Kernel embedding

The kernel embedding of the distribution ${\displaystyle P}$ in ${\displaystyle {\mathcal {H}}}$ (also called the kernel mean or mean map) is given by:^[1]

${\displaystyle \mu _{X}:=\mathbb {E} [k(X,\cdot )]=\mathbb {E} [\varphi (X)]=\int _{\Omega }\varphi (x)\ \mathrm {d} P(x)}$

If ${\displaystyle P}$ allows a square integrable density ${\displaystyle p}$, then ${\displaystyle \mu _{X}={\mathcal {E}}_{k}p}$, where ${\displaystyle {\mathcal {E}}_{k}}$ is the Hilbert–Schmidt integral operator. A kernel is characteristic if the mean embedding ${\displaystyle \mu$ :\{{\text{family of distributions over }}\Omega \}\to {\mathcal {H}}} is injective.^[7] Each distribution can thus be uniquely represented in the RKHS and all statistical features of distributions are preserved by the kernel embedding if a characteristic kernel is used.

Empirical kernel embedding

Given ${\displaystyle n}$ training examples ${\displaystyle \{x_{1},\ldots ,x_{n}\}}$ drawn independently and identically distributed (i.i.d.) from ${\displaystyle P,}$ the kernel embedding of ${\displaystyle P}$ can be empirically estimated as

${\displaystyle {\widehat {\mu }}_{X}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})}$

Joint distribution embedding

If ${\displaystyle Y}$ denotes another random variable (for simplicity, assume the co-domain of ${\displaystyle Y}$ is also ${\displaystyle \Omega }$ with the same kernel ${\displaystyle k}$ which satisfies ${\displaystyle \langle \varphi (x)\otimes \varphi (y),\varphi (x')\otimes \varphi (y')\rangle =k(x,x')k(y,y')}$), then the joint distribution ${\displaystyle P(x,y))}$ can be mapped into a tensor product feature space ${\displaystyle {\mathcal {H}}\otimes {\mathcal {H}}}$ via ^[2]

${\displaystyle {\mathcal {C}}_{XY}=\mathbb {E} [\varphi (X)\otimes \varphi (Y)]=\int _{\Omega \times \Omega }\varphi (x)\otimes \varphi (y)\ \mathrm {d} P(x,y)}$

By the equivalence between a tensor and a linear map, this joint embedding may be interpreted as an uncentered cross-covariance operator ${\displaystyle {\mathcal {C}}_{XY}:{\mathcal {H}}\to {\mathcal {H}}}$ from which the cross-covariance of functions ${\displaystyle f,g\in {\mathcal {H}}}$ can be computed as ^[8]

${\displaystyle \operatorname {Cov} (f(X),g(Y)):=\mathbb {E} [f(X)g(Y)]-\mathbb {E} [f(X)]\mathbb {E} [g(Y)]=\langle f,{\mathcal {C}}_{XY}g\rangle _{\mathcal {H}}=\langle f\otimes g,{\mathcal {C}}_{XY}\rangle _{{\mathcal {H}}\otimes {\mathcal {H}}}}$

Given ${\displaystyle n}$ pairs of training examples ${\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}}$ drawn i.i.d. from ${\displaystyle P}$, we can also empirically estimate the joint distribution kernel embedding via

${\displaystyle {\widehat {\mathcal {C}}}_{XY}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})\otimes \varphi (y_{i})}$

Conditional distribution embedding

Given a conditional distribution ${\displaystyle P(y\mid x),}$ one can define the corresponding RKHS embedding as ^[2]

${\displaystyle \mu _{Y\mid x}=\mathbb {E} [\varphi (Y)\mid X]=\int _{\Omega }\varphi (y)\ \mathrm {d} P(y\mid x)}$

Note that the embedding of ${\displaystyle P(y\mid x)}$ thus defines a family of points in the RKHS indexed by the values ${\displaystyle x}$ taken by conditioning variable ${\displaystyle X}$. By fixing ${\displaystyle X}$ to a particular value, we obtain a single element in ${\displaystyle {\mathcal {H}}}$, and thus it is natural to define the operator

${\displaystyle {\begin{cases}{\mathcal {C}}_{Y\mid X}:{\mathcal {H}}\to {\mathcal {H}}\\{\mathcal {C}}_{Y\mid X}={\mathcal {C}}_{YX}{\mathcal {C}}_{XX}^{-1}\end{cases}}}$

which given the feature mapping of ${\displaystyle x}$ outputs the conditional embedding of ${\displaystyle Y}$ given ${\displaystyle X=x.}$ Assuming that for all ${\displaystyle g\in {\mathcal {H}}:\mathbb {E} [g(Y)\mid X]\in {\mathcal {H}},}$ it can be shown that ^[8]

${\displaystyle \mu _{Y\mid x}={\mathcal {C}}_{Y\mid X}\varphi (x)}$

This assumption is always true for finite domains with characteristic kernels, but may not necessarily hold for continuous domains.^[2] Nevertheless, even in cases where the assumption fails, ${\displaystyle {\mathcal {C}}_{Y\mid X}\varphi (x)}$ may still be used to approximate the conditional kernel embedding ${\displaystyle \mu _{Y\mid x},}$ and in practice, the inversion operator is replaced with a regularized version of itself ${\displaystyle ({\mathcal {C}}_{XX}+\lambda \mathbf {I} )^{-1}}$ (where ${\displaystyle \mathbf {I} }$ denotes the identity matrix).

