laplacian_eigenmaps_functions.ipynb

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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sklearn.neighbors import NearestNeighbors\n",
    "from scipy.sparse import csr_matrix, diags, spdiags\n",
    "from scipy.sparse.linalg import eigsh\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def laplacian_eigenmaps(X, k=2, t=10, sigma=0.3):\n",
    "    # Efficient nearest neighbors search\n",
    "    nbrs = NearestNeighbors(n_neighbors=t + 1).fit(X)\n",
    "    distances, indices = nbrs.kneighbors(X)\n",
    "\n",
    "    m = X.shape[0]\n",
    "\n",
    "    # Efficient weight matrix calculation\n",
    "    S = np.exp(-distances ** 2 / (sigma ** 2))\n",
    "    W = np.zeros((m, m))\n",
    "    for i in range(m):\n",
    "        # Symmetry of W is essential here\n",
    "        W[i, indices[i, 1:]] = S[i, 1:]\n",
    "        W[indices[i, 1:], i] = S[i, 1:]\n",
    "\n",
    "    # Graph Laplacian\n",
    "    D = diags(np.sum(W, axis=1))\n",
    "    L = D - W\n",
    "\n",
    "    # Eigenvalue and eigenvector computation using sparse matrix\n",
    "    num_eigenvalues = k + 1  # +1 for the trivial eigenvector\n",
    "    eigenvalues, eigenvectors = eigsh(L, k=num_eigenvalues, which='SM')\n",
    "    \n",
    "    # Sort eigenvalues and eigenvectors\n",
    "    sorted_indices = np.argsort(eigenvalues)\n",
    "    eigenvalues = eigenvalues[sorted_indices]\n",
    "    eigenvectors = eigenvectors[:, sorted_indices]\n",
    "\n",
    "    # Project data onto the top k eigenvectors\n",
    "    reduced_data = eigenvectors[:, 1:k+1]\n",
    "\n",
    "    return reduced_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def avg_laplacian_eigenmaps(X, k=2, t=10):\n",
    "    # Efficient nearest neighbors search\n",
    "    nbrs = NearestNeighbors(n_neighbors=t + 1).fit(X)\n",
    "    distances, indices = nbrs.kneighbors(X)\n",
    "\n",
    "    m = X.shape[0]\n",
    "    # Initialize an \"average distance matrix\" A\n",
    "    A = np.zeros((m, 1))\n",
    "\n",
    "    for i in range(m):\n",
    "        sum_of_squares = np.sum(np.square(distances[i, 1:]))\n",
    "        A[i, 0] = np.sqrt(sum_of_squares) / t\n",
    "\n",
    "    # Efficient weight matrix calculation\n",
    "    S = -distances ** 2\n",
    "    W = np.zeros((m, m))\n",
    "    for i in range(m):\n",
    "        row = t * A[i] * np.squeeze(A[indices[i, 1:]])\n",
    "        W[i, indices[i, 1:]] = np.exp(S[i, 1:] / row)\n",
    "        W[indices[i, 1:], i] = np.exp(S[i, 1:] / row)\n",
    "\n",
    "    # Graph Laplacian\n",
    "    D = diags(np.sum(W, axis=1))\n",
    "    L = D - W\n",
    "\n",
    "    # Eigenvalue and eigenvector computation using sparse matrix\n",
    "    num_eigenvalues = k + 1  # +1 for the trivial eigenvector\n",
    "    eigenvalues, eigenvectors = eigsh(L, k=num_eigenvalues, which='SM')\n",
    "\n",
    "    # Sort eigenvalues and eigenvectors\n",
    "    sorted_indices = np.argsort(eigenvalues)\n",
    "    eigenvalues = eigenvalues[sorted_indices]\n",
    "    eigenvectors = eigenvectors[:, sorted_indices]\n",
    "\n",
    "    # Project data onto the top k eigenvectors\n",
    "    reduced_data = eigenvectors[:, 1:k+1]\n",
    "\n",
    "    return reduced_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def var_radius_std_laplacian_eigenmaps(X, k=2, max_rad=50, sigma=2):\n",
    "    # Efficient max_rad nearest neighbors search\n",
    "    nbrs = NearestNeighbors(n_neighbors=max_rad + 1).fit(X)\n",
    "    distances, indices = nbrs.kneighbors(X)\n",
    "\n",
    "    m = X.shape[0]\n",
    "    # Initialize an \"average distance matrix\" A\n",
    "    A = np.zeros((m, 1))\n",
    "\n",
    "    # Calculate avg distances based on sample radius max_rad\n",
    "    for i in range(m):\n",
    "        sum_of_squares = np.sum(np.square(distances[i, 1:]))\n",
    "        A[i, 0] = np.sqrt(sum_of_squares) / max_rad\n",
    "\n",
    "    # Calculate average of the avg distance matrix\n",
    "    avg_distance = np.mean(A)\n",
    "\n",
    "    # Efficient weight matrix calculation\n",
    "    S = np.exp(-distances ** 2 / (sigma ** 2))\n",
    "    W = np.zeros((m, m))\n",
    "    for i in range(m):\n",
    "        # Smaller average distance for x_i necessitates larger radius\n",
    "        cur_rad = int(min(max_rad * (avg_distance / A[i, 0]), max_rad))\n",
    "        W[i, indices[i, 1:cur_rad + 1]] = S[i, 1:cur_rad + 1]\n",
    "        W[indices[i, 1:cur_rad + 1], i] = S[i, 1:cur_rad + 1]\n",
    "\n",
    "    # Graph Laplacian\n",
    "    D = diags(np.sum(W, axis=1))\n",
    "    L = D - W\n",
    "\n",
    "    # Eigenvalue and eigenvector computation using sparse matrix\n",
    "    num_eigenvalues = k + 1  # +1 for the trivial eigenvector\n",
    "    eigenvalues, eigenvectors = eigsh(L, k=num_eigenvalues, which='SM')\n",
    "\n",
    "    # Sort eigenvalues and eigenvectors\n",
    "    sorted_indices = np.argsort(eigenvalues)\n",
    "    eigenvalues = eigenvalues[sorted_indices]\n",
    "    eigenvectors = eigenvectors[:, sorted_indices]\n",
    "\n",
    "    # Project data onto the top k eigenvectors\n",
    "    reduced_data = eigenvectors[:, 1:k+1]\n",
    "\n",
    "    return reduced_data"
   ]
  }
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