17-Rapidly Decreasing Singular Values.srt

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让我用麦克风介绍Alex
so let me use the mic to introduce Alex

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在麻省理工学院任教的汤森教授
Townsend who taught here at MIT taught

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线性代数1806非常成功
linear algebra 1806 very successfully

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现在他在康奈尔大学就读了
and now he's at Cornell on the faculty

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仍然非常成功地教学
still teaching very successfully and

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他昨天被邀请来这里
he's he was invited here yesterday for a

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在工程和和他的大事件
big event over in engineering and and he

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同意谈谈一部分
agreed to give a talk about a section of

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如果你看的话，本书第4.33节
the book section 4.33 which if you look

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在它，你会看到他的工作
at it you'll see is all about his work

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现在你可以听到创作者的消息
and now you get to hear from the creator

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自己还好，谢谢谢谢谢谢
himself okay okay thanks thank you thank

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你在这里邀请我，我希望你是
you for inviting me here I hope you're

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今天享受这个课程我想说
enjoying the course today I want to tell

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你有一点为什么会有这么多
you a little about why there are so many

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排名较低的矩阵
matrices that are low ranked in the

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世界
world

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如计算数学家盖尔
so as computational mathematicians Gail

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而我自己也遇到了低级别
and myself we come across low rank

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矩阵一直在我们开始
matrices all the time and we started

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想知道为什么是社区
wondering as a community why why is what

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是关于矩阵，所述
is it about the matrices that the

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我们正在研究什么问题
problems that we're looking at what

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使得低等级矩阵出现在今天
makes low rank matrices appear and today

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我想给你这个故事或者
I want to kind of give you that story or

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至少概述了这个故事
at least an overview of that story so

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对于这个类，X将进一步发展
further for this class X is going to be

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n由n真实矩阵组成，非常方便
n by n real matrix so nice and square

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你已经知道或非常舒服
and you already know or very comfortable

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与矩阵的奇异值，使
with the singular values of a matrix so

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矩阵的奇异值就像你一样
the singular values of a matrix as you

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知道是一系列数字
know are a sequence of numbers that are

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单调不增加告诉
monotonically non-increasing that tell

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我们关于矩阵的各种事情
us all kinds of things about the matrix

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X例如非零数
X for example the number of nonzero

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奇异值告诉我们的等级
singular values tell us the rank of the

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矩阵X和他们你也可能知道
matrix X and they also you probably know

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告诉我们矩阵X的表现如何
tell us how well a matrix X can be

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由低秩矩阵近似所以让
approximated by a low rank matrix so let

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我只是写下你的两个事实
me just write two facts down that you

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已经熟悉了，所以这里是一个
already are familiar with so here's a

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事实上，如果我看一下数量
fact that if I look at the number of

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X中的非零奇异值，所以我是
nonzero singular values in X so I'm

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想象那将是K非零
imagining there's going to be K nonzero

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奇异的价值然后我们可以说几个
singular values then we can say a few

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关于X的事情，例如X的等级
things about X for example X the rank of

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我们知道的X是非零数
X as we know is K the number of nonzero

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奇异值，但我们也知道，从
singular values but we also know from

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我们可以将X分解为的SVD
the SVD that we can decompose X into a

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一级矩阵的总和实际上是一个总和
sum of rank one matrices in fact a sum

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他们的K因为X是等级K我们
of K of them so because X is rank K we

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可以写下低级别的表示
can write down a low rank representation

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对于X，它涉及K这样的术语
for X and it involves K terms like this

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这里的每一个向量都是一个
each one of these vectors here is a

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列向量所以如果我画这个
column vector so if I draw this

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这个家伙看起来像这样
pictorially this guy looks like this

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对，我们有K，所以因为X是
right we have K of them so because X is

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等级好我们可以写X作为K的总和
rank okay we can write X as a sum of K

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排名一个矩阵，我们也有
rank one matrices and we also have

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我们已经知道的最初事实
initial fact that we already know that

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X列空间的维数
the dimension of the column space of X

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等于K，与行相同
is equal to K and the same with the row

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空间
space

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所以X的Col空间等于行
so the Col space of X equals the row

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的X尺寸和空间，他们
space of X the dimension and that they

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所有等于K好，所以有三个
all equal K okay so there are three

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我们可以通过观察确定的事实
facts we can determine from looking at

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这个奇异值的序列
this sequence of singular values of a

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矩阵X当然是奇异值
matrix X of course the singular value

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序列是独一无二的
sequence is unique

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X定义了关于它的单数
X defines that singular about this its

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拥有奇异的价值好吧我们是什么
own singular values okay what we're

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对此感兴趣的是什么使X成为什么
interested in here is what makes X what

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是X的属性，确保
are the properties of X that make sure

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奇异值有很多
that the singular values have a lot of

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我们可以尝试按顺序排列零
zeros in that sequence can we try to

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了解X是什么样的
understand what kind of X makes that

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发生了，我们真的很喜欢矩阵
happen and we really like matrices that

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这里有很多零
have a lot of zeros here for the

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如果X是我们说X，请按照阅读原因
following read reason if X is we say X

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如果以下情况正确，则排名较低
is low rank if the following holds right

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因为我们想把X送到我们家
because we if we wanted to send X to our

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朋友，我们想象X是一张照片
friend we're imagining X is a picture

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其中每个条目就是一个像素
where each entry is a pixel of that

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如果该图像矩阵低的图像
image if that matrix that image was low

