python - 在python中使用cvxpy包构造融合的套索惩罚(Constructing fused lasso penalty with cvxpy package in python)

Question

Introduced by Tibshirani, 2005, encourages sparsity of the coefficients and also sparsity of their differences.

(Tibshirani于2005年提出，鼓励系数的稀疏性以及它们之间差异的稀疏性。)

(article)

(（文章）)

Screenshot of the formula

(公式的屏幕截图)

How can I imitate this with cvxpy package?

(我该如何使用cvxpy软件包来模仿它？)

There are example codes for Lasso and Ridge Penalties seperately.

(分别有套索和岭罚的示例代码。)

I get a general idea how the codes work but there are some fuctions I do not get how I'm supposed to decide which one to use.For example let's compare Lasso and Ridge penalty codes.

(我大致了解了代码的工作原理，但有一些功能我不知道应该如何决定使用哪一个代码，例如让我们比较Lasso和Ridge罚码。)

# Lasso

import cvxpy as cp
import numpy as np
import matplotlib.pyplot as plt

def loss_fn(X, Y, beta):
    return cp.norm2(cp.matmul(X, beta) - Y)**2

def regularizer(beta):
    return cp.norm1(beta)

def objective_fn(X, Y, beta, lambd):
    return loss_fn(X, Y, beta) + lambd * regularizer(beta)

def mse(X, Y, beta):
    return (1.0 / X.shape[0]) * loss_fn(X, Y, beta).value

# Ridge

import cvxpy as cp
import numpy as np
import matplotlib.pyplot as plt

def loss_fn(X, Y, beta):
    return cp.pnorm(cp.matmul(X, beta) - Y, p=2)**2

def regularizer(beta):
    return cp.pnorm(beta, p=2)**2

def objective_fn(X, Y, beta, lambd):
    return loss_fn(X, Y, beta) + lambd * regularizer(beta)

def mse(X, Y, beta):
    return (1.0 / X.shape[0]) * loss_fn(X, Y, beta).value

For both of them we create a loss function but they are created with different commands?

(对于它们两者，我们都创建了损失函数，但是它们是用不同的命令创建的？)

  ask by moli translate from so

Answer 1

From what I understand, you're having trouble expressing the penalty on the differences of coefficients $\sum_k|b_k - b_{k-1}|$.

(据我了解，您很难表达对系数$ \ sum_k | b_k-b_ {k-1} | $的影响。)

To do this, you just need to define the vector difference matrix as follows:

(为此，您只需要定义矢量差矩阵，如下所示：)

def diff_op(shape):
    mat = np.zeros(shape)
    mat[range(2, shape[0]), range(1, shape[1])] = 1
    mat -= np.eye(shape)
    return mat

def difference_pen(beta, epsilon):
    return cvx.sum(cvx.abs((epsilon, diff_op(beta.shape) @ beta))