Snippets
Revised by
Lilian Besson (Naereen)
d357f12
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MIT Licensed (http://lbesson.mit-license.org)
Using N = 1000, T = 5.
For test domains:
- IR ~= [-300.0, 300.0],
- IR+ ~= [ 0, 300.0],
- IN ~= [ 0, 1000],
- IN ~= [ 0, 30] (small to reduce overflow),
Tolerance for small real values = 1e-12.
Tolerance for success rates = 0.01.
And sampling the a_i from uniform in [-50.0, 50.0].
Starting to test 14 kernels...
- Testing the kernel abs_k K(x, y) = < x, y > = x * y :
It has domain = REAL.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel abs_k...
- Testing the kernel one_by_k K(x, y) = 1 / (1 - x * y) :
It has domain = MONEONE.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel one_by_k...
- Testing the kernel power_two_dot_k K(x, y) = 2 ** (x + y) :
It has domain = SMALLINT.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel power_two_dot_k...
- Testing the kernel power_two_sum_k K(x, y) = 2 ** (x * y) :
It has domain = SMALLINT.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel power_two_sum_k...
- Testing the kernel log_k K(x, y) = log(1 + x * y) :
It has domain = REALPLUS.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 96.50%.
===> I guess this kernel log_k K(x, y) = log(1 + x * y) is NOT a positive definite kernel !
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel log_k...
- Testing the kernel exp_k K(x, y) = exp(-(x-y)^2) :
It has domain = REAL.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel exp_k...
- Testing the kernel cos_plus_k K(x, y) = cos(x + y) :
It has domain = REAL.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 46.60%.
===> I guess this kernel cos_plus_k K(x, y) = cos(x + y) is NOT a positive definite kernel !
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel cos_plus_k...
- Testing the kernel cos_minus_k K(x, y) = cos(x - y) :
It has domain = REAL.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel cos_minus_k...
- Testing the kernel min_k K(x, y) = min(x, y) :
It has domain = REALPLUS.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel min_k...
- Testing the kernel max_k K(x, y) = max(x, y) :
It has domain = REALPLUS.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 78.60%.
===> I guess this kernel max_k K(x, y) = max(x, y) is NOT a positive definite kernel !
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel max_k...
- Testing the kernel min_by_max_k K(x, y) = min(x, y) / max(x, y) :
It has domain = REALPLUS.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel min_by_max_k...
- Testing the kernel gcd_k K(x, y) = gcd(x, y) :
It has domain = INTEGER.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 100.00%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel gcd_k...
- Testing the kernel lcm_k K(x, y) = lcm(x, y) :
It has domain = INTEGER.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 58.80%.
===> I guess this kernel lcm_k K(x, y) = lcm(x, y) is NOT a positive definite kernel !
- Tested for definiteness : success rate 99.95%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel lcm_k...
- Testing the kernel gcd_by_lcm_k K(x, y) = gcd(x, y) / lcm(x, y) :
It has domain = INTEGER.
Results of the N = 1000 tests:
- Tested for positiveness : success rate 100.00%.
- Tested for definiteness : success rate 99.95%.
- Tested for symmetry : success rate 100.00%.
Finished for the kernel gcd_by_lcm_k...
Done testing some kernels...
- Apparently, the kernel log_k K(x, y) = log(1 + x * y) is NOT a positive definite kernel : because positivenesswas only observed with probability 96.50%.
- Apparently, the kernel cos_plus_k K(x, y) = cos(x + y) is NOT a positive definite kernel : because positivenesswas only observed with probability 46.60%.
- Apparently, the kernel max_k K(x, y) = max(x, y) is NOT a positive definite kernel : because positivenesswas only observed with probability 78.60%.
- Apparently, the kernel lcm_k K(x, y) = lcm(x, y) is NOT a positive definite kernel : because positivenesswas only observed with probability 58.80%.
- All the other kernel appeared to be positive definite. Now prove it.
Finished for this script 'test_kernel.py'.
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# -*- coding: utf-8; mode: python -*-
"""
A Python 2 script to test kernels, for the Kernel Methods homework.
Source : http://lear.inrialpes.fr/people/mairal/teaching/2015-2016/MVA/
PDF: http://lear.inrialpes.fr/people/mairal/teaching/2015-2016/MVA/fichiers/homework_mva_2016.pdf
- *Date:* Saturday 06 February 2016.
- *Author:* Lilian Besson, for the MVA Master, (C) 2015-16.
- *Licence:* MIT Licence (http://lbesson.mit-license.org).