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排名我们可以将图片发送给我们
rank we could send the picture to our

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朋友有两种方式我们可以送一个
friend in two ways we could send one

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X的每一个条目和我们要做的
every single entry of X and for us to do

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我们将不得不派ñ平方
that we would have to send N squared

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的信息，因为我们不得不
pieces of information because we'd have

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发送每个条目但是如果是
to send every entry but if X is

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我们也可以发送足够低的排名
sufficiently low rank we could also send

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他们是我们的朋友你的矢量u1V1u
them our friend the vectors u u 1 V 1 u

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K到VK
K up to VK

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以及有多少数据
and how many how much pieces of data

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我们要送朋友去吗？
would we have to send our friend to get

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如果我们发送低级别的话，X给他们
X to them if we sent in the low rank

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形成良好这里这里有这个到n
form well here there's to this to n here

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也和这里的数字这个ķ因此其中的
too and here numbers this K of them so

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我们必须发送给K数字和我们
we'd have to send to K n numbers and we

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严格说矩阵秩低中频
strictly say a matrix is low rank if

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将X发送给我们的效率更高
it's more efficient to send X to our

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低级别的朋友比完整的朋友
friend in low rank form than in full

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排名形式好，所以这当然是一个
rank form okay so this of course by a

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很少的计算只是告诉我们
little calculation just shows us that

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如果等级不到一半
provided the rank is less than half the

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我们称之为矩阵的大小
size of the matrix we are calling the

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矩阵
matrix

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我们现在经常在实践中排名
ranked now often in practice we demand

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我们要求K更小
more we demand that K is much smaller

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比这个数字还要多得多
than this number so that it's far more

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高效地发送我们的朋友矩阵
efficient to send our friend the matrix

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X等级低于外国形式
X in low rank form than in foreign form

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好吧所以Kuo筒子架使用了这个词
okay so the Kuo creel use of the word

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低等级就是这种情况
low rank is it's kind of this situation

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但这是对它的严格定义
but this is the strict definition of it

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好吧那么低等级矩阵看起来怎么样
okay so what do low rank matrices look

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喜欢和做那我有一些照片
like and to do that I have some pictures

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为了你我世界上有一些旗帜
for you I have some flags the world

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标志所以这些都是矩阵X
flags so these are all matrices X these

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例子，因为他们是旗帜碰巧
examples because they're Flags happen to

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不是正方形我希望你们都能看到
not be square I hope you can all see

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这是最重要的一点
this but what the top row here are all

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矩阵有极低排名
matrices there are extremely low rank

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例如奥地利国旗，如果你想
for example the Austria flag if you want

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把它发送给你的朋友那个矩阵
to send that to your friend that matrix

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排名第1，所以你所要做的就是
is of Rank 1 so all you have to do is

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向你的朋友发送你需要的两个向量
send your friend two vectors you have to

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告诉你的朋友列空间和
tell your friend the column space and

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该行的空间，这里只有
the row space and there's only the

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尺寸是两者之一
dimensions are one of both for the

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英国国旗你需要发两个
English flag you need to send them two

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列向量和两个行向量u​​1V.
column vectors and two row vectors u 1 V

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1u2和V2，当我们走下这一排
1 u 2 and V 2 and as we go down this row

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他们慢慢充实，充满了等级
they get slowly full and full of rank so

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例如日本国旗很低
the Japanese flag for example is low

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排名但不是我们不是那么小
rank but not us not that small the

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苏格兰国旗基本上是满级
Scottish flag is essentially full rank

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所以发送你的效率非常低效
so it's very inefficient to send your

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朋友是低级别的苏格兰国旗
friend the Scottish flag in low rank

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形成你最好几乎发送
form you're better off sending almost

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每一个条目所以低排名
every single entry so what do low rank

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矩阵看起来像
matrices look like

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好吧，如果矩阵是非常低的等级
well if the matrix is extremely low rank

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比如排名第一，那么当你看到那个
like rank one then when you look at that

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这里的矩阵像旗帜一样
matrix like here like the flag it's a

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与坐标高度一致
highly aligned with the the coordinates

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行和列所以如果是的话
with the rows and columns so if it's

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排名第一，矩阵高度一致
rank one the matrix is highly aligned

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像奥地利国旗，当然还有
like the Austria flag and of course as

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我们在这里添加越来越多的排名了
we adding more and more rank here the

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例如，情况有点模糊
situation gets a bit blurry for example

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一旦我们进入中等级别
once we get into the medium rank

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这是一个圆圈的情况
situation which is a circle it's very

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很难看出圆圈实际上是
hard to see that the circle is actually

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事实上低级别，但我想做什么
a fact low rank but what I wanted to do

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试着理解为什么苏格兰人
was try to understand why the Scottish

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弗兰克苏格兰国旗或对角线
Frank the Scottish flag or diagonal

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模式尤其是一个坏榜样
patterns are particularly a bad example

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为了低排名所以我打算拿下
for low rank so I'm going to take the

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三角旗检查更多
triangular flag to examine that more

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仔细所以三角旗吧
carefully so the triangular flag it

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貌似我拿一个方阵和
looks like I take a square matrix and

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我们在下半部分的颜色还可以
our color in the bottom half okay so

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这个矩阵是in的矩阵
this matrix is the matrix of ones in

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在对角线下面
below the diagonal

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而且我对这个矩阵感兴趣
and I'm interested in this matrix and in

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尤其是它的奇异值尝试
particular its singular values to try to

