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19<main class="docMainContainer_S4Cg docMainContainerEnhanced_j6X2"><div class="container padding-top--md padding-bottom--lg"><div class="row"><div class="col docItemCol_E8dH"><div class="docItemContainer_oCZa"><article><div class="tocCollapsible_bE1U theme-doc-toc-mobile tocMobile_UmiY"><button type="button" class="clean-btn tocCollapsibleButton_EMHn">Содержание этой страницы</button></div><div class="theme-doc-markdown markdown"><header><h1>Обозначения в учебнике</h1></header>
20<h2 class="anchor anchorTargetStickyNavbar_ahd2" id="специальная-терминология">Специальная терминология<a href="#специальная-терминологи�я" class="hash-link" aria-label="Прямая ссылка на Специальная терминология" title="Прямая ссылка на Специальная терминология" translate="no">​</a></h2>
21<ul>
22<li class="">
23<p><strong>Отклик</strong> - целевая величина, которую мы предсказываем.</p>
24</li>
25<li class="">
26<p><strong>Признак</strong> - известное свойство объекта, для которого строится прогноз.</p>
27</li>
28<li class="">
29<p><strong>Вектор признаков</strong> - все известные свойства, записанные в виде вектора.</p>
30</li>
31<li class="">
32<p><strong>Модель</strong> - функция, отображающая вектор признаков в прогнозное значение отклика по некоторому правилу.</p>
33</li>
34<li class="">
35<p><strong>Параметры модели</strong> - параметры, от которых зависит модель и настраиваемые по обучающей выборке. Называются также <strong>весами модели</strong>.</p>
36</li>
37<li class="">
38<p><strong>Гиперпараметры модели</strong> - параметры, от которых зависит модель и выбираемые пользователем либо подбираемые по отдельной валидационной выборке.</p>
39</li>
40<li class="">
41<p><strong>Алгоритм</strong> - то же, что и модель, но с настроенными параметрами.</p>
42</li>
43<li class="">
44<p><strong>Сложность модели</strong> в учебнике обозначает выразительную гибкость или способность модели подстраиваться под обучающие данные. Если рассматривается модель с низкой сложностью, то такая модель называется <strong>простой</strong>. Не путать с вычислительной сложностью/простотой, относящейся к числу элементарных операций, необходимых для вычисления функции.</p>
45</li>
46</ul>
47<h2 class="anchor anchorTargetStickyNavbar_ahd2" id="скаляры-векторы-и-матрицы">Скаляры, векторы и матрицы<a href="#скаляры-векторы-и-матрицы" class="hash-link" aria-label="Прямая ссылка на Скаляры, векторы и матрицы" title="Прямая ссылка на Скаляры, векторы и матрицы" translate="no">​</a></h2>
48<ul>
49<li class="">
50<p>Векторы всегда обозначают векторы-столбцы (а не строки), и обозначаются строчными буквами, такими как <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> или <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">x</mi></mrow><annotation encoding="application/x-tex">\mathbf{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em"></span><span class="mord mathbf">x</span></span></span></span>. Скалярные значения (числа) обозначаются только строчными буквами, как <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>.</p>
51</li>
52<li class="">
53<p>Матрицы обозначаются заглавными буквами, такими как <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>.</p>
54</li>
55<li class="">
56<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≽</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A\succcurlyeq0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.82em;vertical-align:-0.1367em"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel amsrm">≽</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span> - неотрицательно определенная матрица <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>.</p>
57</li>
58</ul>
59<h2 class="anchor anchorTargetStickyNavbar_ahd2" id="специальные-символы">Специальные символы<a href="#специальные-символы" class="hash-l
59ink" aria-label="Прямая ссылка на Специальные символы" title="Прямая ссылка на Специальные символы" translate="no">​</a></h2>
60<ul>
61<li class="">
62<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em">D</span></span></span></span></span></span></span></span></span></span></span> - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span>-мерный вектор. В записи <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>D</mi></msup></mrow><annotation encoding="application/x-tex">x\in\mathbb{R}^D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8413em"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em">D</span></span></span></span></span></span></span></span></span></span></span> означает, что <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> принадлежит множеству <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span>-мерных векторов, т.е. является <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span>-мерным вектором.</p>
63</li>
64<li class="">
65<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord">∀</span></span></span></span> = для любого, например, условие, что синус никогда не превосходит единицу записывается как  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>
65⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn><mtext>  </mtext><mi mathvariant="normal">∀</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin(x)\le 1 \;\forall x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">∀</span><span class="mord mathnormal">x</span></span></span></span>.</p>
66</li>
67<li class="">
68<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span> - общее число признаков, называемое также размерностью признакового пространства.</p>
69</li>
70<li class="">
71<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{x}\in \mathbb{R}^D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathbf">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8413em"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em">D</span></span></span></span></span></span></span></span></span></span></span> - вектор при�знаков, иногда обозначаемый как <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> (при описании нейросетей это может быть тензором, описывающим объект, например, изображение или его внутреннее представление).</p>
