File size: 19,715 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 |
"""
Neighborhood Component Analysis
"""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import sys
import time
from numbers import Integral, Real
from warnings import warn
import numpy as np
from scipy.optimize import minimize
from ..base import (
BaseEstimator,
ClassNamePrefixFeaturesOutMixin,
TransformerMixin,
_fit_context,
)
from ..decomposition import PCA
from ..exceptions import ConvergenceWarning
from ..metrics import pairwise_distances
from ..preprocessing import LabelEncoder
from ..utils._param_validation import Interval, StrOptions
from ..utils.extmath import softmax
from ..utils.multiclass import check_classification_targets
from ..utils.random import check_random_state
from ..utils.validation import check_array, check_is_fitted, validate_data
class NeighborhoodComponentsAnalysis(
ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator
):
"""Neighborhood Components Analysis.
Neighborhood Component Analysis (NCA) is a machine learning algorithm for
metric learning. It learns a linear transformation in a supervised fashion
to improve the classification accuracy of a stochastic nearest neighbors
rule in the transformed space.
Read more in the :ref:`User Guide <nca>`.
Parameters
----------
n_components : int, default=None
Preferred dimensionality of the projected space.
If None it will be set to `n_features`.
init : {'auto', 'pca', 'lda', 'identity', 'random'} or ndarray of shape \
(n_features_a, n_features_b), default='auto'
Initialization of the linear transformation. Possible options are
`'auto'`, `'pca'`, `'lda'`, `'identity'`, `'random'`, and a numpy
array of shape `(n_features_a, n_features_b)`.
- `'auto'`
Depending on `n_components`, the most reasonable initialization
is chosen. If `n_components <= min(n_features, n_classes - 1)`
we use `'lda'`, as it uses labels information. If not, but
`n_components < min(n_features, n_samples)`, we use `'pca'`, as
it projects data in meaningful directions (those of higher
variance). Otherwise, we just use `'identity'`.
- `'pca'`
`n_components` principal components of the inputs passed
to :meth:`fit` will be used to initialize the transformation.
(See :class:`~sklearn.decomposition.PCA`)
- `'lda'`
`min(n_components, n_classes)` most discriminative
components of the inputs passed to :meth:`fit` will be used to
initialize the transformation. (If `n_components > n_classes`,
the rest of the components will be zero.) (See
:class:`~sklearn.discriminant_analysis.LinearDiscriminantAnalysis`)
- `'identity'`
If `n_components` is strictly smaller than the
dimensionality of the inputs passed to :meth:`fit`, the identity
matrix will be truncated to the first `n_components` rows.
- `'random'`
The initial transformation will be a random array of shape
`(n_components, n_features)`. Each value is sampled from the
standard normal distribution.
- numpy array
`n_features_b` must match the dimensionality of the inputs passed
to :meth:`fit` and n_features_a must be less than or equal to that.
If `n_components` is not `None`, `n_features_a` must match it.
warm_start : bool, default=False
If `True` and :meth:`fit` has been called before, the solution of the
previous call to :meth:`fit` is used as the initial linear
transformation (`n_components` and `init` will be ignored).
max_iter : int, default=50
Maximum number of iterations in the optimization.
tol : float, default=1e-5
Convergence tolerance for the optimization.
callback : callable, default=None
If not `None`, this function is called after every iteration of the
optimizer, taking as arguments the current solution (flattened
transformation matrix) and the number of iterations. This might be
useful in case one wants to examine or store the transformation
found after each iteration.
verbose : int, default=0
If 0, no progress messages will be printed.
If 1, progress messages will be printed to stdout.
If > 1, progress messages will be printed and the `disp`
parameter of :func:`scipy.optimize.minimize` will be set to
`verbose - 2`.
random_state : int or numpy.RandomState, default=None
A pseudo random number generator object or a seed for it if int. If
`init='random'`, `random_state` is used to initialize the random
transformation. If `init='pca'`, `random_state` is passed as an
argument to PCA when initializing the transformation. Pass an int
for reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
Attributes
----------
components_ : ndarray of shape (n_components, n_features)
The linear transformation learned during fitting.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
n_iter_ : int
Counts the number of iterations performed by the optimizer.
random_state_ : numpy.RandomState
Pseudo random number generator object used during initialization.
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
See Also
--------
sklearn.discriminant_analysis.LinearDiscriminantAnalysis : Linear
Discriminant Analysis.
sklearn.decomposition.PCA : Principal component analysis (PCA).
References
----------
.. [1] J. Goldberger, G. Hinton, S. Roweis, R. Salakhutdinov.
"Neighbourhood Components Analysis". Advances in Neural Information
Processing Systems. 17, 513-520, 2005.
http://www.cs.nyu.edu/~roweis/papers/ncanips.pdf
.. [2] Wikipedia entry on Neighborhood Components Analysis
https://en.wikipedia.org/wiki/Neighbourhood_components_analysis
Examples
--------
>>> from sklearn.neighbors import NeighborhoodComponentsAnalysis
>>> from sklearn.neighbors import KNeighborsClassifier
>>> from sklearn.datasets import load_iris
>>> from sklearn.model_selection import train_test_split
>>> X, y = load_iris(return_X_y=True)
>>> X_train, X_test, y_train, y_test = train_test_split(X, y,
... stratify=y, test_size=0.7, random_state=42)
>>> nca = NeighborhoodComponentsAnalysis(random_state=42)
>>> nca.fit(X_train, y_train)
NeighborhoodComponentsAnalysis(...)