"""
from __future__ import print_function, division # Python 2 compatibility if needed
import numpy as np
try:
from sympy import lcm, gcd
except:
# if True:
print("Error: sympy is not available, switching back to fractions implementation of lcm/gcd")
from fractions import gcd
def lcm(x, y):
""" lcm(x, y) : lowest common multiple, = x * y / gcd(x, y). """
if gcd(x, y) == 0:
return 0
else:
return (x * y) / gcd(x, y)
#: Number of random tests to do
N = 1000
#: Size of the sum_ij
T = 5
T = 50
#: Lower bound for random number in IR
LOWER_BOUND = -300.0
#: Upper bound for random number in IR
UPPER_BOUND = 300.0
#: Upper bound for random number in IN
UPPER_BOUND_INT = 1000
#: Small upper bound for random number in IN
SMALL_UPPER_BOUND_INT = 30
#: Tolerance for real values tests to zero
REAL_TOLERANCE = 1e-12
#: Tolerance for probabilities check to 100%
PROBA_TOLERANCE = 0.01
#: Standard deviation of the weights
WEIGHT_STD = 100
#: Distribution for weights
# WEIGHT_DISTRIBUTION = "NORMAL"
# WEIGHT_DISTRIBUTION = "UNIFORM_0_1" # Useless!
WEIGHT_DISTRIBUTION = "UNIFORM_-1_1"
#: Map a domain to a random generator on this domain
random_generators = {
"REAL":
lambda: np.random.uniform(LOWER_BOUND, UPPER_BOUND),
"REALPLUS":
lambda: np.random.uniform(0.0, UPPER_BOUND),
"INTEGER":
lambda: np.random.randint(0, UPPER_BOUND_INT),
"SMALLINT":
lambda: np.random.randint(0, SMALL_UPPER_BOUND_INT),
"MONEONE":
lambda: np.random.uniform(-1.0, 1.0)
}
def test_kernel(K, N=N, T=T, verb=False):
"""
Test if a function is a kernel on lots of random values (500 by default):
- test if K(x, y) = K(y, x) (symmetry),
- test if K(x, x) = 0 ==> x = 0 (definiteness),
- test if sum_ij a_i a_j K(x_i, x_j) >= 0 for 'all' t, and a_i in IR, x_i in DOMAIN (positiveness) (for all i,j in range(t), and t drawn in range(T) randomly).
Return numerical values, average of success rate on N = 500 tests.
"""
# Get the domain of K and an appropriate random generator.
domain = K.domain
newvalue = random_generators[domain]
# Random generator for weights
def newweights(t):
if WEIGHT_DISTRIBUTION == "UNIFORM_0_1":
return np.random.rand(t) * WEIGHT_STD
if WEIGHT_DISTRIBUTION == "UNIFORM_-1_1":
return (2*np.random.rand(t) - 1) * (WEIGHT_STD / 2)
if WEIGHT_DISTRIBUTION == "NORMAL":
return WEIGHT_STD * np.random.randn(t)
# Test for symmetry
prop_sym = 0
# Test for definiteness
prop_def = 0
# Test for positiveness
prop_pos = 0
for i in range(N):
x = newvalue()
y = newvalue()
if verb:
print(" - Random values x = {}, y = {}.".format(x, y))
if K(x, y) == K(y, x):
prop_sym += 1
# Test : (A ==> B) <=> (B or not(A))
if (not (abs(K(x, x)) < REAL_TOLERANCE)) or (abs(x) > REAL_TOLERANCE):
prop_def += 1
if (not (abs(K(y, y)) < REAL_TOLERANCE)) or (abs(y) > REAL_TOLERANCE):
prop_def += 1
# Generate lots of samples for checking positiveness
t = np.random.randint(1, T)
X = [newvalue() for i in range(t)]
a = newweights(t)
if verb:
print(" - Random values t = {}, a = {}, X = {}.".format(t, a, X))
if sum(a[i] * a[j] * K(X[i], X[j]) for i in range(t) for j in range(t)) >= 0:
prop_pos += 1
if verb:
print(" - After {} random tests, sym = {:>3}, def = {:>3}, pos = {:>3}.".format(N, prop_sym, prop_def, prop_pos))
# Return the results
return {
"symmetry": (prop_sym / N),
"definiteness": (prop_def / (2*N)),
"positiveness": (prop_pos / N),
}
# %% Tweak to define a new kernel
#: List of all kernels to test, dynamically created by @kernel decorator
all_kernels = []
def kernel(domain="REAL"):
"""
Decorator for a new kernel (just an elegant way to add it to all_kernels).
The parameter 'domain' can be:
- "REAL" for (-oo,+oo) = IR real number,
- "REALPLUS" for [0,+oo) = IR+ positive real number,
- "INTEGER" for [|0,+oo) = IN positive integer,
- "SMALLINT" for [|0,+oo) = IN (small) positive integer,
- "MONEONE" for (-1,+1) = UR the unit open disk of real values.
"""
# global all_kernels
def kernel_decorator(func):
"""
Decorator for a new kernel, with a certain domain.
