在資料科學和機器學習中，我們經常將資料表示為向量和矩陣。在數學和物理中，向量被定義大小和方向的量(例如，距離向量)。然而，通常我們處理的資料不一定會遵循向量的定義，但我們仍然用向量來表示資料。例如，我們可以將人口統計資訊(例如，種族、年齡、性別等)的資料表示為向量，但這並沒有對大小或方向的純幾何解釋。
同樣，在數學中，矩陣是用來表示線性映射的，它被定義為兩個向量空間之間的映射，兩個向量空間保留了向量加法和標量乘法。然而，在資料科學/機器學習中如何使用矩陣的上下文與這種數學定義是不同的。
既然存在這種差異，那麼為什麼向量和矩陣在表示資料時被如此廣泛地使用呢?在本文中，我們將探討解釋這一現象的幾個原因。
在處理資料時，我們通常想要將它們輸入到機器學習模型中，這個過程涉及大量的計算，通常需要對許多數字進行加法和乘法運算。例如，在建立電影推薦系統時，您可能會收集用戶觀看庫中每部電影的長度資料。然後你可以推薦那些平均觀看時間較高的電影。這個平均值是透過所有使用者觀看電影的時間總和除以電影的數量計算出來的。執行這一過程可能會很慢，特別是當用戶和電影數量越來越多時(比如優酷，它擁有超過2.67億的用戶，擁有近2萬部影片)。
然而，電腦科學家已經開發出非常高效的線性代數演算法，向量和矩陣的加法和乘法比傳統的基於元素的加法/乘法要快得多。對於Python，用於科學計算和線性代數的NumPy函式庫提供了更快的速度和效率。再次回顧我們的推薦系統問題，我們可以將每個用戶與一個維度為n的觀看時間向量相關聯，其中n是電影的數量。然後我們的數據將是這些向量的一個矩陣集合，有n行m列，其中n是電影的數量，m是用戶的數量。為了找到要推薦的電影，我們可以沿著行取平均值，找出所有用戶觀看每部電影的平均時間，然後根據平均觀看時間最高的電影進行排序。由於高度最佳化的演算法，用向量和矩陣實現這個問題可以加快計算速度。
為了示範，這裡有一個Python腳本，比較了使用常規 Python 和 NumPy 函式庫（對矩陣和向量進行了最佳化）計算行平均值所需的時間。為了評估計算效率，我們將測量程式運行一個包含 500 部電影和 200 個用戶的資料集所需的時間。
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margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>.<span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>()<br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">total_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">end_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">-</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">start_time</span> # <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">for</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">normal</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">array</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">implementation</span><br><br><br># <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">using</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">NumPy</span><br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_data</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np</span>.<span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">array</span>(<span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">data</span>) # <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">convert</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">data</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">into</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">numpy</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">array</span><br><br><br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_start_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>.<span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>()<br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_average</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np</span>.<span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">mean</span>(<span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_data</span>, <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">axis</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">1</span>) # <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">using</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">numpy</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">mean</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">function</span><br><span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_end_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>.<span style="color: rgb(0, 92, 197); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span>()<br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_total_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">=</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_end_time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">-</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">np_start_time</span> # <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">time</span> <span style="color: rgb(215, 58, 73); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">for</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">numpy</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">array</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">implementation</span><br><br><br><span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">print</span>(<span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">f</span><span style="color: rgb(102, 153, 0); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">"Regular Python: {total_time:4f}; NumPy: {np_total_time:4f}"</span>) # <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">print</span> <span style="color: rgb(89, 89, 89); margin: 0px; padding: 0px; background: none 0% 0% / auto repeat scroll padding-box border-box rgba(0, 0, 0, 0);">results</span>可以將Python程式碼運行十次並對結果取平均值，常規 Python 耗時 9.088 毫秒，而 NumPy 耗時 0.427 毫秒。 NumPy 實作比常規 Python 快大約 20 倍。
更進一步，我們在下面繪製Python和NumPy實現計算平均值的時間，同時將用戶數量從1到1000進行變化，同時保持電影數量為500。
隨著資料量的增加，常規Python和NumPy之間的差異會越來越大。我們還可以透過繪製兩個實現之間的比率來視覺化這一點。
隨著資料量的增加，這個比例也不斷增加，這也證明了使用NumPy可以提高效率。對於非常大的資料來源或複雜的模型,這樣效率更有價值。考慮越來越普遍的大數據領域，往往有數十億到數萬億的數據點。對於深度神經網路模型，它可能由數百萬個節點/參數組成，每個節點/參數的權重和偏差還要進行相乘或相加運算(例如，GPT-3語言模型有超過1750億個參數)。
使用向量/矩陣表示資料的另一個優點是，我們可以利用線性代數和數學工具。一個很好的例子是在電腦視覺中，矩陣被用來描述影像轉換(例如，平移、旋轉、反射、仿射、投影等)。
對於影像旋轉，目標是確定一個函數，用於從影像的每個像素旋轉某個角度。在線性代數中，旋轉矩陣用於旋轉向量/矩陣。透過將圖像表示為矩陣，我們可以利用旋轉矩陣。類似地，還有用於平移、反射和仿射變換的矩陣。
此外，將影像表示為矩陣也有助於進行投影變換，即從一個平面到另一個平面的線的映射。這對於影像拼接和製作全景照片很有用。此外，在處理3D圖形影像時還有進一步的應用。
在處理複雜的資料情況時，使用向量和矩陣表示概念可以更方便、更清晰、更簡潔。我們可以將資料分組到指定的向量或矩陣中，而不是給每個資料點一個名稱。此外，我們也可以使用向量/矩陣約定來表示對資料的運算。
例如，考慮有5個特徵變數的多元線性迴歸的例子。這可以表示為:
使用向量/矩陣，我們可以傳達相同的想法（特徵和特徵上的係數現在是向量）：
注意，這個表示要短得多，而且仍然能捕捉到我們的線性迴歸模型。如果我們有更多的變量，這種表示仍然有效(對於10或1000個特徵變量，它的表達式相同)。此外，向量和矩陣可以用來表示許多資料操作和模型(如邏輯迴歸、隨機森林、神經網路等)。
此外，向量/矩陣的約定在許多領域(如物理、工程、計算等)都非常普遍。這意味著從業者通常都很熟悉，這減少了認知負擔(因為他們不需要學習新的資料/模型約定)。
許多資料和模型的運算以向量/矩陣表示的原因是：用向量和矩陣表示的資料可以實現高效、更快的計算，也可以使用線性代數技術。
以上是為什麼機器學習中的資料要用向量/矩陣來表示？的詳細內容。更多資訊請關注PHP中文網其他相關文章！
