summaries: 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.  This paper extends on the work of deconditional mean embeddings (DMEs) and task transformed Gaussian processes (TTGPs) from Hsu and Ramos (2019) from both an application and theoretical stand point.  From a theoretical stand point, the authors developed the framework around deconditional mean embeddings further by three ways: (1) formulating another elegant way to arrive at deconditional posteriors from Hsu and Ramos (2019), (2) establishing deconditional mean operators (DMOs) as vector-valued regressors in a manner that mirrors Grunewalder et al (2012) which highlights its interpretation as a reconstruction operator, and (3) enhancing guarantees on the convergence rate of deconditional mean operators from what was established in Hsu and Ramos (2019) using the setup and results from  Caponnetto and De Vito (2007), Szabó et al (2016), and Singh et al (2019).  From an application stand point, the authors apply DMEs for refining low resolution spatial fields with high resolution information. This is the special case of DMEs and TTGPs where the task dataset consists of collections of low resolution covariates ($\tilde{y}$) and aggregated targets ($\tilde{z}$), and the transformation dataset consists of collections of (another potentially unmatched set of) low resolution covariates ($y$) each matched with bags of high resolution covariates ($^{b}\bf{x}$). The latter constructions regarding the transformation dataset is what makes this a non-trivial special case due to having bagged high resolution covariates, which allows for a slightly different empirical estimator for the cross-covariance operator. This leads to an alternative conditional mean operator (CMO) which they call the conditional mean shrinkage operator. Finally, the authors then apply their work to toy experiments (swiss roll) and downscaling of atmospheric temperature (CMIP6), where to scale it they also derive a variational formulation to approximate the deconditional posterior.  The paper considers the problem of learning a function  $$f:\mathcal{X}\to \mathbb{R}$$ from the relationship  $$g(y) = \int_x f(x) \mathbb{P}[x|y]dx$$ where the training data consists of two datasets. The first set is  $\mathcal{D}_1 = ( x^b_i, y_i )$ where each $x^b_i$ is a bag of points in $\mathcal{X}$ and $y_i$ is a single point in a space $\mathcal{Y}$, these are empirical samples from some distribution $\mathbb{P}[x|y]\mathbb{P}[y]$. The second dataset is of $\mathcal{D}_2 = (\tilde y_j, \tilde z_i)$ with $\tilde y_j \in \mathcal{Y}$ and $\tilde z_i \in \mathbb{R}$ is a set of input-output pairs to from $\tilde z_i = g(\tilde y_i)$.  Learning models for $f(x)$ from aggregate data, (input bag, output) pairs e.g. $( x^b_i, z_i )$ has been studied, similarly the use of  there are intermediate covariates $y_i \in \mathcal{Y}$ with a dataset of the form $( x^b_i, y_i, z_i )$ has been studied. As I undertsand, the novel constributino of this work is the extension to the case where there are two datasets, $( x^b_i, y_i)$ and $(\tilde y_j, \tilde z_j )$ that both contain points from the domain $\mathcal{Y}$ yet they are not the same points, they are mismatched.  The paper proposes a "conditional mean process", a Gaussian process model that interpolates aggregated data as well as the marginalization operater in order to infer the un-aggregated latent function under this above problem setting.    This paper presents a GP model for refining coarse-grained spatial data, which can be used for handling aggregated data that are unmatched spatially and temporally. The authors also develop the two-stage regression algorithm for downscaling. The effectiveness of the proposed model is demonstrated using synthetic and real-world datasets. The paper proposes a novel Gaussian process approach for settings in which we have two datasets, D_1={(x_i, y_i)}_i=1 ^N and D_2=((y_i,z_i))_i=1^M  that are linked by the mediating variable Y. The datasets are not matched, and we want to learn a function from X to Z. In contrast to existing work, the algorithm generalizes to settings in which the resolution of the two datasets differ.
abstractText: Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an “inverse” of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos [1]. We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
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sections.heading: 1 Introduction
sections.text: 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.
sections.heading: 2 Background Materials
sections.heading: 2.1 Notations
sections.text: 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 ].
sections.heading: 2.2 Conditional and Deconditional Kernel Mean Embeddings
sections.text: 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.
sections.heading: 3 Deconditioning with Gaussian processes
sections.text: In this section, we introduce Conditional Mean Process (CMP), a stochastic process stemming from the conditional expectation of a GP. We provide a characterisation of the CMP and show that the corresponding posterior mean of its integrand is a DME-based estimator. We also derive in Appendix B a variational formulation of our model that scales to large datasets and demonstrate its performance in Section 5.
