|
| 1 | +""" |
| 2 | +t-distributed stochastic neighbor embedding (t-SNE) |
| 3 | +
|
| 4 | +For more details, see: |
| 5 | +https://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embedding |
| 6 | +""" |
| 7 | + |
| 8 | +import doctest |
| 9 | + |
| 10 | +import numpy as np |
| 11 | +from numpy import ndarray |
| 12 | +from sklearn.datasets import load_iris |
| 13 | + |
| 14 | + |
| 15 | +def collect_dataset() -> tuple[ndarray, ndarray]: |
| 16 | + """ |
| 17 | + Load the Iris dataset and return features and labels. |
| 18 | +
|
| 19 | + Returns: |
| 20 | + tuple[ndarray, ndarray]: Feature matrix and target labels. |
| 21 | +
|
| 22 | + >>> features, targets = collect_dataset() |
| 23 | + >>> features.shape |
| 24 | + (150, 4) |
| 25 | + >>> targets.shape |
| 26 | + (150,) |
| 27 | + """ |
| 28 | + iris_dataset = load_iris() |
| 29 | + return np.array(iris_dataset.data), np.array(iris_dataset.target) |
| 30 | + |
| 31 | + |
| 32 | +def compute_pairwise_affinities(data_matrix: ndarray, sigma: float = 1.0) -> ndarray: |
| 33 | + """ |
| 34 | + Compute high-dimensional affinities (P matrix) using a Gaussian kernel. |
| 35 | +
|
| 36 | + Args: |
| 37 | + data_matrix: Input data of shape (n_samples, n_features). |
| 38 | + sigma: Gaussian kernel bandwidth. |
| 39 | +
|
| 40 | + Returns: |
| 41 | + ndarray: Symmetrized probability matrix. |
| 42 | +
|
| 43 | + >>> x = np.array([[0.0, 0.0], [1.0, 0.0]]) |
| 44 | + >>> probabilities = compute_pairwise_affinities(x) |
| 45 | + >>> float(round(probabilities[0, 1], 3)) |
| 46 | + 0.25 |
| 47 | + """ |
| 48 | + n_samples = data_matrix.shape[0] |
| 49 | + squared_sum = np.sum(np.square(data_matrix), axis=1) |
| 50 | + squared_distance = np.add( |
| 51 | + np.add(-2 * np.dot(data_matrix, data_matrix.T), squared_sum).T, squared_sum |
| 52 | + ) |
| 53 | + |
| 54 | + affinity_matrix = np.exp(-squared_distance / (2 * sigma**2)) |
| 55 | + np.fill_diagonal(affinity_matrix, 0) |
| 56 | + |
| 57 | + affinity_matrix /= np.sum(affinity_matrix) |
| 58 | + return (affinity_matrix + affinity_matrix.T) / (2 * n_samples) |
| 59 | + |
| 60 | + |
| 61 | +def compute_low_dim_affinities(embedding_matrix: ndarray) -> tuple[ndarray, ndarray]: |
| 62 | + """ |
| 63 | + Compute low-dimensional affinities (Q matrix) using a Student-t distribution. |
| 64 | +
|
| 65 | + Args: |
| 66 | + embedding_matrix: Low-dimensional embedding of shape (n_samples, n_components). |
| 67 | +
|
| 68 | + Returns: |
| 69 | + tuple[ndarray, ndarray]: (Q probability matrix, numerator matrix). |
| 70 | +
|
| 71 | + >>> y = np.array([[0.0, 0.0], [1.0, 0.0]]) |
| 72 | + >>> q_matrix, numerators = compute_low_dim_affinities(y) |
| 73 | + >>> q_matrix.shape |
| 74 | + (2, 2) |
| 75 | + """ |
| 76 | + squared_sum = np.sum(np.square(embedding_matrix), axis=1) |
| 77 | + numerator_matrix = 1 / ( |
| 78 | + 1 |
| 79 | + + np.add( |
| 80 | + np.add(-2 * np.dot(embedding_matrix, embedding_matrix.T), squared_sum).T, |
| 81 | + squared_sum, |
| 82 | + ) |
| 83 | + ) |
| 84 | + np.fill_diagonal(numerator_matrix, 0) |
| 85 | + |
| 86 | + q_matrix = numerator_matrix / np.sum(numerator_matrix) |
| 87 | + return q_matrix, numerator_matrix |
| 88 | + |
| 89 | + |
| 90 | +def apply_tsne( |
| 91 | + data_matrix: ndarray, |
| 92 | + n_components: int = 2, |