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理解为什么对角线模式不是
understand why diagonal patterns are not

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特别适用于低等级
particularly useful for low rank

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压缩和这个矩阵
compression and this this matrix of all

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有一个非常好的财产，如果
ones has a really nice property that if

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我把它反过来看起来很像
I take its inverse it looks a lot like

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接近鳃最喜欢的矩阵
getting close to gills favorite matrix

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所以如果我采用这个矩阵的逆
so if I take the inverse of this matrix

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它有一个逆，因为它有一个
it has an inverse because it's got ones

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在对角线上然后它的反转是
on the diagonal then its inverse is the

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以下人们熟悉的矩阵
following matrix which people familiar

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有限差分格式将
with finite difference schemes will

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00:11:24,350 --> 00:11:27,315
注意到它和之间的熟悉程度
notice the familiarity between that and

181
00:11:27,320 --> 00:11:30,095
一阶有限差分
the first order finite difference

182
00:11:30,100 --> 00:11:34,005
逼近特别是如果我走了
approximation in particular if I go a

183
00:11:34,010 --> 00:11:35,865
在其中两个中进一步发展
bit further in times two of these

184
00:11:35,870 --> 00:11:41,555
一起做这个然后就是这样
together and do this then this is

185
00:11:41,560 --> 00:11:44,735
基本上是吉尔最喜欢的矩阵
essentially Gil's favourite matrix

186
00:11:44,740 --> 00:11:51,665
除了一个条目恰好是不同的
except one entry happens to be different

187
00:11:51,670 --> 00:11:55,455
最终成为这个矩阵好吧
ends up being this matrix okay which is

188
00:11:55,460 --> 00:11:58,325
非常接近二阶中央
very close to the second order central

189
00:11:58,330 --> 00:12:01,965
有限差分矩阵和人有
finite difference matrix and people have

190
00:12:01,970 --> 00:12:04,545
非常好地研究了矩阵和公正
very well studied that matrix and just

191
00:12:04,550 --> 00:12:06,945
知道这是我的增益值的奇异
know it's I gain values its singular

192
00:12:06,950 --> 00:12:08,955
价值观，他们知道这一切
values they know everything about that

193
00:12:08,960 --> 00:12:11,925
矩阵，你会记得，如果我们
matrix and you'll remember that if we

194
00:12:11,930 --> 00:12:16,365
知道像x这样的矩阵的特征值
know the eigenvalues of a matrix like x

195
00:12:16,370 --> 00:12:19,845
转置x我们知道奇异值
transpose x we know the singular values

196
00:12:19,850 --> 00:12:25,545
因此，这允许我们通过显示
of x so this allows us to show by the

197
00:12:25,550 --> 00:12:28,065
事实上，我们知道这是奇异的
fact that we know that that the singular

198
00:12:28,070 --> 00:12:33,105
这个矩阵的值都不是很
values of this matrix are not very

199
00:12:33,110 --> 00:12:35,595
适合低级别他们都非零
amenable to low rank they're all nonzero

200
00:12:35,600 --> 00:12:41,325
他们甚至没有腐烂所以我得到了
and they don't even decay so I'm getting

201
00:12:41,330 --> 00:12:46,485
这个电话我打电话给吉尔和吉尔说
this phone I rang up Gil and Gil tells

202
00:12:46,490 --> 00:12:53,885
我这些数字
me these numbers

203
00:12:53,890 --> 00:12:56,685
这使我们制定出什么
that allows us to work out exactly what

204
00:12:56,690 --> 00:12:58,455
这个矩阵的奇异值是
the singular values of this matrix are

205
00:12:58,460 --> 00:13:00,405
从连接到有限
from the connection to finite

206
00:13:00,410 --> 00:13:03,675
差异，所以我们可以理解为什么
differences and so we can understand why

207
00:13:03,680 --> 00:13:05,295
看着这个不好看
this is not good by looking at the

208
00:13:05,300 --> 00:13:07,635
奇异值所以第一个奇异
singular values so the first singular

209
00:13:07,640 --> 00:13:10,485
x的值来自此表达式
value of x is from this expression is

210
00:13:10,490 --> 00:13:14,235
将大约像2n个以上
going to be approximately like 2n over

211
00:13:14,240 --> 00:13:17,745
pi又从这个表达中再次出现了
pi and from this expression again the

212
00:13:17,750 --> 00:13:18,615
最后一个人
last guy

213
00:13:18,620 --> 00:13:23,265
x的最后一个角度值是
the last angular value of x is going to

214
00:13:23,270 --> 00:13:27,495
约为1/2，所以这些单数
be approximately 1/2 so these singular

215
00:13:27,500 --> 00:13:29,445
价值都很大，他们没有得到
values are all large they're not getting

216
00:13:29,450 --> 00:13:33,045
如果我绘制这些单数，则接近0
close to 0 if I plotted these singular

217
00:13:33,050 --> 00:13:38,025
图表上的值是正确的，所以这里是
values on a graph right so here's the

218
00:13:38,030 --> 00:13:40,935
第一个奇异值第二个和第二个
first singular value the second and the

219
00:13:40,940 --> 00:13:44,445
N那么图表会是什么样子
N then what would the graph look like

220
00:13:44,450 --> 00:13:51,885
好的情节这些数字除以
well plot these numbers are divided by

221
00:13:51,890 --> 00:13:54,765
这家伙让他们都受到限制
this guy so that they all are bounded