>>> knn = KNeighborsClassifier(n_neighbors=3)
>>> knn.fit(X_train, y_train)
KNeighborsClassifier(...)
>>> print(knn.score(X_test, y_test))
0.933333...
>>> knn.fit(nca.transform(X_train), y_train)
KNeighborsClassifier(...)
>>> print(knn.score(nca.transform(X_test), y_test))
0.961904...
"""
_parameter_constraints: dict = {
"n_components": [
Interval(Integral, 1, None, closed="left"),
None,
],
"init": [
StrOptions({"auto", "pca", "lda", "identity", "random"}),
np.ndarray,
],
"warm_start": ["boolean"],
"max_iter": [Interval(Integral, 1, None, closed="left")],
"tol": [Interval(Real, 0, None, closed="left")],
"callback": [callable, None],
"verbose": ["verbose"],
"random_state": ["random_state"],
}
def __init__(
self,
n_components=None,
*,
init="auto",
warm_start=False,
max_iter=50,
tol=1e-5,
callback=None,
verbose=0,
random_state=None,
):
self.n_components = n_components
self.init = init
self.warm_start = warm_start
self.max_iter = max_iter
self.tol = tol
self.callback = callback
self.verbose = verbose
self.random_state = random_state
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y):
"""Fit the model according to the given training data.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The training samples.
y : array-like of shape (n_samples,)
The corresponding training labels.
Returns
-------
self : object
Fitted estimator.
"""
# Validate the inputs X and y, and converts y to numerical classes.
X, y = validate_data(self, X, y, ensure_min_samples=2)
check_classification_targets(y)
y = LabelEncoder().fit_transform(y)
# Check the preferred dimensionality of the projected space
if self.n_components is not None and self.n_components > X.shape[1]:
raise ValueError(
"The preferred dimensionality of the "
f"projected space `n_components` ({self.n_components}) cannot "
"be greater than the given data "
f"dimensionality ({X.shape[1]})!"
)
# If warm_start is enabled, check that the inputs are consistent
if (
self.warm_start
and hasattr(self, "components_")
and self.components_.shape[1] != X.shape[1]
):
raise ValueError(
f"The new inputs dimensionality ({X.shape[1]}) does not "
"match the input dimensionality of the "
f"previously learned transformation ({self.components_.shape[1]})."
)
# Check how the linear transformation should be initialized
init = self.init
if isinstance(init, np.ndarray):
init = check_array(init)
# Assert that init.shape[1] = X.shape[1]
if init.shape[1] != X.shape[1]:
raise ValueError(
f"The input dimensionality ({init.shape[1]}) of the given "
"linear transformation `init` must match the "
f"dimensionality of the given inputs `X` ({X.shape[1]})."
)
# Assert that init.shape[0] <= init.shape[1]
if init.shape[0] > init.shape[1]:
raise ValueError(
f"The output dimensionality ({init.shape[0]}) of the given "
"linear transformation `init` cannot be "
f"greater than its input dimensionality ({init.shape[1]})."
)
# Assert that self.n_components = init.shape[0]
if self.n_components is not None and self.n_components != init.shape[0]:
raise ValueError(
"The preferred dimensionality of the "
f"projected space `n_components` ({self.n_components}) does"
" not match the output dimensionality of "
"the given linear transformation "
f"`init` ({init.shape[0]})!"
)
# Initialize the random generator
self.random_state_ = check_random_state(self.random_state)
# Measure the total training time
t_train = time.time()
# Compute a mask that stays fixed during optimization:
same_class_mask = y[:, np.newaxis] == y[np.newaxis, :]
# (n_samples, n_samples)
# Initialize the transformation
transformation = np.ravel(self._initialize(X, y, init))
# Create a dictionary of parameters to be passed to the optimizer
disp = self.verbose - 2 if self.verbose > 1 else -1
optimizer_params = {
"method": "L-BFGS-B",
"fun": self._loss_grad_lbfgs,
"args": (X, same_class_mask, -1.0),
"jac": True,
"x0": transformation,
"tol": self.tol,
"options": dict(maxiter=self.max_iter, disp=disp),
"callback": self._callback,
}
# Call the optimizer
self.n_iter_ = 0
opt_result = minimize(**optimizer_params)
# Reshape the solution found by the optimizer
self.components_ = opt_result.x.reshape(-1, X.shape[1])
# Stop timer
t_train = time.time() - t_train
if self.verbose:
cls_name = self.__class__.__name__
# Warn the user if the algorithm did not converge
if not opt_result.success:
warn(
"[{}] NCA did not converge: {}".format(
cls_name, opt_result.message
),
ConvergenceWarning,
)
print("[{}] Training took {:8.2f}s.".format(cls_name, t_train))
return self
def transform(self, X):
"""Apply the learned transformation to the given data.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Data samples.