  For simplicity, we put aside observations bagging in Sections 3.1 and 3.2, our contributions being relevant to the general theory of DMEs. We return to a bagged formalism in Section 3.3 and extend deconditioning to the multiresolution setup inherent to the mediated statistical downscaling application. In what follows, X is a measurable space, Y a Borel space and feature maps kx and `y are Borel-measurable functions for any x ∈ X , y ∈ Y . All proofs and derivations of the paper are included in the appendix.
sections.heading: 3.1 Conditional Mean Process
sections.text: Bayesian quadrature [14, 31, 32] is based on the observation that the integral of a GP with respect to some marginal measure is a Gaussian random variable. We start by probing the implications of integrating with respect to conditional distribution PX|Y=y and considering such integrals as functions of the conditioning variable y. This gives rise to the notion of conditional mean processes. Definition 3.1 (Conditional Mean Process). Let f ∼ GP(m, k) with integrable sample paths, i.e.∫ X |f |dPX < ∞ a.s. The CMP induced by f with respect to PX|Y is defined as the stochastic
  3the slightly unusual notation µX=x|Y is taken from Hsu and Ramos [1] and is meant to contrast the usual conditioning µX|Y =y
  process {g(y) : y ∈ Y} given by g(y) = EX [f(X)|Y = y] = ∫ X f(x) dPX|Y=y(x). (3)
  By linearity of the integral, it is clear that g(y) is a Gaussian random variable for each y ∈ Y . The sample paths integrability requirement ensures g is well-defined almost everywhere. The following result characterizes CMP as a GP on Y . Proposition 3.2 (CMP characterization). Suppose EX [|m(X)|] <∞ and EX [‖kX‖k] <∞ and let (X ′, Y ′) ∼ PXY . Then g is a Gaussian process g ∼ GP(ν, q) a.s. , specified by
  ν(y) = EX [m(X)|Y = y] q(y, y′) = EX,X′ [k(X,X ′)|Y = y, Y ′ = y′] (4) ∀y, y′ ∈ Y . Furthermore, q(y, y′) = 〈µX|Y=y, µX|Y=y′〉k a.s.
  Intuitively, the CMP can be understood as a GP on the conditional means where its covariance q(y, y′) is induced by the similarity between the CMEs at y and y′. Resorting to the kernel ` defined on Y , we can reexpress the covariance using Hilbert space operators as q(y, y′) = 〈CX|Y `y, CX|Y `y′〉k. A natural nonparametric estimate of the CMP covariance thus comes using the CMO estimator from (1) as q̂(y, y′) = `(y,y) (Lyy +NλIN ) −1 Kxx (Lyy +NλIN ) −1 `(y, y′). When m ∈ Hk, the mean function can be written as ν(y) = 〈µX|Y=y,m〉k for which we can also use empirical estimate ν̂(y) = `(y,y) (Lyy +NλIN ) −1 Φ>xm. Finally, one can also quantify the covariance between the CMP g and its integrand f , i.e. Cov(f(x), g(y)) = EX [k(x,X)|Y = y]. Under the same assumptions as Proposition 3.2, this covariance can be expressed using mean embeddings, i.e. Cov(f(x), g(y)) = 〈kx, µX|Y=y〉k and admits empirical estimate k(x,x) (Lyy +NλIN ) −1 `(y, y).
