measuring Statistical Dependence with Hilbert-Schmidt Norm
"Hilbert-Schmist Norm of the Cross-Covariance operator" is proposed as an independence criterion in reproducing kernel Hilbert spaces (RKHSs). The measure is refereed to as Hilbert-Schmidt Independence Criterion, or HSIC. An empirical estimate of this measure is introduced which may be well used in many practical application such as independent Component Analysis (ICA), Maximum Variance Unfolding (MVU), feature extraction, feature selection, ... .
Introduction
In this article, a criterion is introduced which is used to measure the dependence between two multivariate random variables. More specifically, we are looking for an appropriate function of two random variables whose output is zero if the involving random variables are independent and achieves a high value if they are statistically dependent.
If instead of "independence" we were looking for "uncorrelation" the situation would be much easier to handle. Because "correlation" of two random variables involves only expected values, as opposed to concept of independence between two random variables.
However, fortunately, based on the following theorem we still can effectively tackle the problem of independence between two random variables:
Theorem: Two random variables [math]\displaystyle{ \,\!x }[/math] and [math]\displaystyle{ \,\!y }[/math] are independent if and only if any (measurable) function of the two random variables are uncorrelated.
For example, for random variables [math]\displaystyle{ \,\!x }[/math] and [math]\displaystyle{ \,\!y }[/math] to be independent, all pairs [math]\displaystyle{ (x,y),\, (x^2,y),\,(x,e^y), \,(\sin(x),\log(y)),\, ... }[/math] have to be uncorrelated. That is why we need to deal with functions of random variables when examining the independence property. and That is why Hilbert spaces and concepts from "Functional Analysis" are involved in this article (Of course, no one likes to torture him/herself by using complicated mathematical expressions ;)).
In the following, we first give an introductory note on the concepts from functional analysis and Hilbert spaces required in this paper. (No prior knowledge of these concepts is assumed here). Then we introduce the independence measure, namely, Hilbert-Schmidt Independence Criterion and explain why we expect it to work properly. Thirdly, we introduce an statistics which may be used to estimate the HSIC and discuss its convergence behavior. Finally, through some examples, we demonstrate how this criterion may be effectively employed to solve many well-known problems.
Simple introduction to Reproducing Kernel Hilbert Spaces
Let [math]\displaystyle{ \mathcal{F} }[/math] denote the set containing all real-valued, continuous and bounded functions defined on domain [math]\displaystyle{ \,\!\mathcal{X} }[/math]. By real-valued, we mean the output of these functions is always a real number. (Note that this definition makes good sense considering the statement identified by bold font in the previous section.) We may note that this space is linear, as for any [math]\displaystyle{ f,g\in\mathcal{F} }[/math] and [math]\displaystyle{ a,b\in\mathbb{R} }[/math], [math]\displaystyle{ \,\!af+bg }[/math] will be a member of [math]\displaystyle{ \mathcal{F} }[/math], as well. On the other hand, we may define the concept of real-valued inner product between the elements of [math]\displaystyle{ \mathcal{F} }[/math]. The simplest way to do that probably is the familiar:
[math]\displaystyle{ \langle a,b\rangle=\int_{}^{}{f(t)g(t)dt} }[/math]
However, other definitions for the inner product is also possible. The only required conditions is that:
1. [math]\displaystyle{ \langle f,f\rangle\geq0,\,\forall\, f\in\mathcal{F} }[/math] with [math]\displaystyle{ \langle f,f\rangle=0 }[/math] iff [math]\displaystyle{ \,\!f=0 }[/math].
2. [math]\displaystyle{ \langle f,g\rangle=\langle g,f\rangle,\, \forall \,f,g\in \mathcal{F} }[/math].
3. [math]\displaystyle{ \langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle,\,\forall\,f,g,h\in\mathcal{F} }[/math] and all real [math]\displaystyle{ \,\!a }[/math] and [math]\displaystyle{ \,\!b }[/math].