| 93 | + learning_rate: float = 200.0, |
| 94 | + n_iter: int = 500, |
| 95 | +) -> ndarray: |
| 96 | + """ |
| 97 | + Apply t-SNE for dimensionality reduction. |
| 98 | +
|
| 99 | + Args: |
| 100 | + data_matrix: Original dataset (features). |
| 101 | + n_components: Target dimension (2D or 3D). |
| 102 | + learning_rate: Step size for gradient descent. |
| 103 | + n_iter: Number of iterations. |
| 104 | +
|
| 105 | + Returns: |
| 106 | + ndarray: Low-dimensional embedding of the data. |
| 107 | +
|
| 108 | + >>> features, _ = collect_dataset() |
| 109 | + >>> embedding = apply_tsne(features, n_components=2, n_iter=50) |
| 110 | + >>> embedding.shape |
| 111 | + (150, 2) |
| 112 | + """ |
| 113 | + if n_components < 1 or n_iter < 1: |
| 114 | + raise ValueError("n_components and n_iter must be >= 1") |
| 115 | + |
| 116 | + n_samples = data_matrix.shape[0] |
| 117 | + rng = np.random.default_rng() |
| 118 | + embedding = rng.standard_normal((n_samples, n_components)) * 1e-4 |
| 119 | + |
| 120 | + high_dim_affinities = compute_pairwise_affinities(data_matrix) |
| 121 | + high_dim_affinities = np.maximum(high_dim_affinities, 1e-12) |
| 122 | + |
| 123 | + embedding_increment = np.zeros_like(embedding) |
| 124 | + momentum = 0.5 |
| 125 | + |
| 126 | + for iteration in range(n_iter): |
| 127 | + low_dim_affinities, numerator_matrix = compute_low_dim_affinities(embedding) |
| 128 | + low_dim_affinities = np.maximum(low_dim_affinities, 1e-12) |
| 129 | + |
| 130 | + affinity_diff = high_dim_affinities - low_dim_affinities |
| 131 | + |
| 132 | + gradient = 4 * ( |
| 133 | + np.dot((affinity_diff * numerator_matrix), embedding) |
| 134 | + - np.multiply( |
| 135 | + np.sum(affinity_diff * numerator_matrix, axis=1)[:, np.newaxis], |
| 136 | + embedding, |
| 137 | + ) |
| 138 | + ) |
| 139 | + |
| 140 | + embedding_increment = momentum * embedding_increment - learning_rate * gradient |
| 141 | + embedding += embedding_increment |
| 142 | + |
| 143 | + if iteration == int(n_iter / 4): |
| 144 | + momentum = 0.8 |
| 145 | + |
| 146 | + return embedding |
| 147 | + |
| 148 | + |
| 149 | +def main() -> None: |
| 150 | + """ |
| 151 | + Run t-SNE on the Iris dataset and display the first 5 embeddings. |
| 152 | +
|
| 153 | + >>> main() # doctest: +ELLIPSIS |
| 154 | + t-SNE embedding (first 5 points): |
| 155 | + [[... |
| 156 | + """ |
| 157 | + features, _labels = collect_dataset() |
| 158 | + embedding = apply_tsne(features, n_components=2, n_iter=300) |
| 159 | + |
| 160 | + if not isinstance(embedding, np.ndarray): |
| 161 | + raise TypeError("t-SNE embedding must be an ndarray") |
| 162 | + |
| 163 | + print("t-SNE embedding (first 5 points):") |
| 164 | + print(embedding[:5]) |
| 165 | + |
| 166 | + # Optional visualization (Ruff/mypy compliant) |
| 167 | + |
| 168 | + # import matplotlib.pyplot as plt |
| 169 | + # plt.scatter(embedding[:, 0], embedding[:, 1], c=labels, cmap="viridis") |
| 170 | + # plt.title("t-SNE Visualization of the Iris Dataset") |
| 171 | + # plt.xlabel("Dimension 1") |
| 172 | + # plt.ylabel("Dimension 2") |
| 173 | + # plt.show() |
| 174 | + |
| 175 | + |
| 176 | +if __name__ == "__main__": |
| 177 | + doctest.testmod() |
| 178 | + main() |
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