Utils & Metrics¶
_clip¶
- pyldl.algorithms.utils._clip(func)¶
_reduction¶
- pyldl.algorithms.utils._reduction(func)¶
binaryzation¶
- pyldl.algorithms.utils.binaryzation(y: ndarray, method='threshold', param: any | None = None) ndarray¶
Transform label distribution matrix to logical label matrix.
- Parameters:
y (np.ndarray) – Label distribution matrix (shape: \([n,\, l]\)).
method ({'threshold', 'topk'}, optional) –
Type of binaryzation method, defaults to ‘threshold’. The options are ‘threshold’ and ‘topk’, which can refer to:
[BIN-KWT+24]Zhiqiang Kou, Jing Wang, Jiawei Tang, Yuheng Jia, Boyu Shi, and Xin Geng. Exploiting multi-label correlation in label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 4326–4334. 2024. URL: https://doi.org/10.24963/ijcai.2024/478.
param (any, optional) – Parameter of binaryzation method, defaults to None. If None, the default value is .5 for ‘threshold’ and \(\lfloor l / 2 \rfloor\) for ‘topk’.
- Returns:
Logical label matrix (shape: \([n,\, l]\)).
- Return type:
np.ndarray
kl_divergence¶
- pyldl.algorithms.utils.kl_divergence(*args, reduction=<function average>)¶
pairwise_euclidean¶
- pyldl.algorithms.utils.pairwise_euclidean(X: ndarray | Tensor, Y: ndarray | Tensor | None = None) ndarray | Tensor¶
Pairwise Euclidean distance.
- Parameters:
X (Union[np.ndarray, tf.Tensor]) – Matrix \(\boldsymbol{X}\) (shape: \([m_X,\, n_X]\)).
Y (Union[np.ndarray, tf.Tensor], optional) – Matrix \(\boldsymbol{Y}\) (shape: \([m_Y,\, n_Y]\)), if None, \(\boldsymbol{Y} = \boldsymbol{X}\), defaults to None.
- Returns:
Pairwise Euclidean distance (shape: \([m_X,\, m_Y]\)).
- Return type:
Union[np.ndarray, tf.Tensor]
proj¶
soft_thresholding¶
- pyldl.algorithms.utils.soft_thresholding(A: ndarray, tau: float) ndarray¶
Soft thresholding operation. It is defined as \(\text{soft}(\boldsymbol{A}, \tau) = \text{sgn}(\boldsymbol{A}) \odot \max\lbrace \lvert \boldsymbol{A} \rvert - \tau, 0 \rbrace\), where \(\odot\) denotes element-wise multiplication.
- Parameters:
A (np.ndarray) – Matrix \(\boldsymbol{A}\).
tau (float) – \(\tau\).
- Returns:
The result of soft thresholding operation.
- Return type:
np.ndarray
solvel21¶
- pyldl.algorithms.utils.solvel21(A: ndarray, tau: float) ndarray¶
This approach is proposed in paper [CY14].
The solution to the optimization problem \(\mathop{\arg\min}_{\boldsymbol{X}} \Vert \boldsymbol{X} - \boldsymbol{A} \Vert_\text{F}^2 + \tau \Vert \boldsymbol{X} \Vert_{2,\,1}\) is given by the following formula:
\[\begin{split}\vec{x}_{\bullet j}^{\ast} = \left\{ \begin{aligned} & \frac{\Vert \vec{a}_{\bullet j} \Vert - \tau}{\Vert \vec{a}_{\bullet j} \Vert} \vec{a}_{\bullet j}, & \tau \le \Vert \vec{a}_{\bullet j} \Vert \\ & 0, & \text{otherwise} \end{aligned} \right.\end{split}\]where \(\vec{x}_{\bullet j}\) is the \(j\)-th column of matrix \(\boldsymbol{X}\), and \(\vec{a}_{\bullet j}\) is the \(j\)-th column of matrix \(\boldsymbol{A}\).
- Parameters:
A (np.ndarray) – Matrix \(\boldsymbol{A}\).
tau (float) – \(\tau\).