Endowed with the concept of inner product, the set [math]\displaystyle{ \mathcal{F} }[/math] here is, in fact, a Hilbert space. In general, a Hilbert space is a (complete) linear space endowed with an inner product defined on the elements with thw above properties. However, in this article we only deal with those Hilbert spaces whose elements are real-valued continuous bounded functions on [math]\displaystyle{ \mathcal{X} }[/math]). This smooth subtype of Hilbert spaces is called Reproducing Kernel Hilbert Space (RKHS).
To every RKHS we may assign a symmetric positive definite function [math]\displaystyle{ \,\!k }[/math], named kernel, which is of the form [math]\displaystyle{ k:\,\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R} }[/math]. As mentioned above, the RKHSs we consider here share the property that they have all smooth functions on domain [math]\displaystyle{ \mathcal{X} }[/math] as their elements. However, they may have different definitions for the inner product. Two RKHSs with different inner product have different kernel functions as well. It may be shown that there is a unique correspondence between a RKHS and its kernel. In other words, having the Hilbert space, a unique kernel function is assigned to it and inversely, if [math]\displaystyle{ k }[/math] is symmetric positive definite function on a set [math]\displaystyle{ \mathcal{X} }[/math], then there is a unique Hilbert space of functions on [math]\displaystyle{ \mathcal{X} }[/math] for which [math]\displaystyle{ k }[/math] is a reproducing kernel.
The independence measure
Now we turn our attention back to the the problem of measuring independence between two (generally multivariate) random variables [math]\displaystyle{ \,\!x }[/math] and [math]\displaystyle{ \,\!y }[/math]. Induced by the idea described in the previous section, we define Hilbert spaces [math]\displaystyle{ \mathcal{F} }[/math] from [math]\displaystyle{ \mathcal{X} }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] containing all real-valued functions of [math]\displaystyle{ \,\!x }[/math], and Hilbert space [math]\displaystyle{ \mathcal{G} }[/math] from [math]\displaystyle{ \mathcal{Y} }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] containing real-valued functions of [math]\displaystyle{ \,\!y }[/math]. Here, [math]\displaystyle{ \mathcal{X} }[/math] and [math]\displaystyle{ \mathcal{Y} }[/math] denotes the support (set of possible values) of random variables [math]\displaystyle{ \,\!x }[/math] and [math]\displaystyle{ \,\!y }[/math] respectively. As mentioned before, we are particularly interested in the cross-covariance between elements of [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math]:
[math]\displaystyle{ \mathbf{E}_{x,y}[f(x)g(y)]-\mathbf{E}_{x}[f(x)]\mathbf{E}_{y}[g(y)] }[/math]
RKHS Theory
Let [math]\displaystyle{ \mathcal{F} }[/math] be a Hilbert space from [math]\displaystyle{ \mathcal{X} }[/math] to [math]\displaystyle{ \mathbb{R} }[/math]. We assume [math]\displaystyle{ \mathcal{F} }[/math] is a Reproducing Kernel Hilbert Space,i.e., for all [math]\displaystyle{ x\in \mathcal{X} }[/math], the corresponding Dirac evaluation operator [math]\displaystyle{ \delta_x:\mathcal{F} \rightarrow \mathbb{R} }[/math] is a bounded (or equivalently continuous) linear functional. We denote the kernel of this operator by [math]\displaystyle{ k(x,x')=\langle \phi(x)\phi(x') \rangle_{\mathcal{F}} }[/math] where [math]\displaystyle{ k:\mathcal{X}\times \mathcal{X}\rightarrow \mathbb{R} }[/math] is a positive definite function and [math]\displaystyle{ \,\!\phi }[/math] is the feature map of [math]\displaystyle{ \mathcal{F} }[/math]. Similarly, we consider another RKHS named [math]\displaystyle{ \mathcal{G} }[/math] with Domain [math]\displaystyle{ \mathcal{Y} }[/math], kernel [math]\displaystyle{ l(\cdot,\cdot) }[/math] and feature map [math]\displaystyle{ \,\!\psi }[/math]. We assume both [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math] are separable, i.e., they have a complete orthogonal bases.