sections.heading: 3.2 Deconditional Posterior
sections.text: Given independent observations introduced above, D1 = {x,y} and D2 = {ỹ, z̃}, we may now consider an additive noise model with CMP prior on aggregate observations z̃|ỹ ∼ N (g(ỹ), σ2IM ). Let Qỹỹ := q(ỹ, ỹ) be the kernel matrix induced by q on ỹ and let Υ := Cov(f(x), z̃) = Φ>xCX|Y Ψỹ be the cross-covariance between f(x) and z̃. The joint normality of f(x) and z̃ gives[
  f(x) z̃
  ] | y, ỹ ∼ N ([ m(x) ν(ỹ) ] , [ Kxx Υ
  Υ> Qỹỹ + σ 2IM
  ]) . (5)
  Using Gaussian conditioning, we can then readily derive the posterior distribution of the underlying GP field f given the aggregate observations z̃ corresponding to ỹ. The latter can naturally be degenerated if observations are paired, i.e. y = ỹ. This formulation can be seen as an example of the inter-domain GP [33], where we utilise the observed conditional means z̃ as inter-domain inducing features for inference of f . Proposition 3.3 (Deconditional Posterior). Given aggregate observations z̃ with homoscedastic noise σ2, the deconditional posterior of f is defined as the Gaussian process f |z̃ ∼ GP(md, kd) where
  md(x) = m(x) + k > x CX|Y Ψỹ(Qỹỹ + σ 2IM ) −1(z̃− ν(ỹ)), (6) kd(x, x ′) = k(x, x′)− k>x CX|Y Ψỹ(Qỹỹ + σ2IM )−1Ψ>ỹC>X|Y kx′ . (7)
  Substituting terms by their empirical forms, we can define a nonparametric estimate of the md as
  m̂d(x) := m(x) + k(x,x)A(Q̂ỹỹ + σ 2IM ) −1(z̃− ν̂(ỹ))) (8)
  which, when m = 0, reduces to the DME-based estimator in (2) by taking the noise variance σ 2
  N as the inverse regularization parameter. Hsu and Ramos [1] recover a similar posterior mean expression in their Bayesian interpretation of DME. However, they do not link the distributions of f and its CMP, which leads to much more complicated chained inference derivations combining fully Bayesian and MAP estimates, while we naturally recover it using simple Gaussian conditioning.
  Likewise, an empirical estimate of the deconditional covariance is given by
  k̂d(x, x ′) := k(x, x′)− k(x,x)A(Q̂ỹỹ + σ2IM )−1A>k(x, x′). (9)
  Interestingly, the latter can be rearranged to write as the difference between the original kernel and the kernel undergoing conditioning and deconditioning steps k̂d(x, x′) = k(x, x′) − 〈kx, D̂X|Y ĈX|Y kx′〉k. This can be interpreted as a measure of reconstruction quality, which degenerates in the case of perfect deconditioning, i.e. D̂X|Y ĈX|Y = IdHk .
sections.heading: 3.3 Deconditioning and multiresolution data
sections.text: Downscaling application would typically correspond to multiresolution data, with bag dataset bD1 = {(bxj , yj)}Nj=1 where bxj = {x (i) j } nj i=1. In this setup, the mismatch in size between vector concatenations x = [x(1)1 . . . x (nN ) N ] > and y = [y1 . . . yN ] > prevents from readily applying (1) to estimate the CMO and thus infer the deconditional posterior. There is, however, a straightforward approach to alleviate this: simply replicate bag-level covariates yj to match bags sizes nj . Although simple, this method incurs a O(( ∑N j=1 nj) 3) cost due to matrix inversion in (1).
  Alternatively, since bags bxj are sampled from conditional distribution PX|Y=yj , unbiased Monte Carlo estimators of CMEs are given by µ̂X|Y=yj = 1 nj ∑nj i=1 kx(i)j . Let M̂y = [µ̂X|Y=y1 . . . µ̂X|Y=yN ] > denote their concatenation along columns. We can then rewrite the crosscovariance operator as CXY = EY [EX [kX |Y ] ⊗ `Y ] and hence take 1N M̂yΨ > y as an estimate for CXY . Substituting empirical forms into CX|Y = CX|Y C −1 Y Y and applying Woodbury identity, we obtain an alternative CMO estimator that only requires inversion of a N × N matrix. We call it Conditional Mean Shrinkage Operator and define it as
  SĈX|Y := M̂y(Lyy + λNIN ) −1Ψ>y . (10)
  This estimator can be seen as a generalisation of the Kernel Mean Shrinkage Estimator [34] to the conditional case. We provide in Appendix C modifications of (8) and (9) including this estimator for the computation of the deconditional posterior.