Given training examples ${\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\},}$ the empirical kernel conditional embedding operator may be estimated as ^[2]

${\displaystyle {\widehat {C}}_{Y\mid X}={\boldsymbol {\Phi }}(\mathbf {K} +\lambda \mathbf {I} )^{-1}{\boldsymbol {\Upsilon }}^{T}}$

where ${\displaystyle {\boldsymbol {\Phi }}=\left(\varphi (y_{1}),\dots ,\varphi (y_{n})\right),{\boldsymbol {\Upsilon }}=\left(\varphi (x_{1}),\dots ,\varphi (x_{n})\right)}$ are implicitly formed feature matrices, ${\displaystyle \mathbf {K} ={\boldsymbol {\Upsilon }}^{T}{\boldsymbol {\Upsilon }}}$ is the Gram matrix for samples of ${\displaystyle X}$, and ${\displaystyle \lambda }$ is a regularization parameter needed to avoid overfitting.

Thus, the empirical estimate of the kernel conditional embedding is given by a weighted sum of samples of ${\displaystyle Y}$ in the feature space:

${\displaystyle {\widehat {\mu }}_{Y\mid x}=\sum _{i=1}^{n}\beta _{i}(x)\varphi (y_{i})={\boldsymbol {\Phi }}{\boldsymbol {\beta }}(x)}$

where ${\displaystyle {\boldsymbol {\beta }}(x)=(\mathbf {K} +\lambda \mathbf {I} )^{-1}\mathbf {K} _{x}}$ and ${\displaystyle \mathbf {K} _{x}=\left(k(x_{1},x),\dots ,k(x_{n},x)\right)^{T}}$

Convergence of empirical kernel mean to the true distribution embedding

• If ${\displaystyle k}$ is defined such that ${\displaystyle f}$ takes values in ${\displaystyle [0,1]}$ for all ${\displaystyle f\in {\mathcal {H}}}$ with ${\displaystyle \|f\|_{\mathcal {H}}\leq 1}$ (as is the case for the widely used radial basis function kernels), then with probability at least ${\displaystyle 1-\delta }$:^[6]

${\displaystyle \|\mu _{X}-{\widehat {\mu }}_{X}\|_{\mathcal {H}}=\sup _{f\in {\mathcal {B}}(0,1)}\left|\mathbb {E} [f(X)]-{\frac {1}{n}}\sum _{i=1}^{n}f(x_{i})\right|\leq {\frac {2}{n}}\mathbb {E} \left[{\sqrt {\operatorname {tr} K}}\right]+{\sqrt {\frac {\log(2/\delta )}{2n}}}}$

where ${\displaystyle {\mathcal {B}}(0,1)}$ denotes the unit ball in ${\displaystyle {\mathcal {H}}}$ and ${\displaystyle \mathbf {K} =(k_{ij})}$ is the Gram matrix with ${\displaystyle k_{ij}=k(x_{i},x_{j}).}$

• The rate of convergence (in RKHS norm) of the empirical kernel embedding to its distribution counterpart is ${\displaystyle O(n^{-1/2})}$ and does not depend on the dimension of ${\displaystyle X}$.
• Statistics based on kernel embeddings thus avoid the curse of dimensionality, and though the true underlying distribution is unknown in practice, one can (with high probability) obtain an approximation within ${\displaystyle O(n^{-1/2})}$ of the true kernel embedding based on a finite sample of size ${\displaystyle n}$.
• For the embedding of conditional distributions, the empirical estimate can be seen as a weighted average of feature mappings (where the weights ${\displaystyle \beta _{i}(x)}$ depend on the value of the conditioning variable and capture the effect of the conditioning on the kernel embedding). In this case, the empirical estimate converges to the conditional distribution RKHS embedding with rate ${\displaystyle O\left(n^{-1/4}\right)}$ if the regularization parameter ${\displaystyle \lambda }$ is decreased as ${\displaystyle O\left(n^{-1/2}\right),}$ though faster rates of convergence may be achieved by placing additional assumptions on the joint distribution.^[2]

Universal kernels

• Let ${\displaystyle {\mathcal {X}}}$ be a compact metric space and ${\displaystyle C({\mathcal {X}})}$ the set of continuous functions. The reproducing kernel ${\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\rightarrow \mathbb {R} }$ is called universal if and only if the RKHS ${\displaystyle {\mathcal {H}}}$ of ${\displaystyle k}$ is dense in ${\displaystyle C({\mathcal {X}})}$, i.e., for any ${\displaystyle g\in C({\mathcal {X}})}$ and all ${\displaystyle \varepsilon >0}$ there exists an ${\displaystyle f\in {\mathcal {H}}}$ such that ${\displaystyle \|f-g\|_{\infty }\leq \varepsilon }$.^[9] All universal kernels defined on a compact space are characteristic kernels but the converse is not always true.^[10]
• Let ${\displaystyle k}$ be a continuous translation invariant kernel ${\displaystyle k(x,x')=h(x-x')}$ with ${\displaystyle x\in \mathbb {R} ^{b}}$. Then Bochner's theorem guarantees the existence of a unique finite Borel measure ${\displaystyle \mu }$ (called the spectral measure) on ${\displaystyle \mathbb {R} ^{b}}$ such that

