33-Neural Nets and the Learning Function.srt

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数百个麻省理工学院课程的材料
materials from hundreds of MIT courses

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访问位于ocw.mit.edu的麻省理工学院开放式课件
visit MIT opencourseware at ocw.mit.edu

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好的，所以我知道我在哪里
ok so actually I have a I know where

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人们正在从事项目和工作
people are working on projects and

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您不对任何材料负责
you're not responsible for any material

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讲座中的新材料谢谢你
new material in the lectures thank you

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来了，但我确实有一些东西
for coming but I do have something in an

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重要的话题被修订
important topic which is a revised

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关于神经网络构建的版本
version about the construction of neural

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了解我们的基本结构
nets the basic structure that we're

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我们正在努力，以便开放
we're working with so that's on the open

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网页7.1节就是这样
web at section 7.1 that's so

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建立神经网真的是它
construction of neural nets really it's

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学习功能的构建
a construction of the learning function

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f这就是你的功能
f so that's the function that you

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通过梯度下降或优化
optimize by gradient descent or

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随机梯度下降和你
stochastic gradient descent and you

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适用于培训数据，以尽量减少
apply to the training data to minimize

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你只是想到它的损失
the loss you so just thinking about it

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我写的是一种更有条理的方式
in a more organized way because I wrote

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在我知道更多之前的那一节
that section before I knew anything more

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而不是现在如何拼写神经网络
than how to spell neural nets but now

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我考虑的更重要了
I'm thought about it more so the the key

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这一点可以与我所拥有的相比
point may be compared to what I had in

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过去是我现在认为这是
the past is that I now think of this as

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两组变量X的函数
a function of two sets of variables X

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和V.
and V

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所以X是权重，V是
so X are the weights and V are the

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特征向量样本特征
feature vectors the sample feature

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从这样的那些载体来从
vectors from the so those come from the

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如果一次训练数据
training data either one at a time if

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我们正在做随机梯度下降
we're doing stochastic gradient descent

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一次批量大小为1或B
with many batch size one or B at a time

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如果我们做的是小批量的B或者
if we're doing mini batch of size B or

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整个事情整个时代后如
the whole thing a whole epoch at once if

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我们正在进行全面的梯度下降
we're doing full-scale gradient descent

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所以那些是特征向量和
so so those are the feature vectors and

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这些是那些中的数字
these are the numbers in in those in the

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Mait在线性步骤中
Mait in the linear steps where the

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权重所以他们的矩阵AK是
weights so they're the matrices a k that

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你乘以V乘以和
you multiply by multiply V by and also

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你添加的偏向量BK
the bias vectors BK that you add on to

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转移的起源可以的，这些都是
shift the origin okay and these are the

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这些是你的
and these are it's these that you

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优化那些是优化和
optimize those are to optimize and

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什么是整体的结构
what's the structure of the whole the

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整个学习功能和如何
whole of the learning function and how

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你用它来做什么神经
do you use it what what does the neural

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净的样子，所以你把你f的
net look like so you you take f of a

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第一组重量如此f的第一组
first set of weight so f of the first

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权重的设置将A1和B1等等
set of weights would be a1 and b1 so

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这是X部分和实际的部分
that's the X part and the and the actual

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样本矢量样本矢量
sample vector the sample vectors

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在迭代中v为零
v zero in the iteration and this

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产生，然后你做非线性
produces and then you do the nonlinear

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步骤到每个组件并产生
step to each component and that produces

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V一
V one

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所以有一个典型的我可以写出来
so there is a typical I could write out

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这是什么这里有1V0加V1，因此
what this is here a 1 V 0 plus V 1 so

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这两个步骤是线性步骤
that's the two steps are the linear step

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你的输入是V0你采取线性
you the input is V 0 you take the linear

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使用第一个权重a的步骤
step using the using the first weights a

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1和B1然后你采取非线性
1 and B 1 then you take the nonlinear

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步骤，这给你V1这样的
step and that gives you V 1 so that's

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真的比我上面的线条好，所以我会
really better than my line above so I'll

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擦除上面的那条线
erase that line above yep

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这样就可以从V0和V0产生V1
so that produces V 1 from V 0 and the

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第一重，然后是下一级
first weight and then the next level

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输入v1所以我只称这个VK或V.
inputs v1 so I'll just call this VK or V

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K减1，我称之为VK好吧
K minus 1 and I'll call this one VK okay

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所以k等于1到多少
so so k equal to 1 up to however many

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图层URL图层，因此输入为v-0
layers URL layers so the input was v-0

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所以你可以说这个V真的是v-0
so this V is really v-0 you could say

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而你这是神经网络而且这个
and you this is the neural net and this

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是每个输出输出
is the out input and output from each

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层然后VL是最终输出
layer and then V L is the final output

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从最后一层开始，让我们这样做吧
from the final layer so so let's just do

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这里的图片是v-0样本
a picture here here here is v-0 a sample

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矢量或如果我们正在做图像
vector or if we're doing image

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处理它的所有像素
processing it's all the pixels in in the

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图像中的在所述数据和所述
image in the in the in the data and the

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这是一个样本的培训
training from one sample this is

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一个训练样本，然后你
one training sample and then you

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乘以1然后你加上B1和你
multiply by a 1 and you add B 1 and you

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取Riu的那个向量，然后给出
take Riu of that vector and that gives

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你V2V1抱歉给你V1和
you V 2 V 1 sorry that gives you V 1 and

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然后你迭代到最后VL最后
then you iterate to finally V L the last

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你最后没有做真正的uu层
layer you don't do real uu at the last

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层，所以它只是alVL减1加
layer so it's just al V L minus 1 plus

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VL也可能不会做偏置矢量
VL and you may not do a bias vector also

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在那一层，但你可能和那个
at that layer but you might and that's

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这是最后的输出
this is the finally the output so is

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那张照片和照片是这样的
that picture and so that picture is

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更清晰的对我比以前要
clearer for me than it was previously to

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的权重，以便在区分
distinguish between the weights so in

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在梯度下降算法中
the in the gradient descent algorithm

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你正在选择的是这些X.
it's these X's that you're choosing the

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五世的不是这些由给定
V's are not these are given by the

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培训不属于该项目的数据
training data that's not part of the

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优化部分不属于它的一部分
optimization part that's not part of its

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第6章中的X，你找到了
X in chapter 6 where you're finding the

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最佳权重，所以这个X真正的立场
optimal weights so this X really stands

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forX代表所有重量
for X stands for the all the weights

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你计算达AL为L所以它的
that you compute up to a L be L so it's

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这就是所有人的集合
that's that's a collection of all the

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权重和重要部分
weights and the important part for

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实践的应用是实现
applications for practice is to realize

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通常有更多的重量和
that there are often more weights and

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然后在权重中有更多组件
more components in the weights then

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还有在功能组件
there are components in the feature

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在vs中的样本中的向量
vectors in the samples in the vs

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通常X的大小大于
so often the size of X is greater than

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V的大小是一个有趣的
the size of V which is an interesting

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还有那种意想不到的情况
and sort of unexpected situation so

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经常我会写那个经常是X的
often I'll just write that often the X's

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权重X是否在确定之下
are the weights X's are under determined

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因为X的数量超过和
because the number of X's exceeds and

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往往远远超过签证的数量
often far exceeds the number of visa

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基数的数量
number of the cardinality the number of

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这里的权重在A和B中
weights in this is in the A's and B's

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而这些都是在样品中
and the these are in the samples in the

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训练设定好数字
training set the number well the number

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的所有样本的特征
of features of all the samples in the

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训练集，所以我会得到新的
training set so I'll get that new

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第7.1节有希望在本周开始
section 7.1 up hopefully this week on

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在开放的那是开放的设置和
the on the open that's the open set and

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我会在那里给你发电子邮件给你
I'll email to you on stellar is there

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更多我应该说的，你看
more I should say about this you see

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在这里我可以画出画面但是
here the I can draw the picture but of

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当然手绘的图片很远
course a hand-drawn picture is far

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不如机器绘制的图片
inferior to to a machine drawn picture

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但在线图片，但让我这样做
but online picture but let me just do it

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所以训练样本有V.
so there's V the training sample has

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一些组件，然后他们
some components and then they're

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现在乘以这里将是v1
multiplied now here's going to be v1 the

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第一层，可以有一个
first in layer and that can have a

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不同数量的组件
different number of of of components in

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00:11:00,570 --> 00:11:03,295
在第一层不同的数量
in the first layer different number of

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神经元然后每个来自
neurons and then each one comes from

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来自
from the

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结的，所以我不会继续下去，但在这里
eze by so I won't keep going here but

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但你看到的图片就是这样的
but you you see the picture so there's a

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伴侣描述基质的一个
mate that describes a matrix a one that

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告诉你那些权重是多少
tells you what the weights are on those

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然后有一个b1添加了
and then there's a b1 that's added the

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BIOS矢量被添加到所有要获得的内容中
BIOS vector is added to all those to get

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v1
the v1

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所以v1是1V0加B1和
there's so v1 is a 1 V 0 plus B 1 and

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那么这就是现场
then onwards so this this is the spot

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我们手工绘制它很明显
we're drawing it by hand is clearly

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不如任何其他可行的方式
inferior to any other possible way to do

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好吧所以现在我还没有投入
it okay so now I haven't yet put into

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00:12:00,269 --> 00:12:03,774
图片中的损失功能就是这样
the picture the loss function so that's

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要最小化的功能
the function that you want to minimize

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那么损失函数究竟是什么呢？
so what is the loss function so we're

