17_random_Fourier_features_rff.ipynb

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      "cell_type": "markdown",
      "source": [
        "#### The need for data-independent $\\phi(x)$"
      ],
      "metadata": {
        "id": "dLENRj3iObli"
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      "cell_type": "markdown",
      "source": [
        "One of the annoying things we have encountered so far is that the virtual samples $\\phi(x)$ we obtained via eigendecomposition for Bayesian kernel regression is `data-dependent`\n",
        "\n",
        "That is, the virtual samples we get can only approximate the covariance structure over the data points that are used to generate these virtual samples, which can lead to some problems\n",
        "\n",
        "* As shown previously, we have to `correct` the prediction `variance` in the end, as the covariance structure among testing data points is involved in the computation of prediction variance and our virtual samples cannot properly account for it\n",
        "* Of course, if we could generate virtual samples from the kernel matrix over the entire input space (all training and testing points), then the covariance structure among all data points would be properly accounted for. However, this is very inefficient computationally, even if it is not infeasible\n",
        "\n",
        "To take our approach based on weight-space view a bit further, we would like to make virtual samples `data-independent`\n",
        "\n",
        "That is, we want to find a `computationally useful` mapping (as we have shown that the true one is infinite-dimensional) $\\phi(x)$ that we can directly compute for each input $x$, and for every pair of $x_1, x_2$, the mapping preserves the correlation (as defined by `RBF kernel function` for our case)\n",
        "\n",
        "$$\\phi(x_1)^T\\phi(x_2)=e^{-\\frac{\\|x_1-x_2\\|^2}{2\\sigma^2}}$$"
      ],
      "metadata": {
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      "cell_type": "markdown",
      "source": [
        "#### Random features"
      ],
      "metadata": {
        "id": "itJlhKr3UQgr"
      }
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    {
      "cell_type": "markdown",
      "source": [
        "Two great resources\n",
        "* https://random-walks.org/book/papers/rff/rff.html\n",
        "* https://gregorygundersen.com/blog/2019/12/23/random-fourier-features/"
      ],
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      "cell_type": "markdown",
      "source": [
        "Let $\\omega$ be a random $d$-dimensional vector from standard Gaussian distribution $\\omega \\sim N(0, I)$ and let $e(x)=\\exp(i\\omega^Tx)$\n",
        "\n",
        "Let $\\delta = x_1-x_2$, we write out the following expectation\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\mathbb{E}_{\\omega}[e(x_1)e(x_2)^*] & = \\mathbb{E}_{\\omega}[\\exp(i\\omega^Tx_1)\\exp(-i\\omega^Tx_2)] \\\\\n",
        "&=\\mathbb{E}_{\\omega}[\\exp(i\\omega^T(x_1-x_2)] \\\\\n",
        "&=\\int_{R^d}p(\\omega)\\exp(i\\omega^T(x_1-x_2)) d\\omega \\\\\n",
        "&=\\int_{R^d}p(\\omega)\\exp(i\\omega^T\\delta) d\\omega \\\\\n",
        "& \\text{plug in standard Gaussian expression } p(\\omega)\\\\\n",
        "&=(2\\pi)^{-d/2}\\int_{R^d}\\exp(-\\frac{1}{2}\\omega^T\\omega)\\exp(i\\omega^T\\delta) d\\omega \\\\\n",
