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The problem comes from Exercise 6.1 of "Pattern Recognition and Machine Learning" by Christopher M. Bishop:

Consider the dual formulation of the least squares linear regression problem given in Section 6.1. Show that the solution for the components $a_n$ of the vector $\bf a$ can be expressed as a linear combination of the elements of the vector ${\bf\phi}({\bf x}_n)$. Denoting these coefficients by the vector $\bf w$, show that the dual of the dual formulation is given by the original representation in terms of the parameter vector $\bf w$.

Note that the vector ${\bf\phi}({\bf x}_n)=(\phi_1({\bf x}_n),\phi_2({\bf x}_n),\ldots,\phi_M({\bf x}_n))^T$. This exercise has an official solution. I copied it below:

We first of all note that $J({\bf a})$ depends on ${\bf a}$ only through the form $\bf Ka$. Since typically the number $N$ of data points is greater than the number $M$ of basis functions, the matrix ${\bf K} = {\bf\Phi\Phi}^T$ will be rank deficient. There will then be $M$ eigenvectors of $\bf K$ having non-zero eigenvalues, and $N-M$ eigenvectors with eigenvalue zero. We can then decompose ${\bf a} = {\bf a}_{\|}+{\bf a}_{\perp}$ where ${\bf a}_{\|}^T{\bf a}_{\perp}=0$ and ${\bf Ka}_{\perp}={\bf 0}$. Thus the value of ${\bf a}_\perp$ is not determined by $J({\bf a})$. We can remove the ambiguity by setting ${\bf a}_\perp={\bf 0}$, or equivalently by adding a regularizer term $$\frac{\epsilon}{2}{\bf a}_\perp^T{\bf a}_\perp$$ to $J({\bf a})$ where $\epsilon$ is a small positive constant. Then ${\bf a}={\bf a}_\|$ where ${\bf a}_\|$ lies in the span of ${\bf K}={\bf\Phi\Phi}^T$ and hence can be written as a linear combination of the columns of $\bf\Phi$, so that in component notation $$a_n=\sum\limits_{i=1}^M u_i\phi_i({\bf x}_n)$$ or equivalently in vector notation $${\bf a}={\bf\Phi u}.\tag{122}$$ Substituting (122) into (6.7) we obtain $$\begin{aligned} J({\bf u})&=\frac{1}{2}({\bf K\Phi u}-{\bf t})^T({\bf K\Phi u}-{\bf t})+\frac{\lambda}{2}{\bf u}^T{\bf\Phi}^T{\bf K\Phi u}\\ &=\frac{1}{2}({\bf\Phi\Phi}^T{\bf\Phi u}-{\bf t})^T({\bf\Phi\Phi}^T{\bf\Phi u}-{\bf t})+\frac{\lambda}{2}{\bf u}^T{\bf\Phi}^T{\bf\Phi\Phi}^T{\bf\Phi u}.\quad\quad\quad\quad(123)\end{aligned}$$ Since the matrix ${\bf\Phi}^T{\bf\Phi}$ has full rank we can define an equivalent parametrization given by $${\bf w}={\bf\Phi}^T{\bf\Phi u}$$ and substituting this into (123) we recover the original regularized error function (6.2).

Since it is not allowed to ask many questions, this is my first question about the solution:

  1. For the first sentence "... $J({\bf a})$ depends on ${\bf a}$ only through the form $\bf Ka$", how does the second term of (6.7), $\frac{\lambda}{2}{\bf a}^T{\bf Ka}$, depend on $\bf Ka$? Yes, there is a $\bf Ka$ in it, but there is also an $\bf a$ which does not "depend on ${\bf a}$ only through the form $\bf Ka$".
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  • $\begingroup$ Please state what $J({\bf a})$ looks like in your 4 questions. $\endgroup$
    – statmerkur
    Dec 26, 2022 at 8:26

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From the explanation, I understand that the form of $J(\mathbf a)$ that is first mentioned does not have the regularization term in it. So, it should be $$J(\mathbf a)=\frac{1}{2}(\mathbf{Ka-t})^T(\mathbf{Ka-t})$$

which depend on $\mathbf a$ only through $\mathbf{Ka}$. And, they make a point stating that the null space of matrix $\mathbf K$ poses some problems and one approach to deal with it is to add a regularization.

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