{ "cells": [ { "cell_type": "markdown", "metadata": { "hide_input": true, "nbpresent": { "id": "7e5f0ba3-2aab-4514-8a73-9cd9bab4b09e" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix Calculus" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Kindergarten definitions\n", "### Gradient\n", "Let $f(x): \\mathbb{R}^n \\to \\mathbb{R}$, then vector, which contains all first order partial derivatives:\n", "$$\\nabla f(x) = \\dfrac{df}{dx} = \\begin{pmatrix}\n", " \\frac{\\partial f}{\\partial x_1} \\\\\n", " \\frac{\\partial f}{\\partial x_2} \\\\\n", " \\vdots \\\\\n", " \\frac{\\partial f}{\\partial x_n}\n", "\\end{pmatrix}$$\n", "named gradient of $f(x)$. This vector indicates the direction of steepest ascent. Thus, vector $- \\nabla f(x)$ means the direction of the steepest descent of the function in the point. Moreover, the gradient vector is always orthogonal to the contour line in the point.\n", "\n", "![wikipedia's image](./pics/gradient.gif)\n", " \n", "$$\\nabla f(x)^T = \\dfrac{df}{dx^T} = \\left(\\frac{\\partial f}{\\partial x_1}, \\frac{\\partial f}{\\partial x_2}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right)$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Hessian\n", "Let $f(x): \\mathbb{R}^n \\to \\mathbb{R}$, then matrix, containing all the second order partial derivatives:\n", "$$f''(x) = \\dfrac{d(\\nabla f)}{dx^T} = \\dfrac{d\\left(\\nabla f^T\\right)}{dx} = \\begin{pmatrix}\n", " \\frac{\\partial^2 f}{\\partial x_1 \\partial x_1} & \\frac{\\partial^2 f}{\\partial x_1 \\partial x_2} & \\dots & \\frac{\\partial^2 f}{\\partial x_1 \\partial x_n} \\\\\n", " \\frac{\\partial^2 f}{\\partial x_2 \\partial x_1} & \\frac{\\partial^2 f}{\\partial x_2 \\partial x_2} & \\dots & \\frac{\\partial^2 f}{\\partial x_2 \\partial x_n} \\\\\n", " \\vdots & \\vdots & \\ddots & \\vdots \\\\\n", " \\frac{\\partial^2 f}{\\partial x_n \\partial x_1} & \\frac{\\partial^2 f}{\\partial x_n \\partial x_2} & \\dots & \\frac{\\partial^2 f}{\\partial x_n \\partial x_n}\n", "\\end{pmatrix}$$\n", "\n", "But actually, Hessian could be a tensor in such a way: $\\left(f(x): \\mathbb{R}^n \\to \\mathbb{R}^m \\right)$ is just 3d tensor, every slice is just hessian of corresponding scalar function $\\left( H\\left(f_1(x)\\right), H\\left(f_2(x)\\right), \\ldots, H\\left(f_m(x)\\right)\\right)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Jacobian\n", "The extension of the gradient of multidimensional $f(x): \\mathbb{R}^n \\to \\mathbb{R}^m$:\n", "$$f'(x) = \\dfrac{df}{dx^T} = \\begin{pmatrix}\n", " \\frac{\\partial f_1}{\\partial x_1} & \\frac{\\partial f_1}{\\partial x_2} & \\dots & \\frac{\\partial f_1}{\\partial x_n} \\\\\n", " \\frac{\\partial f_2}{\\partial x_1} & \\frac{\\partial f_2}{\\partial x_2} & \\dots & \\frac{\\partial f_2}{\\partial x_n} \\\\\n", " \\vdots & \\vdots & \\ddots & \\vdots \\\\\n", " \\frac{\\partial f_m}{\\partial x_1} & \\frac{\\partial f_m}{\\partial x_2} & \\dots & \\frac{\\partial f_m}{\\partial x_n}\n", "\\end{pmatrix}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Matrix and vector derivatives\n", "### Table\n", "\n", "$$f(x) : X \\to Y; \\;\\;\\;\\;\\;\\;\\;\\; \\frac{\\partial f(x)}{\\partial x} \\in G$$\n", "\n", "\n", "| X | Y | G | Name |\n", "|:-------------------------:|:--------------:|:-------------------------:|:----------------------------------------------:|\n", "| $\\mathbb{R}$ | $\\mathbb{R}$ | $\\mathbb{R}$ | $f'(x)$ (derivative) |\n", "| $\\mathbb{R}^n$ | $\\mathbb{R}$ | $\\mathbb{R^n}$ | $\\dfrac{\\partial f}{\\partial x_i}$ (gradient) |\n", "| $\\mathbb{R}^n$ | $\\mathbb{R}^m$ | $\\mathbb{R}^{n \\times m}$ | $\\dfrac{\\partial f_i}{\\partial x_j}$ (jacobian) |\n", "| $\\mathbb{R}^{m \\times n}$ | $\\mathbb{R}$ | $\\mathbb{R}^{m \\times n}$ | $\\dfrac{\\partial f}{\\partial x_{ij}}$ |" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### The notation of differentials could be useful here\n", "* $dA = 0$\n", "* $d(\\alpha X) = \\alpha (dX)$\n", "* $d(AXB) = A(dX )B$\n", "* $d(X+Y) = dX + dY$\n", "* $d(X^\\top) = (dX)^\\top$\n", "* $d(XY) = (dX)Y + X(dY)$\n", "* $d\\langle X, Y\\rangle = \\langle dX, Y\\rangle+ \\langle X, dY\\rangle$\n", "* $d\\left( \\dfrac{X}{\\phi}\\right) = \\dfrac{\\phi dX - (d\\phi) X}{\\phi^2}$\n", "* $d\\left( \\det X \\right) = \\det X \\langle