docs/tutorials/embed/SplitAE Simulated Data.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Predicting views using SplitAE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import torch\n",
    "from mvlearn.embed import SplitAE\n",
    "import matplotlib.pyplot as plt\n",
    "import sklearn.cross_decomposition\n",
    "plt.style.use(\"ggplot\")\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# cca, previously validated against sklearn CCA\n",
    "def cca(X, Y, regularizationλ=0):\n",
    "\n",
    "    X = X - X.mean(axis=0)\n",
    "    Y = Y - Y.mean(axis=0)\n",
    "    k = min(X.shape[1], Y.shape[1])\n",
    "    covXX = (X.t() @ X) / X.shape[0] + regularizationλ*torch.eye(X.shape[1], device=X.device)\n",
    "    covYY = (Y.t() @ Y) / X.shape[0] + regularizationλ*torch.eye(Y.shape[1], device=X.device)\n",
    "    covXY = (X.t() @ Y) / X.shape[0]\n",
    "\n",
    "    U_x, S_x, V_x = covXX.svd()\n",
    "    U_y, S_y, V_y = covYY.svd()\n",
    "    covXXinvHalf = V_x @ (S_x.sqrt().reciprocal().diag()) @ U_x.t()\n",
    "    covYYinvHalf = V_y @ (S_y.sqrt().reciprocal().diag()) @ U_y.t()\n",
    "    T = covXXinvHalf @ covXY @ covYYinvHalf\n",
    "    U, S, V = T.svd()\n",
    "    A = covXXinvHalf @ U[:, :k]\n",
    "    B = covYYinvHalf @ V[:, :k]\n",
    "    return A.t(), B.t(), S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predicting a held out view with CCA, nonlinear relationship between views"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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    "# The relationship between view1 and view2 is that view2(t) = view1(t) ** 2. \n",
    "# In words, View1(t) is a nonlinear function of View2(t)\n",
    "view1 = np.random.randn(10000, 10)\n",
    "view2 = view1 ** 2\n",
    "# view2 = view1 @ np.random.randn(10, 10)\n",
    "\n",
    "# Let's say now say we have 10,000 points with a view1 but only 5000 of those points have a view 2. So\n",
    "# one obvious goal is to somehow reconstruct the missing view2 data for those points.\n",
    "view1Train = view1[:5000]\n",
    "view2Train = view2[:5000]\n",
    "view1Test = view1[5000:]\n",
    "view2Test = view2[5000:] # these are what we're trying to predict\n",
    "\n",
    "# Let's try and predict view2Test with CCA\n",
    "U, V, S = cca(torch.FloatTensor(view1Train), torch.FloatTensor(view2Train))\n",
    "view1CCs = view1Train @ U.t().numpy()\n",
    "view2CCs = view2Train @ V.t().numpy()\n",
    "covariance = np.mean((view1CCs - view1CCs.mean(axis=0)) * (view2CCs - view2CCs.mean(axis=0)), axis=0)\n",
    "stdprod = np.std(view1CCs, axis=0) * np.std(view2CCs, axis=0)\n",
    "correlations = covariance / stdprod\n",
    "\n",
    "# we can see that the canonical correlations are very low. This means that for any given sample, the\n",
    "# vector of view1 canonical variables will not be close to the vector of view2 canonical variables.\n",
    "# Ideally the canonical correlations would be 1, so that the for each point, each view's canonical variable\n",
    "# has the same vlaue.\n",
    "plt.plot(correlations)\n",
    "plt.title(\"Canonical Correlations\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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     "text": [
      "MSE Loss is  5.096770717383144\n"
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    "# This is how we predict our training data given the canonical variables\n",
    "view1TrainPred = view1CCs @ np.linalg.inv(U.t().numpy())\n",
    "view2TrainPred = view2CCs @ np.linalg.inv(V.t().numpy())\n",
    "assert np.all(view1TrainPred - view1Train < 1e-2)\n",
    "assert np.all(view2TrainPred - view2Train < 1e-2)\n",
    "\n",
    "# This is how we predict View2 from View1 values. Notice the V.t() matrix being used for view1 values.\n",
    "view1TestCCs = view1Test @ U.t().numpy()\n",
    "view2TestPred = view1TestCCs @ np.linalg.inv(V.t().numpy())\n",
    "\n",
    "# Notice that the magnitude of the errors are close to the magnitude of the view2 elements themselves!\n",
    "# these are bad predictions.\n",
    "predictionErrors = np.abs(view2TestPred - view2Test).ravel()\n",
    "plt.hist(predictionErrors)\n",
    "plt.title(\"Prediction Errors\")\n",
    "plt.show()\n",
    "plt.hist(view2.ravel())\n",
    "plt.title(\"View 2 Magnitudes\")\n",
    "plt.show()\n",
    "\n",
    "print(\"MSE Loss is \", np.mean((view2TestPred - view2Test)**2))\n",
    "\n",
    "# If you repeat this experiment with view2 = (some linear combination of the features of view1),\n",
    "# for example view2 = view1 @ np.random.randn(10, 10)\n",
    "# the prediction errors will be zero. This is where CCA exceeds, when the above is true. We will see this \n",
    "# next time we run CCA."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predicting a held out view with SplitAE, nonlinear relationship between views"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Parameter counts: \n",
      "view1Encoder: 1,012\n",
      "view1Decoder: 1,002\n",
      "view2Decoder: 1,002\n"
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     "text": [
      "MSE Loss is  0.052350855326646545\n"
     ]
    }
   ],
   "source": [
    "# Now lets try the same thing with SplitAE!\n",
