX |Y = y ] ∈ Hk . As introduced by Fukumizu et al . [ 27 ] , it is common to formulate conditioning in terms of a Hilbert space operator CX|Y : H ` → Hk called the Conditional Mean Operator ( CMO ) . CX|Y satisfies by definition CX|Y ` y = µX|Y=y and C > X|Y f = EX [ f ( X ) |Y = · ] , ∀f ∈ Hk , where C > X|Y de- notes the adjoint of CX|Y . Plus , the CMO admits expression CX|Y = CXY C −1 Y Y , provided ` y ∈ Range ( CY Y ) , ∀y ∈ Y [ 26 , 27 ] . Song et al . [ 30 ] show that a nonparametric empirical form of the CMO writes ĈX|Y = Φx ( Lyy +NλIN ) −1Ψ > y , ( 1 ) where λ > 0 is some regularisation ensuring the empirical operator is globally defined and bounded . As observed by Grünewälder et al . [ 23 ] , since CX|Y defines a mapping from H ` to Hk , it can be interpreted as the solution to a vector-valued regression problem . This perspective enables derivation of probabilistic convergence bounds on the empirical CMO estimator [ 23 , 25 ] . Deconditional Mean Embeddings Introduced by Hsu and Ramos [ 1 ] as a new class of embeddings , Deconditional Mean Embeddings ( DME ) are natural counterparts of CMEs . While CME µX|Y=y ∈ Hk allows to take the conditional expectation of any f ∈ Hk through inner product EX [ f ( X ) |Y = y ] = 〈f , µX|Y=y〉k , the DME denoted µX=x|Y ∈ H ` solves the inverse problem3 and allows to recover the initial function of which the conditional expectation was taken , through inner product 〈EX [ f ( X ) |Y = · ] , µX=x|Y 〉 ` = f ( x ) . The associated Hilbert space operator , the Deconditional Mean Operator ( DMO ) , is thus defined as the operator DX|Y : Hk → H ` such that D > X|Y EX [ f ( X ) |Y = · ] = f , ∀f ∈ Hk . It admits an expression in terms of CMO and cross-covariance operators DX|Y = ( CX|Y CY Y ) > ( CX|Y CY Y ( CX|Y ) > ) −1 provided ` y ∈ Range ( CY Y ) and kx ∈ Range ( CX|Y CY Y C > X|Y ) , ∀y ∈ Y and ∀x ∈ X . A nonparametric empirical estimate of the DMO using datasets D1 and D2 as described above , is given by D̂X|Y = Ψỹ ( A > KxxA +M IM ) −1 A > Φ > x where > 0 is a regularisation term and A : = ( Lyy + NλI ) −1Lyỹ can be interpreted as a mediation operator . Applying the DMO to expected responses z̃ , Hsu and Ramos [ 1 ] are able to recover an estimate of f as f̂ ( x ) = k ( x , x ) A ( A > KxxA +M IM ) −1 z̃ . ( 2 ) Note that since separate samples ỹ can be used to estimate CY Y , this naturally fits a mediating variables setup where x and the conditional means z̃ are not jointly observed .

Summary: The authors consider the problem of refining low-res (LR) spatial observations using high-res (HR) covariates.   Basically, the problem is as follows. The analyst observes LR outcome Z, HR covariate X, and LR covariate Y from the process Z=E[f(X)|Y]+e. The analyst wishes to recover the function f.  In more detail: the analyst observes two datasets: D1 and D2. D1 consists of N observations (x_j, y_j) where x_j may be a bag.  D2 consists of M observations (y_j, z_j). The function f is modelled as a function in an RKHS (with Bayesian or frequentist approach).  The authors characterize the Bayesian version of this problem and derive the GP posterior. The authors characterize the frequentist version of this problem and prove minimax optimal finite sample rates.