sections.heading: 4 Deconditioning as regression
sections.text: In Section 3.2, we obtain a DMO-based estimate for the posterior mean of f |z̃. When the estimate gets closer to the exact operator, the uncertainty collapses and the Bayesian view meets the frequentist. It is however unclear how the empirical operators effectively converge in finite data size. Adopting an alternative perspective, we now demonstrate that the DMO estimate can be obtained as the minimiser of a two-staged vector-valued regression. This frequentist turn enables us to leverage rich theory of vector-valued regression and establish under mild assumptions a convergence rate on the DMO estimator, with conditions to fulfill minimax optimality in terms of statistical-computational efficiency. In the following, we briefly review CMO’s vector-valued regression viewpoint and construct an analogous regression problem for DMO. We refer the reader to [35] for a comprehensive overview of vector-valued RKHS theory. As for Sections 3.1 and 3.2, this section contributes to the general theory of DMEs and we hence put aside the bag notations.
  Stage 1: Regressing the Conditional Mean Operator As first introduced by Grünewälder et al. [23] and generalised to infinite dimensional spaces by Singh et al. [25], estimating C>X|Y is equivalent to solving a vector-valued kernel ridge regression problem in the hypothesis space of Hilbert-Schmidt operators fromHk toH`, denoted as HS(Hk,H`). Specifically, we may consider the operator-valued kernel defined overHk as Γ(f, f ′) := 〈f, f〉k IdH` . We denoteHΓ theH`-valued RKHS spanned by Γ with norm ‖ · ‖Γ, which can be identified to HS(Hk,H`)4. Singh et al. [25] frame CMO regression as the minimisation surrogate discrepancy Ec(C) := EX,Y [ ‖kX − C>`Y ‖2k ] , to which they substitute an empirical regularised version restricted toHΓ given by
  Êc(C) := 1
  N N∑ i=1 ‖kxi − C>`yi‖2k + λ‖C‖2Γ C ∈ HΓ λ > 0 (11)
  ThisHk-valued kernel ridge regression problem admits a closed-form minimiser which shares the same empirical form as the CMO, i.e. Ĉ>X|Y = arg minC∈HΓ Êc(C) [23, 25].
  4HΓ = Span {Γfh, f ∈ Hk, h ∈ H`} = Span {f ⊗ h, f ∈ Hk, h ∈ H`} = Hk ⊗H` ∼= HS(Hk,H`)
  Stage 2 : Regressing the Deconditional Mean Operator The DMO on the other hand is defined as the operator DX|Y : Hk → H` such that ∀f ∈ Hk, D>X|Y C > X|Y f = f . Since deconditioning corresponds to finding a pseudo-inverse to the CMO, it is natural to consider a reconstruction objective Ed(D) := EY [ ‖`Y −DCX|Y `Y ‖2` ] . Introducing a novel characterization of the DMO, we propose to minimise this objective in the hypothesis space of Hilbert-Schmidt operators HS(Hk,H`) which identifies to HΓ. As per above, we denote ĈX|Y the empirical CMO learnt in Stage 1, and we substitute the loss with an empirical regularised formulation onHΓ
  Êd(D) := 1
  M M∑ j=1 ‖`ỹj −DĈX|Y `ỹj‖2` + ‖D‖2Γ D ∈ HΓ > 0 (12)
  Proposition 4.1 (Empirical DMO as vector-valued regressor). The minimiser of the empirical reconstruction risk is the empirical DMO, i.e. D̂X|Y = arg minD∈HΓ Êd(D)
  Since it requires to estimate the CMO first, minimising (12) can be viewed as a two-staged vector value regression problem.