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选择x2即A和B的全部
choosing x2 that's all the A's and B's

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把损失降到最低功能埃尔好了，
to minimize the loss function el okay so

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这是SRA教授的这一部分
it's this part that Professor SRA's

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讲座是关于他所说的他是el
lecture was about so he he said el is

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通常是所有F的有限总和
often a finite sum over all the of F so

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十六，这是什么？所以这是
what would that be f of X V I so this is

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带权重的输出
the output from the with weights with

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来自样本编号I的X中的权重和if
weights in X from sample number I and if

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我们正在进行批量处理
we're doing batch processing that is

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我们正在做整批一次，然后
we're doing the whole batch at once then

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我们计算所有我和那是
we compute that for all I and that's the

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计算是荒谬的
computation that's ridiculously

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贵，而你去了
expensive and you go instead to

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随机渐变，你只需选择
stochastic gradient and you just choose

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其中一个或那些B.
one of those or B of those

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少数就像30到1208
small number be like 30 to 120 8 of

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这些F但是全尺度渐变
these F's but but full-scale gradient

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下降选择权重X到
descent chooses the weights X to

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现在最小化损失所以法律我
minimize the loss now so the law I

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没有这方面的损失现在还没有出现
haven't got the loss here yet there is

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这个功能损失将是负数
this function the loss would be minus

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00:13:45,129 --> 00:13:51,694
样本II的真实结果不是
the true result from sample I I don't

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写我没有好的记谱
write I haven't got a good notation for

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我愿意接受建议
that I'm open to suggestions

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那我该怎么写空气呢？
so how do I want to write the air

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假设我是这样的话
suppose suppose I'm so that would be if

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这可能是最不正方形
it was least squares I would maybe be

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平衡，这将是一个总和
squaring that so that would be a sum of

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所有的误差平方
squares of errors squared over all the

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样品或如果我做随机
samples or if I'm doing stochastic

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梯度下降我会尽量减少我
gradient descent I would minimize I

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我猜我最小化了这个但是
guess I'm minimizing this but the

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问题是我是否使用整个功能
question is do I use the whole function

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L在每次迭代中或者我只是选择
L in at each iteration or do I just pick

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要查看的样本中的一个或仅两个
one or only B of the samples to look at

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在迭代次数K所以这就是
at iteration number K so so this is the

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这是X的L然后我加起来了
this is the L of X then I've added up

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所有V的所以只是为了保持他们的
over all the V's so just to keep their

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符号直接我有这个功能
notation straight I have this function

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00:14:52,149 --> 00:14:55,975
X和Vs我找到它的输出
of X and V s I find its output

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会是什么V这是什么什么的
what would V what this is what to the

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00:15:00,540 --> 00:15:04,024
神经网络产生它应该是
neural net produces it's supposed to be

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接近我们不希望它的真实
close to the true we don't want it to be

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正是我们，我们不希望这是
exactly we we don't expect this to be

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完全为零，但它可能是因为
exactly zero but it it could be because

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我们有很多权重来实现这一目标
we have lots of weights to achieve that

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00:15:17,399 --> 00:15:21,155
所以这就是无论如何这将是
so that's what anyway that would be the

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00:15:21,160 --> 00:15:23,554
损失我们最小化和B平方
loss we minimize and B squared for

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平方损失，我想我还没有真正
square loss I guess I haven't really

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谈到关于损失的功能
spoken about the about loss functions

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00:15:29,819 --> 00:15:33,184
让我把这些放在这里
let me just put those here

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00:15:33,189 --> 00:15:39,615
实际上我这些都是流行的损失
and actually I these are popular loss

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00:15:39,620 --> 00:15:44,865
功能一是我们所知道的
functions one would be the one we know

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00:15:44,870 --> 00:15:50,444
最好的广场损失和第二号我
best square loss and number two I've

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00:15:50,449 --> 00:15:53,564
从来没有见过它直接用过这个
never seen it used quite this directly

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00:15:53,569 --> 00:16:01,064
将是L一个损失可能是总和
would be the L one loss may be the sum

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00:16:01,069 --> 00:16:07,785
L的一个规范，这是其中的一些
of L one norms this is some of these

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00:16:07,790 --> 00:16:10,425
在你知道的l2范数中误差平方
error squared in the l2 norm you know

220
00:16:10,430 --> 00:16:13,504
一个损失可能是我的一些
one loss could be there's some over I of

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l1损失，但这不是很好
the l1 losses but that's not a well this

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这涉及到具体的其他问题
this comes into specific other problems

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00:16:26,269 --> 00:16:29,504
像套索，还有其他重要的
like lasso and are there other important

224
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你最小化l1规范的问题
problems you're minimizing an l1 norm

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00:16:31,910 --> 00:16:37,425
但现在还没有深入学习和三个
but not in deep learning now and three

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可能是铰链损失可能你们中的一些人
would be hinge loss probably some of you

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00:16:44,720 --> 00:16:48,694
比我更了解公式和
know better than I the formula and the

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00:16:48,699 --> 00:16:51,764
铰链损失背后的背景是它
background behind hinge loss is it for

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00:16:51,769 --> 00:16:55,814
这是四减一一
this is four minus one one

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00:16:55,819 --> 00:17:04,664
分类问题而不是四个
classification problems rather than four

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00:17:04,669 --> 00:17:08,564
这适合回归
that would be appropriate for regression

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00:17:08,569 --> 00:17:15,044
所以这将是回归然后
so this would be for regression and then

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最后对神经最重要
finally the most important for neural

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网是交叉熵损失
nets is cross-entropy loss

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这是神经网络，所以这是
this is for neural nets so this is

236
00:17:35,299 --> 00:17:40,605
真的是我们最常用的损失功能
really our the most used loss function

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00:17:40,610 --> 00:17:44,565
在我们和大多数人的设置中
in the and the set up that we are mostly

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00:17:44,570 --> 00:17:47,355
思考和，我会试着说更多
thinking of and and I'll try to say more

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00:17:47,360 --> 00:17:51,855
关于之前的过程结束，因此是
about that before the course ends so is

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我不知道我没有
that I don't know for me I hadn't got

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00:17:54,919 --> 00:17:58,115
这笔直
this straight

242
00:17:58,120 --> 00:18:03,375
直到重写该部分而且它是
until rewriting that section and it's

243
00:18:03,380 --> 00:18:06,585
现在以更好的形式下注，但评论是
now bet in better form but comments are

244
00:18:06,590 --> 00:18:13,424
欢迎好，所以刚刚完成
welcome okay and so that just completes

245
00:18:13,429 --> 00:18:15,375
我想说什么，你会看到
what I wanted to say and you'll see the

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00:18:15,380 --> 00:18:20,865
新的部分之前的任何评论
new section any comments on that before

247
00:18:20,870 --> 00:18:25,664
我完全没问题
I go to a different topic entirely okay

248
00:18:25,669 --> 00:18:30,164
哦，然后问题让我走了之前
oh and then questions let me before I go

249
00:18:30,169 --> 00:18:32,024
这个话题我会告诉你什么
to this topic which I'll tell you what

250
00:18:32,029 --> 00:18:38,685
它就是它的一小部分
it is it's it's it's a short section in

251
00:18:38,690 --> 00:18:41,894
这本名为距离的距离
the book called distance about distance

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00:18:41,899 --> 00:18:55,005
矩阵和问题是我们是不是
matrices and the question is if we're we

253
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我们在太空中有一堆点
have we have a bunch of points in space

254
00:19:03,289 --> 00:19:10,755
而我们所知道的是我们知道的
and what we know is we know the

255
00:19:10,760 --> 00:19:19,935
点和它之间的距离
distances between the points and it's

256
00:19:19,940 --> 00:19:22,125
方便谈论距离
convenient to talk about distances

257
00:19:22,130 --> 00:19:27,355
平方在这里
squared here

258
00:19:27,360 --> 00:19:31,865
好的，我们怎么知道这些
okay and how would we know these

259
00:19:31,870 --> 00:19:37,524
距离可能是雷达或任何
distances maybe by radar or or any

260
00:19:37,529 --> 00:19:45,424
测量他们可能是传感器
measurement they might be sensors which

261
00:19:45,429 --> 00:19:48,875
我们已经放置了，我们可以衡量
we've placed around and we can measure

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00:19:48,880 --> 00:19:51,304
他们和他们之间的距离
the distances between them and the

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00:19:51,309 --> 00:19:53,544
问题是他们的立场是什么
question is what's their position so

264
00:19:53,549 --> 00:19:59,465
那是啊，这就是问题等等
that's the yeah that's the question so

265
00:19:59,470 --> 00:20:00,965
让我谈谈这个问题
let me talk a little bit about this

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00:20:00,970 --> 00:20:09,235
问题，然后暂停找到位置
question and then pause find positions

267
00:20:09,240 --> 00:20:14,345
太空了，但我不知道
and well in space but I don't know we

268
00:20:14,350 --> 00:20:16,174
提前不知道是否
don't know ahead of time maybe whether

269
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空间是普通的3d空间或
the space is ordinary 3d space or

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00:20:19,779 --> 00:20:23,495
这些是飞机上的传感器还是
whether these are sensors in a plane or

271
00:20:23,500 --> 00:20:24,725
我们是否要走得更高
whether we have to go to higher

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00:20:24,730 --> 00:20:28,205
尺寸我只是把D和我们放在一起
dimensions I'll just put D and and we

273
00:20:28,210 --> 00:20:30,455
我也会说我们也是
also I'll just say then we're also

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00:20:30,460 --> 00:20:39,034
找到D以及这些职位是什么
finding D and what are these positions