72</li>
73<li class="">
74<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{x}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>
74-й объект выборки (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span>-мерный вектор).</p>
75</li>
76<li class="">
77<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>j</mi></msup></mrow><annotation encoding="application/x-tex">x^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span></span></span></span></span></span></span></span> - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.05724em">j</span></span></span></span>-й признак вектора признаков <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">x</mi></mrow><annotation encoding="application/x-tex">\mathbf{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em"></span><span class="mord mathbf">x</span></span></span></span> (скаляр, т.е. число).</p>
78</li>
79<li class="">
80<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span> - истинное значение отклика или целевой переменной, которую мы предсказываем (при описании нейросетей это также может обозначать выход отдельного блока).</p>
81</li>
82<li class="">
83<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.1944em"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em"><span></span></span></span></span></span></span></span></span> - предсказанное значение отклика алгоритмом.</p>
84</li>
85<li class="">
86<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">w</mi></mrow><annotation encoding="application/x-tex">\mathbf{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em"></span><span class="mord mathbf" style="margin-right:0.01597em">w</span></span></span></span> - вектор всех настраиваемых параметров модели по обучающей выборке, наÐ
86·Ñ‹Ð²Ð°ÐµÐ¼Ñ‹Ñ… также весами модели (в контексте обработки текста <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.02691em">w</span></span></span></span> может обозначать слово).</p>
87</li>
88<li class="">
89<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">��​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> - отдельный параметр модели (в контексте обработки текста <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> может обозначать <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>-ое слово).</p>
90</li>
91<li class="">
92<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">w</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\mathbf{w}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7079em"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7079em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathbf" style="margin-right:0.01597em">w</span></span><span style="top:-3.0134em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.1667em"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> - оценка вектора параметров.</p>
93</li>
94<li class="">
95<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span> - число объектов в обучающей выборке.</p>
96</li>
97<li class="">
98<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>
98∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>N</mi><mo>×</mo><mi>D</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X\in\mathbb{R}^{N \times D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em"></span><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8413em"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em">N</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight" style="margin-right:0.02778em">D</span></span></span></span></span></span></span></span></span></span></span></span> - матрица &quot;объекты-признаки&quot;, называемая также матрицей признаков (feature matrix), в которой каждая строка - вектор признаков соответствующего объекта выборки наблюдений. Значения признака для разных объектов идет в соответствующем столбце этой матрицы (в контексте нейросетей это также может быть входная информация об известных объектах в любом, не только векторном, виде).</p>
99</li>
100<li class="">
101<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow><annotation encoding="application/x-tex">Y\in\mathbb{R}^{N}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em"></span><span class="mord mathnormal" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8413em"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em">N</span></span></span></span></span></span></span></span></span></span></span></span> - вектор откликов (желаемых ответов алгоритма), представляющий собой вектор вещественных значений для задачи регрессии и категориальный вектор для задачи классификации (в контексте нейросетей это также может быть информация об ожидаемых откликах в любом, не только вещественном, виде).</p>
102</li>
103<li class="">
104<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.07153em">C</span></span></span></span> - число классов в классификации.</p>
105</li>
106<li class="">
107<p>Классы обозначаются цифрами <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">1,2,...C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">...</span><span class="mord mathnormal" style="margin-right:0.07153em">C</span></span></span></span> в многоклассовой классификации и -1,+1 в бинарной классификации (на два класса).<
107/p>
108</li>
109<li class="">
110<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>^</mo></mover><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}(\hat{y},y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathcal">L</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.1944em"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> - функция потерь на одном объекте.</p>
111</li>
112<li class="">
113<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="bold">w</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi mathvariant="script">L</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>y</mi><mo>^</mo></mover><mi>n</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\mathbf{w})=\frac{1}{N}\sum_{n=1}^{N}\mathcal{L}(\hat{y}_n,y_{n})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em">w</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.3262em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em">N</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathcal">L</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.1944em"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> - средние потери на обучающей выборке (<em>эмпирический риск</em>).</p>
