L1正则化最小二乘法

Question

我想通过L1正则化最小二乘法解决以下拉索优化功能。我为我的项目使用python。

α'* = arg min (||y’–B’α’||_2^2 + λ||α’||_1) 

这里α'*是一个载体。尺寸B’=(m+p)*p,y’=(m+p)*1,α‘=p*1

我无法解决这个方程。请任何人解释eqn和解决这个公式的方法L1正则化最小二乘法。

Answer 1

这是可以与ODL解决的典型问题：

import odl 
import numpy as np 

m = 2 
p = 100 
lam = 0.00001 

# Define B 
B = odl.MatrixOperator(np.random.rand(p, m + p)) 
alpha_true = np.random.rand(m + p) 
y = B(alpha_true) 

# Define functionals 
l2dist = odl.solvers.L2NormSquared(B.range) * (B - y) 
l1 = lam * odl.solvers.L1Norm(B.domain) 
func = l2dist + l1 

# Initial point 
alpha = B.domain.zero() 

# Solve using steepest descent 
odl.solvers.steepest_descent(func, alpha, 
          line_search=0.0003, maxiter=10000, 
          callback=lambda x: print(func(x))) 

如果你想有一个更快的解决方案，你应该看看近端的方法（在ODL也可）。一个这样的例子是FISTA方法：

import odl 
import numpy as np 

m = 2 
p = 100 
lam = 0.00001 

# Define B 
B = odl.MatrixOperator(np.random.rand(p, m + p)) 
alpha_true = np.random.rand(m + p) 
y = B(alpha_true) 

# Define functionals 
l2dist = odl.solvers.L2NormSquared(B.range) * (B - y) 
l1 = lam * odl.solvers.L1Norm(B.domain) 
func = l2dist + l1 

# Use FISTA 
alpha = B.domain.zero() 
odl.solvers.accelerated_proximal_gradient(alpha, l1, l2dist, 
              gamma=0.0001, niter=1000, 
              callback=lambda x: print(func(x)))