275
00:20:39,039 --> 00:20:43,505
这些是XXI的位置所以
these are positions X X I so the

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00:20:43,510 --> 00:20:48,755
XI减去XJ平方之间的距离是
distance between X I minus XJ squared is

277
00:20:48,760 --> 00:20:57,065
是给定的DIJ所以我们给了
is the given D IJ so we're given

278
00:20:57,070 --> 00:20:59,644
它们之间的距离和我们想要的距离
distances between them and we want to

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00:20:59,649 --> 00:21:02,375
找到他们的位置找到他们所以我们
find their positions find their so we

280
00:21:02,380 --> 00:21:04,144
知道距离，我们想找到
know distances and we want to find

281
00:21:04,149 --> 00:21:07,745
这是问题的立场
positions that's the question and it's

282
00:21:07,750 --> 00:21:11,465
只是一个解决的简洁数学问题
just a neat math question that is solved

283
00:21:11,470 --> 00:21:14,904
你会看到一个完整的解决方案
and you'll see a complete solution and

284
00:21:14,909 --> 00:21:19,625
它有很多应用程序，它是
and it has lots of applications and it's

285
00:21:19,630 --> 00:21:22,024
是的它只是一个很好的它是一个很好的
yeah it's just a nice it's a nice

286
00:21:22,029 --> 00:21:24,424
问题所以它占据了一部分
question so it occupies a section of the

287
00:21:24,429 --> 00:21:26,884
但那部分只有两页
book but that section is only two pages

288
00:21:26,889 --> 00:21:29,365
很长一段时间，因为它只是一个简单的
long because it's just a straightforward

289
00:21:29,370 --> 00:21:33,125
给出了这个问题的解决方案
solution to that question given the

290
00:21:33,130 --> 00:21:37,235
距离找到给定的位置
distances find the positions given the

291
00:21:37,240 --> 00:21:42,865
距离找到XS
distances find the XS

292
00:21:42,870 --> 00:21:47,254
好的，所以我会谈论那个让我们
okay so I'm gonna speak about that let

293
00:21:47,259 --> 00:21:52,205
我有一个很好的建议
me I had a suggestion a good suggestion

294
00:21:52,210 --> 00:21:53,705
通过电子邮件
by email

295
00:21:53,710 --> 00:21:57,845
关于项目即将来临的问题
well questions about the projects coming

296
00:21:57,850 --> 00:22:00,154
在项目中开始进入和
in projects are beginning to come in and

297
00:22:00,159 --> 00:22:04,075
至少在一开始我就是这样做的
at least at the beginning I'm I make

298
00:22:04,080 --> 00:22:07,294
在所有情况下开始和结束都很好
well out in all cases beginning and end

299
00:22:07,299 --> 00:22:10,685
我会仔细阅读它们
I'll read them carefully and as long as

300
00:22:10,690 --> 00:22:13,205
我可以退后一步的建议
I can I'll stand back suggestions for

301
00:22:13,210 --> 00:22:18,364
我们的最终重写和我说，所以我想
our final rewrite and as I said so I'd

302
00:22:18,369 --> 00:22:21,004
希望得到一个打印出来是伟大的你
like to get a print out is great you

303
00:22:21,009 --> 00:22:23,044
可以把它留在外面的信封里
could leave it in the envelope outside

304
00:22:23,049 --> 00:22:23,674
我的办公室
my office

305
00:22:23,679 --> 00:22:28,534
但当然在线就是这样
but of course online is it's what

306
00:22:28,539 --> 00:22:30,935
每个人都这样做，这些都是公正的
everybody's doing so those are just

307
00:22:30,940 --> 00:22:34,024
开始进来，他们不是
beginning to come in and they're not if

308
00:22:34,029 --> 00:22:35,914
我们可以从今天起一周内把它们送进去
we can get them in by a week from today

309
00:22:35,919 --> 00:22:40,235
我真的很开心，只是
I'm really really happy yep and just

310
00:22:40,240 --> 00:22:42,845
随时给我发电子邮件我会给我发电子邮件
feel free to email me I would email me

311
00:22:42,850 --> 00:22:46,475
关于项目而不是乔纳森
about projects rather than not Jonathan

312
00:22:46,480 --> 00:22:51,004
而且我认为并非匿名恒星
and not not anonymous stellar I think

313
00:22:51,009 --> 00:22:54,304
你可能最好问问我
you probably better just ask me that

314
00:22:54,309 --> 00:22:57,335
问题那很好，我会试试
question that's that's fine and I'll try

315
00:22:57,340 --> 00:22:59,644
以有用的方式回答你
to answer you in a useful way

316
00:22:59,649 --> 00:23:03,634
是的，所以我总是愿意接受
yeah so that's and I'm always open to

317
00:23:03,639 --> 00:23:07,595
问题所以你可以给我发电子邮件
questions so you could email me like how

318
00:23:07,600 --> 00:23:12,004
该项目应该是我的导师
long should this project be my tutor in

319
00:23:12,009 --> 00:23:15,335
牛津说他喜欢什么
Oxford said something like when he you

320
00:23:15,340 --> 00:23:19,715
知道你在写论文
know you were writing essays that's the

321
00:23:19,720 --> 00:23:22,134
牛津系统是写一篇文章和
Oxford system is to write an essay and

322
00:23:22,139 --> 00:23:25,114
他说只是从它开始的地方开始
he said just start where it starts and

323
00:23:25,119 --> 00:23:27,634
当它结束时就结束了
end when it finishes so that's the

324
00:23:27,639 --> 00:23:29,435
这就是主意
that's the idea

325
00:23:29,440 --> 00:23:33,904
肯定不会长，然后一个
certainly not normally long and then a

326
00:23:33,909 --> 00:23:37,445
问题被提出，我可以问你是否
question was raised and I can ask you if

327
00:23:37,450 --> 00:23:40,774
你有兴趣在
you are interested in that

328
00:23:40,779 --> 00:23:44,254
问题是此后的课程
the question was what courses after this

329
00:23:44,259 --> 00:23:48,245
一个是自然的，有点自然
one are natural are sort of natural to

330
00:23:48,250 --> 00:23:52,085
采取前进，我不知道如何
take to go forward and I don't know how

331
00:23:52,090 --> 00:23:53,115
很多人
many of you

332
00:23:53,120 --> 00:23:55,815
谁想要有时间
who are thinking to take have time to

333
00:23:55,820 --> 00:24:00,405
另外还有麻省理工学院的其他MIT课程
take another MIT other MIT courses in

334
00:24:00,410 --> 00:24:02,865
这个领域的深度学习机器
this area of deep learning machine

335
00:24:02,870 --> 00:24:05,505
学习优化所有主题
learning optimization all the topics

336
00:24:05,510 --> 00:24:09,735
我们这里有人想要带走
we've had here anybody expecting to take

337
00:24:09,740 --> 00:24:12,435
更多的课程只是伸出一只手是啊
more courses just stick up a hand yeah

338
00:24:12,440 --> 00:24:15,045
你已经知道麻省理工学院了
and you already know like what MIT

339
00:24:15,050 --> 00:24:20,415
提供或者使被质疑，认为
offers or so that was the question that

340
00:24:20,420 --> 00:24:23,925
来找我麻省理工学院提供的内容
came to me what does MIT offer in this

341
00:24:23,930 --> 00:24:27,195
方向，我没有抬头看
direction and I haven't looked up to see

342
00:24:27,200 --> 00:24:30,705
roz课程sra的教授人数
the number of professors roz course sra

343
00:24:30,710 --> 00:24:33,615
但在六天之后它就是六点
but after in course six it's six point

344
00:24:33,620 --> 00:24:37,185
数量很高，并且在他的好之后
high number and and after his good

345
00:24:37,190 --> 00:24:39,795
讲座我认为这是必须的
lecture I think that's gotta be

346
00:24:39,800 --> 00:24:44,475
值得的，所以我看了六个课程
worthwhile so I looked on course six I

347
00:24:44,480 --> 00:24:48,765
没找到一个非常非常的研究所
didn't find a really very an institute

348
00:24:48,770 --> 00:24:51,795
广泛的名单可能当然六感觉
wide list maybe course six feels that

349
00:24:51,800 --> 00:24:54,495
他们是研究所，但还有其他
they are the Institute but there's other

350
00:24:54,500 --> 00:25:01,935
周围的课程，但我发现在
courses around and but I found in in the

351
00:25:01,940 --> 00:25:05,475
操作是一个搜索网站哦，看到了
operations were a search site oh are see

352
00:25:05,480 --> 00:25:09,795
运营研究中心让我
the operations Research Center let me

353
00:25:09,800 --> 00:25:13,035
只是把它放在那里以防万一你
just put there this is just in case you

354
00:25:13,040 --> 00:25:17,145
我想考虑其中的任何一个
would like to think about any of these

355
00:25:17,150 --> 00:25:25,875
我写的东西就这样，所以我
things so there as I write that so I

356
00:25:25,880 --> 00:25:28,835
听过Timberners-le的演讲
heard the lecture by Tim berners-lee

357
00:25:28,840 --> 00:25:32,565
这里的其他人认为这一周左右
that others here that this a week or so

358
00:25:32,570 --> 00:25:37,424
之前他创建了网络，所以它很漂亮
ago he created the web so it's a pretty

359
00:25:37,429 --> 00:25:38,985
惊人
amazing

360
00:25:38,990 --> 00:25:43,995
毕竟，这不是AlGore，你知道
it wasn't Al Gore after all and you know