114</li>
115</ul>
116<h2 class="anchor anchorTargetStickyNavbar_ahd2" id="специальные-функции">Специальные функции<a href="#специальные-функции" class="hash-l
116ink" aria-label="Прямая ссылка на Специальные функции" title="Прямая ссылка на Специальные функции" translate="no">​</a></h2>
117<ul>
118<li class="">
119<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">#</mi><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\#[\cdot]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">#</span><span class="mopen">[</span><span class="mord">⋅</span><span class="mclose">]</span></span></span></span> - число элементов, например,</p>
120<ul>
121<li class="">
122<p>#[объектов] = число объектов,</p>
123</li>
124<li class="">
125<p>#[признаков] = число признаков</p>
126</li>
127</ul>
128</li>
129<li class="">
130<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>z</mi><msub><mo stretchy="false">]</mo><mo lspace="0em" rspace="0em">+</mo></msub><mo>=</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">{</mo><mi>z</mi><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[z]_{+}=\max\{z,0\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.04398em">z</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mop">max</span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.04398em">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">0</span><span class="mclose">}</span></span></span></span> - оператор положительной срезки (оставление неотрицательного значения и зануление отрицательного)</p>
131</li>
132<li class="">
133<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">I</mi><mo stretchy="false">[</mo><mtext>условие</mtext><mo stretchy="false">]</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>если условие выполнено</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>если условие не выполнено</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\mathbb{I}[\text{условие}]=\begin{cases}
1341, &amp; \text{если условие выполнено}\\
1350, &amp; \text{если условие не выполнено}
136\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathbb">I</span><span class="mopen">[</span><span class="mord text"><span class="mord cyrillic_fallback">условие</span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">
136{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord cyrillic_fallback">если</span><span class="mord"> </span><span class="mord cyrillic_fallback">условие</span><span class="mord"> </span><span class="mord cyrillic_fallback">выполнено</span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord cyrillic_fallback">если</span><span class="mord"> </span><span class="mord cyrillic_fallback">условие</span><span class="mord"> </span><span class="mord cyrillic_fallback">не</span><span class="mord"> </span><span class="mord cyrillic_fallback">выполнено</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
137</li>
138<li class="">
139<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>sign</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\text{sign}(x)=\begin{cases}
1401, &amp; x\ge0\\
141-1, &amp; x&lt;0
142\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord text"><span class="mord">sign</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
143</li>
144</ul>
145<h2 class="anchor anchorTargetStickyNavbar_ahd2" id="описание-алгоритмов">Описание алгоритмов<a href="#описание-алгоритмов" class="hash-l
145ink" aria-label="Прямая ссылка на Описание алгоритмов" title="Прямая ссылка на Описание алгоритмов" translate="no">​</a></h2>
146<ul>
147<li class="">
148<p>:= означает &quot;присвоить значение&quot;. Например, присвоение переменной <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em"></span><span class="mord mathnormal">t</span></span></span></span> значени�я ноль записывается в виде <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>:</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t:=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span>.</p>
149</li>
150<li class="">
151<p><em>Асимптотическая сложность алгоритмов</em> обозначается как <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>. Например, сложность увеличения всех элементов <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-мерного вектора на единицу равна <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, а асимптотическая сложность увеличения на единицу всех элементов матрицы <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.02778em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
152</li>
153</ul></div></article><nav class="docusaurus-mt-lg pagination-nav" aria-label="Страница документа"></nav></div></div><div class="col col--3"><div class="tableOfContents_FsCt thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#специальная-терминология" class="table-of-contents__link toc-highlight">Специальная терминология</a></li><li><a href="#скаляры-векторы-и-матрицы" class="table-of-contents__link toc-highlight">Скаляры, векторы и матрицы</a></li><li><a href="#специальные-символы" class="table-of-contents__link toc-highlight">Специальные символы</a></li><li><a href="#специальные-функции" class="table-of-contents__link toc-highlight">Специальные функции</a></li><li><a href="#описание-алгоритмов" class="table-of-contents__link toc-highlight">Описание алгоритмов</a></li></ul></div></div></div></div></main></div></div></div><footer class="theme-layout-footer footer footer--dark"><div class="container container-fluid"><div class="footer__bottom text--center">
153<div class="footer__copyright">© 2023-26 <a href="https://victorkitov.github.io/">Виктор Китов.</a>  <a href="https://t.me/deepmachinelearningru">Новости проекта.</a></div></div></div></footer></div>
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