${\displaystyle h(t)=\int _{\mathbb {R} ^{b}}e^{-i\langle t,\omega \rangle }d\mu (\omega ),\quad \forall t\in \mathbb {R} ^{b}.}$

For ${\displaystyle k}$ to be universal it suffices that the continuous part of ${\displaystyle \mu }$ in its unique Lebesgue decomposition ${\displaystyle \mu =\mu _{c}+\mu _{s}}$ is non-zero. Furthermore, if

${\displaystyle d\mu _{c}(\omega )=s(\omega )d\omega ,}$

then ${\displaystyle s}$ is the spectral density of frequencies ${\displaystyle \omega }$ in ${\displaystyle \mathbb {R} ^{b}}$ and ${\displaystyle h}$ is the Fourier transform of ${\displaystyle s}$. If the support of ${\displaystyle \mu }$ is all of ${\displaystyle \mathbb {R} ^{b}}$, then ${\displaystyle k}$ is a characteristic kernel as well.^[11][12][13]

• If ${\displaystyle k}$ induces a strictly positive definite kernel matrix for any set of distinct points, then it is a universal kernel.^[6] For example, the widely used Gaussian RBF kernel

${\displaystyle k(x,x')=\exp \left(-{\frac {1}{2\sigma ^{2}}}\|x-x'\|^{2}\right)}$

on compact subsets of ${\displaystyle \mathbb {R} ^{d}}$ is universal.

Parameter selection for conditional distribution kernel embeddings

• The empirical kernel conditional distribution embedding operator ${\displaystyle {\widehat {\mathcal {C}}}_{Y|X}}$ can alternatively be viewed as the solution of the following regularized least squares (function-valued) regression problem ^[14]

${\displaystyle \min _{{\mathcal {C}}:{\mathcal {H}}\to {\mathcal {H}}}\sum _{i=1}^{n}\left\|\varphi (y_{i})-{\mathcal {C}}\varphi (x_{i})\right\|_{\mathcal {H}}^{2}+\lambda \|{\mathcal {C}}\|_{HS}^{2}}$

where ${\displaystyle \|\cdot \|_{HS}}$ is the Hilbert–Schmidt norm.

• One can thus select the regularization parameter ${\displaystyle \lambda }$ by performing cross-validation based on the squared loss function of the regression problem.

Kernel sum rule

In probability theory, the marginal distribution of ${\displaystyle X}$ can be computed by integrating out ${\displaystyle Y}$ from the joint density (including the prior distribution on ${\displaystyle Y}$)

${\displaystyle Q(X)=\int _{\Omega }P(X\mid Y)\,\mathrm {d} \pi (Y)}$

The analog of this rule in the kernel embedding framework states that ${\displaystyle \mu _{X}^{\pi },}$ the RKHS embedding of ${\displaystyle Q(X)}$, can be computed via

${\displaystyle \mu _{X}^{\pi }=\mathbb {E} [{\mathcal {C}}_{X\mid Y}\varphi (Y)]={\mathcal {C}}_{X\mid Y}\mathbb {E} [\varphi (Y)]={\mathcal {C}}_{X\mid Y}\mu _{Y}^{\pi }}$

where ${\displaystyle \mu _{Y}^{\pi }}$ is the kernel embedding of ${\displaystyle \pi (Y).}$ In practical implementations, the kernel sum rule takes the following form

${\displaystyle {\widehat {\mu }}_{X}^{\pi }={\widehat {\mathcal {C}}}_{X\mid Y}{\widehat {\mu }}_{Y}^{\pi }={\boldsymbol {\Upsilon }}(\mathbf {G} +\lambda \mathbf {I} )^{-1}{\widetilde {\mathbf {G} }}{\boldsymbol {\alpha }}}$

where

${\displaystyle \mu _{Y}^{\pi }=\sum _{i=1}^{\widetilde {n}}\alpha _{i}\varphi ({\widetilde {y}}_{i})}$

is the empirical kernel embedding of the prior distribution, ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{\widetilde {n}})^{T},}$ ${\displaystyle {\boldsymbol {\Upsilon }}=\left(\varphi (x_{1}),\ldots ,\varphi (x_{n})\right)}$, and ${\displaystyle \mathbf {G} ,{\widetilde {\mathbf {G} }}}$ are Gram matrices with entries ${\displaystyle \mathbf {G} _{ij}=k(y_{i},y_{j}),{\widetilde {\mathbf {G} }}_{ij}=k(y_{i},{\widetilde {y}}_{j})}$ respectively.

Kernel chain rule

In probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions

${\displaystyle Q(X,Y)=P(X\mid Y)\pi (Y)}$

The analog of this rule in the kernel embedding framework states that ${\displaystyle {\mathcal {C}}_{XY}^{\pi },}$ the joint embedding of ${\displaystyle Q(X,Y),}$ can be factorized as a composition of conditional embedding operator with the auto-covariance operator associated with ${\displaystyle \pi (Y)}$

${\displaystyle {\mathcal {C}}_{XY}^{\pi }={\mathcal {C}}_{X\mid Y}{\mathcal {C}}_{YY}^{\pi }}$

where

${\displaystyle {\mathcal {C}}_{XY}^{\pi }=\mathbb {E} [\varphi (X)\otimes \varphi (Y)],}$

${\displaystyle {\mathcal {C}}_{YY}^{\pi }=\mathbb {E} [\varphi (Y)\otimes \varphi (Y)].}$