361
00:25:44,000 --> 00:25:47,105
你知道他的名字，他现在就是爵士
do you know his name what he's now Sir

362
00:25:47,110 --> 00:25:55,275
蒂姆伯纳斯利这么双名
Tim berners-lee so that double name

363
00:25:55,280 --> 00:25:57,615
让你怀疑他是来自英格兰
makes you suspect it's he's from England

364
00:25:57,620 --> 00:26:01,424
比他还要他，但我是这样的人
than he is but he so anybody I was going

365
00:26:01,429 --> 00:26:04,905
说他是我让他负责
to say he's I hold him responsible for

366
00:26:04,910 --> 00:26:06,255
这些
these

367
00:26:06,260 --> 00:26:12,435
地址中的过多字母
excessive letters in in the address in

368
00:26:12,440 --> 00:26:17,505
URL我的意思是他让我们都说www
the URL I mean he's made us all say www

369
00:26:17,510 --> 00:26:20,895
一年或找其他方式说
for a year or find some other way to say

370
00:26:20,900 --> 00:26:24,555
它但是说我觉得没关系并不容易
it but it's not easy to say I think okay

371
00:26:24,560 --> 00:26:28,545
无论这是什么操作或
whatever this is the operate the o.r

372
00:26:28,550 --> 00:26:37,185
麻省理工学院的中心，然后是学术界或
Center at MIT and then it's academics or

373
00:26:37,190 --> 00:26:42,885
什么，然后它就像
something and then it's something like

374
00:26:42,890 --> 00:26:45,825
课程设置大约
course offerings that's approximately

375
00:26:45,830 --> 00:26:53,505
权，因为它们适用于
right and since they do apply to

376
00:26:53,510 --> 00:26:56,385
优化，你知道有一个
optimization that's you know there's a

377
00:26:56,390 --> 00:26:58,755
蚂蚁数据分析的标题或
heading of ant data analytics or

378
00:26:58,760 --> 00:27:02,765
统计数据有优化
statistics there's optimization there's

379
00:27:02,770 --> 00:27:06,675
oo-或运营研究其他清单
o-o-or operations research other lists

380
00:27:06,680 --> 00:27:10,695
但是许多课程都很好
but a good list of courses from many

381
00:27:10,700 --> 00:27:12,795
部门担任总裁
departments as president

382
00:27:12,800 --> 00:27:18,185
尤其是如此特别是当然的缘故
especially so especially course sakes

383
00:27:18,190 --> 00:27:21,495
当然15这是哪里
of course 15 which is where the

384
00:27:21,500 --> 00:27:24,195
运营研究中心是课程
operations Research Center are is course

385
00:27:24,200 --> 00:27:27,705
18当然也有其他人
18 and there are others in course too

386
00:27:27,710 --> 00:27:34,905
和其他地方的好耶有人会
and elsewhere yeah well would somebody

387
00:27:34,910 --> 00:27:37,785
想说出你的想法
like to say what course you have in mind

388
00:27:37,790 --> 00:27:41,375
如果你，请在此之后接下一个
to take next after this one if you

389
00:27:41,380 --> 00:27:44,115
看着明年的展望
looked at the looked ahead to next year

390
00:27:44,120 --> 00:27:47,535
什么样的建议
any many suggestions of what looks like

391
00:27:47,540 --> 00:27:51,345
一个好的课程，你知道我只知道我
a good course I I you know I only know I

392
00:27:51,350 --> 00:27:55,575
坐在6.03六点六真的基本
sat in once on 6.03 six the really basic

393
00:27:55,580 --> 00:27:58,635
当然，你想要去
course and you you would want to go

394
00:27:58,640 --> 00:28:04,815
更高的好，我不会有任何可能这是
higher okay I I won't any maybe this is

395
00:28:04,820 --> 00:28:07,725
只是说我有兴趣知道
just to say I'd be interested to know

396
00:28:07,730 --> 00:28:10,005
接下来你的经历是什么
what you do next what your experience is

397
00:28:10,010 --> 00:28:15,495
或者我很乐意提供建议但也许
or I'd be happy to give advice but maybe

398
00:28:15,500 --> 00:28:18,285
我的一般建议是那是一个
my general advice is that that's a

399
00:28:18,290 --> 00:28:19,605
有用的清单
useful list of

400
00:28:19,610 --> 00:28:27,315
课程还可以回到距离
of courses okay back to distance

401
00:28:27,320 --> 00:28:32,405
矩阵没关系，所以这就是问题所在
matrices okay so here's the problem

402
00:28:32,410 --> 00:28:36,165
是的，我可能不得不提高
yep okay I'll probably have to raise

403
00:28:36,170 --> 00:28:40,655
但是我会把它留下一分钟好吧
that but I'll leave it for a minute okay

404
00:28:40,660 --> 00:28:45,255
所以我们知道这些距离，我们想要
so we know these distances and we want

405
00:28:45,260 --> 00:28:46,245
找到出口
to find the exit

406
00:28:46,250 --> 00:28:53,595
所以我们可以这样称呼DIJ吧
so let's call this D IJ maybe so we have

407
00:28:53,600 --> 00:28:56,145
一个D矩阵，我们想找到
a D matrix and we want to find the

408
00:28:56,150 --> 00:28:58,905
位置矩阵让我看看是什么
position matrix let me just see what

409
00:28:58,910 --> 00:29:07,415
符号所以这是3.9节no4.9
notation so this is section 3.9 no 4.9

410
00:29:07,420 --> 00:29:10,545
以前3.9，但章三
previously 3.9 but chapters three and

411
00:29:10,550 --> 00:29:13,335
四个人切换了，也许也许
four got switched so and maybe maybe

412
00:29:13,340 --> 00:29:16,665
实际是的，我认为这是八或
actually yeah I think it's it's eight or

413
00:29:16,670 --> 00:29:22,665
九点或十点其他主题正在尝试
nine or ten it's other topics are trying

414
00:29:22,670 --> 00:29:25,925
找到他们的方式，好吧，这就是
to find their way and okay so that's the

415
00:29:25,930 --> 00:29:29,265
这是在该地段上的参考
that's the reference on the lot on the

416
00:29:29,270 --> 00:29:31,395
网页，我会把这些部分放到
web and I'll get these sections onto

417
00:29:31,400 --> 00:29:35,625
恒星好吧所以问题是我们可以
stellar okay so the question is can we

418
00:29:35,630 --> 00:29:39,455
从距离恢复位置
recover the positions from the distances

419
00:29:39,460 --> 00:29:42,105
其实还有一个问题是
in fact there's also a question are

420
00:29:42,110 --> 00:29:46,005
从给定的位置总是有位置
there always positions from the given

421
00:29:46,010 --> 00:29:50,055
距离所以让我和我提到
distances so let me and and I mentioned

422
00:29:50,060 --> 00:29:52,725
我已经说过几个应用程序
several applications I've already spoken

423
00:29:52,730 --> 00:29:56,975
关于你的无线传感器网络
about wireless sensor networks where you

424
00:29:56,980 --> 00:29:59,535
你可以衡量之间的旅行时间
you can measure travel times between

425
00:29:59,540 --> 00:30:02,625
它们介于传感器和你之间
them between the sensors and then you

426
00:30:02,630 --> 00:30:05,475
然后给你距离
that gives you the distances and then

427
00:30:05,480 --> 00:30:09,615
你使用这个简洁的数学计算
you use this neat little bit of math to

428
00:30:09,620 --> 00:30:12,435
当然找到你的位置
find the positions well of course you

429
00:30:12,440 --> 00:30:16,715
无法找到唯一的位置
can't find the positions uniquely

430
00:30:16,720 --> 00:30:21,045
显然你可以任何僵硬的动作
clearly you could any rigid motion of

431
00:30:21,050 --> 00:30:24,795
如果我有一套，所有的职位
all the positions if I have a set of

432
00:30:24,800 --> 00:30:29,025
定位我将称之为X的那个
positions what am I going to call that X

433
00:30:29,030 --> 00:30:32,715
所以就在这里
so so right here and then

434
00:30:32,720 --> 00:30:40,575
所以我给了D矩阵
so I'm given the D matrix that's

435
00:30:40,580 --> 00:30:44,805
距离和工作是找到X.
distances and the job is to find the X

436
00:30:44,810 --> 00:30:54,335
给出位置和的矩阵
matrix which gives the positions and

437
00:30:54,340 --> 00:30:56,925
我要说的和你在一起
what I'm just going to say and you

438
00:30:56,930 --> 00:30:59,865
已经看到，看到它在你的心中
already saw that saw it in your mind

439
00:30:59,870 --> 00:31:02,655
如果我有一套职位我
that if I have a set of positions I

440
00:31:02,660 --> 00:31:06,555
可以做距离的翻译
could do a translation the distances

441
00:31:06,560 --> 00:31:09,135
不会改变，或者我可以做一个僵硬的
wouldn't change or I could do a rigid

442
00:31:09,140 --> 00:31:16,175
运动其他刚性旋转所以位置
motion other rigid rotation so positions

443
00:31:16,180 --> 00:31:20,565
不是唯一的，但我可以让我来
are not unique but I can make I can come

444
00:31:20,570 --> 00:31:23,375
说把中心更近
closer by saying put the center of

445
00:31:23,380 --> 00:31:26,325
原点上的质心或类似的东西
centroid at the origin or something like

446
00:31:26,330 --> 00:31:28,845
这将取消翻译
that that will take out the translations

447
00:31:28,850 --> 00:31:31,125
至少没关系，所以找到X矩阵
at least okay so find the X matrix

