Regularization Path

def regularization_path(a, b, C, reg=1e-4, itmax=50000):
    """
    Compute regularization path for optimal transport.
    
    This function computes the complete path of transport plans as regularization
    varies from 0 (unregularized) to infinity (uniform transport).
    
    Parameters:
    - a: array-like, source distribution
    - b: array-like, target distribution  
    - C: array-like, cost matrix
    - reg: float, final regularization parameter
    - itmax: int, maximum number of iterations
    
    Returns:
    - gamma_list: list of regularization parameters along path
    - Pi_list: list of corresponding transport plans
    """

def compute_transport_plan(gamma, gamma_list, Pi_list):
    """
    Compute transport plan for specific regularization parameter.
    
    Parameters:
    - gamma: float, target regularization parameter
    - gamma_list: list, regularization parameters from path computation
    - Pi_list: list, transport plans corresponding to gamma_list
    
    Returns:
    - transport plan matrix interpolated to gamma
    """

def fully_relaxed_path(a, b, C, reg=1e-4, itmax=50000):
    """
    Compute fully relaxed regularization path.
    
    Solves the optimal transport problem with both source and target
    marginal constraints relaxed along the regularization path.
    
    Parameters:
    - a: array-like, source distribution
    - b: array-like, target distribution
    - C: array-like, cost matrix
    - reg: float, regularization parameter
    - itmax: int, maximum iterations
    
    Returns:
    - gamma_list: regularization parameters
    - Pi_list: corresponding transport plans
    """

def semi_relaxed_path(a, b, C, reg=1e-4, itmax=50000):
    """
    Compute semi-relaxed regularization path.
    
    Solves optimal transport with only target marginal constraints
    relaxed along the regularization path.
    
    Parameters:
    - a: array-like, source distribution
    - b: array-like, target distribution
    - C: array-like, cost matrix
    - reg: float, regularization parameter
    - itmax: int, maximum iterations
    
    Returns:
    - gamma_list: regularization parameters
    - Pi_list: corresponding transport plans
    """

def recast_ot_as_lasso(a, b, C):
    """
    Recast optimal transport problem as LASSO regression.
    
    Parameters:
    - a: array-like, source distribution
    - b: array-like, target distribution
    - C: array-like, cost matrix
    
    Returns:
    - H: array-like, design matrix for LASSO formulation
    - y: array-like, target vector for LASSO formulation
    - c: array-like, cost vector
    """

def recast_semi_relaxed_as_lasso(a, b, C):
    """
    Recast semi-relaxed optimal transport as LASSO regression.
    
    Parameters:
    - a: array-like, source distribution
    - b: array-like, target distribution
    - C: array-like, cost matrix
    
    Returns:
    - H: array-like, design matrix for semi-relaxed LASSO
    - y: array-like, target vector
    - c: array-like, cost vector
    """

def ot_next_gamma(phi, delta, HtH, Hty, c, active_index, current_gamma):
    """
    Compute next regularization parameter in OT path.
    
    Parameters:
    - phi: array-like, current solution
    - delta: array-like, search direction
    - HtH: array-like, Hessian matrix H^T H
    - Hty: array-like, gradient vector H^T y
    - c: array-like, cost vector
    - active_index: list, indices of active variables
    - current_gamma: float, current regularization parameter
    
    Returns:
    - next_gamma: float, next regularization parameter
    """

def semi_relaxed_next_gamma(phi, delta, HtH, Hty, c, active_index, current_gamma, n):
    """
    Compute next gamma for semi-relaxed path.
    
    Parameters:
    - phi: array-like, current solution
    - delta: array-like, search direction  
    - HtH: array-like, Hessian matrix
    - Hty: array-like, gradient vector
    - c: array-like, cost vector
    - active_index: list, active variable indices
    - current_gamma: float, current regularization
    - n: int, source distribution size
    
    Returns:
    - next_gamma: float, next regularization parameter
    """

def compute_next_removal(phi, delta, current_gamma):
    """
    Compute next variable removal point in path.
    
    Parameters:
    - phi: array-like, current solution
    - delta: array-like, search direction
    - current_gamma: float, current regularization
    
    Returns:
    - removal_gamma: float, regularization at which variable becomes zero
    - removal_index: int, index of variable to remove
    """

def complement_schur(M_current, b, d, id_pop):
    """
    Compute Schur complement for matrix updates.
    
    Parameters:
    - M_current: array-like, current matrix
    - b: array-like, vector for update
    - d: float, diagonal element
    - id_pop: int, index to remove
    
    Returns:
    - updated matrix after Schur complement operation
    """

def construct_augmented_H(active_index, m, Hc, HrHr):
    """
    Construct augmented design matrix H.
    
    Parameters:
    - active_index: list, active variable indices
    - m: int, number of constraints
    - Hc: array-like, constraint matrix
    - HrHr: array-like, reduced Hessian
    
    Returns:
    - H_aug: augmented design matrix
    """

import ot
import ot.regpath
import numpy as np
import matplotlib.pyplot as plt

# Create optimal transport problem
n, m = 50, 60
a = ot.utils.unif(n)
b = ot.utils.unif(m) 
X = np.random.randn(n, 2)
Y = np.random.randn(m, 2)
C = ot.dist(X, Y)

# Compute full regularization path
gamma_list, Pi_list = ot.regpath.regularization_path(a, b, C, reg=1e-3)

# Plot regularization path (sparsity vs regularization)
sparsity = [np.sum(Pi > 1e-8) / Pi.size for Pi in Pi_list]
plt.figure(figsize=(10, 6))
plt.subplot(1, 2, 1)
plt.plot(gamma_list, sparsity, 'b-', linewidth=2)
plt.xlabel('Regularization parameter')
plt.ylabel('Sparsity (fraction of non-zeros)')
plt.title('Regularization Path')

# Compute specific transport plan
gamma_target = 0.01
Pi_target = ot.regpath.compute_transport_plan(gamma_target, gamma_list, Pi_list)

# Compare with semi-relaxed path
gamma_list_semi, Pi_list_semi = ot.regpath.semi_relaxed_path(a, b, C, reg=1e-3)
sparsity_semi = [np.sum(Pi > 1e-8) / Pi.size for Pi in Pi_list_semi]

plt.subplot(1, 2, 2)
plt.plot(gamma_list, sparsity, 'b-', label='Fully constrained', linewidth=2)
plt.plot(gamma_list_semi, sparsity_semi, 'r--', label='Semi-relaxed', linewidth=2)
plt.xlabel('Regularization parameter')
plt.ylabel('Sparsity')
plt.title('Path Comparison')
plt.legend()
plt.show()

import ot.regpath
from ot.regpath import regularization_path, fully_relaxed_path, semi_relaxed_path
from ot.regpath import compute_transport_plan