Add t stochastic neighbour embedding using Iris dataset (#13476)

* Added t-SNE with Iris dataset example

* Added t-SNE with Iris dataset example

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* Update machine_learning/t_stochastic_neighbour_embedding.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* Update machine_learning/t_stochastic_neighbour_embedding.py

Co-authored-by: Christian Clauss <cclauss@me.com>

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Co-authored-by: Christian Clauss <cclauss@me.com>

machine_learning/t_stochastic_neighbour_embedding.py

@@ -0,0 +1,178 @@
+"""
+t-distributed stochastic neighbor embedding (t-SNE)
+
+For more details, see:
+https://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embedding
+"""
+
+import doctest
+
+import numpy as np
+from numpy import ndarray
+from sklearn.datasets import load_iris
+
+
+def collect_dataset() -> tuple[ndarray, ndarray]:
+    """
+    Load the Iris dataset and return features and labels.
+
+    Returns:
+        tuple[ndarray, ndarray]: Feature matrix and target labels.
+
+    >>> features, targets = collect_dataset()
+    >>> features.shape
+    (150, 4)
+    >>> targets.shape
+    (150,)
+    """
+    iris_dataset = load_iris()
+    return np.array(iris_dataset.data), np.array(iris_dataset.target)
+
+
+def compute_pairwise_affinities(data_matrix: ndarray, sigma: float = 1.0) -> ndarray:
+    """
+    Compute high-dimensional affinities (P matrix) using a Gaussian kernel.
+
+    Args:
+        data_matrix: Input data of shape (n_samples, n_features).
+        sigma: Gaussian kernel bandwidth.
+
+    Returns:
+        ndarray: Symmetrized probability matrix.
+
+    >>> x = np.array([[0.0, 0.0], [1.0, 0.0]])
+    >>> probabilities = compute_pairwise_affinities(x)
+    >>> float(round(probabilities[0, 1], 3))
+    0.25
+    """
+    n_samples = data_matrix.shape[0]
+    squared_sum = np.sum(np.square(data_matrix), axis=1)
+    squared_distance = np.add(
+        np.add(-2 * np.dot(data_matrix, data_matrix.T), squared_sum).T, squared_sum
+    )
+
+    affinity_matrix = np.exp(-squared_distance / (2 * sigma**2))
+    np.fill_diagonal(affinity_matrix, 0)
+
+    affinity_matrix /= np.sum(affinity_matrix)
+    return (affinity_matrix + affinity_matrix.T) / (2 * n_samples)
+
+
+def compute_low_dim_affinities(embedding_matrix: ndarray) -> tuple[ndarray, ndarray]:
+    """
+    Compute low-dimensional affinities (Q matrix) using a Student-t distribution.
+
+    Args:
+        embedding_matrix: Low-dimensional embedding of shape (n_samples, n_components).
+
+    Returns:
+        tuple[ndarray, ndarray]: (Q probability matrix, numerator matrix).
+
+    >>> y = np.array([[0.0, 0.0], [1.0, 0.0]])
+    >>> q_matrix, numerators = compute_low_dim_affinities(y)
+    >>> q_matrix.shape
+    (2, 2)
+    """
+    squared_sum = np.sum(np.square(embedding_matrix), axis=1)
+    numerator_matrix = 1 / (
+        1
+        + np.add(
+            np.add(-2 * np.dot(embedding_matrix, embedding_matrix.T), squared_sum).T,
+            squared_sum,
+        )
+    )
+    np.fill_diagonal(numerator_matrix, 0)
+
+    q_matrix = numerator_matrix / np.sum(numerator_matrix)
+    return q_matrix, numerator_matrix
+
+
+def apply_tsne(
+    data_matrix: ndarray,
+    n_components: int = 2,
+    learning_rate: float = 200.0,
+    n_iter: int = 500,
+) -> ndarray:
+    """
+    Apply t-SNE for dimensionality reduction.
+
+    Args:
+        data_matrix: Original dataset (features).
+        n_components: Target dimension (2D or 3D).
+        learning_rate: Step size for gradient descent.
+        n_iter: Number of iterations.
+
+    Returns:
+        ndarray: Low-dimensional embedding of the data.
+
+    >>> features, _ = collect_dataset()
+    >>> embedding = apply_tsne(features, n_components=2, n_iter=50)
+    >>> embedding.shape
+    (150, 2)
+    """
+    if n_components < 1 or n_iter < 1:
+        raise ValueError("n_components and n_iter must be >= 1")
+
+    n_samples = data_matrix.shape[0]
+    rng = np.random.default_rng()
+    embedding = rng.standard_normal((n_samples, n_components)) * 1e-4
+
+    high_dim_affinities = compute_pairwise_affinities(data_matrix)
+    high_dim_affinities = np.maximum(high_dim_affinities, 1e-12)
+
+    embedding_increment = np.zeros_like(embedding)
+    momentum = 0.5
+
+    for iteration in range(n_iter):
+        low_dim_affinities, numerator_matrix = compute_low_dim_affinities(embedding)
+        low_dim_affinities = np.maximum(low_dim_affinities, 1e-12)
+
+        affinity_diff = high_dim_affinities - low_dim_affinities
+
+        gradient = 4 * (
+            np.dot((affinity_diff * numerator_matrix), embedding)
+            - np.multiply(
+                np.sum(affinity_diff * numerator_matrix, axis=1)[:, np.newaxis],
+                embedding,
+            )
+        )
+
+        embedding_increment = momentum * embedding_increment - learning_rate * gradient
+        embedding += embedding_increment
+
+        if iteration == int(n_iter / 4):
+            momentum = 0.8
+
+    return embedding
+
+
+def main() -> None:
+    """
+    Run t-SNE on the Iris dataset and display the first 5 embeddings.
+
+    >>> main()  # doctest: +ELLIPSIS
+    t-SNE embedding (first 5 points):
+    [[...
+    """
+    features, _labels = collect_dataset()
+    embedding = apply_tsne(features, n_components=2, n_iter=300)
+
+    if not isinstance(embedding, np.ndarray):
+        raise TypeError("t-SNE embedding must be an ndarray")
+
+    print("t-SNE embedding (first 5 points):")
+    print(embedding[:5])
+
+    # Optional visualization (Ruff/mypy compliant)
+
+    # import matplotlib.pyplot as plt
+    # plt.scatter(embedding[:, 0], embedding[:, 1], c=labels, cmap="viridis")
+    # plt.title("t-SNE Visualization of the Iris Dataset")
+    # plt.xlabel("Dimension 1")
+    # plt.ylabel("Dimension 2")
+    # plt.show()
+
+
+if __name__ == "__main__":
+    doctest.testmod()
+    main()