  Convergence results Following the footsteps of [24, 25], this perspective enables us to state the performance of the DMO estimate in terms of asymptotic convergence of the objective Ed. As in Caponnetto and De Vito [36], we must restrict the class of probability measure for PXY and PY to ensure uniform convergence even when Hk is infinite dimensional. The family of distribution considered is a general class of priors that does not assume parametric distributions and is parametrized by two variables: b > 1 controls the effective input dimension and c ∈]1, 2] controls functional smoothness. Mild regularity assumptions are also placed on the original spaces X ,Y , their corresponding RKHS Hk,H` and the vector-valued RKHS HΓ. We discuss these assumptions in details in Appendix D. Importantly, whileHk can be infinite dimensional, we nonetheless have to assume the RKHSH` is finite dimensional. In further research, we will relax this assumption. Theorem 4.2 (Empirical DMO Convergence Rate). Denote DPY = arg minD∈HΓ Ed(D). Assume assumptions stated in Appendix D are satisfied. In particular, let (b, c) and (0, c′) be the parameters of the restricted class of distribution for PY and PXY respectively and let ι ∈]0, 1] be the Hölder
  continuity exponent inHΓ. Then, if we choose λ = N− 1 c′+1 , N = M a(c′+1) ι(c′−1) where a > 0, we have the following result,
  • If a ≤ b(c+1)bc+1 , then Ed(D̂X|Y )− Ed(DPY ) = O(M −ac c+1 ) with = M −a c+1
  • If a ≥ b(c+1)bc+1 , then Ed(D̂X|Y )− Ed(DPY ) = O(M −bc bc+1 ) with = M −b bc+1
  This theorem underlines a trade-off between the computational and statistical efficiency with respect to the datasets cardinalities N = |D1|, M = |D2| and the problem difficulty b, c, c′. For a ≤ b(c+1)bc+1 , smaller ameans less samples fromD1 at fixedM and thus computational savings. But it also hampers convergence, resulting in reduced statistical efficiency. At a = b(c+1)bc+1 < 2, convergence rate is a minimax computational-statistical efficiency optimal, i.e. convergence rate is optimal with smallest possible M . We note that at this optimal, N > M and which means less samples are required from D2. a ≥ b(c+1)bc+1 does not improve the convergence rate but only increases the size of D1 and hence the computational cost it bears.
sections.heading: 5 Deconditional Downscaling Experiments
sections.text: We demonstrate and evaluate our CMP-based downscaling approaches on both synthetic experiments and a challenging atmospheric temperature field downscaling problem with unmatched multi-resolution data. We denote the exact CMP deconditional posterior from Section 3 as CMP, the CMP using with efficient shrinkage CMO estimation as S-CMP and the variational formulation as VARCMP. They are compared against VBAGG [9] — which we describe below — and a GP regression [14] baseline (GPR) modified to take bags centroids as the input. Experiments are implemented in PyTorch [37, 38], all code and datasets are made available5 and computational details are provided in Appendix E.
  5https://github.com/shahineb/deconditional-downscaling
  Variational Aggregate Learning VBAGG is introduced by Law et al. [9] as a variational aggregate learning framework to disaggregate exponential family models, with emphasis on the Poisson family. We consider its application to the Gaussian family, which models the relationship between aggregate targets zj and bag covariates {x(i)j }i by bag-wise averaging of a GP prior on the function of interest. In fact, the Gaussian VBAGG corresponds exactly to a special case of CMP on matched data, where the bag covariates are simply one hot encoded indices with kernel `(j, j′) = δ(j, j′) where δ is the Kronecker delta. However, VBAGG cannot handle unmatched data as bag indices do not instill the smoothness that is used for mediation. For fair analysis, we compare variational methods VARCMP and VBAGG together, and exact methods CMP/S-CMP to an exact version of VBAGG, which we implement and refer to as BAGG-GP.
sections.heading: 5.1 Swiss Roll
sections.text: The scikit-learn [39] swiss roll manifold sampling function allows to generate a 3D manifold of points x ∈ R3 mapped with their position along the manifold t ∈ R. Our objective will be to recover t for each point x by only observing t at an aggregate level. In the first experiment, we compare our model to existing weakly supervised spatial disaggregation methods when all high-resolution covariates are matched with a coarse-level aggregate target. We then proceed to withdraw this requirement in a companion experiment.