448
00:31:31,130 --> 00:31:34,125
这是好的工作，我打算去
that's the job okay and I was going to

449
00:31:34,130 --> 00:31:37,095
在我开始之前的形状
before I start on that the shapes of

450
00:31:37,100 --> 00:31:40,545
分子是核的另一种应用
molecules is another application nuclear

451
00:31:40,550 --> 00:31:44,295
磁共振给出了距离
magnetic resonance gives distances gives

452
00:31:44,300 --> 00:31:49,335
D然后我们找到位置MX和
D and then we find the positions M X and

453
00:31:49,340 --> 00:31:51,215
当然那里有一个噪音
of course there's a noise in there and

454
00:31:51,220 --> 00:31:55,065
有时缺少条目和机器
sometimes missing entries and machine

455
00:31:55,070 --> 00:31:58,665
学习也可以简单地描述为
learning could be just described also as

456
00:31:58,670 --> 00:32:01,485
你给了很多分
a you've given a whole lot of points in

457
00:32:01,490 --> 00:32:03,465
空间特征向量在高
space feature vectors in a high

458
00:32:03,470 --> 00:32:05,715
维空间实际上这是一个很大的空间
dimensional space actually this is a big

459
00:32:05,720 --> 00:32:09,255
交易给你很多积分
deal you're given a whole lot of points

460
00:32:09,260 --> 00:32:12,255
在高维空间的哪里
with where the in high dimensional space

461
00:32:12,260 --> 00:32:15,975
那些与他们有关系
and those are related there they're

462
00:32:15,980 --> 00:32:18,105
他们有点聚在一起
there they sort of come together

463
00:32:18,110 --> 00:32:21,215
自然所以他们倾向于适应
naturally so they tend to fit on a

464
00:32:21,220 --> 00:32:25,905
高维空间中的表面低
surface in high dimensional space a low

465
00:32:25,910 --> 00:32:28,035
高维尺寸表面
dimensional surface in high dimensional

466
00:32:28,040 --> 00:32:31,215
空间和真正的很多数学是
space and really a lot of mathematics is

467
00:32:31,220 --> 00:32:35,295
致力于发现低维度
devoted to finding that low dimensional

468
00:32:35,300 --> 00:32:38,565
那个子空间但它可以是弯曲的
that sub space but it could be curved so

469
00:32:38,570 --> 00:32:42,795
子空间真的不是正确的单词
sub space is not the correct word really

470
00:32:42,800 --> 00:32:46,005
歧管是弯曲的歧管本
manifold is curved manifold this

471
00:32:46,010 --> 00:32:50,475
一个甲基几何学家会说这很接近
a meth geometers would say that is close

472
00:32:50,480 --> 00:32:53,835
所有的行星都平滑而接近
to all the planets smooth and close to

473
00:32:53,840 --> 00:32:57,335
所有的点，你可以线性化它
all the points and you can linearize it

474
00:32:57,340 --> 00:33:01,755
你可以把它弄平，然后你
you could flatten it out and then you

475
00:33:01,760 --> 00:33:04,155
有一个大大减少的问题了
have a much reduced problem the

476
00:33:04,160 --> 00:33:06,615
尺寸从原来减少
dimension is reduced from the original

477
00:33:06,620 --> 00:33:10,035
点的维度
dimension of the of the where the points

478
00:33:10,040 --> 00:33:14,265
谎言有很多数据真实
lie with a lot of data to the true

479
00:33:14,270 --> 00:33:17,235
当然，问题的维度
dimension of the problem which of course

480
00:33:17,240 --> 00:33:18,975
如果积分都是直的
if the points were all on a straight

481
00:33:18,980 --> 00:33:20,415
排列问题的真实维度
line the true dimension of the problem

482
00:33:20,420 --> 00:33:21,405
会是一个
would be one

483
00:33:21,410 --> 00:33:27,935
我们所以我们必须发现这一点
well we so we have to discover this that

484
00:33:27,940 --> 00:33:32,835
我们还必须找到尺寸Dok
we also have to find that dimension D ok

485
00:33:32,840 --> 00:33:38,505
好的，我们怎么做才是这样呢
ok so how do we do it so it's a

486
00:33:38,510 --> 00:33:41,595
经典问题它只是一个整洁
classical problem it just has a neat

487
00:33:41,600 --> 00:33:51,905
回答好吧好吧让我们认识一下
answer ok okay so let's recognize the

488
00:33:51,910 --> 00:33:54,345
距离与之间的联系
connection between distances and

489
00:33:54,350 --> 00:34:01,815
位置所以DIJ是它的长度
positions so D IJ is length of it's the

490
00:34:01,820 --> 00:34:04,725
它们之间的平方距离，从而是
square distance between them so that is

491
00:34:04,730 --> 00:34:18,855
XIXI减去XIXJ减去xjxi加x
X I X I minus X I XJ minus xj x i plus x

492
00:34:18,860 --> 00:34:33,124
JX检查确定是正确的确是如此
JX check ok is that right yes ok so

493
00:34:33,129 --> 00:34:37,245
这些都是DIJ的以矩阵和
those are the DI J's in a matrix and

494
00:34:37,250 --> 00:34:47,225
这些是矩阵Dok中的条目
these are entries in the matrix D ok

495
00:34:47,230 --> 00:34:51,574
这些条目
well these entries

496
00:34:51,579 --> 00:34:58,455
只依靠我，他们是相同的
depend only on I they're the same for

497
00:34:58,460 --> 00:35:01,695
每一个J所以这就是这个意志
every J so this is going to be this will

498
00:35:01,700 --> 00:35:04,695
这部分将产生排名第一
this this part will produce a rank one

499
00:35:04,700 --> 00:35:11,324
矩阵，因为事情只依赖于
matrix because things depend only on the

500
00:35:11,329 --> 00:35:17,044
排J但他们的列号不是这样的
row but not on J their column number so

501
00:35:17,049 --> 00:35:31,635
列重复是的，这产生了
columns repeated yeah and this produces

502
00:35:31,640 --> 00:35:35,504
类似的事情只取决于
a similarly something that depends only

503
00:35:35,509 --> 00:35:39,735
在J上只有列号这样的
on J only on the column number so the

504
00:35:39,740 --> 00:35:43,304
行都是一样的，所以这也是一个
rows are all the same so this is also a

505
00:35:43,309 --> 00:35:51,225
排名一个矩阵，所有重复发送
rank one matrix with all repeated sent

506
00:35:51,230 --> 00:35:58,364
所有相同的行，因为如果我改变我
all the same rows because if I change I

507
00:35:58,369 --> 00:36:01,274
这种内在产品没有任何变化
nothing changes in that inner product so

508
00:36:01,279 --> 00:36:08,344
这些是产生的条款
really these are the terms that produce

509
00:36:08,349 --> 00:36:12,915
大多数矩阵的重要部分
most of the matrix the significant part

510
00:36:12,920 --> 00:36:19,905
矩阵好吗，那我们该怎么办呢
of the matrix okay so what do we do with

511
00:36:19,910 --> 00:36:25,835
那些什么呢
those so what what

512
00:36:25,840 --> 00:36:29,715
让我们看看，我没有，我给一个名字
so let's see I've did I give a name for

513
00:36:29,720 --> 00:36:32,735
我想是我正在寻找的矩阵
the matrix that I'm looking for I think

514
00:36:32,740 --> 00:36:40,635
在笔记中我称之为X所以我得到了
in the notes I call it X so so I'm given

515
00:36:40,640 --> 00:36:50,745
给D找到X，我实际上是什么
D given D find X and what I'll actually

516
00:36:50,750 --> 00:36:54,365
发现你可以看到它来到这里
find you can see it coming here is

517
00:36:54,370 --> 00:37:04,425
实际上找到X转置X因为什么
actually find X transpose X because what

518
00:37:04,430 --> 00:37:06,975
我所拥有的只是点积
all I'm what I'm given is dot products

519
00:37:06,980 --> 00:37:15,345
X的所以我想找到
of X's so so I would like to find to

520
00:37:15,350 --> 00:37:18,525
发现所有这些XI
discover out of out of all this what X I

521
00:37:18,530 --> 00:37:22,575
用XJ点缀是正确的
dotted with XJ is that'll be the correct

522
00:37:22,580 --> 00:37:27,345
点积我们称之为矩阵G.
dot product let's call this matrix G for

523
00:37:27,350 --> 00:37:33,645
然后是点积矩阵
the dot product matrix and then then

524
00:37:33,650 --> 00:37:44,145
从G找到X所以这是一个很好的论点
find X from G so this is a nice argument

525
00:37:44,150 --> 00:37:47,655
所以这告诉我的是一堆
so what this tells me is a bunch of some

526
00:37:47,660 --> 00:37:50,535
有关点积的信息，所以这个
information about dot products so this

527
00:37:50,540 --> 00:37:52,635
告诉我关于G的一些事情
is telling me something about the G

528
00:37:52,640 --> 00:37:56,325
矩阵X的转置矩阵X，然后
matrix the X transpose X matrix and then

529
00:37:56,330 --> 00:37:59,205
一旦我知道G，那么这是一个单独的步骤
once I know G then it's a separate step

530
00:37:59,210 --> 00:38:02,415
找到X，当然就是这样
to find X and of course that's this is

531
00:38:02,420 --> 00:38:05,805
如果，X不是唯一的点
the point of which X is not unique if

532
00:38:05,810 --> 00:38:08,715
如果我把旋转放入X.
there's if I put in a rotation into X

533
00:38:08,720 --> 00:38:12,765
然后旋转Q我会看到Q.
then that rotation Q will I'll see a Q