In practical implementations, the kernel chain rule takes the following form

${\displaystyle {\widehat {\mathcal {C}}}_{XY}^{\pi }={\widehat {\mathcal {C}}}_{X\mid Y}{\widehat {\mathcal {C}}}_{YY}^{\pi }={\boldsymbol {\Upsilon }}(\mathbf {G} +\lambda \mathbf {I} )^{-1}{\widetilde {\mathbf {G} }}\operatorname {diag} ({\boldsymbol {\alpha }}){\boldsymbol {\widetilde {\Phi }}}^{T}}$

Kernel Bayes' rule

In probability theory, a posterior distribution can be expressed in terms of a prior distribution and a likelihood function as

${\displaystyle Q(Y\mid x)={\frac {P(x\mid Y)\pi (Y)}{Q(x)}}}$ where ${\displaystyle Q(x)=\int _{\Omega }P(x\mid y)\,\mathrm {d} \pi (y)}$

The analog of this rule in the kernel embedding framework expresses the kernel embedding of the conditional distribution in terms of conditional embedding operators which are modified by the prior distribution

${\displaystyle \mu _{Y\mid x}^{\pi }={\mathcal {C}}_{Y\mid X}^{\pi }\varphi (x)={\mathcal {C}}_{YX}^{\pi }\left({\mathcal {C}}_{XX}^{\pi }\right)^{-1}\varphi (x)}$

where from the chain rule:

${\displaystyle {\mathcal {C}}_{YX}^{\pi }=\left({\mathcal {C}}_{X\mid Y}{\mathcal {C}}_{YY}^{\pi }\right)^{T}.}$

In practical implementations, the kernel Bayes' rule takes the following form

${\displaystyle {\widehat {\mu }}_{Y\mid x}^{\pi }={\widehat {\mathcal {C}}}_{YX}^{\pi }\left(\left({\widehat {\mathcal {C}}}_{XX}\right)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widehat {\mathcal {C}}}_{XX}^{\pi }\varphi (x)={\widetilde {\boldsymbol {\Phi }}}{\boldsymbol {\Lambda }}^{T}\left((\mathbf {D} \mathbf {K} )^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}\mathbf {K} \mathbf {D} \mathbf {K} _{x}}$

where

${\displaystyle {\boldsymbol {\Lambda }}=\left(\mathbf {G} +{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widetilde {\mathbf {G} }}\operatorname {diag} ({\boldsymbol {\alpha }}),\qquad \mathbf {D} =\operatorname {diag} \left(\left(\mathbf {G} +{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widetilde {\mathbf {G} }}{\boldsymbol {\alpha }}\right).}$

Two regularization parameters are used in this framework: ${\displaystyle \lambda }$ for the estimation of ${\displaystyle {\widehat {\mathcal {C}}}_{YX}^{\pi },{\widehat {\mathcal {C}}}_{XX}^{\pi }={\boldsymbol {\Upsilon }}\mathbf {D} {\boldsymbol {\Upsilon }}^{T}}$ and ${\displaystyle {\widetilde {\lambda }}}$ for the estimation of the final conditional embedding operator

${\displaystyle {\widehat {\mathcal {C}}}_{Y\mid X}^{\pi }={\widehat {\mathcal {C}}}_{YX}^{\pi }\left(\left({\widehat {\mathcal {C}}}_{XX}^{\pi }\right)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widehat {\mathcal {C}}}_{XX}^{\pi }.}$

The latter regularization is done on square of ${\displaystyle {\widehat {\mathcal {C}}}_{XX}^{\pi }}$ because ${\displaystyle D}$ may not be positive definite.

Measuring distance between distributions

The maximum mean discrepancy (MMD) is a distance-measure between distributions ${\displaystyle P(X)}$ and ${\displaystyle Q(Y)}$ which is defined as the distance between their embeddings in the RKHS ^[6]

${\displaystyle {\text{MMD}}(P,Q)=\left\|\mu _{X}-\mu _{Y}\right\|_{\mathcal {H}}.}$

While most distance-measures between distributions such as the widely used Kullback–Leibler divergence either require density estimation (either parametrically or nonparametrically) or space partitioning/bias correction strategies,^[6] the MMD is easily estimated as an empirical mean which is concentrated around the true value of the MMD. The characterization of this distance as the maximum mean discrepancy refers to the fact that computing the MMD is equivalent to finding the RKHS function that maximizes the difference in expectations between the two probability distributions

${\displaystyle {\text{MMD}}(P,Q)=\sup _{\|f\|_{\mathcal {H}}\leq 1}\left(\mathbb {E} [f(X)]-\mathbb {E} [f(Y)]\right),}$

a form of integral probability metric.

Kernel two-sample test

Given n training examples from ${\displaystyle P(X)}$ and m samples from ${\displaystyle Q(Y)}$, one can formulate a test statistic based on the empirical estimate of the MMD

${\displaystyle {\begin{aligned}{\widehat {\text{MMD}}}(P,Q)&=\left\|{\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})-{\frac {1}{m}}\sum _{i=1}^{m}\varphi (y_{i})\right\|_{\mathcal {H}}^{2}\\[5pt]&={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}k(x_{i},x_{j})+{\frac {1}{m^{2}}}\sum _{i=1}^{m}\sum _{j=1}^{m}k(y_{i},y_{j})-{\frac {2}{nm}}\sum _{i=1}^{n}\sum _{j=1}^{m}k(x_{i},y_{j})\end{aligned}}}$

to obtain a two-sample test ^[15] of the null hypothesis that both samples stem from the same distribution (i.e. ${\displaystyle P=Q}$) against the broad alternative ${\displaystyle P\neq Q}$.