"""
global all_kernels
# Add a new kernel !
all_kernels += [func]
func.domain = domain
return func
return kernel_decorator
# %% List of Kernels
# Classical example of kernel: linear kernel!
@kernel("REAL")
def abs_k(x, y):
""" K(x, y) = < x, y > = x * y """
return x * y
@kernel("MONEONE")
def one_by_k(x, y):
""" K(x, y) = 1 / (1 - x * y) """
return 1 / (1 - x * y)
# Here we could have an overflow error, so using small integers
# From the slides
@kernel("SMALLINT")
def power_two_dot_k(x, y):
""" K(x, y) = 2 ** (x * y) """
return 2 ** (x * y)
# Here we could have an overflow error, so using small integers
@kernel("SMALLINT")
def power_two_sum_k(x, y):
""" K(x, y) = 2 ** (x + y) """
return 2 ** (x + y)
@kernel("REALPLUS")
def log_k(x, y):
""" K(x, y) = log(1 + x * y) """
return np.log(1 + x * y)
@kernel("REAL")
def exp_k(x, y):
""" K(x, y) = exp(-(x-y)^2) """
return np.exp(-(x-y)**2)
@kernel("REAL")
def cos_plus_k(x, y):
""" K(x, y) = cos(x + y) """
return np.cos(x + y)
@kernel("REAL")
def cos_minus_k(x, y):
""" K(x, y) = cos(x - y) """
return np.cos(x - y)
@kernel("REALPLUS")
def min_k(x, y):
""" K(x, y) = min(x, y) """
return min(x, y)
@kernel("REALPLUS")
def max_k(x, y):
""" K(x, y) = max(x, y) """
return max(x, y)
@kernel("REALPLUS")
def min_by_max_k(x, y):
""" K(x, y) = min(x, y) / max(x, y) """
if abs(max(x, y)) < REAL_TOLERANCE:
return 0.0
else:
return min(x, y) / max(x, y)
@kernel("INTEGER")
def gcd_k(x, y):
""" K(x, y) = gcd(x, y) """
return gcd(x, y)
@kernel("INTEGER")
def lcm_k(x, y):
""" K(x, y) = lcm(x, y) """
return lcm(x, y)
@kernel("INTEGER")
def gcd_by_lcm_k(x, y):
""" K(x, y) = gcd(x, y) / lcm(x, y) """
if abs(lcm(x, y)) < REAL_TOLERANCE:
return 0.0
else:
return gcd(x, y) / lcm(x, y)
# %% Testing all the kernels
def main(kernels_to_test=all_kernels, T=T, verb=False):
"""
Test a lot of different kernels.
"""
nb = len(all_kernels)
print("\nStarting to test {} kernels...".format(nb))
# Store results
results = [1] * nb
not_kernels = []
# Start
for i, K in enumerate(all_kernels):
print("\n- Testing the kernel {} {}:".format(K.__name__, K.__doc__))
print(" It has domain = {}.".format(K.domain))
result = test_kernel(K, N=N, T=T, verb=verb)
print(" Results of the N = {} tests:".format(N))
# bad_test = None
# min_rate = None
for test, rate in result.items():
print(" - Tested for {:^12} : success rate {:.2%}.".format(test, rate))
if rate < 1.0 - PROBA_TOLERANCE:
print(" ===> I guess this kernel {} {} is NOT a positive definite kernel !".format(K.__name__, K.__doc__))
not_kernels.append({'K': K, 'test': test, 'rate': rate})
# bad_test, min_rate = test, rate
print(" Finished for the kernel {}...".format(K.__name__))
results[i] = result
print("\n\nDone testing some kernels...")
return results, not_kernels
if __name__ == '__main__':
print("\nScript to test if functions are kernels, by Lilian Besson (C) 2016")
print("MIT Licensed (http://lbesson.mit-license.org)")
print("\n\nUsing N = {}, T = {}.".format(N, T))
print("For test domains:")
print(" - IR ~= [{}, {}],".format(LOWER_BOUND, UPPER_BOUND))
print(" - IR+ ~= [ 0, {}],".format(UPPER_BOUND))
print(" - IN ~= [ 0, {}],".format(UPPER_BOUND_INT))
print(" - IN ~= [ 0, {}] (small to reduce overflow),".format(SMALL_UPPER_BOUND_INT))
print("Tolerance for small real values = {}.".format(REAL_TOLERANCE))
print("Tolerance for success rates = {}.".format(PROBA_TOLERANCE))
if WEIGHT_DISTRIBUTION == "UNIFORM_0_1":
print("And sampling the a_i from uniform in [0, {}].".format(WEIGHT_STD))
if WEIGHT_DISTRIBUTION == "UNIFORM_-1_1":
print("And sampling the a_i from uniform in [-{}, {}].".format(WEIGHT_STD/2, WEIGHT_STD/2))
if WEIGHT_DISTRIBUTION == "NORMAL":
print("And sampling the a_i from a Gaussian of mean 0 and std {}.".format(WEIGHT_STD))
# all_kernels = [absk]
# results, not_kernels = main(kernels_to_test=all_kernels, verb=True)
results, not_kernels = main()
for item in not_kernels:
K = item['K']
test = item['test']
rate = item['rate']
print("\n- Apparently, the kernel {} {} is NOT a positive definite kernel : because {} was only observed with probability {:.2%}.".format(K.__name__, K.__doc__, test, rate))
print("\n- All the other kernel appeared to be positive definite. Now prove it.")
print("\nFinished for this script 'test_kernel.py'.")
# End of test_kernel.py
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