534
00:38:12,770 --> 00:38:15,465
转置Q它会消失所以
transpose Q and it'll disappear so the

535
00:38:15,470 --> 00:38:19,095
所以我可以自由旋转X.
so the I'm free to rotate the X's

536
00:38:19,100 --> 00:38:20,805
因为这不会改变点
because that doesn't change the dot

537
00:38:20,810 --> 00:38:23,655
产品，但我想要的是G
product but so it's G that I want to

538
00:38:23,660 --> 00:38:26,505
知道，这告诉我关于G的事情
know and this tells me something about G

539
00:38:26,510 --> 00:38:30,875
这告诉我一些关于G和
and this tells me something about G and

540
00:38:30,880 --> 00:38:34,685
那就是这样，但这就是我必须要做的
so does that but that's what I have to

541
00:38:34,690 --> 00:38:39,285
看到那些告诉我的是什么
see so what is what do those tell me

542
00:38:39,290 --> 00:38:42,895
让我们看看让我写下我的内容
let's see let me write down what I have

543
00:38:42,900 --> 00:38:50,665
在这里所以，所以让我们说一个
here so so the the the so let's say a

544
00:38:50,670 --> 00:39:00,325
以VII为内部的对角矩阵
diagonal matrix with VII as the inner

545
00:39:00,330 --> 00:39:07,885
产品XI将XI我们认为
product X I would X I that we that we're

546
00:39:07,890 --> 00:39:10,195
从这里因此让部分信息
getting partial information from here so

547
00:39:10,200 --> 00:39:12,735
所以，我正在介绍那个
so is that okay I'm introducing that

548
00:39:12,740 --> 00:39:15,835
符号，因为现在这样
notation because this is now going to

549
00:39:15,840 --> 00:39:21,175
告诉我，我的D矩阵就是这样
tell me that my D matrix is so what is

550
00:39:21,180 --> 00:39:24,765
这就是对角线矩阵
that so this is the diagonal matrix

551
00:39:24,770 --> 00:39:29,625
也许这只是我应该说的一个载体
maybe it's just a vector I should say

552
00:39:29,630 --> 00:39:41,515
是的，是的，所以我可以写下来
yeah yeah yep so can I write down the

553
00:39:41,520 --> 00:39:44,455
方程那这就是它的
equation that that that's that it's

554
00:39:44,460 --> 00:39:47,845
基本在这里，然后哦，然后我们会
fundamental here and then oh then we'll

555
00:39:47,850 --> 00:39:50,385
找出它意味着什么，所以它是一个
figure out what it means so it's a

556
00:39:50,390 --> 00:39:57,595
点积的G的等式
equation for G for the dot product

557
00:39:57,600 --> 00:40:00,565
矩阵好吧让我为此腾出空间
matrix okay let me make space for that

558
00:40:00,570 --> 00:40:06,295
等式，所以这就是我相信我们
equation so here's the I believe that we

559
00:40:06,300 --> 00:40:08,575
我可以得到点积矩阵
can get the dot product matrix which I'm

560
00:40:08,580 --> 00:40:16,435
按照这个来调用G
calling G as as according to this it's

561
00:40:16,440 --> 00:40:22,795
减去D矩阵的1/2加1/2的1/2
minus 1/2 of the D matrix plus 1/2 of

562
00:40:22,800 --> 00:40:33,055
那些时间D对角线D和
the ones times the D the diagonal D and

563
00:40:33,060 --> 00:40:40,135
它加上了深度的1/2
it's plus 1/2 of the depth of the

564
00:40:40,140 --> 00:40:49,285
时间和曾经这些是矩阵
the times and once these that's a matrix

565
00:40:49,290 --> 00:40:53,455
与恒定行相同
with the same with constant rows that

566
00:40:53,460 --> 00:40:57,595
矩阵就是这就是这个
matrix is that this here is this is

567
00:40:57,600 --> 00:41:01,045
从有这个未来是一个矩阵
coming from there this is a matrix with

568
00:41:01,050 --> 00:41:11,125
总是在同一列，让我看不到
always the same columns so let me see no

569
00:41:11,130 --> 00:41:12,955
我没有得到那些正确的
I haven't got haven't got those right

570
00:41:12,960 --> 00:41:17,515
但我的意思是我希望这些排名第一
yet I mean I want these to be rank one

571
00:41:17,520 --> 00:41:20,445
矩阵，所以我就是这个
matrices so I've it's this one that's

572
00:41:20,450 --> 00:41:36,115
让我解决这个问题一个一个一个一个时间d
let me fix that one one one one times D

573
00:41:36,120 --> 00:41:39,405
转置所以它的列时间排和
transpose so it's column times row and

574
00:41:39,410 --> 00:41:48,835
这个也是列时间行
this one is also column times row with

575
00:41:48,840 --> 00:41:54,865
这里的d现在没事了，让我看看那个
the D here okay now let me look at that

576
00:41:54,870 --> 00:41:59,475
这个家伙的每一行都是合适的
properly so every row in this guy is a

577
00:41:59,480 --> 00:42:02,475
多个一一一
multiple of one one one one

578
00:42:02,480 --> 00:42:04,855
那是什么告诉我所有的
so what does that telling me that all

579
00:42:04,860 --> 00:42:07,725
所有列都是相同的列
columns are the same all columns are

580
00:42:07,730 --> 00:42:12,505
这部分是反映这些
this this part is is reflecting these

581
00:42:12,510 --> 00:42:15,985
列重复的地方
ones where the columns are repeated this

582
00:42:15,990 --> 00:42:18,415
一个是反映这个行的位置
one is reflecting this where the rows

583
00:42:18,420 --> 00:42:21,475
重复D只是D的集合
are repeated the D is just the set of D

584
00:42:21,480 --> 00:42:27,295
让我们称之为di，这就是
numbers let's call that di and this is

585
00:42:27,300 --> 00:42:38,035
DJ所以我和这里都是D矩阵
DJ so all I'm and here's the D matrix

586
00:42:38,040 --> 00:42:41,975
所以这部分D的一部分
so part of the so so part of the D

587
00:42:41,980 --> 00:42:46,295
矩阵是这是这个坑和这个
matrix is is these is this pit and this

588
00:42:46,300 --> 00:42:50,795
现在就是这一点，每个都给一个排名
bit each giving a rank one now it's this

589
00:42:50,800 --> 00:42:52,925
我必须要了解的部分
part that I have to understand

590
00:42:52,930 --> 00:42:56,855
所以，当你检查时，让我来
so let me while you're checking on that

591
00:42:56,860 --> 00:43:14,405
让我好吧，让我们看看我们在哪里
let me okay yep let's just see where we

592
00:43:14,410 --> 00:43:18,035
如果这是真的，如果这是真的我是
are if this is true if this is true I'm

593
00:43:18,040 --> 00:43:24,665
给出D矩阵然后这些点
given the D matrix and then these dot

594
00:43:24,670 --> 00:43:29,675
我能找到的产品，所以我可以找到这些
products I can find so I can find these

595
00:43:29,680 --> 00:43:31,355
换句话说，这是关键
so in other words this is the key

596
00:43:31,360 --> 00:43:37,745
方程式告诉我那就是那个
equation that tells me that's that's the

597
00:43:37,750 --> 00:43:39,995
关键方程，它要来只
key equation and it's going to come just

598
00:43:40,000 --> 00:43:43,715
来自那个简单的身份
from that simple identity just from

599
00:43:43,720 --> 00:43:45,575
我们这个学期检查每个学期
checking each term this term we

600
00:43:45,580 --> 00:43:48,065
确定我们确定的上一学期
identified that last term we identified

601
00:43:48,070 --> 00:43:50,945
现在这个词当然是B好吧
it now this term is B well of course

602
00:43:50,950 --> 00:43:58,415
它是D所以我有两个，我
it's D so I have two of those and I'm

603
00:43:58,420 --> 00:44:02,375
将其中的一半用于
going to take half of half of that to

604
00:44:02,380 --> 00:44:12,035
得到DI认为是的是的，我们将我们
get D I think yep yep and we'll we'll

605
00:44:12,040 --> 00:44:21,425
看起来如此，是的
look so so yeah

606
00:44:21,430 --> 00:44:24,945
所以我想我没有马上看到原因
so I guess I'm not seeing right away why

607
00:44:24,950 --> 00:44:27,845
这一半在这里
this half is in here

608
00:44:27,850 --> 00:44:30,585
但那是，但我认为我做得对
but that's but I think I had it right

609
00:44:30,590 --> 00:44:33,555
并且有一个原因，但你看到了
and there's there's a reason but you see

610
00:44:33,560 --> 00:44:35,955
该矩阵这个X的转置X
that this matrix this X transpose X

611
00:44:35,960 --> 00:44:38,865
矩阵来自这些等级1
matrix is coming from these Rank 1

612
00:44:38,870 --> 00:44:42,165
地方和这些作品
places and these pieces which are the

613
00:44:42,170 --> 00:44:50,445
跨产品哦，哦，我知道我看到什么
the cross-product oh oh I see I see what

614
00:44:50,450 --> 00:44:52,305
真的是那个等式真的
really what that equation is really

615
00:44:52,310 --> 00:44:58,325
说是D矩阵就是这个
saying is that the the D matrix is this

616
00:44:58,330 --> 00:45:01,545
如果我只是阅读并翻译
if I just read that along and translate

617
00:45:01,550 --> 00:45:05,225
它并把它放在矩阵的语言是这个
it and put it in matrix language is this