Density estimation via kernel embeddings

Although learning algorithms in the kernel embedding framework circumvent the need for intermediate density estimation, one may nonetheless use the empirical embedding to perform density estimation based on n samples drawn from an underlying distribution ${\displaystyle P_{X}^{*}}$. This can be done by solving the following optimization problem ^[6][16]

${\displaystyle \max _{P_{X}}H(P_{X})}$ subject to ${\displaystyle \|{\widehat {\mu }}_{X}-\mu _{X}[P_{X}]\|_{\mathcal {H}}\leq \varepsilon }$

where the maximization is done over the entire space of distributions on ${\displaystyle \Omega .}$ Here, ${\displaystyle \mu _{X}[P_{X}]}$ is the kernel embedding of the proposed density ${\displaystyle P_{X}}$ and ${\displaystyle H}$ is an entropy-like quantity (e.g. Entropy, KL divergence, Bregman divergence). The distribution which solves this optimization may be interpreted as a compromise between fitting the empirical kernel means of the samples well, while still allocating a substantial portion of the probability mass to all regions of the probability space (much of which may not be represented in the training examples). In practice, a good approximate solution of the difficult optimization may be found by restricting the space of candidate densities to a mixture of M candidate distributions with regularized mixing proportions. Connections between the ideas underlying Gaussian processes and conditional random fields may be drawn with the estimation of conditional probability distributions in this fashion, if one views the feature mappings associated with the kernel as sufficient statistics in generalized (possibly infinite-dimensional) exponential families.^[6]

Measuring dependence of random variables

A measure of the statistical dependence between random variables ${\displaystyle X}$ and ${\displaystyle Y}$ (from any domains on which sensible kernels can be defined) can be formulated based on the Hilbert–Schmidt Independence Criterion ^[17]

${\displaystyle {\text{HSIC}}(X,Y)=\left\|{\mathcal {C}}_{XY}-\mu _{X}\otimes \mu _{Y}\right\|_{{\mathcal {H}}\otimes {\mathcal {H}}}^{2}}$

and can be used as a principled replacement for mutual information, Pearson correlation or any other dependence measure used in learning algorithms. Most notably, HSIC can detect arbitrary dependencies (when a characteristic kernel is used in the embeddings, HSIC is zero if and only if the variables are independent), and can be used to measure dependence between different types of data (e.g. images and text captions). Given n i.i.d. samples of each random variable, a simple parameter-free unbiased estimator of HSIC which exhibits concentration about the true value can be computed in ${\displaystyle O(n(d_{f}^{2}+d_{g}^{2}))}$ time,^[6] where the Gram matrices of the two datasets are approximated using ${\displaystyle \mathbf {A} \mathbf {A} ^{T},\mathbf {B} \mathbf {B} ^{T}}$ with ${\displaystyle \mathbf {A} \in \mathbb {R} ^{n\times d_{f}},\mathbf {B} \in \mathbb {R} ^{n\times d_{g}}}$. The desirable properties of HSIC have led to the formulation of numerous algorithms which utilize this dependence measure for a variety of common machine learning tasks such as: feature selection (BAHSIC ^[18]), clustering (CLUHSIC ^[19]), and dimensionality reduction (MUHSIC ^[20]).

HSIC can be extended to measure the dependence of multiple random variables. The question of when HSIC captures independence in this case has recently been studied:^[21] for more than two variables

• on ${\displaystyle \mathbb {R} ^{d}}$: the characteristic property of the individual kernels remains an equivalent condition.
• on general domains: the characteristic property of the kernel components is necessary but not sufficient.

Kernel belief propagation

Belief propagation is a fundamental algorithm for inference in graphical models in which nodes repeatedly pass and receive messages corresponding to the evaluation of conditional expectations. In the kernel embedding framework, the messages may be represented as RKHS functions and the conditional distribution embeddings can be applied to efficiently compute message updates. Given n samples of random variables represented by nodes in a Markov random field, the incoming message to node t from node u can be expressed as

${\displaystyle m_{ut}(\cdot )=\sum _{i=1}^{n}\beta _{ut}^{i}\varphi (x_{t}^{i})}$

if it assumed to lie in the RKHS. The kernel belief propagation update message from t to node s is then given by ^[2]

${\displaystyle {\widehat {m}}_{ts}=\left(\odot _{u\in N(t)\backslash s}\mathbf {K} _{t}{\boldsymbol {\beta }}_{ut}\right)^{T}(\mathbf {K} _{s}+\lambda \mathbf {I} )^{-1}{\boldsymbol {\Upsilon }}_{s}^{T}\varphi (x_{s})}$

where ${\displaystyle \odot }$ denotes the element-wise vector product, ${\displaystyle N(t)\backslash s}$ is the set of nodes connected to t excluding node s, ${\displaystyle {\boldsymbol {\beta }}_{ut}=\left(\beta _{ut}^{1},\dots ,\beta _{ut}^{n}\right)}$, ${\displaystyle \mathbf {K} _{t},\mathbf {K} _{s}}$ are the Gram matrices of the samples from variables ${\displaystyle X_{t},X_{s}}$, respectively, and ${\displaystyle {\boldsymbol {\Upsilon }}_{s}=\left(\varphi (x_{s}^{1}),\dots ,\varphi (x_{s}^{n})\right)}$ is the feature matrix for the samples from ${\displaystyle X_{s}}$.