  5.1.1 Direct matching
  Experimental Setup Inspired by the experimental setup from Law et al. [9], we regularly split space along height B − 1 times as depicted in Figure 2 and group together manifold points within each height level, hence mixing together points with very different positions on the manifold. We obtain bags of samples {(bxj , tj)}Bj=1 where the jth bag contains nj points bxj = {x(i)j } nj i=1 and their corresponding targets tj = {t(i)j } nj i=1. We then construct bag aggregate targets by taking noisy bag targets average zj := 1nj ∑nj i=1 t (i) j + εj , where εj ∼ N (0, σ2). We thus obtain matched weakly supervised bag learning dataset D◦ = {(bxj , zj)}Bj=1. Since each bag corresponds to a height-level, the center altitude of each height split yj ∈ R is a natural candidate bag-level covariate that informs on relative positions of the bags. We can
  augment the above dataset as D = {(bxj , yj , zj)}Bj=1. Using these bag datasets, we now wish to downscale aggregate targets zj to recover the unobserved manifold locations {tj}Bj=1 and be able to query the target at any previously unseen input x.
  Models We use a zero-mean prior on f and choose a Gaussian kernel for k and `. Inducing points location is initialized with K-means++ procedure for VARCMP and VBAGG such that they spread evenly across the manifold. For exact methods, kernel hyperparameters and noise variance σ2 are learnt on D by optimising the marginal likelihood. For VARCMP, they are learnt jointly with variational distribution parameters by maximising an evidence lower bound objective. While CMP-based methods can leverage bag-level covariates yj , baselines are restricted to learn from D◦. Adam optimiser [40] is used in all experiments.
  Results We test models against unobserved groundtruth {tj}Bj=1 by evaluating the root mean square error (RMSE) to the posterior mean. Table 1 shows that CMP, S-CMP and VARCMP outperform their corresponding counterparts i.e. BAGG-GP and VBAGG, with statistical significance confirmed by a Wilcoxon signed-rank test in Appendix E. Most notably, this shows that the additional knowledge on bag-level dependence instilled by ` is reflected even in a setting where each bag is matched with an aggregate target.
sections.heading: 5.1.2 Indirect matching
sections.text: Experimental Setup We now impose indirect matching through mediating variable yj . We randomly select N = bB2 c bags which we consider to be the N first ones without loss of generality and split D into D1 = {(bxj , yj)}Nj=1 and D2 = {(ỹj , z̃j)} B−N j=1 = {(yN+j , zN+j)} B−N j=1 , such that no pair of covariates bag bxj and aggregate target z̃j are jointly observed.
  Models CMP-based methods are naturally able to learn from this setting and are trained by independently drawing samples from D1 and D2. Baseline methods however require bags of covariates to be matched with an aggregate bag target. To remedy this, we place a separate prior g ∼ GP(0, `) and fit regression model z̃j = g(ỹj) + εj over D2. We then use the predictive posterior mean to augment the first dataset as D′1 = {( bxj ,E[g(yj)|D2] )}N j=1
  . This dataset can then be used to train BAGG-GP, VBAGG and GPR.
  Results For comparison, we use the same evaluation as in the direct matching experiment. Table 1 underlines an anticipated drop in RMSE for all models, but we observe that BAGG-GP and VBAGG suffer most from the mediated matching of the dataset while CMP and VARCMP report best scores by a substantial margin. This highlights how using a separate prior on g to mediate D1 and D2 turns out to be suboptimal in contrast to using the prior naturally implied by CMP. While it is more computationally efficient than CMP, we observe a relative drop in performance for S-CMP.
sections.heading: 5.2 Mediated downscaling of atmospheric temperature
sections.text: Given the large diversity of sources and formats of remote sensing and model data, expecting directly matched observations is often unrealistic [41]. For example, two distinct satellite products will often provide low and high resolution imagery that can be matched neither spatially nor temporally [2, 3, 4, 5]. Climate simulations [19, 20, 21] on the other hand provide a comprehensive coarse resolution coverage of meteorological variables that can serve as a mediating dataset.
  For the purpose of demonstration, we create an experimental setup inspired by this problem using Coupled Model Intercomparison Project Phase 6 (CMIP6) [19] simulation data. This grants us access to groundtruth high-resolution covariates to facilitate model evaluation.
