26-Structure of Neural Nets for Deep Learning.srt

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the following content is provided under

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will help MIT OpenCourseWare continue to

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offer high quality educational resources

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for free

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捐款或查看额外的捐款
to make a donation or to view additional

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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，所以这是重要的一天
ok so this is an important day and

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星期五是重要的一天，希望你
Friday was an important day I I hope you

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享受教授的Fras太棒了
enjoyed professor's Fras terrific

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演讲，就像我做你可能
lecture as much as I did you probably

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看见我记笔记疯狂的
saw me taking notes like mad for the

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要写入有关节现在
section that's now to be written about

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随机梯度下降所以他和他
stochastic gradient descent so and he he

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如果你记得并且答应了一个定理
promised a theorem if you remember and

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没有时间，所以他要去
there wasn't time and so he was going to

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发送给我或仍然要发送
send it to me or still is going to send

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对我来说，我会报告我没有得到它
it to me I'll report I haven't got it

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但我会这样，但我会把它带到课堂上
yet so I'll but I'll bring it to class

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等待和希望看到和那个
wait and see hopefully about and that'll

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让我们有机会回顾随机性
give us a chance to review stochastic

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梯度下降中心算法
gradient descent the central algorithm

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深度学习然后今天就是这样
of deep learning and then this today is

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关于深层的中心结构
about the central structure of deep

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神经网和你们中的一些人会知道
neural nets and some of you will know

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已经是他们如何联系的了
already what they how they're connected

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什么是什么功能f
what what the what the function f the

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学习功能它可以把它叫做
learning function it could call it the

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那是学习功能
learning function that that's

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构建了整个系统的目标
constructed the whole system is aiming

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在构建这个函数f
at constructing this function f which

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然后学习训练数据
learns the training data and then

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将它应用于测试数据和
applying it to the test data and the

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奇迹是它在这方面做得很好
miracle is that it does so well in in

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实践那就是改变了
practice that's what has transformed

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深入学习如此重要
deep learning into such a important

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应用程序，以便使，所以这是
application so so so this is

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这虽然第七章已经开始了
this though Chapter seven has been up

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几个月来吉祥物mit.edu斜线
for months on the mascot mit.edu slash

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从数据视觉中学习，我会添加它
learning from data sight and I'll add it

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太棒了，因为那就是你要去的地方
to stellar because that's where you'll

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正在寻找它好然后
be looking for it okay and then the

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第二个反向传播的方式
second the backpropagation the way to

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我可能会计算梯度
compute the gradient I'll probably

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今天达到了这个想法，你会看到
reached that idea today and you'll see

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什么是连锁规则，但它是怎么回事
what it's the chain rule but how is it

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组织好，所以这是什么
organized okay so so what's the

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结构是什么样的深层计划
structure what's the plan for deep

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神经网络从这里开始就是好的开始
neural nets good start starting here so

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我们所拥有的是培训数据，所以我们拥有
what we have is training data so we have

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向量x1到X我应该使用什么
vectors x1 to X what should I use for

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那些样本的培训数量
the number of training of samples that

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我们在训练中接受了培训
we have in the training the training

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数据好Sadie数据确定和每个
data well Sadie for data ok and each

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矢量那些被称为特征向量
vector those are called feature vectors

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所以每个都等于特征向量
so equals feature vectors so each one

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每个X都是呃有这样的特征
each X is uh has like m features so

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也许它可能是我的符号不是
maybe it maybe I my notation isms isn't

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这里很热，我有很多
so hot here I have a whole lot of

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向量让我不要使用下标
vectors let me not use the subscript for

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那些正确的矢量功能
those right away so vectors feature

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向量和每个向量可能都有
vectors and each vector has got maybe so

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我们说m的功能就像我们一样
we say m features like if we were

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测量身高，年龄和体重
measuring height and age and weight and

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所以这些工作就是特色
so on those would be features so the job

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是神经网络的工作
is of the the job of the neural network

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是创造一个和
is to create a and

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我们打算也许我们分类
and we're going to classify maybe we're

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要对男女老少进行分类
going to classify men and women or boys

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和女孩
and girls

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所以我们让它成为一个分类
so our let's make it a classification

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问题只是一个二进制所以
problem a just a binary so the

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分类问题是我们应该做的
classification problem is what should we

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00:04:48,020 --> 00:04:56,205
说减去一个或一个或类似的
say minus one or one which is sort of or

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零或一个或男孩或女孩或猫或狗
zero or one or boy or girl or cat or dog

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或卡车或汽车或反正只有两个
or truck or car or anyway just two

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课程到两个班，所以我要去
classes to two class so I'm just gonna

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做两级分类所以我们
do two class classification so the we

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知道训练数据属于哪个类
know which class the training data is in

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对于每个向量X，我们知道正确的
for each vector X we know the right

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回答所以我们要创建一个函数
answer so we want to create a function

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给出正确的答案，然后
that gives the right answer and then

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我们将在其他数据上使用该功能
we'll use that function on other data

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人们都知道，所以让我写下来
people know so let me write that down

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创建一个X的函数f，以便
create a function f of X that so that

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获得大部分的课程都是正确的
that gets most of gets the class correct

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换句话说，X的f应该是负数
in other words f of X should be negative

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当分类是什么时候
for when when the classification is

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X的减1和f应为正
minus 1 and f of X should be positive

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00:06:06,340 --> 00:06:11,865
当分类加1和as时
when the classification is plus 1 and as

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我们知道我们不一定得到
we know we don't necessarily have to get

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每X个样品都可能正确
every X every sample right that may be

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如果有一些样品过度拟合
overfitting if there's some sample

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得到那个真的很奇怪
that's just truly weird by getting that

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我们正在寻找
right we're going to be looking for

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测试集中的真正奇怪的数据和
truly weird data in the test set and and

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这不是我们想要的，我们希望有一个好主意
that's not a good idea we want we want

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我们试图发现的规则
the rule that we're trying to discover

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该规则的
the rule the

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涵盖几乎所有情况，但不是每个
covers almost all cases but not every

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疯狂怪异的情况好吧这是我们的工作
crazy weird case okay so that's our job

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创建一个X的函数f
to create a function f of X that is

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几乎所有的训练都是正确的
correct on almost all of the training

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数据是的
data yeah

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所以之前我画的图片
so before I draw the picture of the

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网络让我记得记住
network let me just remember remember to

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00:07:16,479 --> 00:07:24,095
提到我不知道的网站游乐场
mention the site playground I don't know

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如果你已经看过那么我会
if you've looked at that so I'm gonna

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问你在tensorflowdotorg的游乐场
ask you playground at tensorflow dot org

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有多少人知道该网站或已经搞砸了
how many know that site or have messed

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只有几个没关系好吧
with it just just a few okay okay so

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它不是一个非常复杂的网站
it's not a very sophisticated site it's

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只有四个例子和例子
got only four examples for examples and

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是的，所以这样一个例子是一个
and the yeah so the so one example is a

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很多积分都有蓝色B表示
whole lot of points there are blue B for

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蓝色里面有一堆点
blue inside a bunch of points that are

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另一套旧的橙色
another set that are old for orange

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橙色，但没关系，所以这两个
orange but okay so those are the two

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类橙色和蓝色所以点X.
classes orange and blue so the points X

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并且特征向量是
and the the feature vector is is the

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特征向量在这里只是X.
feature vector is here is just the X

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坐标的特征是X.
what the coordinates features are the X

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点的Y坐标和我们的工作
Y coordinates of the point and our job

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是找到一个积极的功能
is to find a function that's positive on

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这些点和负面的
these points and negative on those

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分那么有一个简单的模型问题
points so there's a simple model problem

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和
and

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我建议也只是部分是，如果
I recommend well just partly it's if

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你是深度学习的专家
you're a expert in deep learning this is

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对于孩子，但在道德上我在这里
for children but but morally here I

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00:09:05,670 --> 00:09:09,834
当然在玩这个教训
certainly learned from playing in this

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操场上，所以你设置了你，你
playground so you you set the you you

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设置步长您设置还是设置
set the step size do you set it or does

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它说没有，是的，我想你可以
it say no yeah yeah I guess you can

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改变它我没有我不认为我
change it I haven't I don't think I've

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改变它你还说什么哦你
changed it what else do you said oh you

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00:09:31,450 --> 00:09:37,505
你设置了非线性激活
you set the the nonlinear activation the

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00:09:37,510 --> 00:09:44,704
非线性激活函数有效
nonlinear activation function active

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我会说功能，让我走吧
I'll say function and let me just go

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在这里，说什么功能的人
over here and say what functions people

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现在大多使用激活
now mostly use the the activation

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函数被称为riilu发音
function is called riilu pronounced

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不同的方式我不知道我们得到了什么
different ways I don't know how we got

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00:10:04,450 --> 00:10:07,475
进入这个功能的疯狂的事情
into that crazy thing for this function

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00:10:07,480 --> 00:10:14,795
这是零和X所以函数riilu
that is zero and X so the function riilu

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00:10:14,800 --> 00:10:17,735
作为X的函数是最大的
as a function of X is the maximum the

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00:10:17,740 --> 00:10:24,125
大于0和X所以它产生了
larger of 0 and X so it produces the

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要点是它不是线性的和重点
point is it's not linear and the point

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如果我们不允许非线性
is that if we didn't allow non-linearity

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在这里的某个地方
in here somewhere

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我们甚至无法解决这个游乐场
we couldn't even solve this playground

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问题因为如果我们的分类器是
problem because if our classifiers were

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所有线性分类器，如支持
all linear classifiers like support

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矢量机我无法分开
vector machines I couldn't separate the

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蓝色的橙色与飞机我是
blue from the orange with a plane I it's

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以某种方式创造了一些非线性
got to somehow create some nonlinear

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函数可能是函数
function which is maybe the function is

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努力成为一个好的功能
trying to be a good a good function