Thus, if the incoming messages to node t are linear combinations of feature mapped samples from ${\displaystyle X_{t}}$, then the outgoing message from this node is also a linear combination of feature mapped samples from ${\displaystyle X_{s}}$. This RKHS function representation of message-passing updates therefore produces an efficient belief propagation algorithm in which the potentials are nonparametric functions inferred from the data so that arbitrary statistical relationships may be modeled.^[2]

Nonparametric filtering in hidden Markov models

In the hidden Markov model (HMM), two key quantities of interest are the transition probabilities between hidden states ${\displaystyle P(S^{t}\mid S^{t-1})}$ and the emission probabilities ${\displaystyle P(O^{t}\mid S^{t})}$ for observations. Using the kernel conditional distribution embedding framework, these quantities may be expressed in terms of samples from the HMM. A serious limitation of the embedding methods in this domain is the need for training samples containing hidden states, as otherwise inference with arbitrary distributions in the HMM is not possible.

One common use of HMMs is filtering in which the goal is to estimate posterior distribution over the hidden state ${\displaystyle s^{t}}$ at time step t given a history of previous observations ${\displaystyle h^{t}=(o^{1},\dots ,o^{t})}$ from the system. In filtering, a belief state ${\displaystyle P(S^{t+1}\mid h^{t+1})}$ is recursively maintained via a prediction step (where updates ${\displaystyle P(S^{t+1}\mid h^{t})=\mathbb {E} [P(S^{t+1}\mid S^{t})\mid h^{t}]}$ are computed by marginalizing out the previous hidden state) followed by a conditioning step (where updates ${\displaystyle P(S^{t+1}\mid h^{t},o^{t+1})\propto P(o^{t+1}\mid S^{t+1})P(S^{t+1}\mid h^{t})}$ are computed by applying Bayes' rule to condition on a new observation).^[2] The RKHS embedding of the belief state at time t+1 can be recursively expressed as

${\displaystyle \mu _{S^{t+1}\mid h^{t+1}}={\mathcal {C}}_{S^{t+1}O^{t+1}}^{\pi }\left({\mathcal {C}}_{O^{t+1}O^{t+1}}^{\pi }\right)^{-1}\varphi (o^{t+1})}$

by computing the embeddings of the prediction step via the kernel sum rule and the embedding of the conditioning step via kernel Bayes' rule. Assuming a training sample ${\displaystyle ({\widetilde {s}}^{1},\dots ,{\widetilde {s}}^{T},{\widetilde {o}}^{1},\dots ,{\widetilde {o}}^{T})}$ is given, one can in practice estimate

${\displaystyle {\widehat {\mu }}_{S^{t+1}\mid h^{t+1}}=\sum _{i=1}^{T}\alpha _{i}^{t}\varphi ({\widetilde {s}}^{t})}$

and filtering with kernel embeddings is thus implemented recursively using the following updates for the weights ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\dots ,\alpha _{T})}$ ^[2]

${\displaystyle \mathbf {D} ^{t+1}=\operatorname {diag} \left((G+\lambda \mathbf {I} )^{-1}{\widetilde {G}}{\boldsymbol {\alpha }}^{t}\right)}$

${\displaystyle {\boldsymbol {\alpha }}^{t+1}=\mathbf {D} ^{t+1}\mathbf {K} \left((\mathbf {D} ^{t+1}K)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}\mathbf {D} ^{t+1}\mathbf {K} _{o^{t+1}}}$

where ${\displaystyle \mathbf {G} ,\mathbf {K} }$ denote the Gram matrices of ${\displaystyle {\widetilde {s}}^{1},\dots ,{\widetilde {s}}^{T}}$ and ${\displaystyle {\widetilde {o}}^{1},\dots ,{\widetilde {o}}^{T}}$ respectively, ${\displaystyle {\widetilde {\mathbf {G} }}}$ is a transfer Gram matrix defined as ${\displaystyle {\widetilde {\mathbf {G} }}_{ij}=k({\widetilde {s}}_{i},{\widetilde {s}}_{j+1}),}$ and ${\displaystyle \mathbf {K} _{o^{t+1}}=(k({\widetilde {o}}^{1},o^{t+1}),\dots ,k({\widetilde {o}}^{T},o^{t+1}))^{T}.}$

Support measure machines

The support measure machine (SMM) is a generalization of the support vector machine (SVM) in which the training examples are probability distributions paired with labels ${\displaystyle \{P_{i},y_{i}\}_{i=1}^{n},\ y_{i}\in \{+1,-1\}}$.^[22] SMMs solve the standard SVM dual optimization problem using the following expected kernel

${\displaystyle K\left(P(X),Q(Z)\right)=\langle \mu _{X},\mu _{Z}\rangle _{\mathcal {H}}=\mathbb {E} [k(x,z)]}$

which is computable in closed form for many common specific distributions ${\displaystyle P_{i}}$ (such as the Gaussian distribution) combined with popular embedding kernels ${\displaystyle k}$ (e.g. the Gaussian kernel or polynomial kernel), or can be accurately empirically estimated from i.i.d. samples ${\displaystyle \{x_{i}\}_{i=1}^{n}\sim P(X),\{z_{j}\}_{j=1}^{m}\sim Q(Z)}$ via