X^{-\\top}, dX \\rangle $\n", "* $d \\text{tr } X = I$\n", "* $df(g(x)) = \\dfrac{df}{dg} \\cdot dg(x)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "First order approximation\n", "\n", "$$\n", "df(x) = \\langle \\nabla f(x), dx\\rangle\n", "$$\n", "\n", "Second order approximation\n", "\n", "$$\n", "d^2f(x) = \\langle \\nabla^2 f(x) dx_1, dx_2\\rangle = \\langle H_f(x) dx_1, dx_2\\rangle\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Examples\n", "\n", "All examples are divided by 2 groups: firstly we calculate derivatives manually, secondly we do it in differential notations\n", "\n", "### Example 1.1\n", "Calculate $\\nabla f(x)$, if $f(x) = c^Tx$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "* $f(x) = \\sum\\limits_{i=1}^n c_i x_i$\n", "* $\\dfrac{\\partial f(x)}{\\partial x_k} = c_k \\rightarrow \\nabla f(x) = c$ " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 1.2\n", "Calculate $\\nabla f(x)$, if $f(x) = \\dfrac{1}{2}x^TAx + b^Tx + c$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "* $f(x) = \\sum\\limits_{i=1}^n\\left[ \\dfrac{1}{2}x_i \\left(\\sum\\limits_{j=1}^n a_{ij}x_j \\right) + b_ix_i + c_i\\right] = \\dfrac{1}{2} \\sum\\limits_{i,j=1}^n\\left[ x_i a_{ij}x_j \\right] + \\sum\\limits_{i=1}^n b_ix_i + c_i$\n", "* $\\dfrac{\\partial f(x)}{\\partial x_i} = \\dfrac{1}{2}\\sum\\limits_{i,j=1}^n \\left(a_{ij} + a_{ji}\\right)x_i + b_i \\rightarrow \\nabla f(x) = \\dfrac{1}{2} \\left( A + A^T \\right)x + b$ " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 1.3\n", "Calculate $\\nabla f(x), f''(x)$, if $f(x) = -e^{-x^Tx}$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Решение: \n", "\n", "* $f(x) = - e ^{-\\sum\\limits_i x_i^2}$\n", "* Let's calculate one component of gradient vector:\n", "$$\\dfrac{\\partial f(x)}{\\partial x_k} = - e ^{-\\sum\\limits_i x_i^2} \\cdot \\left( \\dfrac{\\partial (-\\sum\\limits_i x_i^2)}{\\partial x_k} \\right) = e ^{-\\sum\\limits_i x_i^2} \\cdot 2x_k$$\n", " Thus, we can state, that $\\nabla f(x) = 2 e^{-x^Tx} \\cdot x$\n", "* Absolutely the same logic could be \n", "$$g_k = \\dfrac{\\partial f(x)}{\\partial x_k} \\rightarrow H_{k,p} = \\dfrac{\\partial g_k}{\\partial x_p}$$\n", "$$H_{k,p} = - \\left( e ^{-\\sum\\limits_i x_i^2} \\cdot 2x_p\\right) 2x_k + 2 e ^{-\\sum\\limits_i x_i^2} \\dfrac{\\partial x_k}{\\partial x_p} = 2 e ^{-\\sum\\limits_i x_i^2} \\cdot \\left( \\dfrac{\\partial x_k}{\\partial x_p} - 2x_px_k\\right)$$\n", "\n", "* Finally, $f''(x) = H_{f(x)} = 2e^{-x^Tx} \\left( E - 2 xx^T\\right)$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 2.1\n", "Calculate $\\nabla f(x)$, if $f(x) = \\dfrac{1}{2} \\|Ax - b\\|^2_2$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "$$\n", "f(x) = \\dfrac{1}{2} \\langle Ax-b, Ax-b\\rangle\n", "$$\n", "\n", "$$\n", "d f(x) = \\dfrac{1}{2} 2 \\langle Ax-b, Adx\\rangle\n", "$$\n", "\n", "$$\n", "d f(x) = \\langle A^\\top(Ax-b), dx\\rangle \\;\\;\\;\\; \\to \\nabla f(x) = A^\\top (Ax-b)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 2.2\n", "Calculate $\\nabla f(x)$, if $f(x) = \\ln \\langle Ax, x \\rangle$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "$$\n", "df(x) = \\dfrac{d\\langle Ax, x \\rangle}{\\langle Ax, x \\rangle} = \\dfrac{Ad\\langle x, x \\rangle}{\\langle Ax, x \\rangle} = \\dfrac{2A \\langle x, dx \\rangle}{\\langle Ax, x \\rangle} = \\dfrac{2\\langle Ax, dx \\rangle}{\\langle Ax, x \\rangle}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 2.3\n", "Calculate $\\nabla f(X)$, if $f(X) = \\text{tr } AX$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "$$\n", "f(X) = \\langle A^\\top, X\\rangle\n", "$$\n", "\n", "$$\n", "d f(X) = d \\langle A^\\top, X\\rangle = \\langle A^\\top, dX\\rangle\n", "$$\n", "\n", "$$\n", "\\nabla f(X) = A^\\top\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Example 2.4\n", "Calculate $\\nabla f(X)$, if $f(X) = \\langle S, X\\rangle - \\log \\det X$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Solution:\n", "\n", "$$\n", "df(X) = \\langle S, dX \\rangle - \\dfrac{d(\\det X)}{\\det X}\n", "$$\n", "\n", "$$\n", "df(X) = \\langle S, dX \\rangle - \\dfrac{\\det X \\langle X^{-\\top}, dX\\rangle}{\\det X}\n", "$$\n", "\n", "Suppose, that $\\det X \\neq 0$\n", "\n", "$$\n", "df(X) = \\langle 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