Replace V with Vt and add a note about the error in ch 8

08_dimensionality_reduction.ipynb

@@ -101,6 +101,22 @@
     "## PCA using SVD decomposition"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note: the `svd()` function returns `U`, `s` and `Vt`, where `Vt` is equal to $\\mathbf{V}^T$, the transpose of the matrix $\\mathbf{V}$. Earlier versions of the book mistakenly said that it returned `V` instead of `Vt`. Also, Equation 8-1 should actually contain $\\mathbf{V}$ instead of $\\mathbf{V}^T$, like this:\n",
+    "\n",
+    "$\n",
+    "\\mathbf{V} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\mid & \\mid & & \\mid \\\\\n",
+    "  \\mathbf{c_1} & \\mathbf{c_2} & \\cdots & \\mathbf{c_n} \\\\\n",
+    "  \\mid & \\mid & & \\mid\n",
+    "\\end{pmatrix}\n",
+    "$"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 3,
@@ -110,9 +126,9 @@
    "outputs": [],
    "source": [
     "X_centered = X - X.mean(axis=0)\n",
-    "U, s, V = np.linalg.svd(X_centered)\n",
+    "U, s, Vt = np.linalg.svd(X_centered)\n",
-    "c1 = V.T[:, 0]\n",
+    "c1 = Vt.T[:, 0]\n",
-    "c2 = V.T[:, 1]"
+    "c2 = Vt.T[:, 1]"
    ]
   },
   {
@@ -135,7 +151,7 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "np.allclose(X_centered, U.dot(S).dot(V))"
+    "np.allclose(X_centered, U.dot(S).dot(Vt))"
    ]
   },
   {
@@ -146,7 +162,7 @@
    },
    "outputs": [],
    "source": [
-    "W2 = V.T[:, :2]\n",
+    "W2 = Vt.T[:, :2]\n",
     "X2D = X_centered.dot(W2)"
    ]
   },
@@ -290,7 +306,7 @@
    },
    "outputs": [],
    "source": [
-    "X3D_inv_using_svd = X2D_using_svd.dot(V[:2, :])"
+    "X3D_inv_using_svd = X2D_using_svd.dot(Vt[:2, :])"
    ]
   },
   {
@@ -338,7 +354,7 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "V[:2]"
+    "Vt[:2]"
    ]
   },
   {
@@ -1673,7 +1689,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Oh no! Training is actually 3 times slower now! How can that be? Well, as we saw in this chapter, dimensionality reduction does not always lead to faster training time: it depends on the dataset, the model and the training algorithm. See figure 8-6 (the `manifold_decision_boundary_plot*` plots above). If you try a softmax classifier instead of a random forest classifier, you will find that training time is reduced by a factor of 3 when using PCA. Actually, we will do this in a second, but first let's check the precision of the new random forest classifier."
+    "Oh no! Training is actually more than twice slower now! How can that be? Well, as we saw in this chapter, dimensionality reduction does not always lead to faster training time: it depends on the dataset, the model and the training algorithm. See figure 8-6 (the `manifold_decision_boundary_plot*` plots above). If you try a softmax classifier instead of a random forest classifier, you will find that training time is reduced by a factor of 3 when using PCA. Actually, we will do this in a second, but first let's check the precision of the new random forest classifier."
    ]
   },
   {
@@ -1824,7 +1840,9 @@
   {
    "cell_type": "code",
    "execution_count": 88,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "from sklearn.datasets import fetch_mldata\n",
@@ -1842,7 +1860,9 @@
   {
    "cell_type": "code",
    "execution_count": 89,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "np.random.seed(42)\n",
@@ -1864,7 +1884,9 @@
   {
    "cell_type": "code",
    "execution_count": 90,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "from sklearn.manifold import TSNE\n",
@@ -2303,21 +2325,21 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 2",
+   "display_name": "Python 3",
    "language": "python",
-   "name": "python2"
+   "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
     "name": "ipython",
-    "version": 2
+    "version": 3
    },
    "file_extension": ".py",
    "mimetype": "text/x-python",
    "name": "python",
    "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython2",
+   "pygments_lexer": "ipython3",
-   "version": "2.7.12"
+   "version": "3.5.2"
   },
   "nav_menu": {
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