Broadcasting

[shahin1009]

Using broadcasting to subtract the average face from the training faces, makes the code more readable.
X = trainingFaces - np.tile(avgFace,(trainingFaces.shape[1],1)).T
to:
X = trainingFaces - avgFace.reshape(-1,1).

avgFace.reshape(-1,1) is an array with shape=(32256,1). For conducting the subtraction, numpy would broadcast this array to (322561,n) so the shape become equal to trainingFaces.shape.

[shahin1009]

cvxpy pakage can manage the L1-norm optimization much faster than SLSQP method in scipy. The process of defining the problem is the same but I add comments to make it more readable.

more detail on CVXPY pakage:
https://pypi.org/project/cvxpy/

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

# defining parameters same as in the previous example
n = 1000 # dimension of s
p = 200 # number of measurements, dim(y)
Theta = np.random.randn(p,n)
y = np.random.randn(p)

# Creating cvxpy variables(the variable that we want to optimize)
s = cp.Variable(n) # s is the variable that we want to optimize

# Creating constraints (y = Theta * s)
constr = [Theta @ s == y] # y = Theta * s

# creating objective
obj = cp.norm(s,1) # minimize the L1 norm of s

# Solving the problem
prob = cp.Problem(cp.Minimize(obj),constr)
prob.solve()

s_L1_cvxpy = s.value # s_L1_cvxpy is the optimized solution
s_L2 = np.linalg.pinv(Theta) @ y # L2 solution from before