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This code provides a templated function pav() that solves for x^* where:

  • x^* = argmin_x ||y - x||^2
  • s.t. components of x are ordered...(1)

It uses a linear time implementation of the pool adjacent vioaltors algorithm (PAV).

You can get the effect of replacing ||x - y||^2 by any other Bregman divergence, by using the transformation: inv[grad phi] (x^*). The most common case is:

  • min_x 1/2 ||y - x||_W^2
  • s.t. components of x are ordered.

If W is a diagonal matrix diag(w), the solution is obtained as x^* / sqrt(w). Thus one need not write a special minimization code for the weighted version of the problem. All that is needed is to transform the minimizer of (1) appropriately. This transformation property holds with wider generality than just for weighted squared Euclidean distances.

Other variations that this code solves are:

lbound_margin_pav:

  • argmin_x ||x - y||^2
  • s.t. x_0 >= m_0
  • s.t. x_i - x_{i+1} <= -m_i forall i in [1, n)

ubound_margin_pav:

  • argmin_x ||x - y||^2
  • s.t. x_{n-1} <= -m_{n-1}
  • s.t. x_{i+1} - x_i >= m_i forall i in [0, n-1)

lbound_maxmargin_pav:

  • argmin_x ||x - y||^2 - <C,m>
  • s.t. x_0 >= m_0
  • s.t. x_i - x_{i+1} <= -m_i forall i in [1, n)
  • s.t. m >= 0

ubound_maxmargin_pav:

  • argmin_x ||x - y||^2 - <C,m>
  • s.t. x_{n-1} <= -m_{n-1}
  • s.t. x_{i+1} - x_i >= m_i forall i in [0, n-1)
  • s.t. m >= 0