Hilbert-Schmidt Norm
For a linear operator [math]\displaystyle{ C:\mathcal{G}\rightarrow \mathcal{F} }[/math], provided the sum converges, the Hilbert-Schmidt (HS) norm is defined as:
[math]\displaystyle{ \|C\|^2_{HS}:=\sum_{i,j}\langle Cv_i,u_j \rangle^2_{\mathcal{F}} }[/math]
where [math]\displaystyle{ \,\!u_j }[/math] and [math]\displaystyle{ \,\!v_i }[/math] are orthogonal bases of [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math], respectively. It is easy to see that the Frobenius norm on matrices may be considered a spacial case of this norm.
Hilbert-Schmidt Operator
A Hilbert-Schmidt Operator is a linear operator for which the Hilbert-Schmidt norm (introduced above) exists.
Tensor Product Operator
We may employ any [math]\displaystyle{ f\in \mathcal{F} }[/math] and [math]\displaystyle{ g\in \mathcal{G} }[/math] to define a tensor product operator [math]\displaystyle{ f\otimes g:\mathcal{G}\rightarrow\mathcal{F} }[/math] as follows:
[math]\displaystyle{ (f\otimes g)h:=f\langle g,h\rangle_{\mathcal{G}} \quad }[/math] for all [math]\displaystyle{ h\in\mathcal{G} }[/math]
Using the definition of HS norm introduced above, we can simply show the norm of [math]\displaystyle{ f\otimes g }[/math] equals
[math]\displaystyle{ \|f\otimes g\|^2_{HS}=\|f\|^2_{\mathcal{F}}\; \|g\|^2_{\mathcal{G}} }[/math]
Cross-Covariance Operator
Mean
Mean elements of [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math] are defined as those elements of these spaces for which
[math]\displaystyle{ \langle\mu_x,f \rangle_{\mathcal{F}}=\mathbf{E}_x[\langle\phi(x),f \rangle_{\mathcal{F}}]=\mathbf{E}_x[f(x)] }[/math]
[math]\displaystyle{ \langle\mu_y,g \rangle_{\mathcal{G}}=\mathbf{E}_y[\langle\psi(y),g \rangle_{\mathcal{G}}]=\mathbf{E}_y[g(x)] }[/math]
Based on this, [math]\displaystyle{ \|\mu_x\|^2_{\mathcal{F}} }[/math] may be calculated by applying expectation twice as follows:
[math]\displaystyle{ \|\mu_x\|^2_{\mathcal{F}}=\mathbf{E}_{x,x'}[\langle \phi(x),\phi(x')\rangle_{\mathcal{F}}]=\mathbf{E}_{x,x'}[k(x,x')] }[/math]
Cross-covariance Operator
Now we are in aposition to define the cross-covariance operator as follows
[math]\displaystyle{ C_{xy}:=\underbrace{\mathbf{E}_{x,y}[\phi(x)\otimes\psi(y)]}_{:=\tilde{C}_{xy}}-\underbrace{\mu_x\otimes\mu_y}_{:=M_{xy}} }[/math]
We will use [math]\displaystyle{ \tilde{C}_{xy} }[/math] and [math]\displaystyle{ \,\!M_{xy} }[/math] as the basis of our measure of dependence.
Hilbert-Schmidt Independence Criterion
Definition (HSIC) Given separable RKHSs [math]\displaystyle{ \mathcal{F} }[/math], [math]\displaystyle{ \mathcal{G} }[/math] and a joint probability [math]\displaystyle{ p_{x\,\!y} }[/math], we define the Hilbert-Schmidt Independence Criterion(HSIC) as the squared HS-norm of the associated cross-covariance operator [math]\displaystyle{ C_{x\,\!y} }[/math]
[math]\displaystyle{ \text{HSIC}(p_{xy},\mathcal{F},\mathcal{G}):=\|C_{xy}\|^2_{HS} }[/math]
HSIC in terms of kernels
To compute HSIC we need to express it in terms of kernel functions. It can be shown that this can be achieved via the following identity:
[math]\displaystyle{ \text{HSIC}(p_{xy},\mathcal{F},\mathcal{G})=\mathbf{E}_{x,x',y,y'}[k(x,x')l(y,y')]+\mathbf{E}_{x,x'}[k(x,x')]\mathbf{E}_{y,y'}[l(y,y')]-2\mathbf{E}_{x,y}[\mathbf{E}_{x'}[k(x,x')]\mathbf{E}_{y'}[l(y,y')]] }[/math]
Because the data is random, the kernels are random so we use expected values.
