Consider the robust regression problem

$\text{minimize} ~\sum_{i=1}^m\text{huber}(a_i^T x - b_i),$

where

\text{huber}(x) = \left\{\begin{aligned} &(1/2)x^2 & |x| \leq 1 \\\ &|x| - (1/2) & |x| > 1 \end{aligned} \right.

which has the graph form representation

\begin{aligned} &\text{minimize} & & \sum_{i=1}^m\text{huber}(y_i - b_i) \\ & \text{subject to} & & y = A x. \end{aligned}

or equivalently

\begin{aligned} &\text{minimize} & & f(y) + g(x) \\ & \text{subject to} & & y = A x, \end{aligned}

where

$f_i(y_i) = \text{huber}(y_i - b_i), ~~\text{ and } ~~g_j(x_j) = 0.$

### MATLAB Code

 1 2 3 4 5 6 7 8 9 10 11 % Generate Data A = randn(100, 10); b = randn(100, 1); % Populate f and g f.h = kHuber; f.b = b; g.h = kZero; % Solve x = pogs(A, f, g); 

A similar example can be found in the file

 1 /examples/matlab/huber_fit.m 
 .

### R Code

 1 2 3 4 5 6 7 8 9 10 # Generate Data A = matrix(rnorm(100 * 10), 100, 10) b = rnorm(100) # Populate f and g f = list(h = kHuber(), b = b) g = list(h = kZero()) # Solve solution = pogs(A, f, g) 

A similar example can be found in the file

 1 /examples/r/huber_fit.R 
 .

### C++ Code

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 #include #include #include "pogs.h" int main() { // Generate Data size_t m = 100, n = 10; std::vector A(m * n); std::vector b(m); std::vector x(n); std::vector y(m); std::default_random_engine generator; std::normal_distribution n_dist(0.0, 1.0); for (unsigned int i = 0; i < m * n; ++i) A[i] = n_dist(generator); for (unsigned int j = 0; j < n; ++j) b[i] = n_dist(generator); // Populate f and g PogsData pogs_data(A.data(), m, n); pogs_data.x = x.data(); pogs_data.y = y.data(); pogs_data.f.reserve(m); for (unsigned int i = 0; i < m; ++i) pogs_data.f.emplace_back(kHuber, 1.0, b[i]); pogs_data.g.reserve(n); for (unsigned int i = 0; i < n; ++i) pogs_data.g.emplace_back(kZero); // Solve Pogs(&pogs_data); } 

A similar example can be found in the file

 1 /examples/cpp/huber_fit.cpp 
 .