  Experimental Setup We collect monthly mean 2D atmospheric fields simulation from CMIP6 data [42, 43] for the following variables: air temperature at cloud top (T), mean cloud top pressure (P), total cloud cover (TCC) and cloud albedo (α). First, we collocate TCC and α onto a HR latitudelongitude grid of size 360×720 to obtain fine grained fields (latitudeHR, longitudeHR, altitudeHR, TCCHR, αHR) augmented with a static HR surface altitude field. Then we collocate P and T onto a LR grid of size 21×42 to obtain coarse grained fields (latitudeLR, longitudeLR, PLR, TLR). We denote by B the number of low resolution pixels.
  Our objective is to disaggregate TLR to the HR fields granularity. We assimilate the jth coarse temperature pixel to an aggregate target zj := TLRj corresponding to bag j. Each bag includes HR covariates bxj = {x(i)j } nj i=1 := {(latitude HR(i) j , longitude HR(i) j , altitude HR(i) j , TCC HR(i) j , α HR(i) j )} nj i=1. To emulate unmatched observations, we randomly select N = bB2 c of the bags { bxj}Nj=1 and keep the opposite half of LR observations {zN+j}B−Nj=1 , such that there is no single aggregate bag target that corresponds to one of the available bags. Finally, we choose the pressure field PLR as the mediating variable. We hence compose a third party low resolution field of bag-level covariates yj := (latitudeLRj , longitude LR j , P LR j ) which can separately be matched with both above sets to obtain datasets D1 = {(xj , yj)}Nj=1 and D2 = {(ỹj , z̃j)} B−N j=1 = {(yN+j , zN+j)} B−N j=1 .
  Models Setup We only consider variational methods to scale to large number of pixels. VARCMP is naturally able to learn from indirectly matched data. We use a Matérn-1.5 kernel for rough spatial covariates (latitude, longitude) and a Gaussian kernel for atmospheric covariates (P, TCC, α) and surface altitude. k and ` are both taken as sums of Matérn and Gaussian kernels, and their hyperparameters are learnt along with noise variance during training. A high-resolution noise term is also introduced, with details provided in Appendix E. Inducing points locations are uniformly initialized across the HR grid. We replace GPR with an inducing point variational counterpart VARGPR [15]. Since baseline methods require a matched dataset, we proceed as with the unmatched swiss roll experiment and fit a GP regression model g with kernel ` on D2 and then use its predictive posterior mean to obtain pseudo-targets for the bags of HR covariates from D1.
  Results Performance is evaluated by comparing downscaling deconditional posterior mean against original high resolution field THR available in CMIP6 data [43], which we emphasise is never observed. We use random Fourier features [44] approximation of kernel k to scale kernel evaluation to the HR covariates grid during testing. As reported in Table 2, VARCMP substantially outperforms both baselines with statistical significance provided in Appendix E. Figure 3 shows the reconstructed image with VARCMP, plots for other methods are included in the Appendix E. The model resolves statistical patterns from HR covariates into coarse resolution temperature pixels, henceforth reconstructing a faithful HR version of the temperature field.
sections.heading: 6 Discussion
sections.text: We introduced a scalable Bayesian solution to the mediated statistical downscaling problem, which handles unmatched multi-resolution data. The proposed approach combines Gaussian Processes with the framework of deconditioning using RKHSs and recovers previous approaches as its special cases. We provided convergence rates for the associated deconditioning operator. Finally, we demonstrated the advantages over spatial disaggregation baselines in synthetic data and in a challenging atmospheric fields downscaling problem.
  In future work, exploring theoretical guarantees of the computationally efficient shrinkage formulation in a multi-resolution setting and relaxing finite dimensionality assumptions for the convergence rate will have fruitful practical and theoretical implications. Further directions also open up to quantify uncertainty over the deconditional posterior since it is computed using empirical estimates of the CMP covariance. This may be problematic if the mediating variable undergoes covariate shift between the two datasets.
sections.heading: Acknowledgments
sections.text: SLC is supported by the EPSRC and MRC through the OxWaSP CDT programme EP/L016710/1. SB is supported by the EU’s Horizon 2020 research and innovation programme under Marie SkłodowskaCurie grant agreement No 860100. DS is supported in partly by Tencent AI lab and partly by the Alan Turing Institute (EP/N510129/1).
title: Deconditional Downscaling with Gaussian Processes