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可能是r和theta的函数
would be a function of r and theta maybe

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00:11:02,610 --> 00:11:08,855
也许r减5所以也许是距离
maybe r minus 5 so maybe the distance

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00:11:08,860 --> 00:11:10,805
那个让我们假设的那个
out to the that let's suppose that

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距离是5然后r减5
distance is 5 then r minus 5 will be

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对蓝调不利，因为R很小
negative on the Blues because R is small

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我们的负5将是正面的
and our minus 5 will be positive on the

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橘子因为R更大而且
oranges because R is bigger and

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因此我们会有正确的迹象
therefore we will have the right signs

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小于0大于0，它会
less than 0 greater than 0 and it'll

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将此数据归类为此
classify this the this data this

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训练数据是的，所以它必须这样做
training data yeah so so it has to do

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这不是一件难事
that this is not a hard one to do there

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我说两个是四个例子
are four examples as I say two are

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琐事它解决了它找到了一个好处
trivial it solves it finds a good

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功能很好是啊我忘记了
function well yeah I've forgotten

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他们是如此微不足道，他们却没有
they're so trivial they've they don't

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不应该提及然后这是
shouldn't be mentioned and then this is

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00:12:02,829 --> 00:12:09,334
中等测试然后硬测试
the medium test and then the hard test

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00:12:09,339 --> 00:12:13,324
当你有橘子时，你有一种
is when you have oranges you have a sort

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00:12:13,329 --> 00:12:18,394
橙子螺旋和里面你有
of spiral of oranges and inside you have

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00:12:18,399 --> 00:12:23,074
一个煮熟的蓝色缎面螺旋
a spiral of blue satin that was a cooked

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同比饲料，所以你会这么说的
up by a feed so you would so that the

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系统正试图找到一个功能
system is trying to find a function

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这在一个螺旋上是正面的
that's positive on one spiral and

184
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另一方面也是负面的
negative on the other spiral and that

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需要相当多的时间
takes quite a bit of time many many

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00:12:44,220 --> 00:12:50,255
时代我学会了一个时代
epochs I learned what an epoch is did

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00:12:50,260 --> 00:12:52,535
你知道什么是一个时代，我没有
you know what an epoch is I I didn't

188
00:12:52,540 --> 00:12:53,915
知道是不是只是一个花哨的词
know whether it was just a fancy word

189
00:12:53,920 --> 00:12:57,274
用于计算渐变中的步数
for counting the steps in gradient

190
00:12:57,279 --> 00:13:01,715
血统，但它计算它的数量
descent but it counts it counts the

191
00:13:01,720 --> 00:13:05,885
步骤可以，但一个时代就是
steps all right but one epoch is the

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00:13:05,890 --> 00:13:09,965
与大小匹配的步数
number of steps that matches the size of

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00:13:09,970 --> 00:13:12,035
训练数据，如果你有一个
the training data so if you have a

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00:13:12,040 --> 00:13:16,145
百万个样本我们正在评分普通
million samples we're grading ordinary

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00:13:16,150 --> 00:13:18,215
渐渐下降你会做一个
gradient descent you would be doing a

196
00:13:18,220 --> 00:13:20,695
百万你有一百万
million you have a million by

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00:13:20,700 --> 00:13:25,165
当然随机的每一步问题
in problem per step of course stochastic

198
00:13:25,170 --> 00:13:28,435
梯度下降只做一个小批量
gradient descent just does a mini batch

199
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1或32或者其他什么，但无论如何，如果我们
of 1 or 32 or something but anyway if we

200
00:13:33,540 --> 00:13:39,505
是的，所以这是你的数量
had yeah so so it's the number of you

201
00:13:39,510 --> 00:13:42,235
必须做足够的迷你批次
have to do enough mini batches so that

202
00:13:42,240 --> 00:13:44,785
你所涵盖的总数是
the total number you've covered is the

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00:13:44,790 --> 00:13:49,975
相当于一个完整的通过
equivalent of one full run through the

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00:13:49,980 --> 00:13:55,615
训练数据，这很有趣
training data and that was interesting

205
00:13:55,620 --> 00:13:58,915
点没有你在拿起点
point did you pick up that point that in

206
00:13:58,920 --> 00:14:02,005
随机梯度下降你可以
stochastic gradient descent you could

207
00:14:02,010 --> 00:14:04,075
或者你可以做迷你
either do it you could either do a mini

208
00:14:04,080 --> 00:14:08,265
批量然后将它们放回汤中
batch and then put them back in the soup

209
00:14:08,270 --> 00:14:13,135
因此与更换或者你可以只
so with replacement or you could just

210
00:14:13,140 --> 00:14:18,445
从奇迹中提取数据
put your data in some order from wonder

211
00:14:18,450 --> 00:14:23,755
zillion所以这里是第一个X.
zillion so here's here's of a first X

212
00:14:23,760 --> 00:14:28,435
然后更多的X，然后只是和
and then more more X's and then just and

213
00:14:28,440 --> 00:14:30,835
随机化订单，所以你必须
randomize the order so you'd have to

214
00:14:30,840 --> 00:14:33,595
随机化顺序为随机
randomize the order for for stochastic

215
00:14:33,600 --> 00:14:36,175
梯度下降是合理的
gradient descent to be reasonable and

216
00:14:36,180 --> 00:14:38,485
然后拿一个小批量和迷你
then take a mini batch and the mini

217
00:14:38,490 --> 00:14:40,285
在迷你批次中批量批量生产
batch in a mini batch in a mini batch

218
00:14:40,290 --> 00:14:42,085
当你到达底部时，你已经到了
and when you get to the bottom you've

219
00:14:42,090 --> 00:14:44,455
完成了一个时代，然后你就完成了
finished one epoch and then you'd

220
00:14:44,460 --> 00:14:47,365
也许如果你可能再次随机
probably randomize again maybe if you

221
00:14:47,370 --> 00:14:50,035
如果你愿意的话，如果你这样做的话
were if you were doing it if you wanted

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00:14:50,040 --> 00:14:54,535
过着正确的生活并经历迷你
to live right and go through the mini

223
00:14:54,540 --> 00:14:57,655
再批次，可能做一千
batches again and probably do a thousand

224
00:14:57,660 --> 00:15:03,715
无论如何，所以我还没说
times yeah anyway so I haven't said yet

225
00:15:03,720 --> 00:15:07,105
你做了什么X的这个是什么样的但是
what you do what this f of X is like but

226
00:15:07,110 --> 00:15:09,175
你可以在屏幕上看到它
you can sort of see it on the screen

227
00:15:09,180 --> 00:15:14,185
因为它创造了这个功能
because as it creates this function f it

228
00:15:14,190 --> 00:15:18,685
有点情节和你在上面看到的
kind of plots it and what you see on the

229
00:15:18,690 --> 00:15:23,005
屏幕是为此设置的零
screen is the is the zero set for that

230
00:15:23,010 --> 00:15:26,215
功能，所以它是完美的
function so it perfect would be for it

231
00:15:26,220 --> 00:15:28,555
如果我有另一个，那就去零
to go through zero if I had another

232
00:15:28,560 --> 00:15:30,985
颜色哦，我有另一个
color oh I do have another

233
00:15:30,990 --> 00:15:33,205
看这是第一次整体
look this is the first time the whole

234
00:15:33,210 --> 00:15:38,815
学期蓝色已经出现好了所以如果
semester blue has appeared okay so so if

235
00:15:38,820 --> 00:15:42,325
如果该功能在那里是积极的
it if the function was positive there in

236
00:15:42,330 --> 00:15:45,075
在这部分关于蓝调和
the in this part on the Blues and

237
00:15:45,080 --> 00:15:50,605
在该地区以外的负面
negative outside that region for the

238
00:15:50,610 --> 00:15:52,765
橙子，这正是我们想要的
oranges that would be just what we want

239
00:15:52,770 --> 00:15:53,335
对
right

240
00:15:53,340 --> 00:15:55,225
那就是这个小小的东西
that would be what this little

241
00:15:55,230 --> 00:15:58,075
游乐场网站正在创建和上
playground site is creating and on the

242
00:15:58,080 --> 00:16:00,855
屏幕你会看到它你会看到这个
screen you'll see it you'll see this

243
00:16:00,860 --> 00:16:05,515
这条曲线，它穿过零，以便
this curve where it crosses zero so that

244
00:16:05,520 --> 00:16:07,525
假设它穿过零的曲线
curve where it crosses zero is supposed

245
00:16:07,530 --> 00:16:10,615
分开两套一套是
to separate the two sets one set is

246
00:16:10,620 --> 00:16:12,925
积极的一套是负面的
positive one set is negative where it's

247
00:16:12,930 --> 00:16:16,225
零介于两者之间，重点在于它
zero is in between and the point is it's

248
00:16:16,230 --> 00:16:20,065
不是直线，因为我们有
not a straight line because we've got

249
00:16:20,070 --> 00:16:24,355
这个非线性函数就是这个
this nonlinear function this this this

250
00:16:24,360 --> 00:16:33,415
是非线性的，它允许我们拥有
is nonlinear and it allows us to to have

251
00:16:33,420 --> 00:16:38,455
功能就像我们的减五等等
functions like our minus five and so at

252
00:16:38,460 --> 00:16:41,965
五点那就是功能所在
at five that's where the function would

253
00:16:41,970 --> 00:16:43,975
是零，你会看到，在
be zero and that you'll see that on the

254
00:16:43,980 --> 00:16:48,625
屏幕你可能只是去帐篷或
screen you might just go to tents or

255
00:16:48,630 --> 00:16:50,395
当然，在tensorflow的游乐场
playground at tensorflow of course

256
00:16:50,400 --> 00:16:53,995
tensorflow是一个很大很大的系统，这是
tensorflow is a big big system this is

257
00:16:54,000 --> 00:16:59,295
他们孩子的部门但是我
the their child's department but it's I

