# What does representer theorem in machine learning tells us?

In reference to the Representer Theorem in machine learning, Why this is so important? Somehow, this theorem justifies the importance of Kernels in machine learning, i.e. the Kernel trick - a more computationally effective way to "shuttle" data between different dimensions. What does representer theorem in machine learning tells us?

$$f(x) = \vec{w}\cdot\phi\{\vec{x}\} + b = \sum_{i=1}^\ell\alpha_i\mathcal{K}(\vec{x},\vec{x}_i) + b$$
Where $$\vec{x}_i$$ is a training example, $$\phi\{\cdot\}$$ is the non-linear transformation for which the kernel function $$\mathcal{K}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$$ gives us the inner product, i.e. $$\mathcal{K}(\vec{x},\vec{x}') = \left<\phi\{\vec{x}\}\cdot\phi\{\vec{x}\}\right>$$. Now the reason this is important is that we know that we can never get a better model by adding any more terms to this expansion, so if we have one term for each training pattern, then we know can obtain the optimal solution given by the primal parameters.