${\displaystyle {\widehat {K}}(X,Z)={\frac {1}{nm}}\sum _{i=1}^{n}\sum _{j=1}^{m}k(x_{i},z_{j})}$

Under certain choices of the embedding kernel ${\displaystyle k}$, the SMM applied to training examples ${\displaystyle \{P_{i},y_{i}\}_{i=1}^{n}}$ is equivalent to a SVM trained on samples ${\displaystyle \{x_{i},y_{i}\}_{i=1}^{n}}$, and thus the SMM can be viewed as a flexible SVM in which a different data-dependent kernel (specified by the assumed form of the distribution ${\displaystyle P_{i}}$) may be placed on each training point.^[22]

Domain adaptation under covariate, target, and conditional shift

The goal of domain adaptation is the formulation of learning algorithms which generalize well when the training and test data have different distributions. Given training examples ${\displaystyle \{(x_{i}^{\text{tr}},y_{i}^{\text{tr}})\}_{i=1}^{n}}$ and a test set ${\displaystyle \{(x_{j}^{\text{te}},y_{j}^{\text{te}})\}_{j=1}^{m}}$ where the ${\displaystyle y_{j}^{\text{te}}}$ are unknown, three types of differences are commonly assumed between the distribution of the training examples ${\displaystyle P^{\text{tr}}(X,Y)}$ and the test distribution ${\displaystyle P^{\text{te}}(X,Y)}$:^[23][24]

1. Covariate shift in which the marginal distribution of the covariates changes across domains: ${\displaystyle P^{\text{tr}}(X)\neq P^{\text{te}}(X)}$
2. Target shift in which the marginal distribution of the outputs changes across domains: ${\displaystyle P^{\text{tr}}(Y)\neq P^{\text{te}}(Y)}$
3. Conditional shift in which ${\displaystyle P(Y)}$ remains the same across domains, but the conditional distributions differ: ${\displaystyle P^{\text{tr}}(X\mid Y)\neq P^{\text{te}}(X\mid Y)}$. In general, the presence of conditional shift leads to an ill-posed problem, and the additional assumption that ${\displaystyle P(X\mid Y)}$ changes only under location-scale (LS) transformations on ${\displaystyle X}$ is commonly imposed to make the problem tractable.

By utilizing the kernel embedding of marginal and conditional distributions, practical approaches to deal with the presence of these types of differences between training and test domains can be formulated. Covariate shift may be accounted for by reweighting examples via estimates of the ratio ${\displaystyle P^{\text{te}}(X)/P^{\text{tr}}(X)}$ obtained directly from the kernel embeddings of the marginal distributions of ${\displaystyle X}$ in each domain without any need for explicit estimation of the distributions.^[24] Target shift, which cannot be similarly dealt with since no samples from ${\displaystyle Y}$ are available in the test domain, is accounted for by weighting training examples using the vector ${\displaystyle {\boldsymbol {\beta }}^{*}(\mathbf {y} ^{\text{tr}})}$ which solves the following optimization problem (where in practice, empirical approximations must be used) ^[23]

${\displaystyle \min _{{\boldsymbol {\beta }}(y)}\left\|{\mathcal {C}}_{{(X\mid Y)}^{\text{tr}}}\mathbb {E} [{\boldsymbol {\beta }}(y)\varphi (y^{\text{tr}})]-\mu _{X^{\text{te}}}\right\|_{\mathcal {H}}^{2}}$ subject to ${\displaystyle {\boldsymbol {\beta }}(y)\geq 0,\mathbb {E} [{\boldsymbol {\beta }}(y^{\text{tr}})]=1}$

To deal with location scale conditional shift, one can perform a LS transformation of the training points to obtain new transformed training data ${\displaystyle \mathbf {X} ^{\text{new}}=\mathbf {X} ^{\text{tr}}\odot \mathbf {W} +\mathbf {B} }$ (where ${\displaystyle \odot }$ denotes the element-wise vector product). To ensure similar distributions between the new transformed training samples and the test data, ${\displaystyle \mathbf {W} ,\mathbf {B} }$ are estimated by minimizing the following empirical kernel embedding distance ^[23]

${\displaystyle \left\|{\widehat {\mu }}_{X^{\text{new}}}-{\widehat {\mu }}_{X^{\text{te}}}\right\|_{\mathcal {H}}^{2}=\left\|{\widehat {\mathcal {C}}}_{(X\mid Y)^{\text{new}}}{\widehat {\mu }}_{Y^{\text{tr}}}-{\widehat {\mu }}_{X^{\text{te}}}\right\|_{\mathcal {H}}^{2}}$

In general, the kernel embedding methods for dealing with LS conditional shift and target shift may be combined to find a reweighted transformation of the training data which mimics the test distribution, and these methods may perform well even in the presence of conditional shifts other than location-scale changes.^[23]