258
00:16:59,300 --> 00:17:01,945
以为它很不错然后继续
thought it was pretty good and then on

259
00:17:01,950 --> 00:17:05,425
这个网站你决定了多少层
this site you you decide how many layers

260
00:17:05,430 --> 00:17:08,095
每个神经元中会有多少神经元
there will be how many neurons in each

261
00:17:08,100 --> 00:17:11,245
图层，以便您创建结构
layer so you create the structure that

262
00:17:11,250 --> 00:17:15,655
我即将画画，你不会
I'm about to draw and and you won't be

263
00:17:15,660 --> 00:17:21,325
能够解决这两个问题
able to get to solve this two problem to

264
00:17:21,330 --> 00:17:24,675
找到一个学习该数据的函数f
find a function f that learns that data

265
00:17:24,680 --> 00:17:29,275
没有多少层和一个数字
without a number of layers and a number

266
00:17:29,280 --> 00:17:32,575
如果你不给它足够的神经元
of neurons if you don't give it enough

267
00:17:32,580 --> 00:17:36,715
你会看到它挣扎但它不会
you'll see it struggling but it won't it

268
00:17:36,720 --> 00:17:40,135
只是它创造了零集
just it's creating that the the zero set

269
00:17:40,140 --> 00:17:43,345
试图遵循这一点，但它有它
tries to follow this but it but it has

270
00:17:43,350 --> 00:17:45,285
它在某个时候放弃了
it gives up at some point

271
00:17:45,290 --> 00:17:48,025
这个不需要太多层次
this one doesn't take too many layers

272
00:17:48,030 --> 00:17:51,255
而这两个琐碎的例子就是
and the two trivial examples just are

273
00:17:51,260 --> 00:17:55,885
只有少数神经元可以正常工作
just a few neurons do the job okay so

274
00:17:55,890 --> 00:18:00,115
现在对一个人进行一点评论
now so that's a little comment on on one

275
00:18:00,120 --> 00:18:02,815
网站，如果你知道其他人的父亲
website if others father if you know

276
00:18:02,820 --> 00:18:05,065
我应该知道的其他网站
other websites that I should know and

277
00:18:05,070 --> 00:18:08,515
应该引起注意你能发送吗？
should call attention to could you send

278
00:18:08,520 --> 00:18:10,915
我发了一封电子邮件我知道我不是
me an email I you know I'm just not

279
00:18:10,920 --> 00:18:15,355
知道一切，但在那里或如果
aware of everything but out there or if

280
00:18:15,360 --> 00:18:21,355
你知道一个好的卷积
you know a good a good convolutional

281
00:18:21,360 --> 00:18:24,915
神经网络CNN可用于
neural net CNN that is available to

282
00:18:24,920 --> 00:18:30,415
练习你可以给它的地方
practice on where you could give it the

283
00:18:30,420 --> 00:18:34,105
训练设置就是我在说什么
training set that's what I'm talking

284
00:18:34,110 --> 00:18:40,165
在这里，我很高兴知道，因为我
about here I'd be glad to know because I

285
00:18:40,170 --> 00:18:41,875
我只是不知道这一切
I just don't know all that

286
00:18:41,880 --> 00:18:44,455
我应该没问题，那是什么呢
that I should okay so what does the

287
00:18:44,460 --> 00:18:50,005
功能看起来和我说的一样线性
function look like well as I say linear

288
00:18:50,010 --> 00:18:52,645
不会这样做，但线性是一个
isn't going to do it but linear is a

289
00:18:52,650 --> 00:18:56,275
它的很重要的一部分的这
very important part of it of the of this

290
00:18:56,280 --> 00:18:59,515
函数f所以函数f确实有
function f so the function f really has

291
00:18:59,520 --> 00:19:04,705
形式很好，所以我们从这里开始
the form well so we've got we start here

292
00:19:04,710 --> 00:19:10,165
用一个二三三四的矢量是
with a vector of one two three four M is

293
00:19:10,170 --> 00:19:16,155
五是这是矢量X.
five this is the this is the vector X

294
00:19:16,160 --> 00:19:22,015
五个组件好吧让我擦除
five components okay so let me erase

295
00:19:22,020 --> 00:19:24,205
现在好吗
that now okay

296
00:19:24,210 --> 00:19:34,735
那么我们就有了第一层
so then we have layer one with some

297
00:19:34,740 --> 00:19:41,665
分数让我们说N一个是六
number of points let's say N one is six

298
00:19:41,670 --> 00:19:47,725
神经元让我这个简单的我会
neurons let me make this simple I'll

299
00:19:47,730 --> 00:19:50,425
只是有一个层，然后我会
just have that one layer and then I'll

300
00:19:50,430 --> 00:19:52,435
有输出这将是输出
have the output this will be the output

301
00:19:52,440 --> 00:19:55,085
层
layer

302
00:19:55,090 --> 00:19:58,985
它只是一个数字所以
and it's just gonna be one number so

303
00:19:58,990 --> 00:20:03,004
每一个所以我会有一个矩阵
each of these so I'm gonna have a matrix

304
00:20:03,009 --> 00:20:09,664
一个让我从这一个人的意志
a one that takes me from this a one will

305
00:20:09,669 --> 00:20:13,134
这将是5乘5乘5
be this will be 5 by no six by five

306
00:20:13,139 --> 00:20:17,014
因为我想要六个输出和五个输出
because I want six outputs and five

307
00:20:17,019 --> 00:20:19,865
输入六乘五矩阵，所以我有30
inputs six by five matrix so I have 30

308
00:20:19,870 --> 00:20:27,935
在那里选择权重等等
weights to choose there and so the the

309
00:20:27,940 --> 00:20:32,815
所散发出来的y是要为y1
the y that comes out is gonna be y 1

310
00:20:32,820 --> 00:20:38,335
将是X0的1倍，因此X0是
will be a 1 times X 0 so X 0 is the is

311
00:20:38,340 --> 00:20:42,744
具有五个组件的特征向量
the feature vector with five components

312
00:20:42,749 --> 00:20:46,264
所以它是一个纯线性的东西
so it's so that's a purely linear thing

313
00:20:46,269 --> 00:20:53,164
但我们也想要一个偏移函数
but we also want a an offset function

314
00:20:53,169 --> 00:20:55,774
偏移向量所以这是一个载体，这是
offset vector so that's a vector this is

315
00:20:55,779 --> 00:20:59,465
然后这就是即将到来的
then this is that was a y that's coming

316
00:20:59,470 --> 00:21:03,485
out有六个组件，一个是六个组件
out has six components the a one is six

317
00:21:03,490 --> 00:21:08,014
由五个X化为乌有是五经之一，
by five the X naught was five by one and

318
00:21:08,019 --> 00:21:10,624
那么当然这是六经之一，所以
then of course this is six by one so

319
00:21:10,629 --> 00:21:14,284
这些都是权重
these are the these are the weights well

320
00:21:14,289 --> 00:21:16,654
这些都是这些我会称之为全部
these are these yeah I'll call them all

321
00:21:16,659 --> 00:21:26,105
这样计算重量的权重
weight weights to compute so in this so

322
00:21:26,110 --> 00:21:29,674
这些都是通常连接的图片
these are connected usual picture is to

323
00:21:29,679 --> 00:21:34,445
展示这些所有这些联系我会
show these all these connections I'll

324
00:21:34,450 --> 00:21:38,975
只是把它们中的一些放在这里我们
just put in some of them so in here we

325
00:21:38,980 --> 00:21:46,914
有三十六个参数
have thirty plus six parameters

326
00:21:46,919 --> 00:21:49,804
三十六个参数然后我要去
thirty-six parameters and then I'm going

327
00:21:49,809 --> 00:21:52,924
关闭这将是一个非常
to close this it's gonna be a very

328
00:21:52,929 --> 00:22:01,855
浅薄的东西，这将只是公正
shallow thing so that will be just just

329
00:22:01,860 --> 00:22:07,914
一点一点，是的，没关系
one by six yeah just okay

330
00:22:07,919 --> 00:22:10,595
对，所以我们只得到一个输出
right so we're just getting one output

331
00:22:10,600 --> 00:22:13,885
所以这只是一个向量
so it's that's just a vector at this

332
00:22:13,890 --> 00:22:19,115
最后一点，但当然是整体
final point but of course the the whole

333
00:22:19,120 --> 00:22:21,635
深度神经网络的想法就是你
idea of deep neural Nets is that you

334
00:22:21,640 --> 00:22:26,135
有很多层，所以你有36个
have many layers so you have that 36

335
00:22:26,140 --> 00:22:28,294
更现实，就像数十个
more realistically as in the tens of

336
00:22:28,299 --> 00:22:31,445
成千上万，你有多次
thousands and you have it multiple times

337
00:22:31,450 --> 00:22:39,185
而这个想法似乎是第一个
and the idea seems to be that the first

338
00:22:39,190 --> 00:22:41,914
你可以分类的那个层
layer that you can sort of separate what

339
00:22:41,919 --> 00:22:47,885
第一层了解数据和来自
layer one learns about the data and from

340
00:22:47,890 --> 00:22:49,865
第二层了解数据
what layer two learns about the data

341
00:22:49,870 --> 00:22:54,904
层有一个由是一个明显
layer one there's a one apparently by

342
00:22:54,909 --> 00:22:58,895
只是照顾这个计算
just looking after the computation this

343
00:22:58,900 --> 00:23:02,885
了解了一些有关的基本事实
learned some some basic facts about the

344
00:23:02,890 --> 00:23:06,715
这将学习下一个a到的数据
data this learns the next next a to

345
00:23:06,720 --> 00:23:12,174
会在这里进一步了解更多
which would go in here would learn more

346
00:23:12,179 --> 00:23:15,395
细节然后三个会学到更多
detail and then a three would learn more

347
00:23:15,400 --> 00:23:18,515
细节，所以我们会有一些
details so we would have a number of