618
00:45:05,230 --> 00:45:13,655
111112D让我们说
1 1 1 1 the 1 2 D for let's say

619
00:45:13,660 --> 00:45:18,945
转置是这个秩1矩阵和
transpose is this Rank 1 matrix and the

620
00:45:18,950 --> 00:45:25,215
另一种是DS倍11，其是
other one is the DS times 1 1 which is

621
00:45:25,220 --> 00:45:28,005
转换那个然后另一个
the transpose of that and then the other

622
00:45:28,010 --> 00:45:31,575
一个是交叉产品的减2
one was a minus 2 of the cross product

623
00:45:31,580 --> 00:45:35,535
矩阵，所以当我写那个
matrices yep so when I write that

624
00:45:35,540 --> 00:45:39,015
矩阵语言中的方程式我得到了
equation in matrix language I just get

625
00:45:39,020 --> 00:45:43,485
那个，现在当我解决x
that and now when I solve for x

626
00:45:43,490 --> 00:45:49,925
哦，它只是2减2X转置XI
oh it's 2 minus 2 X transpose X I just

627
00:45:49,930 --> 00:45:53,655
是的，对不起
yeah yep sorry

628
00:45:53,660 --> 00:45:58,845
交叉产品XS所以我有一套
cross products the XS so I had one set

629
00:45:58,850 --> 00:46:00,885
交叉产品然后这是
of cross products and then this is the

630
00:46:00,890 --> 00:46:03,375
与此相同所以我减去其中的2个
same as this so I have minus 2 of them

631
00:46:03,380 --> 00:46:06,555
所以现在我只是重写那个
so now I'm just rewriting the that when

632
00:46:06,560 --> 00:46:08,535
我重写了我所做的那个等式
I rewrite that equation I have that do

633
00:46:08,540 --> 00:46:12,015
你知道我把它放在这边我
you see that I put that on this side I

634
00:46:12,020 --> 00:46:16,155
把D放在这里作为减去DI除以
put D over here as a minus D I divide by

635
00:46:16,160 --> 00:46:20,325
2，然后是这样的公式，以便
2 and then that's the formula so

636
00:46:20,330 --> 00:46:26,475
最终这个简单的身份只是
ultimately this simple identity just

637
00:46:26,480 --> 00:46:29,985
看着因为这些碎片是如此
looked at because these pieces were so

638
00:46:29,990 --> 00:46:32,835
简单只是排名一个和这个
simple just rank one pieces and this

639
00:46:32,840 --> 00:46:34,255
这些作品是
these pieces were

640
00:46:34,260 --> 00:46:36,445
实际上我们想要X转置X.
actually what we want the X transpose X

641
00:46:36,450 --> 00:46:41,755
那个方程告诉我们D的G
pieces the G that equation told us the D

642
00:46:41,760 --> 00:46:49,555
我们知道这一切都很清楚，所以这是什么
we know all this is known well so what's

643
00:46:49,560 --> 00:46:52,555
真的是，D和这个已知的是什么
really yeah what's known is D and this

644
00:46:52,560 --> 00:46:55,315
这所以现在我们有公式
and this so now we have the equation for

645
00:46:55,320 --> 00:47:03,135
x转置X是D减去1/2
x transpose X is minus 1/2 of D minus

646
00:47:03,140 --> 00:47:13,885
-这些排名很抱歉让它看起来
- these rank ones sorry to make it look

647
00:47:13,890 --> 00:47:17,665
如果我记得RogeroutT，那就太乱了
messy if I remember that Roger out T

648
00:47:17,670 --> 00:47:20,005
说起去年春天也
talking about a last spring also the

649
00:47:20,010 --> 00:47:25,975
代数变得慌乱，但我们得到了它
algebra got flustered but so we get it

650
00:47:25,980 --> 00:47:31,555
所以我们知道X现在转换X矩阵
so we know X transpose X that matrix now

651
00:47:31,560 --> 00:47:33,925
你能为我们做四分钟吗？
can you just can we just do four minutes

652
00:47:33,930 --> 00:47:43,665
今天结束时线性代数的给出
of linear algebra at the end today given

653
00:47:43,670 --> 00:47:56,665
X转置X找到X这是nbynhow
X transpose X find X this is n by n how

654
00:47:56,670 --> 00:48:00,805
你会做，你能做到这一点会
would you do that could you do it would

655
00:48:00,810 --> 00:48:06,175
只有一个X知道什么是这样的
there be just one X know what so so any

656
00:48:06,180 --> 00:48:09,895
如果你有一个X倍数
if you had if you had one X multiply

657
00:48:09,900 --> 00:48:12,115
通过正交旋转
that by a rotation by an orthogonal

658
00:48:12,120 --> 00:48:15,415
矩阵你还有另外一个所以这个
matrix you'd have another one so so this

659
00:48:15,420 --> 00:48:18,705
你是在寻找X到正交
is you're finding X up to an orthogonal

660
00:48:18,710 --> 00:48:20,215
转型，但你怎么样
transformation but how would you

661
00:48:20,220 --> 00:48:22,825
实际上做得很好我们知道什么
actually do that well what do we know

662
00:48:22,830 --> 00:48:27,075
关于这个矩阵X转置X.
about this matrix X transpose X

663
00:48:27,080 --> 00:48:30,355
对称清晰，特别是我们
symmetric clearly and what we especially

664
00:48:30,360 --> 00:48:35,695
知道它也是正面的或半正面的
know is that it's also positive or semi

665
00:48:35,700 --> 00:48:41,295
确定所以这是半确定的
definite so this is semi definite

666
00:48:41,300 --> 00:48:44,305
所以我给了一个半定矩阵
so I'm given a semi definite matrix and

667
00:48:44,310 --> 00:48:47,335
我想找到一个平方根你可以
I want to find a square root you could

668
00:48:47,340 --> 00:48:50,995
说矩阵是X转置X和
say that matrix is the X transpose X and

669
00:48:51,000 --> 00:48:53,515
我想找到XI认为有两个
I want to find X I think there are two

670
00:48:53,520 --> 00:48:57,325
领先的候选人有很多
leading candidates there are many

671
00:48:57,330 --> 00:49:00,925
候选人，因为如果你找到一个
candidates because if you find one then

672
00:49:00,930 --> 00:49:10,105
然后任何QX都没关系，因为如果我
then any QX is okay yeah because if I

673
00:49:10,110 --> 00:49:12,445
把Q转置Q放在那里就是了
put a Q transpose Q in there it's the

674
00:49:12,450 --> 00:49:16,765
身份好吧所以一种方法是使用本征
identity okay so one way is to use eigen

675
00:49:16,770 --> 00:49:24,985
X转置x和其它的值
values of x transpose x and the other

676
00:49:24,990 --> 00:49:30,445
方法是在X上使用消除
way would be to use elimination on X

677
00:49:30,450 --> 00:49:37,795
移调X以便使用，如果我使用的话
transpose X so so for use so if I use

678
00:49:37,800 --> 00:49:40,885
如果我找到本征，则X的本征值
eigen values of X if I find the eigen

679
00:49:40,890 --> 00:49:44,095
X的值转置X然后我正在写
values of X transpose X then I'm writing

680
00:49:44,100 --> 00:49:46,825
因为它是对称的
this as it's a symmetric

681
00:49:46,830 --> 00:49:49,435
肯定地把它写成Qlambda
positive-definite writing it as Q lambda

682
00:49:49,440 --> 00:49:52,545
Q转置正确，这是根本
Q transpose right that's the fundamental

683
00:49:52,550 --> 00:49:55,315
最重要的定理在附近
most important theorem in the near

684
00:49:55,320 --> 00:49:58,255
代数你可以说是对称的
algebra you could say that a symmetric

685
00:49:58,260 --> 00:50:02,715
正半正定矩阵
positive semi definite matrix has

686
00:50:02,720 --> 00:50:05,635
更大的本征值大于等于零
greater eigen values greater equal zero

687
00:50:05,640 --> 00:50:09,685
和特征向量是正交如此
and eigen vectors that are orthogonal so

688
00:50:09,690 --> 00:50:12,745
现在如果我知道什么是好的X呢
now if I know that what's a good X then

689
00:50:12,750 --> 00:50:21,405
把X变为什么，所以我得到了
take X to be what so I've got the

690
00:50:21,410 --> 00:50:23,815
特征值和X的特征向量
eigenvalues and eigenvectors of X

691
00:50:23,820 --> 00:50:26,065
转置X而我正在寻找X.
transpose X and I'm looking for an X

692
00:50:26,070 --> 00:50:29,755
这将是有效的，一个想法只是为了
that will work and one idea is just to

693
00:50:29,760 --> 00:50:32,725
取相同的特征值特征向量
take the same eigen values eigen vectors

694
00:50:32,730 --> 00:50:36,505
并采取征的平方根
and take the square roots of the eigen

695
00:50:36,510 --> 00:50:40,585
值因为如果我跟踪我是否相乘
values because if I multiply if I track

696
00:50:40,590 --> 00:50:43,135
该对称的现在这是等于X
that symmetric now this is equal to X

697
00:50:43,140 --> 00:50:46,875
颠倒
transpose

698
00:50:46,880 --> 00:50:52,255
如果我这样那就是一个正方形
and if I so that was a that's a square

699
00:50:52,260 --> 00:50:54,595
根符号或λ的二分之一
root symbol or a lambda to the one-half

700
00:50:54,600 --> 00:50:57,655
我可以，我可以这么说，当我乘
I could I could say so when I multiply

