How to interpret basis function that yields vector in machine learning algorithm?

I'm struggling to understand what $\phi(x_{N+1})$ is in this excerpt of an algorithm (namely Linear Bayesian Regression embedded in other algorithm):

$c_i = \gamma_i / \sum^L_{j} \gamma_j$

$V_i^{N+1} = ((V_i^N)^{-1}+\beta \phi(x_{N+1})^Tc_i \phi(x_{N+1}))^{-1}$

$\theta_i^{N+1} =V_i ^ {N+1} ((V_i^N)^{-1}\theta_i^N+\beta \phi(x_{N+1})^Tc_i r)$

The authors say:

"...where $\phi(x_i)$ denotes the feature vector of $x_i$ (...) The choice of $\phi(x)$ depends on the task. In our experiments however, a Gaussian basis function where the center is given by the augmented state $x_i$ proved to be a good choice"

I know what a Gaussian basis function is, but how does it yield a vector? I think that users with statistics and machine learning background will probably understand right away what $\phi(x)$ is here, but let me know if you need more context.

• If this is the paper, the first reference to $\phi$ in your quote above references $x$ without a subscript; might $x_i$ reference a row or column vector within a matrix $x$? In that case $\phi(x)$ might be the vector of $\phi$ applied to each row or column. – Sean Easter Apr 30 '14 at 14:26
• Thanks for your help. Yes, that is the paper. Each $x_i$ is a vector with several state variables corresponding to the state where the system is in the instant $i$. What you say may or may not make sense, but the real question is what is $\phi(x_N)$ - It is a vector, but of what? – jmacedo Apr 30 '14 at 14:50
• What is $x_N$ in that case? – Sean Easter Apr 30 '14 at 15:26
• @SeanEaster It is a vector with several variables that characterize the augmented state of the system at moment $N$ . I now realize that the version of the paper you linked is not as detailed as this one. The point of the whole algorithm is to learn to mix a series of motion primitives to attain a new task in robotics. The weights given to each primitive $i$ ($\gamma_i$) are calculated from $\theta$ (see the page 5 of the doc), which is the vector that needs to be learned. – jmacedo May 1 '14 at 14:55
• This is indeed perplexing. $\phi_i(\boldsymbol {x})$ seems to reference the Gaussian radial basis function with center at $\boldsymbol x_i$ applied to $\boldsymbol x$, which would indeed (usually) produce a scalar. Might the authors' general process be to extract a feature vector for each primitive, which in the specific case of the Gaussian basis is one-dimensional? – Sean Easter May 1 '14 at 19:56