Universal Approximation Toy Basis

Fit a one-dimensional function with a fixed ReLU-hinge basis. Build the basis matrix $B_{ij}=\max (x_i-t_j,0)B\_\{ij\}=\backslash max(x\_i-t\_j,0)$ for the given knots $t_{j}t\_j$, solve the least-squares problem for the weights, and return both the fitted values and the weights.

Implement solve(X, y, knots).

Returns. Return a tuple (fitted, weights): the fitted values $BwBw$ (length $nn$) and the weights (length = number of knots). Rank-deficient bases use the minimum-norm least-squares solution (np.linalg.lstsq).

Examples

Example 1

Input:

solve([0, 1, 2], [0, 1, 2], [0])

Output:

([0.0, 1.0, 2.0], [1.0])

Example 2

Input:

solve([0, 1, 2], [0, 0, 1], [0])

Output:

([0.0, 0.4, 0.8], [0.4])