348
00:23:18,520 --> 00:23:23,424
层和它的构造
layers and it's that construction that

349
00:23:23,429 --> 00:23:27,304
已经使神经网络成功但我
has made neural net successful but I

350
00:23:27,309 --> 00:23:30,304
还没完，因为现在是
haven't finished because right now it's

351
00:23:30,309 --> 00:23:33,544
我现在只有线性的
only linear right now I just have

352
00:23:33,549 --> 00:23:36,455
现在我在这里喝两杯酒
alcohol at a two in here right now I

353
00:23:36,460 --> 00:23:38,615
只会有一个矩阵乘法
would just have a matrix multiplication

354
00:23:38,620 --> 00:23:41,825
应用1然后应用2但是在
apply a 1 and then apply a 2 but in

355
00:23:41,830 --> 00:23:50,945
之间是一个一个接一个的动作
between there's a is a one by one action

356
00:23:50,950 --> 00:23:57,695
在这个功能上
on each by this function

357
00:23:57,700 --> 00:24:03,015
以便该函数对该数字起作用
so that function acts on that number to

358
00:24:03,020 --> 00:24:05,205
再给那个号码还是给
give that number back again or to give

359
00:24:05,210 --> 00:24:09,255
零，所以在那里有真正的哦
zero so in in there in there is real ooh

360
00:24:09,260 --> 00:24:13,215
就在那里，它变得真实
right in there in this it becomes real

361
00:24:13,220 --> 00:24:18,255
哦，每六个六个副本的反应
ooh on each six six copies of reacting

362
00:24:18,260 --> 00:24:21,255
这六个数字中的每一个都是正确的
on each of those six numbers right

363
00:24:21,260 --> 00:24:28,365
所以真正的x1来自于y1
so that really x1 comes from y1 by

364
00:24:28,370 --> 00:24:33,465
将riilu应用于它然后给予它
applying riilu to it then that gives the

365
00:24:33,470 --> 00:24:36,945
X所以这里是线性的Y.
X so here are the Y's from the linear

366
00:24:36,950 --> 00:24:40,125
一部分，这里有这就是为什么那些在X
part and here are the X that's why ones

367
00:24:40,130 --> 00:24:43,155
这是一个矢量y1，线性来自于
that's a vector y1 the linear from just

368
00:24:43,160 --> 00:24:47,595
线性加一个仿射图线性加
a linear plus an affine map linear plus

369
00:24:47,600 --> 00:24:48,435
不变
constant

370
00:24:48,440 --> 00:24:52,985
这很好，然后下一步是
that's a fine and then the next step is

371
00:24:52,990 --> 00:24:56,235
逐个组件我们应用它
component by component we apply this

372
00:24:56,240 --> 00:25:02,265
功能，我们得到X-X是什么和
function and we get X - X what yeah and

373
00:25:02,270 --> 00:25:05,205
然后再一次又一次这样做
then do it again and again and again so

374
00:25:05,210 --> 00:25:07,935
你看到我的功能如何？
do you see that the function how do i

375
00:25:07,940 --> 00:25:14,505
现在描述x的函数f所以
describe now the function f of x so the

376
00:25:14,510 --> 00:25:22,745
学习功能依赖于
learning function which depends on the

377
00:25:22,750 --> 00:25:32,265
关于a和b的权重是如此
on the weights on the a's and b's is so

378
00:25:32,270 --> 00:25:38,345
我从XI开始申请F1a1
I start with an X I apply F 1 a 1 to it

379
00:25:38,350 --> 00:25:42,045
是的，让我这样做，这是什么
yeah let me do this so what what this is

380
00:25:42,050 --> 00:25:45,645
函数fX的XF的打算
the function f of X f of X is going to

381
00:25:45,650 --> 00:25:55,875
是F3，假设是X12的F1的F2
be F 3 let's say of F 2 of F 1 of X 1 2

382
00:25:55,880 --> 00:25:59,835
3个括号没错，所以这是一个链条
3 parentheses right okay so it's a chain

383
00:25:59,840 --> 00:26:03,405
你可以说F是正确的
you could say F is a what's the right

384
00:26:03,410 --> 00:26:07,275
如果我采取一系列功能的话
word for a chain of functions if I take

385
00:26:07,280 --> 00:26:10,005
函数的函数是
a function of a function which is the

386
00:26:10,010 --> 00:26:11,515
我使用单词链的原因
reason I use the word chain

387
00:26:11,520 --> 00:26:14,515
链规则给出了导数
the chain rule gives the derivative so a

388
00:26:14,520 --> 00:26:18,195
一个函数的函数功能
function of a function of a function

389
00:26:18,200 --> 00:26:22,465
这就是所谓的作曲创作
that's called composition composing

390
00:26:22,470 --> 00:26:26,895
功能所以这是一个作文我
function so this is a composition I

391
00:26:26,900 --> 00:26:29,605
不知道是否有标准符号
don't know if there's a standard symbol

392
00:26:29,610 --> 00:26:33,015
从F1开始并做一些
for starting with F 1 and do some

393
00:26:33,020 --> 00:26:36,505
组成和做一些组成和
composition and do some composition and

394
00:26:36,510 --> 00:26:39,415
那些什么是那些单独的F那些
now what what are those separate F those

395
00:26:39,420 --> 00:26:43,675
单独的FS是或者那样的
separate FS are the or the so the

396
00:26:43,680 --> 00:26:50,515
单独的FS是向量的F1
separate FS are the F 1 of a vector

397
00:26:50,520 --> 00:26:53,725
将调整真实的部分
would be in tunes the real part the

398
00:26:53,730 --> 00:27:02,005
非线性部分1X0加B1所以2
nonlinear part of a 1 X 0 plus B 1 so 2

399
00:27:02,010 --> 00:27:07,705
部分你做线性或仿射地图
parts you do the linear or affine map on

400
00:27:07,710 --> 00:27:12,105
你的特征向量，然后你
your feature vector and then you

401
00:27:12,110 --> 00:27:16,585
您应用该组件的组件
component by component you apply that

402
00:27:16,590 --> 00:27:20,065
非线性函数，它需要一些
nonlinear function and it took some

403
00:27:20,070 --> 00:27:22,705
年中非线性函数之前
years before that nonlinear function

404
00:27:22,710 --> 00:27:28,465
成为人们想象的最爱
became a big favorite people imagine

405
00:27:28,470 --> 00:27:30,535
这对我来说更重要
that it was better it was important to

406
00:27:30,540 --> 00:27:33,895
原来有顺畅的功能
have a smooth function so the original

407
00:27:33,900 --> 00:27:37,075
功能是sigmoids
functions were sigmoids

408
00:27:37,080 --> 00:27:44,845
喜欢s曲线，但当然它
like s curves but but of course it

409
00:27:44,850 --> 00:27:46,675
原来，试验表明，
turned out that experiment showed that

410
00:27:46,680 --> 00:27:49,195
这样做的效果更好是的
this worked even better yes so that

411
00:27:49,200 --> 00:27:52,555
将是F1然后F2将具有
would be F 1 and then F 2 would have the

412
00:27:52,560 --> 00:27:54,625
相同的形式和F3将具有相同的形式
same form and F 3 would have the same

413
00:27:54,630 --> 00:27:58,195
因此，这可能有36个重量和
form so maybe this had 36 weights and

414
00:27:58,200 --> 00:28:00,505
下一个会有另一个号码
the next one would have another number

415
00:28:00,510 --> 00:28:02,815
和下一个号码所以我们是
and the next another number so we're

416
00:28:02,820 --> 00:28:05,455
让你变得相当复杂
getting you get quite complicated

417
00:28:05,460 --> 00:28:09,895
你知道，由作曲组成的功能
functions by composition by you know

418
00:28:09,900 --> 00:28:13,015
喜欢X的正弦或正弦的正弦
like e to the sine of X or e to the sine

419
00:28:13,020 --> 00:28:15,775
X的对数或类似的东西
of the logarithm of X or things like

420
00:28:15,780 --> 00:28:17,795
那
that

421
00:28:17,800 --> 00:28:21,335
纯粹的群众问你能得到什么
pure masses asked could you get what

422
00:28:21,340 --> 00:28:23,435
功能可以让你知道尝试
functions can you get you know try to

423
00:28:23,440 --> 00:28:25,895
现在想起他们都是什么样的
think of them all now what kind of

424
00:28:25,900 --> 00:28:28,535
功能我们在这里有什么可以说的
functions do we have here what can I say

425
00:28:28,540 --> 00:28:32,195
关于X的f作为数学函数
about f of X as a function as a math

426
00:28:32,200 --> 00:28:36,095
人是什么样的功能
person what what kind of a function is

427
00:28:36,100 --> 00:28:41,915
因此它是由矩阵和
it so it's created out of matrices and

428
00:28:41,920 --> 00:28:45,235
线性或仿射图中的向量
vectors out of a linear or affine map

429
00:28:45,240 --> 00:28:50,315
然后是非线性的
followed by a nonlinear by that

430
00:28:50,320 --> 00:28:54,845
特别是非线性函数那么
particular nonlinear function so what

431
00:28:54,850 --> 00:28:57,365
我有一种功能
kind of a function is it well I've

432
00:28:57,370 --> 00:28:59,735
虽然写下了这些话
written though with those words down up

433
00:28:59,740 --> 00:29:05,335
这里在X的f将是一个
here at f of X is going to be a

434
00:29:05,340 --> 00:29:09,925
连续分段线性函数
continuous piecewise linear function

435
00:29:09,930 --> 00:29:15,695
因为每一步都是连续的
because every step is continuous that's

436
00:29:15,700 --> 00:29:18,635
连续函数线性函数
a continuous function linear functions

437
00:29:18,640 --> 00:29:20,345
是连续的功能所以我们当然
are continuous function so we certainly

438
00:29:20,350 --> 00:29:23,375
有我们正在做的一个组成
have we're taking a composition of

439
00:29:23,380 --> 00:29:25,265
连续功能，所以它是连续的
continuous function so it's continuous