        "&=(2\\pi)^{-d/2}\\int_{R^d}\\exp\\left(-\\frac{1}{2}\\omega^T\\omega+i\\omega^T\\delta\\right)\n",
        "d\\omega \\\\\n",
        "&=(2\\pi)^{-d/2}\\int_{R^d}\\exp\\left(-\\frac{1}{2}\\left(\\omega^T\\omega-2i\\omega^T\\delta-\\delta^T\\delta\\right)-\\frac{1}{2}\\delta^T\\delta\\right)\n",
        "d\\omega \\\\\n",
        "&=(2\\pi)^{-d/2}\\exp\\left(-\\frac{1}{2}\\delta^T\\delta\\right)\\int_{R^d}\\exp\\left(-\\frac{1}{2}\\left(\\omega^T\\omega-2i\\omega^T\\delta-\\delta^T\\delta\\right)\\right)\n",
        "d\\omega \\\\\n",
        "&=(2\\pi)^{-d/2}\\exp\\left(-\\frac{1}{2}\\delta^T\\delta\\right)\\int_{R^d}\\exp\\left(-\\frac{1}{2}(\\omega-i\\delta)^T(\\omega-i\\delta)\\right)\n",
        "d\\omega \\\\\n",
        "& \\text{the integral is unnormalized Gaussian} \\\\\n",
        "&=(2\\pi)^{-d/2} \\exp\\left(-\\frac{1}{2}\\delta^T\\delta\\right) (2\\pi)^{d/2} \\\\\n",
        "&=\\exp\\left(-\\frac{1}{2}\\delta^T\\delta\\right) \\\\\n",
        "&=\\exp\\left(-\\frac{1}{2}\\|x_1-x_2\\|^2\\right)\n",
        "\\end{align*}$$\n",
        "\n",
        "We see that this is the RBF kernel we want!\n",
        "\n",
        "Now, we need to have this approximated by `dot product` of a mapping"
      ],
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      "cell_type": "markdown",
      "source": [
        "We know that we can approximate the integral above\n",
        "\n",
        "$$\\int_{R^d}p(\\omega)\\exp(i\\omega^T(x_1-x_2)) d\\omega$$\n",
        "\n",
        "using `Monte Carlo` method\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\mathbb{E}_{\\omega}[e(x_1)e(x_2)^*] & = \\mathbb{E}_{\\omega}[\\exp(i\\omega^Tx_1)\\exp(-i\\omega^Tx_2)] \\\\\n",
        "&=\\mathbb{E}_{\\omega}[\\exp(i\\omega^T(x_1-x_2)] \\\\\n",
        "&=\\int_{R^d}p(\\omega)\\exp(i\\omega^T(x_1-x_2)) d\\omega \\\\\n",
        "& \\approx \\frac{1}{m}\\sum_{j=1}^m \\exp(i\\omega_j^T(x_1-x_2)) \\\\\n",
        "&=\\frac{1}{\\sqrt{m}}\\begin{bmatrix}\\exp(i\\omega_1^Tx_1)\\\\\\exp(i\\omega_2^Tx_1)\\\\ \\vdots \\\\ \\exp(i\\omega_m^Tx_1)\\end{bmatrix}^T\\frac{1}{\\sqrt{m}}\\begin{bmatrix}\\exp(-i\\omega_1^Tx_2)\\\\\\exp(-i\\omega_2^Tx_2)\\\\ \\vdots \\\\ \\exp(-i\\omega_m^Tx_2)\\end{bmatrix}\n",
        "\\end{align*}$$\n",
        "\n",
        "We are getting close as the final expression is now in dot product of two vectors, but we still need to get rid of the imaginary part"
      ],
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    {
      "cell_type": "markdown",
      "source": [
        "Note that both $\\omega$ and our kernel function results are `real`, therefore, we have (with Euler's formula)\n",
        "\n",
        "$$\\boxed{\\exp(i\\omega^T(x_1-x_2))=\\cos(\\omega^T(x_1-x_2))}$$\n",
        "\n",
        "If we define $z(x)=\\sqrt{2} \\cos(\\omega^Tx+b), \\, b\\sim \\text{Uniform}(0, 2\\pi)$\n",
        "\n",
        "and with the following\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\cos(x+y) + \\cos(x-y) &= \\cos(x+y)+\\cos(x+(-y))\\\\\n",
        "&= (\\cos x \\cos y- \\sin x \\sin y)+(\\cos x \\cos (-y) -\\sin x \\sin (-y)) \\\\\n",
        "&=2 \\cos x \\cos y\n",
        "\\end{align*}$$\n",
        "\n",
        "we can show that\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\mathbb{E}_{\\omega}[z(x_1)z(x_2)] & = \\mathbb{E}_{\\omega}[\\sqrt{2} \\cos(\\omega^Tx_1+b)\\sqrt{2} \\cos(\\omega^Tx_2+b)] \\\\\n",
