# Writing a piecewise linear function as a sum of ReLU functions

Suppose I have a piecewise linear function $$f(x) = \sum^n_{i=1}a_i\phi_i(x)$$, where $$\{\phi_i\}_{i=1}^n$$ is a finite dimension space of dimension $$n-1$$, in particular I am interested in the functions of the form: $$\phi_{j}(x)=\left\{\begin{array}{ll}\frac{x-x_{j-1}}{h}, & \text { if } x_{j-1} \leq x \leq x_{j} \\ \frac{x_{j+1}-x}{h}, & \text { if } x_{j} \leq x \leq x_{j+1} \\ 0, & \text { otherwise }\end{array}\right.$$

Rough illustration:

Now, suppose I am working with the ReLU activation function $$\sigma(x)=max(0,x)$$ and I am trying to find the coefficients $$\alpha,W$$ and biases $$b$$ such that the function $$u_\theta(x) = \sum^n_{i=1} \alpha_i \sigma(w_ix+b_i)$$ is exactly our function $$f(x)$$.

In other words, for a given piecewise linear function $$f(x) = \sum^n_{i=1}a_i\phi_i(x)$$, what are the coefficients $$\alpha,W,b$$ such that $$\sum^n_{i=1}a_i\phi_i(x)=\sum^n_{i=1} \alpha_i \sigma(w_ix+b_i)$$?