222
00:13:54,770 --> 00:13:59,055
由于介于1和0的
between 1 and 0 because of the

223
00:13:59,060 --> 00:14:01,035
归一化，因为我除以西格玛
normalization because I divided by Sigma

224
00:14:01,040 --> 00:14:05,025
X中的1个，所以我们可以绘制它们和它们
1 of X and so we can plot them and they

225
00:14:05,030 --> 00:14:12,295
看起来会像这样的东西
will look like this kind of thing

226
00:14:12,300 --> 00:14:16,525
好的，这个数字恰好在这里
okay this number happens to be here

227
00:14:16,530 --> 00:14:20,445
他们在那里成为超过4米的PI
where they they come to be PI over 4 M

228
00:14:20,450 --> 00:14:23,215
好吧，这是我把这个数字除以
okay which is me dividing this number by

229
00:14:23,220 --> 00:14:27,655
这个号码大概还可以
this number approximately okay so

230
00:14:27,660 --> 00:14:30,175
三角形图案非常糟糕
triangular patterns are extremely bad

231
00:14:30,180 --> 00:14:33,835
对于低等级我们需要的东西或我们最少
for low rank we need things or we least

232
00:14:33,840 --> 00:14:35,875
直觉地认为我们需要的东西
intuitively think that we need things

233
00:14:35,880 --> 00:14:39,265
与行和列对齐但是
aligned with the rows and columns but

234
00:14:39,270 --> 00:14:42,595
圆形情况恰好也是
the circle case is happens to also be

235
00:14:42,600 --> 00:14:48,445
低等级，所以发生了什么事
low rank and so what happened to the

236
00:14:48,450 --> 00:14:56,295
日本国旗为什么是日本国旗
Japanese flag why is the Japanese flag

237
00:14:56,300 --> 00:15:02,065
方便低排名这是
convenient for low rank well it's the

238
00:15:02,070 --> 00:15:03,865
事实上它是一个圆圈而且有很多
fact that it's a circle and there's lots

239
00:15:03,870 --> 00:15:06,745
如果我试着，那么圆圈中的对称性
of symmetry in a circle so if I try to

240
00:15:06,750 --> 00:15:10,675
看一下圆圈的等级吧
look at the rank of a circle the

241
00:15:10,680 --> 00:15:17,205
日本国旗然后我可以约束这个级别
Japanese flag then I can bound this rank

242
00:15:17,210 --> 00:15:22,195
通过将日本国旗分解成
by decomposing the Japanese flag into

243
00:15:22,200 --> 00:15:25,105
两件事情所以这将是少
two things so this is going to be less

244
00:15:25,110 --> 00:15:29,725
超过或等于两个等级的等级
than or equal to the rank of sum of two

245
00:15:29,730 --> 00:15:32,425
矩阵，我会这样做的
matrices and I'll do it so that the

246
00:15:32,430 --> 00:15:34,435
分解工作我有
decomposition works out I have the

247
00:15:34,440 --> 00:15:38,065
我要削减排名第一
circle I'm going to cut out a rank one

248
00:15:38,070 --> 00:15:40,945
生活在这个中间的一块
piece that lives in the middle of this

249
00:15:40,950 --> 00:15:48,625
圈好了，我要剪掉一个
circle okay and I'm gonna cut out a

250
00:15:48,630 --> 00:15:54,085
从那个圆圈的内部广场
square from the interior of that circle

251
00:15:54,090 --> 00:15:57,025
好吧，我当然可以搞清楚了
okay and I can figure out of course the

252
00:15:57,030 --> 00:15:58,825
排名只是这些的总和为界
rank is just bounded by the sum of those

253
00:15:58,830 --> 00:16:02,545
两个级别这个人受到排名的限制
two ranks this guy is bounded by rank

254
00:16:02,550 --> 00:16:05,005
一个因为它与...高度一致
one because it's highly aligned with the

255
00:16:05,010 --> 00:16:15,975
因此，这个家伙被排名第一
grid so this guy is bounded by rank one

256
00:16:15,980 --> 00:16:27,985
这件事是加一个，现在我有
this thing yeah plus one and now I have

257
00:16:27,990 --> 00:16:30,445
试图了解这个的等级
to try to understand the rank of this

258
00:16:30,450 --> 00:16:34,435
现在这件作品有很多
piece now this piece has lots of

259
00:16:34,440 --> 00:16:38,365
对称性例如我们知道
symmetry for example we know that the

260
00:16:38,370 --> 00:16:40,735
该矩阵的等级是维度
rank of that matrix is the dimension of

261
00:16:40,740 --> 00:16:43,645
海岸柱空间和
the Coast the column space and the

262
00:16:43,650 --> 00:16:46,915
行空间的维度，所以当我们
dimension of the row space so when we

263
00:16:46,920 --> 00:16:48,925
因为对称性而看这个矩阵
look at this matrix because of symmetry

264
00:16:48,930 --> 00:16:53,455
如果我把这个矩阵分成两半
if I divide this matrix in half along

265
00:16:53,460 --> 00:16:56,635
列左侧的所有列
the columns all the columns on the left

266
00:16:56,640 --> 00:16:59,935
出现这样的列权
appear on the right so the columns for

267
00:16:59,940 --> 00:17:02,005
例如，这个矩阵的等级是
example the rank of this matrix is the

268
00:17:02,010 --> 00:17:04,824
与该矩阵的排名相同，因为
same as the rank of that matrix because

