I'm reading a paper about image classification. According to the paper, it says
For a given nonlinear mapping $\Phi (x)$, the input data space $\mathbb{R}^{n}$ can be mapped into the feature space $\cal H$:
$$\Phi : \mathbb{R}^{n} \mapsto \cal H$$
$$x \mapsto \Phi(x)$$
Question:
What does this mean? Is $x_i$ a vector or a matrix? Can somewone give an example how $\Phi$ might look like?
In the formula you are asking about $x_i$ is the i-th coordinate vector of the input space (which is $\mathbb{R}^n$). The $y_j$ should probably be $x_j$, that's a typo.
The feature map is an arbitrary map into the feature space $H$. For example, if $H$ is finite dimensional (say, of dimension $m$), you can pick an orthonormal basis for it, and think of each component of $\Phi$ in that basis as a feature. This is what scalar features are. In this case you have complete freedom to choose $m$ arbitrary scalar valued functions on $\mathbb{R}^n$ as your scalar features (i.e. the components of your feature map). One option is to take as features all the monomials of degree at most $d$ in original coordinates, scaled in some way. For a particular choice of scalings, this is equivalent to using the polynomial kernel. In practice, one often starts with a kernel, and leaves $\Phi$ and $H$ implicit; in most cases the corresponding $H$ is infinite-dimensional, and the corresponding $\Phi$ is non-linear (and mapping into the infinite-dimensional $H$).