14 - 6 - Reconstruction from Compressed Representation (4 min).srt

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In some of the earlier videos,
在以前的视频中，（字幕翻译：中国海洋大学，黄海广，haiguang2000@qq.com）

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I was talking about PCA as
我谈论PCA

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a compression algorithm where you
作为压缩算法。

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may have say, a thousand dimensional
在那里你可能需要把1000维

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data and compress it
的数据压缩

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to a hundred dimensional feature back
一百维特征，

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there, or have three dimensional
或具有三维

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data and compress it to a two dimensional representation.
数据压缩到一二维表示。

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So, if this is a
所以，如果这是一个

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compression algorithm, there should
压缩算法，应该

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be a way to go back from
能回到

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this compressed representation, back to
这个压缩表示，回

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an approximation of your original high dimensional data.
到你原有的高维数据的一种近似。

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So, given z(i), which maybe
所以，给定的Z（i），这可能

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a hundred dimensional, how do
一百维，怎么办

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you go back to your original
你回到你原来的

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representation x(i), which was maybe a thousand dimensional?
表示x（i），这可能是一千维的数组？

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In this video, I'd like to
这个视频，我想

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describe how to do that.
描述如何去做。

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In the PCA algorithm, we may have an example like this.
在PCA算法，我们可能有一个这样的样本。

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So maybe that's my example x1
也许那是那是我们的样本x1，

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and maybe that's my example x2.
也许那是我的样本X2。

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And what we do
我们做的是，

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is, we take these examples and
我们把这些样本

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we project them onto this one dimensional surface.
投射到这一个维平面。

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And then now we need
然后现在我们需要

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to use only a real number,
只使用一个实数，

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say z1, to specify the
比如Z1，指定

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location of these points after
这些点的位置后

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they've been projected onto this one dimensional surface. So
他们被投射到这一个三维曲面。所以

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, given a point
，给定一个点

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like this, given a point z1,
这样，给定一个点Z1，

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how can we go back to
我们怎么能回去

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this original two-dimensional space?
这个原始的二维空间？

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And in particular, given the
特别是，给定

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point z, which is an
Z点，这是一个

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r, can we map
R，我们可以映射

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this back to some approximate
这些来回到一些近似

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representation x and r2
表示X和R2

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of whatever the original value of the data was?
的任何数据的原始值？

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So, whereas z equals U
而z等于

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reduced transverse x, if you
UTreduced x，如果你

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want to go in the opposite
想去相反的

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direction, the equation for
方向，方程

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that is, we're going
变为，

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to write x approx equals
x approx 等于

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U reduce times z.
U reduce乘以 z

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Again, just to check the dimensions,
只是为了检查维度，

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here U reduce is
这里U reduce 为

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going to be an n by k
N*K

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dimensional vector, z is
维向量，Z

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going to be a k by 1 dimensional vector.
变成k阶一维向量。

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So, we multiply these out and that's going to be n by one.
所以，我们将这些相乘，将是一个N。

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So x approx is going
所以x approx

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to be an n dimensional vector.
是一个N维向量。

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And so the intent of PCA,
这就是PCA的意图，

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that is, the square projection error
方块上的投影误差

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is not too big, is that
不太大，

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this x approx will be
x approx将会

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close to whatever was
接近

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the original value of x
x的初始值

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that you had used to derive z in the first place.
你有用于导出Z放在第一位。

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To show a picture of what this looks like, this is what it looks like.
这张图片，看起来就是它的样子。

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What you get back of this
当你回到

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procedure are points that lie
程序点

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on the projection of that onto the green line.
的投影到绿色线。

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So to take our early example,
所以采取我们早期的样本，

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if we started off with
如果我们开始

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this value of x1, and got
这个值x1，并得到了

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this z1, if you plug
Z1，如果你用

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z1 through this formula to get
Z1通过这个公式得到

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x1 approx, then this
x1 approx，那么这个

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point here, that will be
点在这里，那将是

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x1 approx, which is
x1 approx，这

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going to be r2 and so.
将变为R2等。

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And similarly, if you
同样地，如果你

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do the same procedure, this will
做同样的程序，这将

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be x2 approx.
是x2 approx。

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And you know, that's a pretty
如你所知，这是一个漂亮的

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decent approximation to the original data.
与原始数据相当相似。

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So, that's how you
所以，这就是你

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go back from your low dimensional
从低维

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representation z back to
表示Z回到

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an uncompressed representation of
未压缩的表示

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the data we get back an
我们得到的数据的一个

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the approxiamation to your original
之间你的原始

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data x, and we
数据X，我们

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also call this process reconstruction
也把这个过程称为重建

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of the original data.
原始数据。

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When we think of trying to reconstruct
当我们认为试图重建

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the original value of x from the compressed representation.
从压缩表示x的初始值。

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So, given an unlabeled data
所以，给定未标记的数据

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set, you now know how to
集，您现在知道如何

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apply PCA and take
应用PCA和带

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your high dimensional features x and
你的高维特征X和

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map it to this
映射到这

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lower dimensional representation z, and
的低维表示Z，和

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from this video, hopefully you now
这个视频，希望你现在

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also know how to take
也知道如何采取

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these low representation z and
这些低维表示Z

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map the backup to an approximation
映射到备份到一个近似

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of your original high dimensional data.
你原有的高维数据。

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Now that you know how to
现在你知道如何

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implement in applying PCA, what
实施应用PCA，

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we will like to do next is to
我们将要做的事是

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talk about some of the
谈论

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mechanics of how to
一些技术在

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actually use PCA well,
实际使用PCA很好，

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and in particular, in the next
特别是，在接下来的

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video, I like to talk
视频中，我想谈一谈

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about how to choose K, which is,
关于如何选择K，这是，

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how to choose the dimension
如何选择维度

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of this reduced representation vector z.
这减少表示的向量Z。