269
00:17:04,829 --> 00:17:07,944
我没有改变列空间没关系
I didn't change the column space okay

270
00:17:07,949 --> 00:17:11,324
现在我又去了，沿着行划分
now I go again and divide along the rows

271
00:17:11,329 --> 00:17:16,405
现在这个道路的维度
and now the road dimension of this

272
00:17:16,410 --> 00:17:19,165
矩阵与上半部分相同
matrix is the same as the top half

273
00:17:19,170 --> 00:17:21,204
因为我消灭了那些我没有的东西
because as I wipe out those I didn't

274
00:17:21,209 --> 00:17:22,555
更改行空间的维度
change the dimension of the row space

275
00:17:22,560 --> 00:17:25,005
因为行的顶部是相同的
because the rows are the same top bottom

276
00:17:25,010 --> 00:17:27,954
所以这就成了那个等级
and so this becomes the rank of that

277
00:17:27,959 --> 00:17:31,435
那里的小小矩阵因为
tiny little matrix there and because

278
00:17:31,440 --> 00:17:34,935
它很小，不会有更高的等级
it's small it won't have to larger rank

279
00:17:34,940 --> 00:17:40,195
所以这肯定比我少
so this is definitely less than if I

280
00:17:40,200 --> 00:17:44,965
把它分成一个小家伙吧
divide that up a little guy here it

281
00:17:44,970 --> 00:17:50,305
看起来像那个加上那里的另一个人
looks like that plus the other guy there

282
00:17:50,310 --> 00:17:55,825
看起来像
looks like

283
00:17:55,830 --> 00:18:02,725
加一个当然是行
that plus one and so of course the rows

284
00:18:02,730 --> 00:18:06,495
列是该矩阵的行空间
the column the row space of this matrix

285
00:18:06,500 --> 00:18:08,665
不能很高，因为这是一个
cannot be very high because this is a

286
00:18:08,670 --> 00:18:11,275
非常薄的矩阵有很多零
very thin matrix there's lots of zeros

287
00:18:11,280 --> 00:18:14,125
在那个矩阵中只有几个等等
in that matrix only a few ones and so

288
00:18:14,130 --> 00:18:15,985
你可以继续做一些技巧
you can go along and do a bit of trick

289
00:18:15,990 --> 00:18:19,105
试图弄清楚有多少行
to try to figure out what how many rows

290
00:18:19,110 --> 00:18:23,125
在这个矩阵中非零并且有点
are nonzero in this matrix and a bit of

291
00:18:23,130 --> 00:18:26,395
技巧告诉你它取决于
trick tells you well it depends on the

292
00:18:26,400 --> 00:18:29,815
这个原始圆的半径，如果我
radius of this original circle so if I

293
00:18:29,820 --> 00:18:32,155
本作的原始半径R
make the original radius R of this

294
00:18:32,160 --> 00:18:35,725
日本国旗然后你的约束
Japanese flag then the bound that you

295
00:18:35,730 --> 00:18:39,465
最终得到的将是这个矩阵R.
end up getting will be for this matrix R

296
00:18:39,470 --> 00:18:43,975
1个减去平方根2超过2此
1 minus square root 2 over 2 for this

297
00:18:43,980 --> 00:18:46,675
家伙有点更好的确保
guy a bit of trig better make sure

298
00:18:46,680 --> 00:18:48,985
这是一个整数，然后再来一次
that's an integer and then again here

299
00:18:48,990 --> 00:18:51,135
它是相同的，但对于列空间
it's the same but for the column space

300
00:18:51,140 --> 00:18:56,575
所以这就是我正在做的好事
so this is me just doing trick ok and

301
00:18:56,580 --> 00:18:58,345
它恰好在排名上受到约束
that's a bound on the rank it happens to

302
00:18:58,350 --> 00:19:00,525
非常好，如果你锻炼身体
be extremely good and if you work out

303
00:19:00,530 --> 00:19:03,175
这个等级是什么，并试图回顾
what that rank is and try to look back

304
00:19:03,180 --> 00:19:05,335
你会发现它非常有效率
you'll find it's extremely efficient to

305
00:19:05,340 --> 00:19:08,185
送日本国旗给你的朋友在
send the Japanese flag to your friend in

306
00:19:08,190 --> 00:19:11,605
低等级形式，因为它不是满级
low rank form because it's not full rank

307
00:19:11,610 --> 00:19:13,855
因为这些数字太小了
because these numbers are so small so

308
00:19:13,860 --> 00:19:15,745
这此出来是像
this this comes out to be like

309
00:19:15,750 --> 00:19:21,865
大约比1/2小得多
approximately 1/2 so much smaller than

310
00:19:21,870 --> 00:19:24,505
你会期待什么，因为记住了
what you would expect because remember a

311
00:19:24,510 --> 00:19:27,415
圆几乎是反版
circle is almost the anti version of

312
00:19:27,420 --> 00:19:29,845
与网格对齐但是是的
aligned with the grid but yeah it's

313
00:19:29,850 --> 00:19:34,125
仍然低排名好
still low rank ok

314
00:19:34,130 --> 00:19:39,505
现在我们提出了大多数矩阵
now most matrices that we come up with

315
00:19:39,510 --> 00:19:42,325
在计算数学中并不完全是一个
in computational math are not exactly a

316
00:19:42,330 --> 00:19:47,835
有限等级它们具有数值等级
finite rank they are of numerical rank