        "& = \\mathbb{E}_{\\omega,b}[\\cos(\\omega^T(x_1+x_2)+2b)] + \\mathbb{E}_{\\omega}[\\cos(\\omega^T(x_1-x_2))] \\\\\n",
        "& \\text{law of total expectation} \\\\\n",
        "&=\\mathbb{E}_{\\omega}[\\mathbb{E}_{b}[\\cos(\\omega^T(x_1+x_2)+2b)|\\omega]]+ \\mathbb{E}_{\\omega}[\\cos(\\omega^T(x_1-x_2))] \\\\\n",
        "&=\\mathbb{E}_{\\omega}[\\frac{1}{2\\pi}\\int_0^{2\\pi} \\cos(\\omega^T(x_1+x_2)+2b) \\,db]+ \\mathbb{E}_{\\omega}[\\cos(\\omega^T(x_1-x_2))] \\\\\n",
        "& \\text{integral in first term is zero due to periodicity of cosine} \\\\\n",
        "&=\\boxed{\\mathbb{E}_{\\omega}[\\cos(\\omega^T(x_1-x_2))]}\n",
        "\\end{align*}$$"
      ],
      "metadata": {
        "id": "Y-uY6iXHdOhe"
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    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Random Fourier features (RFFs)"
      ],
      "metadata": {
        "id": "cPNIFMcVnsoB"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now we can define our $\\phi(x)$ as\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\phi(x) &=\\frac{1}{\\sqrt{m}}\\begin{bmatrix}z(x; \\omega_1, b_1) \\\\ \\vdots \\\\z(x; \\omega_m, b_m) \\end{bmatrix}\\\\\n",
        "&= \\begin{bmatrix}\\sqrt{\\frac{2}{m}}\\cos(\\omega_1^Tx+b_1) \\\\ \\vdots \\\\\\sqrt{\\frac{2}{m}}\\cos(\\omega_m^Tx+b_m)\\end{bmatrix}\n",
        "\\end{align*}$$"
      ],
      "metadata": {
        "id": "b5AeonE5oe8E"
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    },
    {
      "cell_type": "markdown",
      "source": [
        "We can show that\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\phi(x_1)^T\\phi(x_2)&=\\begin{bmatrix}\\sqrt{\\frac{2}{m}}\\cos(\\omega_1^Tx_1+b_1) &\\cdots &\\sqrt{\\frac{2}{m}}\\cos(\\omega_m^Tx_1+b_m)\\end{bmatrix}\\begin{bmatrix}\\sqrt{\\frac{2}{m}}\\cos(\\omega_1^Tx_2+b_1) \\\\ \\vdots \\\\\\sqrt{\\frac{2}{m}}\\cos(\\omega_m^Tx_2+b_m)\\end{bmatrix} \\\\\n",
        "&= \\frac{1}{m}\\sum_{i=1}^m\\left[\\sqrt{2}\\cos(\\omega_i^Tx_1+b_i)\\right]\\left[\\sqrt{2}\\cos(\\omega_i^Tx_2+b_i)\\right] \\\\\n",
        "&\\approx \\mathbb{E}_{\\omega}[\\cos(\\omega^T(x_1-x_2))]\\\\\n",
        "&=\\exp\\left(-\\frac{1}{2}\\|x_1-x_2\\|^2\\right)\n",
        "\\end{align*}$$\n",
        "\n",
        "\n",
        "where\n",
        "\n",
        "* $\\omega_i$ follows `Gaussian distribution` $N(0, I)$ (vector or scalar, scale covariance by $\\frac{1}{\\sigma_k}$ if $\\sigma_k\\neq 1$)\n",
        "* $b_i$ follows `uniform distribution` over $(0, 2\\pi)$ (scalar)\n",
        "* $m$ is a parameter indicating the number of RFFs to construct $\\phi(\\cdot)$"
      ],
      "metadata": {
        "id": "szAYTzQYl1pT"
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    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Matrix notation"
      ],
      "metadata": {
        "id": "ldE7URL0qRT7"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "If we have $n$ data points $x_1, \\cdots, x_n\\in\\mathbf{R}^d$, then their RFFs can be expressed compactly as\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\begin{bmatrix}\\phi(x_1)^T \\\\ \\vdots \\\\ \\phi(x_n)^T\\end{bmatrix}\n",
