## 2 Basic theory
### Brief Introduction to the \(\phi\)-Projection Pursuit Methodology (\(\phi\)-PP)
#### Functioning of the Algorithm  
Let \(f\) be a density on \(\mathbb{R}^{d}\). We consider an instrumental density \(g\) with the same mean and variance as \(f\). We start with performing the \(D_{\phi}(g,f)=0\) test; should this test turn out to be positive, then \(f=g\) and the algorithm stops, otherwise, the first step of our algorithm consists in defining a vector \(a_{1}\) and a density \(g^{(1)}\) by  
\[a_{1}\ =\ arg\inf_{a\in\mathbb{R}^{d}_{*}}\ D_{\phi}\left(g\frac{f_{a}}{g_{a}},f \right)\ \text{and}\ g^{(1)}=g\frac{f_{a_{1}}}{g_{a_{1}}} \tag{2.2}\]  
where \(\mathbb{R}^{d}_{*}\) is the set of non-null vectors of \(\mathbb{R}^{d}\) and \(f_{a}\) (resp. \(g_{a}\)) stands for the density of \(a^{\top}X\) (resp. \(a^{\top}Y\)) when \(f\) (resp. \(g\)) is the density of \(X\) (resp. \(Y\)).  
In our second step, we replace \(g\) with \(g^{(1)}\) and we repeat the first step, and so on. By iterating this process, we end up obtaining a sequence \((a_{1},a_{2},...)\) of vectors in \(\mathbb{R}^{d}_{*}\) and a sequence of densities \(g^{(i)}\).  
\(\mathcal{R}emark\): **2.2**.: _First, to obtain an approximation of \(f\), we stop our algorithm when the divergence equals zero, i.e., we stop when \(D_{\phi}(g^{(j)},f)=0\) since it implies \(g^{(j)}=f\) with \(j\leq d\), or when our algorithm reaches the \(d^{th}\) iteration, i.e., we approximate \(f\) with \(g^{(d)}\)._  
_Second, we get \(D_{\phi}(g^{(0)},f)\geq D_{\phi}(g^{(1)},f)\geq.....\geq 0\) with \(g^{(0)}=g\)._  
_Finally, the specific form of the relationship (2.2) implies that we deal with M-estimation. We can therefore state that our method is robust--see Sections 6, [PERSON] [19], [PERSON] [20] as well as [PERSON] [21]._  
The main steps of the present algorithm have been summarized in Table 1.  
At present, let us study the following example:  
\(\mathcal{E}xample\) **2.2**.: _Let \(f\) be a density defined on \(\mathbb{R}^{3}\) by \(f(x_{1},x_{2},x_{3})=n(x_{1},x_{2})h(x_{3})\), with \(n\) being a bi-dimensional Gaussian density, and \(h\) being a non-Gaussian density. Let us also consider \(g\), a Gaussian density with the same mean and variance as \(f\)._  
_Since \(g(x_{1},x_{2}/x_{3})=n(x_{1},x_{2})\), we have \(D_{\phi}(g\frac{f_{3}}{g_{3}},f)=D_{\phi}(n.f_{3},f)=D_{\phi}(f,f)=0\) as \(f_{3}=h\), i.e., the function \(a\mapsto D_{\phi}(g\frac{f_{0}}{g_{a}},f)\) reaches zero for \(e_{3}=(0,0,1)^{\prime}\), where \(f_{3}\) and \(g_{3}\) are the third marginal densities of \(f\) and \(g\) respectively. We therefore obtain \(g(x_{1},x_{2}/x_{3})=f(x_{1},x_{2}/x_{3})\)._  
To recapitulate our method, if \(D_{\phi}(g,f)=0\), we derive \(f\) from the relationship \(f=g\); whenever a sequence \((a_{i})_{i=1,..,j}\), \(j<d\), of vectors in \(\mathbb{R}^{d}_{*}\) defining \(g^{(j)}\) and such that \(D_{\phi}(g^{(j)},f)=0\) exists, then \(f(./a_{i}^{\top}x,1\leq i\leq j)=g(./a_{i}^{\top}x,1\leq i\leq j)\), _i.e._, \(f\) coincides with \(g\) on the complement of the vector subspace generated by the family \(\{a_{i}\}_{i=1,..,j}\)--see also Section 3 for a more detailed explanation.  
In the remaining of our study of the algorithm, after having clarified the choice of \(g\), we will consider the statistical solution to the representation problem, assuming that \(f\) is unknown and that \(X_{1}\), \(X_{2}\),... \(X_{m}\) are i.i.d. with density \(f\). We will provide asymptotic results pertaining to the family of optimizing vectors \(a_{k,m}\)--which we will define more precisely below--as \(m\) goes to infinity. Our results also prove that the empirical representation scheme converges towards the theoretical one.