    "splitae = SplitAE(hidden_size=32, num_hidden_layers=1, embed_size=20, training_epochs=50, batch_size=32, learning_rate=0.01, print_info=False, print_graph=True)\n",
    "splitae.fit([view1Train, view2Train], validationXs=[view1Test, view2Test])\n",
    "\n",
    "# (I'm using the test data to see validation loss, in a real case the validation set is held out data and the test set is unknown / not used until the end)\n",
    "embeddings, reconstructedView1, predictedView2 = splitae.transform([view1Test])\n",
    "predictionErrors = np.abs(predictedView2 - view2Test).ravel()\n",
    "plt.hist(predictionErrors)\n",
    "plt.title(\"Prediction Errors\")\n",
    "plt.show()\n",
    "plt.hist(view2.ravel())\n",
    "plt.title(\"View 2 Magnitudes\")\n",
    "plt.show()\n",
    "\n",
    "print(\"MSE Loss is \", np.mean((predictedView2 - view2Test)**2))\n",
    "\n",
    "# The bins near 0 are a bit deceiving on the histograms, but the loss shows it all -- with splitAE we can \n",
    "# predict our view2 from view1 with much higher accuracy than CCA.\n",
    "# The tradeoff here was hyperparameter tuning -- I had to get the embed size right, the number of hidden layers right\n",
    "# (too big, and the loss will converge to something higher), and train for the right amount of time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predicting a held out view with CCA, linear relationship between views, few data points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MSE Loss is  2.517854473585315e-11\n"
     ]
    }
   ],
   "source": [
    "# Lets say instead of 5000 input points we only have 50 train points and 50 test points. And that this time,\n",
    "# we have a generally linear relationship.\n",
    "view1 = np.random.randn(100, 10)\n",
    "view2 = view1 @ np.random.randn(10, 10)\n",
    "\n",
    "view1Train = view1[:50]\n",
    "view2Train = view2[:50]\n",
    "view1Test = view1[50:]\n",
    "view2Test = view2[50:] # these are what we're trying to predict\n",
    "\n",
    "U, V, S = cca(torch.FloatTensor(view1Train), torch.FloatTensor(view2Train))\n",
    "view1TestCCs = view1Test @ U.t().numpy()\n",
    "view2TestPred = view1TestCCs @ np.linalg.inv(V.t().numpy())\n",
    "print(\"MSE Loss is \", np.mean((view2TestPred - view2Test)**2))\n",
    "\n",
    "# CCA achieves a loss of ~0. Can splitAE achieve the same?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predicting a held out view with SplitAE, linear relationship between views, few data points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Parameter counts: \n",
      "view1Encoder: 2,068\n",
      "view1Decoder: 2,058\n",
      "view2Decoder: 2,058\n"
     ]
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     "text": [
      "MSE Loss for test data  5.59877941169036\n",
      "MSE Loss for train data  0.06715857622620428\n",
      "MSE Loss when predicting mean 8.489032078377859\n"
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    "splitae = SplitAE(hidden_size=32, num_hidden_layers=2, embed_size=20, training_epochs=500, batch_size=10, learning_rate=0.01, print_info=False, print_graph=True)\n",
    "splitae.fit([view1Train, view2Train], validationXs=[view1Test, view2Test])\n",
    "embeddings, reconstructedView1, predictedView2 = splitae.transform([view1Test]) # using test data\n",
    "\n",
    "print(\"MSE Loss for test data \", np.mean((predictedView2 - view2Test)**2))\n",
    "embeddings, reconstructedView1, predictedView2 = splitae.transform([view1Train]) # using training data\n",
    "print(\"MSE Loss for train data \", np.mean((predictedView2 - view2Train)**2))\n",
    "print(\"MSE Loss when predicting mean\", np.mean((0 - view2Train)**2))\n",
    "\n",
    "# Clearly we have overfit, and from the graph we can see that we have done so within the first dozen epochs.\n",
    "# Our test error is almost as bad a just predicting the mean. Can further tuning the parameters s.t.\n",
    "# we don't overfit allow us to match CCA performance?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Parameter counts: \n",
      "view1Encoder: 220\n",
      "view1Decoder: 210\n",
      "view2Decoder: 210\n"
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     "text": [
      "MSE Loss for test data  0.00038991122173028984\n"
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   "source": [
    "splitae = SplitAE(hidden_size=32, num_hidden_layers=0, embed_size=20, training_epochs=500, batch_size=10, learning_rate=0.01, print_info=False, print_graph=True)\n",
    "splitae.fit([view1Train, view2Train], validationXs=[view1Test, view2Test])\n",
    "embeddings, reconstructedView1, predictedView2 = splitae.transform([view1Test]) # using test data\n",
    "print(\"MSE Loss for test data \", np.mean((predictedView2 - view2Test)**2))\n",
    "\n",
    "# Luckily, by converting our model to a linear one (i.e. numHiddenLayers=0, so no activations are performed) \n",
    "# we have once again predicted the test data correctly.\n",
    "# But the trade-off here is clear. CCA has performed maybe 10 matrix operations. SplitAE has performed at least \n",
    "# 500*2 = 1000 equivalent matrix operations. \n",
    "# Using %%timeit, \n",
    "# - CCA takes ~600us to predict view2Test. \n",
    "# - SplitAE takes ~4.5s (7,000x slower) to predict view2Test "
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