317
00:19:47,840 --> 00:19:54,985
所以我就这么定义了
and so I just defined that so the

318
00:19:54,990 --> 00:19:58,345
矩阵的数值排名非常
numerical rank of a matrix it's very

319
00:19:58,350 --> 00:20:00,025
除了我们允许之外，类似于排名
similar to the rank except we allow

320
00:20:00,030 --> 00:20:02,155
我们自己有点摆动的空间
ourselves a little bit of wiggle room

321
00:20:02,160 --> 00:20:05,155
当我们定义它，从而使量
when we define it and so that amount of

322
00:20:05,160 --> 00:20:09,445
wiggleroom将是一个名为的参数
wiggle room will be a parameter called

323
00:20:09,450 --> 00:20:12,415
洞叫Epsilon好吧那是一个
hole called Epsilon okay that's a

324
00:20:12,420 --> 00:20:14,245
我正在考虑将epsilon视为一种
tolerance I'm thinking of epsilon as a

325
00:20:14,250 --> 00:20:17,815
宽容是摆动的量
tolerance that's the amount of wiggle

326
00:20:17,820 --> 00:20:23,395
房间我会给自己好的，我们
room I'm gonna give myself okay and we

327
00:20:23,400 --> 00:20:26,565
说数字排名
say that the numerical rank

328
00:20:26,570 --> 00:20:29,065
我会在那里放一个epsilon来表示一个
I'll put an epsilon there to denote a

329
00:20:29,070 --> 00:20:35,065
如果K是第一个，则数值等级为K.
numerical rank is K if K is the first

330
00:20:35,070 --> 00:20:37,315
奇异值或最后一个奇异
singular value or the last singular

331
00:20:37,320 --> 00:20:39,685
以下是epsilon以上的值
value above epsilon in the following

332
00:20:39,690 --> 00:20:41,755
感觉我正在复制上面的定义
sense I'm copying the definition above

333
00:20:41,760 --> 00:20:45,085
但如果使用epsilon而不是零
but with epsilon is instead of zeros if

334
00:20:45,090 --> 00:20:50,985
这个奇异值小于epsilon
this singular value is less than epsilon

335
00:20:50,990 --> 00:20:54,825
相对而且案件不是
relatively and the case one was not

336
00:20:54,830 --> 00:20:59,035
下面所以k加1是第一个单数
below so k plus 1 is the first singular

337
00:20:59,040 --> 00:21:01,795
这个亲戚的价值低于epsilon
value below epsilon in this relative

338
00:21:01,800 --> 00:21:09,385
感觉当然是0x的等级如果
sense so of course the rank of 0x if

339
00:21:09,390 --> 00:21:11,785
这被定义为与它相同
that was defined it's the same as the

340
00:21:11,790 --> 00:21:15,145
X的等级所以这只是允许的
rank of X so this is just allowing

341
00:21:15,150 --> 00:21:18,145
我们自己有些摆动，但这是
ourselves some wiggle room but this is

342
00:21:18,150 --> 00:21:19,675
实际上我们更感兴趣的是
actually what we're interested more in

343
00:21:19,680 --> 00:21:22,135
在实践中我不一定要
in practice I don't want to necessarily

344
00:21:22,140 --> 00:21:25,015
把我的朋友发送到确切的标志
send my friend the flag to exact

345
00:21:25,020 --> 00:21:27,205
我真的很高兴
precision I would actually be happy to

346
00:21:27,210 --> 00:21:30,325
向我的朋友发送最多16位数的旗帜
send my friend the flag up to 16 digits

347
00:21:30,330 --> 00:21:32,545
例如，它们不是精确的
of precision for example they're not

348
00:21:32,550 --> 00:21:33,895
要告诉之间的区别
going to tell the difference between

349
00:21:33,900 --> 00:21:36,565
那两个旗帜，如果我可以逃脱
those two flags and if I can get away

350
00:21:36,570 --> 00:21:39,985
压缩矩阵更多
with compressing the matrix a lot more

351
00:21:39,990 --> 00:21:41,485
一旦我有回旋余地的一点点
once I have a little bit of wiggle room

352
00:21:41,490 --> 00:21:48,865
那将是一件好事，所以我们也是如此
that would be a good thing so what so we

353
00:21:48,870 --> 00:21:59,005
从方舟知道一个金元那个
know from the ark a kin-yuen that the

354
00:21:59,010 --> 00:22:01,705
奇异的价值告诉我们我们能做得多好
singular values tell us how well we can

355
00:22:01,710 --> 00:22:05,335
通过低秩矩阵近似X.
approximate X by a low rank matrix in

356
00:22:05,340 --> 00:22:10,545
特别是我们知道k加1
particular we know that the k plus 1

357
00:22:10,550 --> 00:22:14,755
x的奇异值告诉我们X的好坏
singular value of x tells us how well X

358
00:22:14,760 --> 00:22:19,335
可以通过秩K矩阵来近似
can be approximated by a rank K matrix

359
00:22:19,340 --> 00:22:22,215
好吧，例如当排名是
okay for example when the rank was

360
00:22:22,220 --> 00:22:27,304
正好K西格玛K加1是0和
exactly K the Sigma K plus 1 was 0 and

361
00:22:27,309 --> 00:22:29,924
然后我们发现这是0
then this came out to be 0 and we found

362
00:22:29,929 --> 00:22:33,374
这里X恰好是一个等级K矩阵
there X was exactly a rank K matrix here