Discussion
One of the methods to find dependence between functions in RKHS, as proposed by Gretton et al ,the constarined covariance (COCO) was to use the largest singular value of the cross-covariance operator, which behaves identically to the correlation operator at independence and no regularization is required.
In this paper, the concept of COCO is extended by using the entire spectrum of the cross-covariance operator to determine when all its singular values are zero, rather than looking only at the largest singular value.
Empirical Criterion
Definition (Empirical HSIC)
Let [math]\displaystyle{ Z:=\{(x_1,y_1),\cdots,(x_m,y_m)\}\subseteq \mathcal{X}\times\mathcal{Y} }[/math] be a series of [math]\displaystyle{ \,\!m }[/math] independent observations drawn from [math]\displaystyle{ p_{x\,\!y} }[/math]. An estimator of HSIC, is given by
[math]\displaystyle{ \text{HSIC}(Z,\mathcal{F},\mathcal{G}):=(m-1)^{-2}\textbf{tr}(KHLH) }[/math]
where [math]\displaystyle{ H, K, L\in\mathbb{R}^{m\times m}, K_{ij}:=k(x_i,x_j), L_{ij}:=l(y_i,y_j) \,\, \text{and} \, \, H_{ij}:=\delta_{ij}-m^{-1} }[/math].
Bias of Estimator
It may bee shown that the bias of the above empirical estimation is of the order [math]\displaystyle{ \,\!O(m^{-1}) }[/math]:
Theorem: Let [math]\displaystyle{ \mathbf{E}_Z }[/math] denote the expectation taken over [math]\displaystyle{ \,\!m }[/math] independent copies [math]\displaystyle{ \,\!(x_i,y_i) }[/math] drawn from [math]\displaystyle{ p_{\,\!xy} }[/math]. Then
[math]\displaystyle{ \text{HSIC}(p_{xy},\mathcal{F},\mathcal{G})=\mathbf{E}_Z[\text{HSIC}(Z,\mathcal{F},\mathcal{G})]+O(m^{-1}) }[/math].
Bound on Empirical HSIC
Theorem: Assume that [math]\displaystyle{ \,\!k }[/math] and [math]\displaystyle{ \,\!l }[/math] are bounded almost everywhere by 1, and are non-negative. Then for [math]\displaystyle{ \,\!m\gt 1 }[/math] and all [math]\displaystyle{ \,\!\delta\gt 0 }[/math], with probablity at least [math]\displaystyle{ \,\!1-\delta }[/math], for all [math]\displaystyle{ \,\!p_{xy} }[/math] :
[math]\displaystyle{ |\text{HSIC}(p_{xy},\mathcal{F},\mathcal{G})-\text{HSIC}(Z,\mathcal{F},\mathcal{G})|\leq\sqrt{\frac{log(6/\delta)}{\alpha^2m}}+\frac{C}{m} }[/math]
where [math]\displaystyle{ \,\!\alpha\gt 0.24 }[/math] and [math]\displaystyle{ \,\!C }[/math] are constants.
Independence Test using HSIC
Theorem: Denote by [math]\displaystyle{ \mathcal{F} }[/math], [math]\displaystyle{ \mathcal{G} }[/math] RKHSs with universal
kernels [math]\displaystyle{ \,\!k }[/math], [math]\displaystyle{ \,\!l }[/math] on the compact domains [math]\displaystyle{ \mathcal{X} }[/math] and [math]\displaystyle{ \mathcal{Y} }[/math] respectively. We assume without
loss of generality that [math]\displaystyle{ \|f\|_{\infty}\leq 1 }[/math] and [math]\displaystyle{ \|g\|_{\infty}\leq 1 }[/math] for all [math]\displaystyle{ f \in \mathcal{F} }[/math] and [math]\displaystyle{ g \in \mathcal{G} }[/math]. Then
[math]\displaystyle{ \|C_{xy}\|_{HS} =0 }[/math] if and only if [math]\displaystyle{ \,\!x }[/math] and [math]\displaystyle{ \,\!y }[/math] are independent.
