L1 min and L2 min of (exponential - x) are both very flat 2017-03-31 Mar

exp-L1-L2.py

#!/usr/bin/env python2
""" min_x av |exp - x| at 0.7  -- W Least_absolute_deviations, L1
    min_x rms( exp - x ) at 1  -- least squares, L2
    are both very flat
    which might explain why L1 minimization with IRLS doesn't work very well.
"""
    # goo "L1 minimization" irls
    # different L1 min problems: sparsity, outliers

from __future__ import division
import sys
import numpy as np

import etc.numpyutil as nu

np.set_printoptions( threshold=20, edgeitems=10, linewidth=140,
        formatter = dict( float = lambda x: "%.2g" % x ))  # float arrays %.2g

nx = 100
seed = 1

# to change these params, run this.py a=1 b=None 'c = ...'  in sh or ipython
for arg in sys.argv[1:]:
    exec( arg )

np.random.seed( seed )
print "nx %d  seed %d" % (nx, seed)

#...............................................................................
yrandom = np.sort( np.random.exponential( size=nx ))
ylogu = - np.log( np.linspace( .5 / nx, 1 - .9 / nx, nx ))[::-1]  # ~ exp, brokenstick.py
for y in [yrandom, ylogu]:
    print "y: av %.3g  %s" % (y.mean(), nu.quantiles( y ))

#...............................................................................
for x in np.arange( .6, 1.1 + .01, .05 ):
    av1 = nu.av1( y - x )
    av2 = nu.av2( y - x )
    print "exp - %-5.2g: av1 %-5.3g  av2 %-5.3g" % (x, av1, av2)