{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "In this notebook, we introduce different strategies for normalizing a feature representation. Parts of the notebook follow Section 3.1.2.1 and Section 2.2.3.3 of [Müller, FMP, Springer 2015].\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Formal Definition of a Norm \n", "\n", "In previous notebooks, we introduced various feature representations including [spectral features](../C2/C2_STFT-Basic.html), [log-frequency spectral features](../C3/C3S1_SpecLogFreq-Chromagram.html), and [chroma features](../C3/C3S1_SpecLogFreq-Chromagram.html). Such features are typically elements of an Euclidean space of some dimension $K\\in\\mathbb{N}$. In the following, we denote this **feature space** by $\\mathcal{F}=\\mathbb{R}^K$. Often one needs a measure that assigns to a feature a size or some kind of length, which leads us to the notion of a **norm**. In mathematical terms, given a vector space (e.g., $\\mathcal{F}=\\mathbb{R}^K$), a norm is a nonnegative function $p:\\mathcal{F} \\to \\mathbb{R}_{\\geq 0}$ that satisfies three properties:\n", "\n", "* Triangle inequality: $p(x + y) \\leq p(x) + p(y)$ for all $x,y\\in\\mathcal{F}$.\n", "\n", "* Positive scalability: $p(\\alpha x) = |\\alpha| p(x)$ for all $x\\in\\mathcal{F}$ and $\\alpha\\in\\mathbb{R}$. \n", "\n", "* Positive definiteness: $p(x) = 0$ if an only if $x=0$.\n", "\n", "The number $p(x)$ for a vector $x\\in\\mathcal{F}$ is called the **length** of the vector. Furthermore, a vector with $p(x)=1$ is also called **unit vector**. Note that there is a large number of different norms. In the following, we only consider the vector space $\\mathcal{F}=\\mathbb{R}^K$ and discuss three specific norms that play a role in the FMP notebooks." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Euclidean Norm\n", "\n", "The most commonly used norm is the **Euclidean norm** (or $\\ell^2$-norm). This norm is often denoted by $\\|\\cdot\\|_2$ and defined by\n", "\n", "$$\n", " \\|x\\|_2 = \\sqrt{\\langle x\\mid x\\rangle} = \\Big(\\sum_{k=1}^K x(k)^2\\Big)^{1/2}\n", "$$\n", "\n", "for a vector $x=(x(1),x(2),\\ldots,x(K))^\\top \\in\\mathbb{R}^K$. The Euclidean norm $\\|x\\|_2$ gives the usual distance from the origin $(0,0)$ to the point $x$. The set of unit vectors with regard to the Euclidean norm forms a **unit sphere** denoted as $S^{K-1}\\subset\\mathbb{R}^K$. In the case of $K=2$, the unit sphere is the unit circle $S^1$ with origin $(0,0)$. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "execution": { "iopub.execute_input": "2024-02-15T08:51:12.127721Z", "iopub.status.busy": "2024-02-15T08:51:12.127470Z", "iopub.status.idle": "2024-02-15T08:51:14.851266Z", "shell.execute_reply": "2024-02-15T08:51:14.850587Z" } }, "outputs": [ { "data": { "image/png": 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\n", 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