363
00:22:33,379 --> 00:22:35,355
因为我们有摆动的房间
because we have the wiggle room the

364
00:22:35,360 --> 00:22:37,845
epsilon我们得到的近似值不是
epsilon we get an approximation not an

365
00:22:37,850 --> 00:22:43,575
确切地说，这告诉我们我们有多好
exact so this is telling us how well we

366
00:22:43,580 --> 00:22:49,294
可以通过秩K矩阵近似X.
can approximate X by a rank K matrix

367
00:22:49,299 --> 00:22:52,335
好的，这就是奇异值
okay that's what the singular values are

368
00:22:52,340 --> 00:22:57,644
告诉我们，这样我们就可以尝试了
telling us and so this allows us to try

369
00:22:57,649 --> 00:22:59,894
我们最好压缩矩阵但使用
our best to compress matrices but use

370
00:22:59,899 --> 00:23:04,575
低秩近似而不是做
low-rank approximation rather than doing

371
00:23:04,580 --> 00:23:08,744
确切的事情，当然还有一个
things exactly and of course on a

372
00:23:08,749 --> 00:23:10,575
我们使用浮点时的计算机
computer when we're using floating-point

373
00:23:10,580 --> 00:23:14,235
算术或计算机，因为我们
arithmetic or on a computer because we

374
00:23:14,240 --> 00:23:17,864
总是将数字舍入到最近的16
always round numbers to the nearest 16

375
00:23:17,869 --> 00:23:21,225
如果epsilon是16位数，则为数字
digit number if epsilon was 16 digits

376
00:23:21,230 --> 00:23:23,294
你的电脑将不能告诉
your computer wouldn't be able to tell

377
00:23:23,299 --> 00:23:29,565
X或X之间的差异等级K.
the difference between X or X the rank K

378
00:23:29,570 --> 00:23:32,534
如果这个数字是近似值
approximation if this number was

379
00:23:32,539 --> 00:23:35,565
满意这表达你的电脑
satisfied this expression your computer

380
00:23:35,570 --> 00:23:37,605
会认为x和XK是一样的
would think of x and XK as the same

381
00:23:37,610 --> 00:23:40,364
矩阵因为它不可避免地会变圆
matrix because it would inevitably round

382
00:23:40,369 --> 00:23:46,244
这对Epsilon内的epsilon都没关系
both to epsilon within Epsilon okay so

383
00:23:46,249 --> 00:23:49,004
什么样的矩阵在数字上
what kind of matrices are numerically of

384
00:23:49,009 --> 00:24:01,175
降低
lower

385
00:24:01,180 --> 00:24:06,165
当然所有低级矩阵都是
of course all low rank matrices are

386
00:24:06,170 --> 00:24:09,165
数字上的低等级因为绝对
numerically of low rank because absolute

387
00:24:09,170 --> 00:24:16,545
摆动的房间只能帮助你
the wiggle room can only help you but

388
00:24:16,550 --> 00:24:18,335
它的远不止于此
it's far more than that

389
00:24:18,340 --> 00:24:20,895
有很多满级矩阵
there are many full rank matrices

390
00:24:20,900 --> 00:24:22,845
没有任何单数的矩阵
matrices that don't have any singular

391
00:24:22,850 --> 00:24:25,245
值为0但是单数
values that are 0 but the singular

392
00:24:25,250 --> 00:24:28,695
价值迅速衰减到零
values decay rapidly to zero that are

393
00:24:28,700 --> 00:24:31,365
具有低数值的满秩矩阵
full rank matrices with low numerical

394
00:24:31,370 --> 00:24:34,155
排名因为摆动空间所以
rank because of the wiggle room so for

395
00:24:34,160 --> 00:24:39,015
例如这里是典型的矩阵
example here is the classic matrix that

396
00:24:39,020 --> 00:24:44,145
如果我给你这个，这适合这个政权
fits this regime if I give you this this

397
00:24:44,150 --> 00:24:51,435
被称为希尔伯特矩阵，这是一个
is called the Hilbert matrix this is a

398
00:24:51,440 --> 00:24:53,775
碰巧有极端的矩阵
matrix that happens to have extremely

399
00:24:53,780 --> 00:24:57,825
低数值排名，但实际上
low numerical rank but it's actually

400
00:24:57,830 --> 00:25:01,245
满级，这意味着我可以
full rank which means that I can

401
00:25:01,250 --> 00:25:06,135
通过秩K矩阵近似h2
approximate h2 by a rank K matrix where

402
00:25:06,140 --> 00:25:09,075
K非常小，但为什么呢
K is quite small very well but why did

403
00:25:09,080 --> 00:25:11,055
你给我一些摆动的空间，但它是
you give me some wiggle room but it's

404
00:25:11,060 --> 00:25:13,755
从某种意义上说，不是低等级矩阵
not a low rank matrix in the sense that

405
00:25:13,760 --> 00:25:16,095
如果epsilon为0，那么你不允许
if epsilon was 0 here you didn't allow

406
00:25:16,100 --> 00:25:17,835
我是一个单数的蠕动室
me the wriggle room all the singular

407
00:25:17,840 --> 00:25:21,575
这个矩阵的值是正的
values of this matrix are positive so

408
00:25:21,580 --> 00:25:26,655
它的低数值秩但不是的
it's of low numerical rank but not of

409
00:25:26,660 --> 00:25:29,085
它不是另一个低等级矩阵
it's not a low rank matrix the other

410
00:25:29,090 --> 00:25:32,445
经典案例中的经典案例
classical case the classic example which

