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拓撲數據分析 Análise topológica de dados Топологический анализ данных Топологічний аналіз даних Topological data analysis
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拓撲數據分析（Topological Data Analysis ；縮寫作TDA），是應用數學當中一門在數據集的分析用上了拓撲學的新技術領域，主要用於數據挖掘和計算機視覺理論研究。要從多維度、不完整和雜訊多的數據集中提取訊息，一般也具有挑戰性的。 拓撲數據分析的主要問題有： * 如何從低維度的表示去獲得高維度的結構; * 如何將離散單位添加到全局結構中。 人腦可以輕易從低維度的私人數據構建整體結構。例如，從每隻眼睛中的平面圖像中獲取物體的三維形狀並不困難。公共結構的創建也通過將時間片段中的離散組合成連續圖像來執行。例如，電視圖像在技術上是個別點的陣列，然而，它們被認為是單個場景。 拓撲數據分析提供了一種總體框架，以對所選擇的特定度量不敏感的方式分析這些數據，並提供降低維數和對噪聲的魯棒性。以下為當中的主要方法： * 根據接近度參數，通過一些單純複合物系列替換一組數據元素，例如：在數據的點雲中尋找出同調竹4部分。 * 通過這些拓撲結構，特別是穩定同源性的新理論。 * 將數據集的穩定同源性轉碼為參數化版本的Betti數字，以下稱為條形碼。 Em matemática aplicada, a análise topológica de dados (TDA, na abreviatura do nome em inglês, topological data analysis) é uma abordagem para a análise de conjuntos de dados por meio de técnicas da topologia. A extração de informações de conjuntos de dados de dimensão alta, incompletos e com ruídos é um desafio. A TDA fornece uma estrutura geral para analisar esses dados de maneira insensível à métrica específica escolhida e fornece redução de dimensionalidade e robustez ao ruído. Além disso, ela herda funtorialidade, um conceito fundamental da matemática moderna, de sua natureza topológica, o que lhe permite adaptar-se às novas ferramentas matemáticas. Топологический анализ данных — новая область теоретических исследований для задач анализа данных (Data mining) и компьютерного зрения. Основные вопросы: 1. * Как из низкоразмерных представлений получать структуры высоких размерностей; 2. * Как дискретные единицы складываются в глобальные структуры. Основной метод топологического анализа данных: In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.
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one being the study of homological invariants of data one individual data sets, and the other is the use of homological invariants in the study of databases where the data points themselves have geometric structure.
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In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools. The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields. Moreover, its mathematical foundation is also of theoretical importance. The unique features of TDA make it a promising bridge between topology and geometry. Топологический анализ данных — новая область теоретических исследований для задач анализа данных (Data mining) и компьютерного зрения. Основные вопросы: 1. * Как из низкоразмерных представлений получать структуры высоких размерностей; 2. * Как дискретные единицы складываются в глобальные структуры. Человеческий мозг легко строит представление об общей структуре по частным данным низких размерностей.Ему, например, не составляет труда получить трехмерную форму объекта по плоским изображениям в каждом глазу. Создание общей структуры также производится при объединении дискретных во времени фрагментов в образ. Так, например, телевизионное изображение технически является массивом отдельных точек, который, однако, воспринимается как единая сцена. Основной метод топологического анализа данных: 1. * Замена набора элементов данных некоторым семейством симплициальных комплексов в соответствии с параметром близости. 2. * Анализ этих топологических комплексов с помощью алгебраической топологии, а конкретно новой теорией персистентных гомологий. 3. * Перекодировка устойчивой гомологии набора данных в параметризованную версию чисел Бетти, называемую баркодом. 拓撲數據分析（Topological Data Analysis ；縮寫作TDA），是應用數學當中一門在數據集的分析用上了拓撲學的新技術領域，主要用於數據挖掘和計算機視覺理論研究。要從多維度、不完整和雜訊多的數據集中提取訊息，一般也具有挑戰性的。 拓撲數據分析的主要問題有： * 如何從低維度的表示去獲得高維度的結構; * 如何將離散單位添加到全局結構中。 人腦可以輕易從低維度的私人數據構建整體結構。例如，從每隻眼睛中的平面圖像中獲取物體的三維形狀並不困難。公共結構的創建也通過將時間片段中的離散組合成連續圖像來執行。例如，電視圖像在技術上是個別點的陣列，然而，它們被認為是單個場景。 拓撲數據分析提供了一種總體框架，以對所選擇的特定度量不敏感的方式分析這些數據，並提供降低維數和對噪聲的魯棒性。以下為當中的主要方法： * 根據接近度參數，通過一些單純複合物系列替換一組數據元素，例如：在數據的點雲中尋找出同調竹4部分。 * 通過這些拓撲結構，特別是穩定同源性的新理論。 * 將數據集的穩定同源性轉碼為參數化版本的Betti數字，以下稱為條形碼。 Em matemática aplicada, a análise topológica de dados (TDA, na abreviatura do nome em inglês, topological data analysis) é uma abordagem para a análise de conjuntos de dados por meio de técnicas da topologia. A extração de informações de conjuntos de dados de dimensão alta, incompletos e com ruídos é um desafio. A TDA fornece uma estrutura geral para analisar esses dados de maneira insensível à métrica específica escolhida e fornece redução de dimensionalidade e robustez ao ruído. Além disso, ela herda funtorialidade, um conceito fundamental da matemática moderna, de sua natureza topológica, o que lhe permite adaptar-se às novas ferramentas matemáticas. A motivação inicial é estudar a forma dos dados. A TDA combinou a topologia algébrica e outras ferramentas da matemática pura para permitir o estudo matematicamente rigoroso da "forma". A ferramenta principal é a homologia persistente, uma adaptação da homologia para dados de nuvem de pontos. A homologia persistente foi aplicada a muitos tipos de dados em muitas áreas. Além disso, sua base matemática também é de importância teórica. As características exclusivas da TDA fazem dela uma ponte promissora entre topologia e geometria.
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