440
00:29:25,270 --> 00:29:29,435
部分是线性的
and it's piecewise linear because part

441
00:29:29,440 --> 00:29:31,715
它是线性的，部分是线性的
of it is linear and part of it is

442
00:29:31,720 --> 00:29:37,675
分段线性所以这是一些
piecewise linear so so this is some

443
00:29:37,680 --> 00:29:48,055
连续分段线性函数
continuous piecewise linear function

444
00:29:48,060 --> 00:29:55,985
来自XX和M的功能
function from from of X X and M

445
00:29:55,990 --> 00:30:01,994
尺寸没关系
dimensions okay

446
00:30:01,999 --> 00:30:07,435
所以我有一个小小的数学问题
so one one little math question which I

447
00:30:07,440 --> 00:30:12,134
认为有助于向迈克理解
think helps to understand to Mike like

448
00:30:12,139 --> 00:30:18,414
吞下一连串的想法
to swallow the idea of a chain of of the

449
00:30:18,419 --> 00:30:20,514
我们在这里有一种线性链
kind of chain we have here of linear

450
00:30:20,519 --> 00:30:28,494
其次是riilu所以这就是我的
followed by riilu so so here's my

451
00:30:28,499 --> 00:30:30,624
问题这是一个问题，我要
question this is a question I'm going to

452
00:30:30,629 --> 00:30:31,134
问
ask

453
00:30:31,139 --> 00:30:34,195
顺便说一句，回传就是
and by the way back propagation is

454
00:30:34,200 --> 00:30:36,685
当Z相当肯定会来
certainly going to come when Z rather

455
00:30:36,690 --> 00:30:39,475
比今天这是一个重要话题
than today that's a major topic in

456
00:30:39,480 --> 00:30:44,294
本身让我继续这个
itself so let me keep going with this

457
00:30:44,299 --> 00:30:48,294
功能你可以得到任何功能
function could you get any function

458
00:30:48,299 --> 00:30:51,774
无论如何这种方式你都不会得到
whatsoever this way well no you only get

459
00:30:51,779 --> 00:30:54,284
连续分段线性函数
continuous piecewise linear functions

460
00:30:54,289 --> 00:30:59,215
这是一个有趣的案例让我
that's a it's an interesting case let me

461
00:30:59,220 --> 00:31:02,514
只是问你一个练习
just ask you well one of the exercises

462
00:31:02,519 --> 00:31:06,024
如果我连续两次，我会说
says if if I took two continuous

463
00:31:06,029 --> 00:31:08,924
分段线性函数你有吗？
piecewise linear functions do you have a

464
00:31:08,929 --> 00:31:12,835
接下来的20分钟是企图
the next 20 minutes are an attempt to

465
00:31:12,840 --> 00:31:16,044
给我们的图形的图片
give us a picture of the graph of a

466
00:31:16,049 --> 00:31:19,605
分段线性函数那么什么
piecewise linear function so what what

467
00:31:19,610 --> 00:31:23,215
比如说两个变量的函数
say in say a function of two variables

468
00:31:23,220 --> 00:31:29,424
所以我的M等于2，我画了它
so I have M equal to two and I draw its

469
00:31:29,429 --> 00:31:32,455
graph好帮我画出它的图表
graph okay help me to draw its graph so

470
00:31:32,460 --> 00:31:37,605
这将是f1x1x2和f的图形
this would be a graph of f of x1 x2 and

471
00:31:37,610 --> 00:31:39,894
它将是连续的
it's going to be continuous and

472
00:31:39,899 --> 00:31:42,174
分段线性所以它的图形是什么
piecewise linear so what does its graph

473
00:31:42,179 --> 00:31:44,994
看起来就是那个问题
look like that's that's the question

474
00:31:44,999 --> 00:31:47,485
什么是分段的图表呢？
what's the graph of a piecewise then

475
00:31:47,490 --> 00:31:51,985
你的功能看起来很好
your function looks like well it's got

476
00:31:51,990 --> 00:31:56,965
变化之间的扁平碎片
flat pieces right in between the change

477
00:31:56,970 --> 00:32:02,294
从美杜莎分段，这意味着它
from a Medusa piecewise that means it's

478
00:32:02,299 --> 00:32:05,715
得到不同的东西，但在一块之内
got different pieces but within a piece

479
00:32:05,720 --> 00:32:10,764
它是线性的，旁边的部分适合
it's linear and the pieces fit next to

480
00:32:10,769 --> 00:32:12,595
因为它是相互适应的
fit with each other because it's

481
00:32:12,600 --> 00:32:14,815
连续所以我视觉上
continuous so I visually

482
00:32:14,820 --> 00:32:20,065
这就是折纸这就好了
is that the well it's like origami this

483
00:32:20,070 --> 00:32:25,164
折纸理论几乎是正确的
is the theory of origami almost so right

484
00:32:25,169 --> 00:32:27,504
折纸，你拿一个扁平的东西和你
origami you take a flat thing and you

485
00:32:27,509 --> 00:32:32,754
沿着直的折叠折叠它所以
fold it along straight folds so so

486
00:32:32,759 --> 00:32:34,764
与折纸有什么不同可能不是
what's different from origami maybe not

487
00:32:34,769 --> 00:32:37,455
许多
much

488
00:32:37,460 --> 00:32:41,995
也许折纸允许比我们更多
well maybe origami allows more than we

489
00:32:42,000 --> 00:32:44,845
允许你在这里折纸
allow here origami would allow you to

490
00:32:44,850 --> 00:32:48,205
将它折叠起来然后折纸就可以了
fold it up and over so origami would

491
00:32:48,210 --> 00:32:52,164
因为你，给你一个多值的东西
give you a multivalued thing because you

492
00:32:52,169 --> 00:32:54,085
知道它有一个顶部和底部和
know it's got a top and a bottom and

493
00:32:54,090 --> 00:32:57,975
其他折叠，这只是出去
other folds this is just going out to

494
00:32:57,980 --> 00:33:03,595
扁平的无限和问题
infinity in flat pieces and the question

495
00:33:03,600 --> 00:33:06,685
将会有多少件让我问
will be how many pieces so let me ask

496
00:33:06,690 --> 00:33:10,105
你这个问题我有多少件
you that question how many pieces do I

497
00:33:10,110 --> 00:33:15,024
你知道我的意思吗？
have you see what I mean by a piece so

498
00:33:15,029 --> 00:33:17,185
所以我想，有一个图表
so I'm thinking of a graph that has

499
00:33:17,190 --> 00:33:20,664
这些扁平的碎片和它们相连
these flat pieces and they're connected

500
00:33:20,669 --> 00:33:24,685
沿直的直边和那些
along straight straight edges and those

501
00:33:24,690 --> 00:33:29,065
直边来从实际
straight edges come from from the real

502
00:33:29,070 --> 00:33:31,705
操作得好，有两个
ooh operation that well that's got two

503
00:33:31,710 --> 00:33:35,095
实际上我们可以在一天内完成
pieces actually we could do in one day

504
00:33:35,100 --> 00:33:37,195
有一天我们可以数一数
and one day we could count the number of

505
00:33:37,200 --> 00:33:39,955
件很容易，所以会是什么
pieces pretty easily so what would what

506
00:33:39,960 --> 00:33:41,784
将是一个分段线性让我放
would be a piecewise linear let me put

507
00:33:41,789 --> 00:33:44,695
它在这里的这一边，并删除它-
it over here on this side and erase it -

508
00:33:44,700 --> 00:33:47,275
好的
okay

509
00:33:47,280 --> 00:33:52,525
所以这里的枫树是连续的
so here's a maple one a continuous

510
00:33:52,530 --> 00:33:55,405
分段线性
piecewise linear

511
00:33:55,410 --> 00:34:03,294
我只是绘制它的图表就好了
f I'll just draw its graph so okay so

512
00:34:03,299 --> 00:34:09,715
它是直件直件
it's got straight pieces straight pieces

513
00:34:09,720 --> 00:34:13,235
就像这样，是的，你知道它是一个
like so yeah you've got the idea it's a

514
00:34:13,240 --> 00:34:15,545
有时人们说破线型
broken line type sometimes people say

515
00:34:15,550 --> 00:34:18,035
虚线，但我不确定
broken line but that I'm never sure

516
00:34:18,040 --> 00:34:20,845
这是一个很好的说明
that's a good description of this

517
00:34:20,850 --> 00:34:23,795
分段线性连续所以它是
piecewise linear continuous so it's

518
00:34:23,800 --> 00:34:28,164
连续因为这些碎片相遇而且
continuous because the pieces meet and

519
00:34:28,169 --> 00:34:32,495
它的分段线性显然是可以的
it's piecewise linear obviously okay so

520
00:34:32,500 --> 00:34:36,845
这就是我所拥有的那种画面
that's the kind of picture I have for a

521
00:34:36,850 --> 00:34:40,174
现在我的一个变量的功能
function of one variable now that my

522
00:34:40,179 --> 00:34:43,835
问题是一种帮助尝试
question is as an aid to try to

523
00:34:43,840 --> 00:34:49,265
在2d中可视化这个功能就是看
visualize this function in 2d is to see

524
00:34:49,270 --> 00:34:52,144
如果我们可以计算这些碎片，看看我们是否可以
if we can count the pieces see if we can

525
00:34:52,149 --> 00:34:54,755
算上件是和该
count the pieces yes and that for that

526
00:34:54,760 --> 00:34:58,895
所以我在这个笔记中找到了它
so that's in this notes I found it in a

527
00:34:58,900 --> 00:35:05,765
五位作者的论文来自于
paper by five authors for a from for a

528
00:35:05,770 --> 00:35:12,485
会议和所以实际上整个世界
meeting and so actually the whole world

529
00:35:12,490 --> 00:35:19,895
神经网络就是会议
of neural nets it's it's the conferences