Domain generalization via invariant feature representation

Given N sets of training examples sampled i.i.d. from distributions ${\displaystyle P^{(1)}(X,Y),P^{(2)}(X,Y),\ldots ,P^{(N)}(X,Y)}$, the goal of domain generalization is to formulate learning algorithms which perform well on test examples sampled from a previously unseen domain ${\displaystyle P^{*}(X,Y)}$ where no data from the test domain is available at training time. If conditional distributions ${\displaystyle P(Y\mid X)}$ are assumed to be relatively similar across all domains, then a learner capable of domain generalization must estimate a functional relationship between the variables which is robust to changes in the marginals ${\displaystyle P(X)}$. Based on kernel embeddings of these distributions, Domain Invariant Component Analysis (DICA) is a method which determines the transformation of the training data that minimizes the difference between marginal distributions while preserving a common conditional distribution shared between all training domains.^[25] DICA thus extracts invariants, features that transfer across domains, and may be viewed as a generalization of many popular dimension-reduction methods such as kernel principal component analysis, transfer component analysis, and covariance operator inverse regression.^[25]

Defining a probability distribution ${\displaystyle {\mathcal {P}}}$ on the RKHS ${\displaystyle {\mathcal {H}}}$ with

${\displaystyle {\mathcal {P}}\left(\mu _{X^{(i)}Y^{(i)}}\right)={\frac {1}{N}}\qquad {\text{ for }}i=1,\dots ,N,}$

DICA measures dissimilarity between domains via distributional variance which is computed as

${\displaystyle V_{\mathcal {H}}({\mathcal {P}})={\frac {1}{N}}\operatorname {tr} (\mathbf {G} )-{\frac {1}{N^{2}}}\sum _{i,j=1}^{N}\mathbf {G} _{ij}}$

where

${\displaystyle \mathbf {G} _{ij}=\left\langle \mu _{X^{(i)}},\mu _{X^{(j)}}\right\rangle _{\mathcal {H}}}$

so ${\displaystyle \mathbf {G} }$ is a ${\displaystyle N\times N}$ Gram matrix over the distributions from which the training data are sampled. Finding an orthogonal transform onto a low-dimensional subspace B (in the feature space) which minimizes the distributional variance, DICA simultaneously ensures that B aligns with the bases of a central subspace C for which ${\displaystyle Y}$ becomes independent of ${\displaystyle X}$ given ${\displaystyle C^{T}X}$ across all domains. In the absence of target values ${\displaystyle Y}$, an unsupervised version of DICA may be formulated which finds a low-dimensional subspace that minimizes distributional variance while simultaneously maximizing the variance of ${\displaystyle X}$ (in the feature space) across all domains (rather than preserving a central subspace).^[25]

Distribution regression

In distribution regression, the goal is to regress from probability distributions to reals (or vectors). Many important machine learning and statistical tasks fit into this framework, including multi-instance learning, and point estimation problems without analytical solution (such as hyperparameter or entropy estimation). In practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between sets of points. Distribution regression has been successfully applied for example in supervised entropy learning, and aerosol prediction using multispectral satellite images.^[26]

Given ${\displaystyle {\left(\{X_{i,n}\}_{n=1}^{N_{i}},y_{i}\right)}_{i=1}^{\ell }}$ training data, where the ${\displaystyle {\hat {X_{i}}}:=\{X_{i,n}\}_{n=1}^{N_{i}}}$ bag contains samples from a probability distribution ${\displaystyle X_{i}}$ and the ${\displaystyle i^{\text{th}}}$ output label is ${\displaystyle y_{i}\in \mathbb {R} }$, one can tackle the distribution regression task by taking the embeddings of the distributions, and learning the regressor from the embeddings to the outputs. In other words, one can consider the following kernel ridge regression problem ${\displaystyle (\lambda >0)}$

${\displaystyle J(f)={\frac {1}{\ell }}\sum _{i=1}^{\ell }\left[f\left(\mu _{\hat {X_{i}}}\right)-y_{i}\right]^{2}+\lambda \|f\|_{{\mathcal {H}}(K)}^{2}\to \min _{f\in {\mathcal {H}}(K)},}$

where

${\displaystyle \mu _{{\hat {X}}_{i}}=\int _{\Omega }k(\cdot ,u)\,\mathrm {d} {\hat {X}}_{i}(u)={\frac {1}{N_{i}}}\sum _{n=1}^{N_{i}}k(\cdot ,X_{i,n})}$

with a ${\displaystyle k}$ kernel on the domain of ${\displaystyle X_{i}}$-s ${\displaystyle (k:\Omega \times \Omega \to \mathbb {R} )}$, ${\displaystyle K}$ is a kernel on the embedded distributions, and ${\displaystyle {\mathcal {H}}(K)}$ is the RKHS determined by ${\displaystyle K}$. Examples for ${\displaystyle K}$ include the linear kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=\langle \mu _{P},\mu _{Q}\rangle _{{\mathcal {H}}(k)}\right]}$, the Gaussian kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=e^{-\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}^{2}/(2\sigma ^{2})}\right]}$, the exponential kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=e^{-\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}/(2\sigma ^{2})}\right]}$, the Cauchy kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=\left(1+\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}^{2}/\sigma ^{2}\right)^{-1}\right]}$, the generalized t-student kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=\left(1+\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}^{\sigma }\right)^{-1},(\sigma \leq 2)\right]}$, or the inverse multiquadrics kernel ${\displaystyle \left[K(\mu _{P},\mu _{Q})=\left(\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}^{2}+\sigma ^{2}\right)^{-{\frac {1}{2}}}\right]}$.