Returns
-------
X_embedded: ndarray of shape (n_samples, n_components)
The data samples transformed.
Raises
------
NotFittedError
If :meth:`fit` has not been called before.
"""
check_is_fitted(self)
X = validate_data(self, X, reset=False)
return np.dot(X, self.components_.T)
def _initialize(self, X, y, init):
"""Initialize the transformation.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The training samples.
y : array-like of shape (n_samples,)
The training labels.
init : str or ndarray of shape (n_features_a, n_features_b)
The validated initialization of the linear transformation.
Returns
-------
transformation : ndarray of shape (n_components, n_features)
The initialized linear transformation.
"""
transformation = init
if self.warm_start and hasattr(self, "components_"):
transformation = self.components_
elif isinstance(init, np.ndarray):
pass
else:
n_samples, n_features = X.shape
n_components = self.n_components or n_features
if init == "auto":
n_classes = len(np.unique(y))
if n_components <= min(n_features, n_classes - 1):
init = "lda"
elif n_components < min(n_features, n_samples):
init = "pca"
else:
init = "identity"
if init == "identity":
transformation = np.eye(n_components, X.shape[1])
elif init == "random":
transformation = self.random_state_.standard_normal(
size=(n_components, X.shape[1])
)
elif init in {"pca", "lda"}:
init_time = time.time()
if init == "pca":
pca = PCA(
n_components=n_components, random_state=self.random_state_
)
if self.verbose:
print("Finding principal components... ", end="")
sys.stdout.flush()
pca.fit(X)
transformation = pca.components_
elif init == "lda":
from ..discriminant_analysis import LinearDiscriminantAnalysis
lda = LinearDiscriminantAnalysis(n_components=n_components)
if self.verbose:
print("Finding most discriminative components... ", end="")
sys.stdout.flush()
lda.fit(X, y)
transformation = lda.scalings_.T[:n_components]
if self.verbose:
print("done in {:5.2f}s".format(time.time() - init_time))
return transformation
def _callback(self, transformation):
"""Called after each iteration of the optimizer.
Parameters
----------
transformation : ndarray of shape (n_components * n_features,)
The solution computed by the optimizer in this iteration.
"""
if self.callback is not None:
self.callback(transformation, self.n_iter_)
self.n_iter_ += 1
def _loss_grad_lbfgs(self, transformation, X, same_class_mask, sign=1.0):
"""Compute the loss and the loss gradient w.r.t. `transformation`.
Parameters
----------
transformation : ndarray of shape (n_components * n_features,)
The raveled linear transformation on which to compute loss and
evaluate gradient.
X : ndarray of shape (n_samples, n_features)
The training samples.
same_class_mask : ndarray of shape (n_samples, n_samples)
A mask where `mask[i, j] == 1` if `X[i]` and `X[j]` belong
to the same class, and `0` otherwise.
Returns
-------
loss : float
The loss computed for the given transformation.
gradient : ndarray of shape (n_components * n_features,)
The new (flattened) gradient of the loss.
"""
if self.n_iter_ == 0:
self.n_iter_ += 1
if self.verbose:
header_fields = ["Iteration", "Objective Value", "Time(s)"]
header_fmt = "{:>10} {:>20} {:>10}"
header = header_fmt.format(*header_fields)
cls_name = self.__class__.__name__
print("[{}]".format(cls_name))
print(
"[{}] {}\n[{}] {}".format(
cls_name, header, cls_name, "-" * len(header)
)
)
t_funcall = time.time()
transformation = transformation.reshape(-1, X.shape[1])
X_embedded = np.dot(X, transformation.T) # (n_samples, n_components)
# Compute softmax distances
p_ij = pairwise_distances(X_embedded, squared=True)
np.fill_diagonal(p_ij, np.inf)
p_ij = softmax(-p_ij) # (n_samples, n_samples)
# Compute loss
masked_p_ij = p_ij * same_class_mask
p = np.sum(masked_p_ij, axis=1, keepdims=True) # (n_samples, 1)
loss = np.sum(p)
# Compute gradient of loss w.r.t. `transform`
weighted_p_ij = masked_p_ij - p_ij * p
weighted_p_ij_sym = weighted_p_ij + weighted_p_ij.T
np.fill_diagonal(weighted_p_ij_sym, -weighted_p_ij.sum(axis=0))
gradient = 2 * X_embedded.T.dot(weighted_p_ij_sym).dot(X)
# time complexity of the gradient: O(n_components x n_samples x (
# n_samples + n_features))
if self.verbose:
t_funcall = time.time() - t_funcall
values_fmt = "[{}] {:>10} {:>20.6e} {:>10.2f}"
print(
values_fmt.format(
self.__class__.__name__, self.n_iter_, loss, t_funcall
)
)
sys.stdout.flush()
return sign * loss, sign * gradient.ravel()
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.target_tags.required = True
return tags
@property
def _n_features_out(self):
"""Number of transformed output features."""
return self.components_.shape[0]
|