530
00:35:19,900 --> 00:35:21,635
每隔几年是那
every couple of years that that

531
00:35:21,640 --> 00:35:25,715
每个人都准备提交超过
everybody prepares for submits more than

532
00:35:25,720 --> 00:35:28,205
一张纸，所以它是一种分段的
one paper so it's kind of a piecewise

533
00:35:28,210 --> 00:35:33,575
线性会议，而这些都是在
linear conference and those are the the

534
00:35:33,580 --> 00:35:37,145
大型会议没关系，所以这就是后面
big conferences okay so this is the back

535
00:35:37,150 --> 00:35:39,185
传播部分，我想看看
propagation section and I want to look

536
00:35:39,190 --> 00:35:46,895
在这好吧所以我要表现出来
at this at the okay so I'm gonna show

537
00:35:46,900 --> 00:35:49,235
Kleinberg和其他人的这篇论文
this paper by Kleinberg and for others

538
00:35:49,240 --> 00:35:53,015
克莱因伯格是一个计算机科学专业的学生
Kleinberg is a he's a computer science

539
00:35:53,020 --> 00:35:56,104
指导康奈尔，他是这里的博士
guide Cornell he was a PhD from here in

540
00:35:56,109 --> 00:36:00,325
数学，他是一个非常
math and he's a very

541
00:36:00,330 --> 00:36:05,165
并且重要的人不是这样
and significant person on not on not so

542
00:36:05,170 --> 00:36:09,535
很多神经网络就像
much on neural networks as just the

543
00:36:09,540 --> 00:36:14,375
整个计算机的整个部分
whole this whole part of computer

544
00:36:14,380 --> 00:36:17,975
科学是正确的，无论如何他们和其他
science right so anyway they and other

545
00:36:17,980 --> 00:36:20,075
人们都问这个同样的
people to have have asked this same

546
00:36:20,080 --> 00:36:23,975
问题假设你认为我在
problem suppose you have suppose I'm in

547
00:36:23,980 --> 00:36:29,255
两个变量，假设是什么
two variables and suppose so what are

548
00:36:29,260 --> 00:36:33,965
你现在想象的是表面的
you imagining now for the surface of the

549
00:36:33,970 --> 00:36:38,375
X的f和它的原因图
graph of this of f of X and why it has

550
00:36:38,380 --> 00:36:42,205
这些线条折叠线
these it has these lines fold lines

551
00:36:42,210 --> 00:36:48,965
好吧，我想它有折叠线，使
right I'm thinking it has fold lines so

552
00:36:48,970 --> 00:36:50,555
我开始可以从一个完整的开始
I start could start with a complete

553
00:36:50,560 --> 00:36:53,075
飞机和我沿着一条线折叠
plane and I fold it along along one line

554
00:36:53,080 --> 00:36:57,515
所以现在就像RayLu是一个半场比赛
so now it's like Ray Lu is one play half

555
00:36:57,520 --> 00:36:59,885
飞机他们要去另一半
plane they're going to a different half

556
00:36:59,890 --> 00:37:03,515
每个人都带着它飞行，现在我
plane there everybody with it and now I

557
00:37:03,520 --> 00:37:08,315
采取表面哪个功能
take that function that surface which

558
00:37:08,320 --> 00:37:11,584
只有两部分，我放入另一部分
just has two parts and I put in another

559
00:37:11,589 --> 00:37:13,315
折
fold

560
00:37:13,320 --> 00:37:17,155
好吧，我现在有多少部分
okay how many parts have I got now I

561
00:37:17,160 --> 00:37:21,844
我认为四点是正确的四个部分是的
think four am i right four parts yes

562
00:37:21,849 --> 00:37:25,584
因为这与此不同
because this will be different from this

563
00:37:25,589 --> 00:37:28,805
因为它沿着那条线折叠了
because it was folded along that line so

564
00:37:28,810 --> 00:37:31,675
这些将是他们的四个不同的部分
these will be four different pieces they

565
00:37:31,680 --> 00:37:34,354
它们在中心具有相同的价值
they have the same value at the center

566
00:37:34,359 --> 00:37:37,084
在那里和他们匹配沿着
there and this and they match along the

567
00:37:37,089 --> 00:37:41,045
线条，但有平坦的数量
lines but there's so the number of flat

568
00:37:41,050 --> 00:37:44,405
这件事是四件，所以这就是
pieces is four for this so that's with

569
00:37:44,410 --> 00:37:47,555
两折，现在我只想问你
two folds and now I just want to ask you

570
00:37:47,560 --> 00:37:51,385
M折叠有多少件
with M folds how many pieces are there

571
00:37:51,390 --> 00:37:56,825
我可以达到三倍，所以我会
can I get up to three folds so I'm gonna

572
00:37:56,830 --> 00:37:59,825
寻找褶皱的数量，以便让我
look for the number of folds so let me

573
00:37:59,830 --> 00:38:03,114
让我用得到的符号也许[R
let me just use a get a notation maybe R

574
00:38:03,119 --> 00:38:10,845
R为平坦件数的数
R is the number of number of flat pieces

575
00:38:10,850 --> 00:38:23,215
而M是我的X维度
and M is the dimension of X in my

576
00:38:23,220 --> 00:38:28,975
图片是2和n是多少
picture it's 2 and n is the number of

577
00:38:28,980 --> 00:38:34,885
折叠让我再说一遍
folds so let me say it again

578
00:38:34,890 --> 00:38:41,905
我坐飞机，我会折起来平面
I'm taking a plane I'll fold it plane

579
00:38:41,910 --> 00:38:44,665
因为维度是2
because the dimension was 2

580
00:38:44,670 --> 00:38:51,625
我会折叠它折叠n倍
I'll fold it fold it n times how many

581
00:38:51,630 --> 00:38:52,165
件
pieces

582
00:38:52,170 --> 00:39:05,295
这将是多少扁平件
how many flat pieces so this would be

583
00:39:05,300 --> 00:39:09,265
这将是一个中心步骤
this would be a central step in

584
00:39:09,270 --> 00:39:13,345
了解他们的功能有多接近
understanding how close their function

585
00:39:13,350 --> 00:39:15,085
你知道你有什么自由
you know what freedom you have in the

586
00:39:15,090 --> 00:39:19,195
例如，函数f可以吗？
function f for example can you

587
00:39:19,200 --> 00:39:23,005
通过逼近任意连续函数
approximate any continuous function by

588
00:39:23,010 --> 00:39:26,035
通过采取这些功能之一
one of these functions f by taking

589
00:39:26,040 --> 00:39:28,915
足够的折叠似乎是答案
enough folds seems like the answer

590
00:39:28,920 --> 00:39:31,315
应该是肯定的，这是肯定的
should be yes and it is yes so that's

591
00:39:31,320 --> 00:39:35,545
对于纯粹的数学而言，这是一个很重要的
the that's one big for pure math that's

592
00:39:35,550 --> 00:39:39,445
一个问题就是这一类
one question is the is this class of

593
00:39:39,450 --> 00:39:43,135
功能普遍，所以普遍性
functions universal so the universality

594
00:39:43,140 --> 00:39:47,114
定理就是说任何一个
theorem would be to say that any any

595
00:39:47,119 --> 00:39:51,354
函数正弦X无论如何
function sine X whatever could be

596
00:39:51,359 --> 00:39:56,155
近似尽可能接近一个
approximated as close as you like by one

597
00:39:56,160 --> 00:39:59,755
这些家伙有足够的折叠和结束
of these guys with enough fold and over

598
00:39:59,760 --> 00:40:03,885
在这里，我们有点做得更多
here we're kind of making it more

599
00:40:03,890 --> 00:40:05,965
数字，我们将计算
numerical we're going to count the

600
00:40:05,970 --> 00:40:08,545
件数，只是怎么看
number of pieces to just to see how

601
00:40:08,550 --> 00:40:11,335
很快就会成长，所以会发生什么
quickly do they grow so what happens

602
00:40:11,340 --> 00:40:14,935
在这里，所以我现在有四个折叠
here so I have four folds now this right

603
00:40:14,940 --> 00:40:19,645
现在我有n等于我mm这里是两个
now I have n equal to I mm is two here

604
00:40:19,650 --> 00:40:22,614
在这张照片中
in this in this picture

605
00:40:22,619 --> 00:40:24,864
而我正试图绘制表面和
and I'm trying to draw the surface and

606
00:40:24,869 --> 00:40:28,195
在这里，我接受了两个，我接受了
here I've put in two did I take in yeah

607
00:40:28,200 --> 00:40:32,034
两折，现在我要去
two folds and now I'm gonna go up to

608
00:40:32,039 --> 00:40:35,754
三折好吧让我折叠起来
three folds okay so let me fold it along

609
00:40:35,759 --> 00:40:43,364
那条线我现在有多少件
that line how many pieces have I got now

610
00:40:43,369 --> 00:40:49,945
让我们看看我能算出那些碎片吗？
let's see can I count those pieces is it

611
00:40:49,950 --> 00:40:54,774
如果我这样做，那么什么是公式呢？
seven so what is a formula what if I do

612
00:40:54,779 --> 00:41:01,044
另一个结果发生了什么，我们做了什么
another fold what happens what end we do

613
00:41:01,049 --> 00:41:09,294
另一个文件夹是的，也许是的
another folder yeah yeah maybe that's

614
00:41:09,299 --> 00:41:13,945
会是它这是一个很好的问题
gonna be it it's a kind of nice question

615
00:41:13,950 --> 00:41:15,925
因为这是可视化的
because it is care to visualize this

616
00:41:15,930 --> 00:41:22,104
事情还好，所以发生了多少怎么样
thing okay so what happened how many how

617
00:41:22,109 --> 00:41:25,464
如果我投入，那些线路中的很多都会是
many of those lines will be if I put in

618
00:41:25,469 --> 00:41:31,494
第四线多少呀多少新
a fourth line how many yeah how many new