701
00:50:57,660 --> 00:51:01,525
我只是X转置X是
that I'm just really X transpose X is

702
00:51:01,530 --> 00:51:05,475
当我把它放在一边时，只是x平方
just x squared here when I square it the

703
00:51:05,480 --> 00:51:09,445
Q转置Q乘以自身给出
Q transpose Q multiplies itself to give

704
00:51:09,450 --> 00:51:12,235
身份是lambdax的平方根
the identity the square root of lambda x

705
00:51:12,240 --> 00:51:14,965
lambda的平方根是那些
times the square root of lambda is those

706
00:51:14,970 --> 00:51:17,335
是给出lambda的对角矩阵
are diagonal matrices that give lambda

707
00:51:17,340 --> 00:51:20,725
我得到了正确的答案，所以有一种方法
and I get the right answer so one way is

708
00:51:20,730 --> 00:51:24,385
用几句话来说就是平方根
in a few words take the square roots of

709
00:51:24,390 --> 00:51:26,125
特征值并保持
the eigenvalues and keep the

710
00:51:26,130 --> 00:51:28,995
特征向量，这是特征值
eigenvectors so that's the eigenvalue

711
00:51:29,000 --> 00:51:31,855
施工，这样就产生了X.
construction so that's producing an X

712
00:51:31,860 --> 00:51:35,745
这是对称的正半确定
that is symmetric positive semi definite

713
00:51:35,750 --> 00:51:39,145
这可能是你想要的东西
that might be what you want it's a

714
00:51:39,150 --> 00:51:40,975
很少工作，因为你在计算
little work because you're computing

715
00:51:40,980 --> 00:51:43,645
这样做的特征值和特征向量
eigenvalues and eigenvectors to do it

716
00:51:43,650 --> 00:51:47,425
但现在我认为这是一个选择
but that's one choice now I believe that

717
00:51:47,430 --> 00:51:50,325
淘汰会给我们另一种选择
elimination would give us another choice

718
00:51:50,330 --> 00:51:53,085
所以消除会产生什么
so elimination produces what

719
00:51:53,090 --> 00:51:56,005
这个因素化仍然是我们的
factorization of this this is still our

720
00:51:56,010 --> 00:51:59,995
对称正定矩阵如果
symmetric positive definite matrix if

721
00:52:00,000 --> 00:52:02,335
你消灭了你
you do elimination on that you you

722
00:52:02,340 --> 00:52:06,645
通常期望L下三角形
usually expect L a lower triangular

723
00:52:06,650 --> 00:52:13,405
时间Z的枢轴次数是上部的
times Z the the pivots times u the upper

724
00:52:13,410 --> 00:52:15,885
三角这就是通常的结果
triangular that's the usual result of of

725
00:52:15,890 --> 00:52:19,435
消除LD你我正在考虑
elimination LD u I'm factoring out the

726
00:52:19,440 --> 00:52:22,675
枢轴所以他们在对角线上的那些
pivots so they're ones on the diagonals

727
00:52:22,680 --> 00:52:26,365
L和U但现在如果它是对称的
of L and U but now if it's symmetric

728
00:52:26,370 --> 00:52:32,035
矩阵怎么了我们通过消除拉链
matrix what's up we zip by elimination

729
00:52:32,040 --> 00:52:38,995
关于这是一个1806琐碎的一点
regarding that as a 1806 trivial bit of

730
00:52:39,000 --> 00:52:41,455
线性代数但当然它是高度的
linear algebra but of course it's highly

731
00:52:41,460 --> 00:52:44,425
重要的是这里的情况如何
important so what's the situation here

732
00:52:44,430 --> 00:52:49,185
当基体是实际上对称的
when the matrix is actually symmetric

733
00:52:49,190 --> 00:52:52,465
U所以我想看看
the U so I want something to look

734
00:52:52,470 --> 00:52:54,715
对称我该如何看待
symmetric how do I make that look

735
00:52:54,720 --> 00:52:58,255
对称的U被L取代
symmetric the U gets replaced by L

736
00:52:58,260 --> 00:53:00,325
如果我转移
transpose if I

737
00:53:00,330 --> 00:53:05,365
如果我，如果我正在做一个积极的
if I if I'm working on a positive

738
00:53:05,370 --> 00:53:07,555
明确说明正定矩阵
definite say positive definite matrix

739
00:53:07,560 --> 00:53:13,215
然后我得到积极的支点和L和
then I get positive pivots and L and

740
00:53:13,220 --> 00:53:15,895
下三角形和上三角形或
lower triangular and upper triangular or

741
00:53:15,900 --> 00:53:18,505
所以现在是什么彼此调换
transposes of each other so now what is

742
00:53:18,510 --> 00:53:19,285
然后
then

743
00:53:19,290 --> 00:53:24,715
什么是X就像那样
then what's the X it's just like that

744
00:53:24,720 --> 00:53:31,735
我将使用DL转置的L平方根
I'll use L square root of D L transpose

745
00:53:31,740 --> 00:53:37,435
是的，等一下，是什么
is that right ooh wait a minute what's

746
00:53:37,440 --> 00:53:41,575
不，因为我不会工作
up no that's not gonna work because I

747
00:53:41,580 --> 00:53:44,695
我没有L转置L我有Q
don't have L transpose L where I had Q

748
00:53:44,700 --> 00:53:48,775
转置Q很好不抱歉让我们得到
transpose Q was good no sorry let's get

749
00:53:48,780 --> 00:53:53,605
完全擦除X部分应该
that totally erased the X part should

750
00:53:53,610 --> 00:53:57,385
只是dL转置的平方根
just be a square root of d L transpose

751
00:53:57,390 --> 00:54:01,105
这就是X现在是我们的
that's that's the X is now our

752
00:54:01,110 --> 00:54:03,775
三角矩阵的平方根
triangular matrix the square root of the

753
00:54:03,780 --> 00:54:07,765
枢轴和L转置部分现在
pivots and the L transpose part and now

754
00:54:07,770 --> 00:54:11,035
当我做X转置X然后你看到
when I do X transpose X then you you see

755
00:54:11,040 --> 00:54:13,825
X转置X正确到达X.
X transpose X coming correctly X

756
00:54:13,830 --> 00:54:18,835
转置将是L将是L转置
transpose will be L will be L transpose

757
00:54:18,840 --> 00:54:21,265
转置会给我L平方根
transpose will give me the L square root

758
00:54:21,270 --> 00:54:23,095
d的d的次平方根会给
of D times square root of D will give

759
00:54:23,100 --> 00:54:25,675
D然后L转置是正确的
the D and then the L transpose is right

760
00:54:25,680 --> 00:54:31,435
所以这叫做我试着写的
so this is called the do I try to write

761
00:54:31,440 --> 00:54:34,945
在这里，这是我今天最后一个字
it here this is my last word for today

762
00:54:34,950 --> 00:54:40,405
乔列斯基那些都是这样的
the cholesky those are the this is the

763
00:54:40,410 --> 00:54:43,035
命名的胆甾醇分解
cholesky factorization named after

764
00:54:43,040 --> 00:54:47,055
实际上是法国士兵的法国人
french guy who a French soldier actually

765
00:54:47,060 --> 00:54:51,645
所以LDL转置是胆怯的，那就是
so LD L transpose is cholesky and that's

766
00:54:51,650 --> 00:54:56,365
这很容易计算得更快
that's easy to compute much faster to

767
00:54:56,370 --> 00:54:58,195
计算比平方的本征值
compute than the eigen value of square

768
00:54:58,200 --> 00:55:00,745
根，但这个方根是三角形的
root but this square root is triangular

769
00:55:00,750 --> 00:55:04,045
这个平方根是对称的
this square root is symmetric those are

770
00:55:04,050 --> 00:55:07,045
的两个排序线性代数的条
the two sort of pieces of linear algebra

771
00:55:07,050 --> 00:55:10,485
寻找减少事物的东西
to find things to reduce things to

772
00:55:10,490 --> 00:55:13,395
三角形或减少它们
triangular form or to reduce them to

773
00:55:13,400 --> 00:55:15,915
用对称矩阵连接它们
to connect them with symmetric matrices

774
00:55:15,920 --> 00:55:19,245
好的，谢谢你今天的关注
okay thank you for attention today so

775
00:55:19,250 --> 00:55:25,725
今天我们做了距离矩阵和
today we did the distance matrices and

776
00:55:25,730 --> 00:55:30,545
这是获得X和X的最后一步
this was the final step to get the X and

777
00:55:30,550 --> 00:55:35,355
也越来越重要了
also getting the more most important was

778
00:55:35,360 --> 00:55:38,205
获得神经网络的结构
to get the structure of a neural net

779
00:55:38,210 --> 00:55:41,505
直分离五世的样本
straight separating the V's the sample

780
00:55:41,510 --> 00:55:45,555
从X的向量权重可以
vectors from the X's the weights okay so

781
00:55:45,560 --> 00:55:49,995
星期五我有一位志愿者在说话
Friday I've got one volunteer to talk

782
00:55:50,000 --> 00:55:52,245
关于一个项目，我很拼命
about a project and I'm desperately

783
00:55:52,250 --> 00:55:56,205
寻找更多请发给我一个
looking for more please just send me an

784
00:55:56,210 --> 00:56:00,765
电邮你在你的它会是
email you're in your your it'll be

785
00:56:00,770 --> 00:56:02,740
appreciated or I'll send you an email if

786
00:56:00,770 --> 00:56:06,700
赞赏或我会发送电子邮件，如果

787
00:56:02,750 --> 00:56:06,700
必要的，谢谢
necessary okay thanks