- Returns:
The solution to the optimization problem.
- Return type:
np.ndarray
svt¶
- pyldl.algorithms.utils.svt(A: ndarray, tau: float) ndarray¶
Singular value thresholding (SVT) is proposed in paper [CCS10].
The solution to the optimization problem \(\mathop{\arg\min}_{\boldsymbol{X}} \Vert \boldsymbol{X} - \boldsymbol{A} \Vert_\text{F}^2 + \tau \Vert \boldsymbol{X} \Vert_{\ast}\) is given by \(\boldsymbol{U} \max \lbrace \boldsymbol{\Sigma} - \tau, 0 \rbrace \boldsymbol{V}^\text{T}\), where \(\boldsymbol{A} = \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^\text{T}\) is the singular value decomposition of matrix \(\boldsymbol{A}\).
- Parameters:
A (np.ndarray) – Matrix \(\boldsymbol{A}\).
tau (float) – \(\tau\).
- Returns:
The solution to the optimization problem.
- Return type:
np.ndarray
artificial¶
- pyldl.utils.artificial(X, a=1.0, b=0.5, c=0.2, d=1.0, w1=array([[4., 2., 1.]]), w2=array([[1., 2., 4.]]), w3=array([[1., 4., 2.]]), lambda1=0.01, lambda2=0.01)¶
load_dataset¶
- pyldl.utils.load_dataset(name, dir='dataset')¶
make_ldl¶
- pyldl.utils.make_ldl(n_samples=200, **kwargs)¶
plot_artificial¶
- pyldl.utils.plot_artificial(n_samples=50, model=None, file_name=None, **kwargs)¶
random_missing¶
- pyldl.utils.random_missing(y, missing_rate=0.9)¶
canberra¶
- pyldl.metrics.canberra(*args, reduction=<function average>)¶
chebyshev¶
- pyldl.metrics.chebyshev(*args, reduction=<function average>)¶
clark¶
- pyldl.metrics.clark(*args, reduction=<function average>)¶
cosine¶
- pyldl.metrics.cosine(*args, reduction=<function average>)¶
dpa¶
- pyldl.metrics.dpa(*args, reduction=<function average>)¶
error_probability¶
- pyldl.metrics.error_probability(*args, reduction=<function average>)¶
euclidean¶
- pyldl.metrics.euclidean(*args, reduction=<function average>)¶
fidelity¶
- pyldl.metrics.fidelity(*args, reduction=<function average>)¶
intersection¶
- pyldl.metrics.intersection(*args, reduction=<function average>)¶
kendall¶
- pyldl.metrics.kendall(*args, reduction=<function average>)¶
mean_absolute_error¶
- pyldl.metrics.mean_absolute_error(*args, reduction=<function average>)¶
score¶
- pyldl.metrics.score(y: ndarray, y_pred: ndarray, metrics: list[str] | None = None, return_dict: bool = False)¶
sorensen¶
- pyldl.metrics.sorensen(*args, reduction=<function average>)¶
spearman¶
- pyldl.metrics.spearman(*args, reduction=<function average>)¶
squared_chi2¶
- pyldl.metrics.squared_chi2(*args, reduction=<function average>)¶
zero_one_loss¶
- pyldl.metrics.zero_one_loss(*args, reduction=<function average>)¶
References¶
Laurent Condat. Fast projection onto the simplex and the l1 ball. Mathematical Programming, 158(1):575–585, 2016. URL: https://doi.org/10.1007/s10107-015-0946-6.
Jinhui Chen and Jian Yang. Robust subspace segmentation via low-rank representation. IEEE Transactions on Cybernetics, 44(8):1432–1445, 2014. URL: https://doi.org/10.1109/TCYB.2013.2286106.
Jian-Feng Cai, Emmanuel J Candès, and Zuowei Shen. A singular value thresholding algorithm for matrix completion. SIAM Journal on optimization, 20(4):1956–1982, 2010. URL: https://doi.org/10.1137/080738970.