619
00:41:31,499 --> 00:41:33,984
折叠我是否创造了那种
folds do I create that's kind of the

620
00:41:33,989 --> 00:41:35,724
问题，我假设第四
question and I'm assuming that fourth

621
00:41:35,729 --> 00:41:38,484
线不经过任何的这些
line doesn't go through any of these

622
00:41:38,489 --> 00:41:41,984
指出它处于一般位置
points it's sort of in general position

623
00:41:41,989 --> 00:41:45,834
是的，所以我做了第四行
yeah so I put in a fourth line done it

624
00:41:45,839 --> 00:41:49,695
那没关系，所以这里发生了什么
there it is okay so what happened here

625
00:41:49,700 --> 00:41:54,175
有多少新人创造了多少
how many new ones did it create how many

626
00:41:54,180 --> 00:41:59,724
新的创造了它让我做到了
new ones did it create let me make that

627
00:41:59,729 --> 00:42:01,885
一个绿色因为我有所区别
one green because I'm distinguishing

628
00:42:01,890 --> 00:42:05,155
这是那个在之后添加的人
that's the guy that's added after the

629
00:42:05,160 --> 00:42:07,915
原来我们有七个我们有
original we had we had seven we had

630
00:42:07,920 --> 00:42:13,284
七件，现在我们已经有了
seven pieces and now we now we've got

631
00:42:13,289 --> 00:42:16,614
更多是7它不是一个
more was it seven it was wasn't it one

632
00:42:16,619 --> 00:42:20,244
二三四有五六七但现在
two three four five six seven but now

633
00:42:20,249 --> 00:42:22,794
我有多少件或有多少件
how many pieces have I got or how many

634
00:42:22,799 --> 00:42:27,594
这样做了多少件呢
how many pieces that did this did this

635
00:42:27,599 --> 00:42:30,625
新线创造了我们想要的东西
new line create that's what we want to

636
00:42:30,630 --> 00:42:32,334
构建它使用递归
build it up use a recursion

637
00:42:32,339 --> 00:42:34,785
这件新品有多少件
how many pieces did this new new

638
00:42:34,790 --> 00:42:39,075
这条新线折叠创造了一条
well this new line folded created one

639
00:42:39,080 --> 00:42:41,415
新作品中有一件新作品
new piece there right one new piece

640
00:42:41,420 --> 00:42:44,235
那里有一件新作品
there one new piece there one new piece

641
00:42:44,240 --> 00:42:51,635
那么有四件新作品还可以
there so there are four new pieces okay

642
00:42:51,640 --> 00:42:55,365
是的，所以有一些公式是正确的
yes so there's some formula right that's

643
00:42:55,370 --> 00:42:58,095
要告诉我们，现在会怎么样
gonna tell us that and now what would

644
00:42:58,100 --> 00:43:00,705
下一个创造得很好我现在有了
the next one create well I've got now I

645
00:43:00,710 --> 00:43:04,485
有一两三四行，所以现在我
have one two three four lines so now I'm

646
00:43:04,490 --> 00:43:07,125
要通过第五行和那个
gonna put through a fifth line and that

647
00:43:07,130 --> 00:43:09,405
我会创造出一大堆作品
will create a whole bunch of pieces I'm

648
00:43:09,410 --> 00:43:12,465
失去了失去这一点的线索
losing that losing the thread of this

649
00:43:12,470 --> 00:43:16,575
争论，但你是正确的，是的
argument but you're onto it right yeah

650
00:43:16,580 --> 00:43:28,775
所以任何建议都没关系，是的
so any any suggestions yeah okay yes

651
00:43:28,780 --> 00:43:31,425
那是对的，所以有一个
there that's right so there's a

652
00:43:31,430 --> 00:43:33,345
我想知道的递归公式
recursion formula that I want to know

653
00:43:33,350 --> 00:43:37,625
我从Kleinberg的论文中学到了这一点
and I learned it from Kleinberg paper

654
00:43:37,630 --> 00:43:41,835
然后我们有一个补充
and then and then we have an addition to

655
00:43:41,840 --> 00:43:44,055
这样递归将告诉我如何
do so the recursion will tell me how

656
00:43:44,060 --> 00:43:46,635
每一个新的都会增加
much it goes up each with each new

657
00:43:46,640 --> 00:43:50,085
功能，那么我们必须添加好了，
function and then we have to add okay so

658
00:43:50,090 --> 00:43:51,705
递归公式让我写出来
the recursion formula let me write that

659
00:43:51,710 --> 00:43:55,565
下
down

660
00:43:55,570 --> 00:44:03,945
所以这是我们现在和我想要的
so this is our of and and M now I'd like

661
00:44:03,950 --> 00:44:06,825
找到一个公式，它的数量
to find a formula for it's the number of

662
00:44:06,830 --> 00:44:11,055
M中的扁平件具有M维
flat pieces in M with an M dimensional

663
00:44:11,060 --> 00:44:13,695
表面很好，我们把他们当作两个人
surface well we're taking em to be two

664
00:44:13,700 --> 00:44:21,195
和n折叠，让我们等于1
and n folds and let's just so an equal 1

665
00:44:21,200 --> 00:44:24,405
23让我们写下来的内容
2 3 let's let's write down the what the

666
00:44:24,410 --> 00:44:27,315
我们知道的数字如何
numbers we know with with one fold how

667
00:44:27,320 --> 00:44:35,865
到目前为止，很多东西太好了
many pieces too good so far so good with

668
00:44:35,870 --> 00:44:39,255
一个折有两块所以这
one fold there were two pieces so this

669
00:44:39,260 --> 00:44:44,175
是计数R然后两次折叠
is the count R and then with two folds

670
00:44:44,180 --> 00:44:46,895
我们有多少人已经超越了这一点
how many we've gone past that point so

671
00:44:46,900 --> 00:44:48,425
我们能回来吗
can we get back

672
00:44:48,430 --> 00:44:54,175
只有那两个是好的谢谢
to just those two was it for okay thanks

673
00:44:54,180 --> 00:44:57,935
现在，当我把第三折怎么样
now when I put in that third fold how

674
00:44:57,940 --> 00:45:00,935
我没有绿线
many did I have without the green line

675
00:45:00,940 --> 00:45:08,585
然而七是七，当时
yet seven was it seven and when the

676
00:45:08,590 --> 00:45:11,705
第四个窗口进去，绿色去了
fourth window went in that green went

677
00:45:11,710 --> 00:45:15,685
有多少人我在这张照片变得如此
how many have I got in this picture so

678
00:45:15,690 --> 00:45:18,515
问题是我有多少新人
the question is how many new ones did I

679
00:45:18,520 --> 00:45:21,425
创造我想那样那条线
create I guess so that that that line

680
00:45:21,430 --> 00:45:24,095
被切成那块的那块
got chopped into that piece that piece

681
00:45:24,100 --> 00:45:26,705
那块那块四片为
that piece that piece four pieces for

682
00:45:26,710 --> 00:45:29,045
新系列四件新品
the new line four pieces for the new

683
00:45:29,050 --> 00:45:33,595
然后每一行都像
line and then each of those pieces like

684
00:45:33,600 --> 00:45:37,685
添加了一个扁平的位，因为它
added added a flat bit because it that

685
00:45:37,690 --> 00:45:41,435
从这里到这里分开这些
piece from here to here separated these

686
00:45:41,440 --> 00:45:43,955
两个以前只是一件
two which were previously just one piece

687
00:45:43,960 --> 00:45:46,775
他们跟着它做了一件平板
one flat piece they followed it on that

688
00:45:46,780 --> 00:45:49,235
我把它折叠在一起我在那里折叠了
line I folded on this I folded there I

689
00:45:49,240 --> 00:45:55,655
认为它上升了四到十一个
think it went up by four to eleven so

690
00:45:55,660 --> 00:45:58,175
现在我们只需要猜一个公式，
now we just have to guess a formula that

691
00:45:58,180 --> 00:46:01,235
给那些匹配那些数字和
gives those matches those numbers and

692
00:46:01,240 --> 00:46:04,505
当然我们真的应该猜测一下
then of course we really should guess it

693
00:46:04,510 --> 00:46:10,595
对于任何M和任何N，我会写下来
for any M and any N and I'll write down

694
00:46:10,600 --> 00:46:17,075
他们发现它的公式就是它
the formula that they found it's it

695
00:46:17,080 --> 00:46:19,595
每个都涉及二项式数
involves binomial numbers every

696
00:46:19,600 --> 00:46:21,775
世界上的一切都涉及到
everything in the world involves

697
00:46:21,780 --> 00:46:24,965
二项式数，因为它们满足
binomial numbers because they satisfy

698
00:46:24,970 --> 00:46:33,225
你能想到的每一个身份
every identity you could think of

699
00:46:33,230 --> 00:46:39,734
所以这里是他们的公式R.
so here's their formula R of with

700
00:46:39,739 --> 00:46:43,435
enfolds和我们在M维度，所以我们
enfolds and we're in M dimensions so we

701
00:46:43,440 --> 00:46:45,445
相信早，我们的想法有
believe early and our thinking had m

702
00:46:45,450 --> 00:46:50,334
等于2，但我们应该知道成长
equal to 2 but we should you know grow

703
00:46:50,339 --> 00:46:53,725
起来让我们成为5维，所以我们
up and get em to be 5 dimensional so we

704
00:46:53,730 --> 00:46:56,725
有一个5维让我们不要想
have a 5 dimensional let's not think

705
00:46:56,730 --> 00:47:01,334
关于那个好，所以事实证明它是
about that ok so so it turns out it's

706
00:47:01,339 --> 00:47:16,375
二项式数和0＆1up2和Mso
binomial numbers and 0 & 1 up 2 and M so

707
00:47:16,380 --> 00:47:21,564
所以对于m等于2这是我的照片
so for m equal to 2 which is my picture