Based on this result, to maximize the dependence between two kernels we need to increase the value of the empirical estimate, i.e., [math]\displaystyle{ \,\!\textbf{tr}(KHLH) }[/math].
It can be shown that if even one of the kernels [math]\displaystyle{ \,\!K }[/math] or [math]\displaystyle{ \,\!L }[/math] is already centered, we may drop the centering matrices [math]\displaystyle{ \,\!H }[/math] and simply use the objective function [math]\displaystyle{ \,\!\textbf{tr}(KL) }[/math].
Kernel Canonical Correlation Analysis (CCA)
In Classical CCA linear mapping [math]\displaystyle{ a^{T}X }[/math] and [math]\displaystyle{ b^{T}Y }[/math] are obtained, respectively for random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], such that the linear projections have maximum correlation.
Kernel CCA extends this method to estimate functions [math]\displaystyle{ f\in \mathcal{F} }[/math] and [math]\displaystyle{ g\in \mathcal{G} }[/math] such that the correlation of [math]\displaystyle{ f(X) }[/math] and [math]\displaystyle{ g(Y) }[/math] is maximized.
to find [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] the following optimization problem must be solved:
[math]\displaystyle{ max_{(f \in \mathcal{F} , g \in \mathcal{G})(f \neq 0 , g \neq 0)} \frac{Cov(f(x),g(Y))}{(Var(f(X))_{1/2} (Var(g(y))_{1/2}} }[/math]
For finite independent sample [math]\displaystyle{ {(X_{1},Y_{1}),(X_{2},Y_{2}),...,(X_{n},Y_{n})} }[/math] estimators for functions [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] can be obtained by optimizing the empirical version of the optimization problem:
[math]\displaystyle{ max_{(f \in \mathcal{F} , g \in \mathcal{G})(f \neq 0 , g \neq 0)} \frac{\widehat{Cov}(f(x),g(Y))}{(\widehat{Var}(f(X)+\epsilon_{n}||f||_{\mathcal{F}}^{2})_{1/2} (\widehat{Var}(g(y)+\epsilon_{n}||g||_{\mathcal{G}}^{2})_{1/2}} }[/math]
for which:
[math]\displaystyle{ \widehat{Cov}(f(x),g(Y))= \frac {1}{n} \sum_{i=1}^{n} \left( f(X_{i})- \sum_{j=1}^{n}f(X_{j}) \right) \left( g(Y_{i})- \sum_{j=1}^{n}g(Y_{j}) \right) }[/math]
[math]\displaystyle{ \widehat{Var}(f(X))= \frac {1}{n} \sum_{i=1}^{n} \left( f(X_{i})- \sum_{j=1}^{n}f(X_{j}) \right)^{2} }[/math]
[math]\displaystyle{ \widehat{Var}(g(Y))= \frac {1}{n} \sum_{i=1}^{n} \left( g(Y_{i})- \sum_{j=1}^{n}g(Y_{j}) \right)^{2} }[/math]
Representation of Kernel CCA by Cross-Covariance Operator
Optimization problem for kernel CCA can be formulated using Cross-Covariance Operator for [math]\displaystyle{ (X,Y) }[/math]
[math]\displaystyle{ Sup_{(f \in \mathcal{F} , g \in \mathcal{G})} \langle g,\Sigma_{XY} f \rangle_{\mathcal{G}} }[/math] subject to [math]\displaystyle{ \langle f,\Sigma_{XX} f \rangle_{\mathcal{F}}=1 }[/math] and [math]\displaystyle{ \langle g,\Sigma_{YY} g \rangle_{\mathcal{G}}=1 }[/math]
Solution of this problem is the eigenfunction of the largest eigenvalue of the following eigenproblem
[math]\displaystyle{ \Sigma_{YX} f = \rho_{1} \Sigma_{YY} g }[/math]
[math]\displaystyle{ \Sigma_{YX} g = \rho_{1} \Sigma_{XX} f }[/math]