        "&= \\begin{bmatrix}\\begin{bmatrix}\\sqrt{\\frac{2}{m}}\\cos(\\omega_{1}^Tx_1+b_1) &\\cdots &\\sqrt{\\frac{2}{m}}\\cos(\\omega_{m}^Tx_1+b_m)\\end{bmatrix} \\\\ \\vdots \\\\ \\begin{bmatrix}\\sqrt{\\frac{2}{m}}\\cos(\\omega_{1}^Tx_n+b_1) &\\cdots &\\sqrt{\\frac{2}{m}}\\cos(\\omega_{m}^Tx_n+b_m)\\end{bmatrix}\\end{bmatrix} \\\\\n",
        "&=\\sqrt{\\frac{2}{m}}\\cos \\left(\\begin{bmatrix}\\begin{bmatrix}(\\omega_{1}^Tx_1+b_1) &\\cdots &(\\omega_{m}^Tx_1+b_m)\\end{bmatrix} \\\\ \\vdots \\\\ \\begin{bmatrix}(\\omega_{1}^Tx_n+b_1) &\\cdots &(\\omega_{m}^Tx_n+b_m)\\end{bmatrix}\\end{bmatrix}\\right)\\\\\n",
        "&=\\sqrt{\\frac{2}{m}}\\cos  \\left(\\begin{bmatrix}x_1^T \\\\ \\vdots \\\\x_n^T\\end{bmatrix}_{n \\times d}\\begin{bmatrix}\\omega_1 & \\cdots &\\omega_m\\end{bmatrix}_{d \\times m}+\\begin{bmatrix}b_1 & \\cdots &b_m \\end{bmatrix}_{1 \\times m}\\right)\n",
        "\\end{align*}$$\n",
        "\n",
        "where the final line uses `broadcasting`"
      ],
      "metadata": {
        "id": "DYCHJnK3qTN8"
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    },
    {
      "cell_type": "markdown",
      "source": [
        "#### We can compare true RBF kernel function and approximation using RFFs"
      ],
      "metadata": {
        "id": "DURcAI0n0vG-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "np.set_printoptions(formatter={'float': '{: 0.4f}'.format})\n",
        "plt.style.use('dark_background')"
      ],
      "metadata": {
        "id": "LAakHns80qMP"
      },
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def rbf_rff_comparison(idx, d=1, m=150, n=100, sigma=1.):\n",
        "    x = np.linspace(0, 6, n) # n: number of data\n",
        "    omega = np.random.normal(0, 1/sigma, (d, m)) # d: data dim, m: number of RFFs\n",
        "    b = np.random.uniform(0, 2*np.pi, (1, m))\n",
        "\n",
        "    # RFFs\n",
        "    phi = np.sqrt(2/m) * np.cos(x.reshape(-1, 1) @ omega + b)\n",
        "    print(phi.shape)\n",
        "\n",
        "    # Anchor point\n",
        "    x_1 = x[idx]\n",
        "\n",
        "    # Compute RBF function centered at x_1\n",
        "    rbf_function = np.exp(-0.5 * (x - x_1)**2 / sigma**2)\n",
        "    print(rbf_function.shape)\n",
        "\n",
        "    # Compute approximation based on RFFs\n",
        "    rff_approximation = phi @ phi[idx].T\n",
        "    print(rff_approximation.shape)\n",
        "\n",
        "    # Plot both\n",
        "    plt.figure(figsize=(8,4))\n",
        "    plt.plot(x, rbf_function, 'm', linewidth=2, label='RBF function')\n",
        "    plt.plot(x, rff_approximation, 'c', linewidth=2, label='RFF approximation')\n",
        "    plt.legend()\n",
        "    plt.show()"
      ],
      "metadata": {
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      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(42)\n",
        "\n",
        "rbf_rff_comparison(idx=100//2, d=1, m=400, n=300, sigma=0.3)"
      ],
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      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
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          "text": [
            "(300, 400)\n",
            "(300,)\n",
            "(300,)\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 800x400 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "7XPiS2ueyqiY"
      },
      "execution_count": 3,
      "outputs": []
    }
  ]
}