April 2020
Intermediate to advanced
816 pages
18h 32m
Chinese
Content preview from 精通機器學習
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210
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第八章：降維
維數災難
因為我們都習慣在三維空間中生活
1
，
所以我們無法直覺地想像高維空間
，
我們的大腦就
連基本的
4D
超立方體也難以描繪
（
見圖
8-1），
更不用說在
1000
維空間裡面被折彎的
200
維橢球體了
。
圖
8-1
點
、
線段
、
正方形
、
立方體
，
和
tesseract（0D
到
4D
超立方體
）
2
事實上
，
在高維空間中
，
很多事物都有截然不同的行為
。
例如
，
當你在一個單位正方形
（1 × 1
矩形
）
中隨機選取一點時
，
那個點只有
0.4%
的機率位於距邊界
0.001
之內的地方
（
換句話說
，
隨機一點靠任一維度
「
非常
」
近的機率非常低
）。
但是在
10,000
維的單位
超立方體中
，
這個機率大於
99.999999%。
在高維超立方體裡面
，
大部分的點都非常接近
邊界
3
。
此外還有一個更麻煩的差異
：
如果你在單位正方形裡面隨便選兩點
，
這兩點的距離平均
大約是
0.52。
如果你在單位
3D
立方體中隨便選兩點
，
平均距離大約是
0.66。
但如果你
在
1,000,000
維的超立方體裡面隨便選兩點呢
？
信不信由你
，
平均距離大約是
408.25（
大
約
）！
這跟直覺不一樣
：
在同一個單位超立方體裡面的兩點怎麼會相距這麼
遠
？
其實
，
在高維空間裡面有很多空間
，
因此
，
高維資料組有非常稀疏的風險
：
大部分的
訓練實例很有可能彼此相距極遠
。
這也代表新實例可能會離任何訓練實例極遠
，
做出來的
預測比低維空間不可靠
，
因為它們都要採取更大規模的外推法
（extrapolation）。
總之
，
訓
練組的維數越多
，
過擬它的風險就越大
。
1
好啦
，
如果你把時間算進去的話
，
有
4
維
，
如果你很懂弦理論 ...
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O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.