708
00:47:21,569 --> 00:47:30,804
它结束了0加N1加n2和
it's ends and 0 plus N 1 plus n 2 and

709
00:47:30,809 --> 00:47:34,465
和和是什么这那是什么成
and and what are these what is that into

710
00:47:34,470 --> 00:47:37,885
意思是例如那是什么
mean for example what's that that's a

711
00:47:37,890 --> 00:47:42,584
二项式数我不知道你是不是
binomial number I don't know if you're

712
00:47:42,589 --> 00:47:46,104
一些人热衷于二项式数字
keen on binomial numbers some people

713
00:47:46,109 --> 00:47:48,715
他们的一生都是二项式的
their whole lives go into binomial

714
00:47:48,720 --> 00:47:52,584
数字所以它就像是它
numbers so so it's something like is it

715
00:47:52,589 --> 00:47:58,344
它是因子除以n减2
is it n factorial divided by n minus 2

716
00:47:58,349 --> 00:48:06,084
factorial+2factorial我认为是
factorial + 2 factorial I think that's

717
00:48:06,089 --> 00:48:08,395
什么是数字意味着这就是
what that number means that's the

718
00:48:08,400 --> 00:48:11,725
二项式数，所以这就是我上去
binomial number so this would be I go up

719
00:48:11,730 --> 00:48:15,385
在这一点上，我希望得到
to so at this point I'm hoping to get

720
00:48:15,390 --> 00:48:18,925
答案7我想我现在就说
the answer 7 I think right now I've put

721
00:48:18,930 --> 00:48:22,554
在我已经达到了我的意思
in I've gone up to am equal to I mean

722
00:48:22,559 --> 00:48:27,234
艾美认为我已经走了-所以我
Emmy put that I've gone up to - so I've

723
00:48:27,239 --> 00:48:32,425
是的所以我觉得我很明显
yeah so I think yeah I've obviously

724
00:48:32,430 --> 00:48:39,205
允许3个3削减和他们是当
allowed for 3 3 cuts and they are when

725
00:48:39,210 --> 00:48:43,385
我们只有3是7，所以就是这样
we had just 3 was 7 so is that

726
00:48:43,390 --> 00:48:47,765
是这样所以现在我正在接受他们
is this so so this is now I'm taking em

727
00:48:47,770 --> 00:48:55,075
到b3，我希望答案是希望
to b3 and I'm hoping for hope answer is

728
00:48:55,080 --> 00:49:01,655
七，所以什么是3什么是3所以我补充说
seven so what is 3 what is 3 so I add

729
00:49:01,660 --> 00:49:06,155
这三件事就是二项式
these 3 things so what is 3 the binomial

730
00:49:06,160 --> 00:49:09,455
2号被遗忘怎么说
number 3 with 2 forgotten how to say

731
00:49:09,460 --> 00:49:12,275
我很惭愧地承认3选2
that I'm ashamed to admit 3 choose 2

732
00:49:12,280 --> 00:49:13,175
谢谢
thanks

733
00:49:13,180 --> 00:49:15,185
我知道有一个好方法，所以3是什么
I knew there was a good way so what is 3

734
00:49:15,190 --> 00:49:22,325
选择2好把它放在3和2它在
choose 2 well put it in 3 and 2 it's in

735
00:49:22,330 --> 00:49:23,015
已经
there already

736
00:49:23,020 --> 00:49:28,595
所以这将超过1倍2
so it would be 6 over 1 times 2 this

737
00:49:28,600 --> 00:49:36,785
将是3将是3和什么是3
would be 3 would that be 3 and what is 3

738
00:49:36,790 --> 00:49:42,815
选择1你怎么知道你
choose 1 how do you know that you

739
00:49:42,820 --> 00:49:44,935
可能是对mm-hm
probably right mm-hm

740
00:49:44,940 --> 00:49:48,935
3我想是的哦，是的，可能是一个
3 I think yeah oh yeah probably a

741
00:49:48,940 --> 00:49:51,865
定理，如果这些加到3
theorem that that if these add to 3

742
00:49:51,870 --> 00:49:54,845
是的，所以我正在做N等于3
yeah so so this I'm doing N equals 3

743
00:49:54,850 --> 00:49:57,965
好的，所以我们是的，我同意这是3和
okay so we yeah I agree that's 3 and

744
00:49:57,970 --> 00:49:59,945
那么n到0
what about n to 0

745
00:49:59,950 --> 00:50:03,035
什么是你必须忍受0
what's that you have to live with 0

746
00:50:03,040 --> 00:50:07,835
因子不是0因子
factorial by 0 factorial is by no means

747
00:50:07,840 --> 00:50:16,705
0什么是0因子1是的我
0 what so what is 0 factorial 1 yeah I

748
00:50:16,710 --> 00:50:18,995
记得当我是本科生的时候
remember when I was an undergraduate

749
00:50:19,000 --> 00:50:26,435
我赢了赌注或数学，但他没有
having a bet or math I won but he didn't

750
00:50:26,440 --> 00:50:31,835
还清，所以是的，所以3这是3
pay off so yeah so so 3 this is 3

751
00:50:31,840 --> 00:50:34,685
阶乘超过3阶乘以零
factorial over 3 factorial times zero

752
00:50:34,690 --> 00:50:37,415
战士所以6次超过6次1
fighters so it's 6 over 6 times 1 so

753
00:50:37,420 --> 00:50:40,825
不是一个是啊所以1比3和3比赛7
isn't one yeah so 1 in 3 and 3 makes 7

754
00:50:40,830 --> 00:50:44,105
所以它证明了他们的公式
so it proves their formula

755
00:50:44,110 --> 00:50:46,825
干得好，证明了公式但是
well done quite proved the formula but

756
00:50:46,830 --> 00:50:50,075
他们通过归纳证明这一点的方式
their way to prove it is by an induction

757
00:50:50,080 --> 00:50:53,935
如果你喜欢这个，我会的
and I'll just if you like this this

758
00:50:53,940 --> 00:50:56,535
东西
stuff the

759
00:50:56,540 --> 00:51:01,515
您使用归纳的递归
the recursion that you use induction on

760
00:51:01,520 --> 00:51:03,975
这就是我们现在所做的事情
which is just what we did now what we

761
00:51:03,980 --> 00:51:06,965
在这里做这里在四个号码进来
did here here comes in at number four

762
00:51:06,970 --> 00:51:11,475
它切断了哦耶，然后我们
and it cuts through oh yeah and then we

763
00:51:11,480 --> 00:51:14,445
只计算那里的四件
just counted the the four pieces there

764
00:51:14,450 --> 00:51:19,395
所以是的他们让我告诉你
so yeah so they let me just tell you

765
00:51:19,400 --> 00:51:23,355
他们是N和M的数字
what they are of N and M the number

766
00:51:23,360 --> 00:51:26,415
我们正在寻找的是我们的号码
we're looking for is the number that we

767
00:51:26,420 --> 00:51:32,055
有一个减少一个，所以这是
had with one less cut so that's the

768
00:51:32,060 --> 00:51:36,465
以前的扁平件数加上
previous count of flat pieces plus the

769
00:51:36,470 --> 00:51:40,425
数加那是数
number of the plus the number that was

770
00:51:40,430 --> 00:51:43,005
这里是为作品的数量
here was for the number of pieces that

771
00:51:43,010 --> 00:51:49,215
削减那个，这是我们的n减1米
cut that and that's our of n minus 1 M

772
00:51:49,220 --> 00:51:56,535
减去1是的，是的，我不会去
minus 1 yeah and yeah I won't go go

773
00:51:56,540 --> 00:52:00,765
更进一步，但这就是那个
further times up but this the and that

774
00:52:00,770 --> 00:52:04,095
递归规则在in中被证明
rule for recursion is proved in the in

775
00:52:04,100 --> 00:52:08,595
第7.1节摘自论文
the section 7.1 taken from the paper by

776
00:52:08,600 --> 00:52:12,945
克莱因伯格和其他人是的，我想是的
Kleinberg and others yeah so ok I think

777
00:52:12,950 --> 00:52:15,525
这是我不知道你的感受
this is I don't know what you feel this

778
00:52:15,530 --> 00:52:19,635
对我来说这就像给了我一个更好的
for me this like gave me a better

779
00:52:19,640 --> 00:52:22,395
觉得我明白了什么
feeling that I was understanding what

780
00:52:22,400 --> 00:52:28,425
我们在这里有这样的功能
kind of functions we had here and so

781
00:52:28,430 --> 00:52:34,035
然后问题是与这个家庭
then the question is to with this family

782
00:52:34,040 --> 00:52:39,045
我们想要选择两个功能我们
of functions we want to choose two we

783
00:52:39,050 --> 00:52:40,845
想要选择A和权重
want to choose the A's and the weights

784
00:52:40,850 --> 00:52:46,605
A公司和B的给予匹配
the A's and the B's to give to to match

785
00:52:46,610 --> 00:52:50,925
培训数据，以便我们有一个
the training data and so that we have a

786
00:52:50,930 --> 00:52:55,185
在最小化总亏损问题
problem in in minimizing the total loss

787
00:52:55,190 --> 00:53:00,015
我们有一个梯度下降
and we have a we have a gradient descent

788
00:53:00,020 --> 00:53:01,815
问题所以我们必须找到梯度
problem so we have to find the gradient

789
00:53:01,820 --> 00:53:05,175
所以这是周三的工作星期三工作
so that's Wednesdays job Wednesdays job

790
00:53:05,180 --> 00:53:08,235
是找到F的梯度
is to find the gradient of F

791
00:53:08,240 --> 00:53:11,485
好吧，那反向传播很好
okay and that's backpropagation good

792
00:53:11,490 --> 00:53:15,020
thank you very much that's that's 7.1 is

793
00:53:11,490 --> 00:53:17,390
非常感谢那就是7.1

794
00:53:15,030 --> 00:53:17,390
DONE
done