411
00:25:32,450 --> 00:25:34,905
在这方面激发了很多研究
motivated a lot of the research in this

412
00:25:34,910 --> 00:25:38,205
区域是矩阵上的随机区域，所以这里是
area was the random on matrix so here is

413
00:25:38,210 --> 00:25:49,425
vandermonde矩阵是N乘n版本
the vandermonde matrix an N by n version

414
00:25:49,430 --> 00:25:56,385
我认为消费税是真实的
of it I think of the excise is real this

415
00:25:56,390 --> 00:26:01,145
非常多
is very much

416
00:26:01,150 --> 00:26:03,645
这是出现时的矩阵
this is the matrix that comes up when

417
00:26:03,650 --> 00:26:05,675
你试着做多项式插值
you try to do polynomial interpolation

418
00:26:05,680 --> 00:26:11,235
在实际情况下，这是一个非常糟糕的
at real points this is an extremely bad

419
00:26:11,240 --> 00:26:13,415
矩阵处理因为它是
matrix to deal with because it's

420
00:26:13,420 --> 00:26:16,785
在数字上低等级，通常是你
numerically low rank and often you

421
00:26:16,790 --> 00:26:18,555
实际上想要解决一个线性系统
actually want to solve a linear system

422
00:26:18,560 --> 00:26:22,515
用这个矩阵和数字低等级
with this matrix and numerical low rank

423
00:26:22,520 --> 00:26:24,555
暗示这是非常难的
implies that it's extremely hard to

424
00:26:24,560 --> 00:26:30,435
因此这是低等级数值低
invert so this is low rank numerical low

425
00:26:30,440 --> 00:26:31,785
排名并不总是对你好
rank is not always good for you

426
00:26:31,790 --> 00:26:39,995
好的，我们经常希望我们想要逆
okay often we want we want the inverse

427
00:26:40,000 --> 00:26:47,705
存在，但很难，因为
which exists but it's difficult because

428
00:26:47,710 --> 00:27:05,145
因为他很低，所以人们有
because he has low okay so people have

429
00:27:05,150 --> 00:27:06,675
一直试图理解为什么这些
been trying to understand why these

430
00:27:06,680 --> 00:27:10,605
矩阵在数值上是低等级的
matrices are numerically of low rank for

431
00:27:10,610 --> 00:27:15,065
多年和经典的原因
a number of years and the classic reason

432
00:27:15,070 --> 00:27:17,775
为什么有这么多的低阶矩阵
why there are so many low rank matrices

433
00:27:17,780 --> 00:27:21,105
因为世界是平稳的
is because the the world is smooth as

434
00:27:21,110 --> 00:27:24,755
人们说他们说世界是平稳的
people say they say the world is smooth

435
00:27:24,760 --> 00:27:30,765
这就是为什么数值矩阵低的原因
that's why matrices of numerical low

436
00:27:30,770 --> 00:27:36,945
排名并说明我的观点
rank and to illustrate that point I will

437
00:27:36,950 --> 00:27:38,315
举个例子
do an example

438
00:27:38,320 --> 00:27:40,845
这是经典的理解
this is classically understood

439
00:27:40,850 --> 00:27:49,635
一个名叫Reid1983的男人就是这样
a man called Reid 1983 and that's this

440
00:27:49,640 --> 00:27:51,885
这就是他的理由，我有一个
this is what his reason was I have a

441
00:27:51,890 --> 00:27:54,345
JohnReid的照片他不是他不是
picture of John Reid he doesn't he's not

442
00:27:54,350 --> 00:27:56,745
非常有名，所以我想尽量确保
very famous so I like try to make sure

443
00:27:56,750 --> 00:28:00,175
他的照片四处传播
his picture gets around

444
00:28:00,180 --> 00:28:01,795
它正在弹钢琴就像一个人
it's playing the piano that's like one

445
00:28:01,800 --> 00:28:03,055
那能找到的唯一的照片
of the only pictures that could find of

446
00:28:03,060 --> 00:28:06,925
他这么干嘛，为什么呢
him so what is in this reason why do

447
00:28:06,930 --> 00:28:10,515
人们说这很好，这是一个例子
people say this well here's an example

448
00:28:10,520 --> 00:28:13,255
如果我采取一个说明它
that illustrates it if I take a

449
00:28:13,260 --> 00:28:20,065
两个变量的多项式和I的变量
polynomial in two variables and I for

450
00:28:20,070 --> 00:28:21,805
例如，这是两个多项式
example this is a polynomial of two

451
00:28:21,810 --> 00:28:25,615
变量和我的X矩阵来自
variables and my X matrix comes from

452
00:28:25,620 --> 00:28:29,685
采样多项式是整数
sampling that polynomial are integers

453
00:28:29,690 --> 00:28:39,685
例如，这个矩阵然后那个矩阵
for example this matrix then that matrix

454
00:28:39,690 --> 00:28:45,595
碰巧在数学上排名很低
happens to be of low rank mathematically

455
00:28:45,600 --> 00:28:50,365
低等级与epsilon等于零的原因
of low rank with epsilon equals zero why

456
00:28:50,370 --> 00:28:53,395
如果我把X写成一个就好了
is that well if I write down X as a as

457
00:28:53,400 --> 00:28:55,195
就矩阵而言，您可以轻松看到
in terms of matrices you can easily see

458
00:28:55,200 --> 00:28:58,105
它所以这是由所有的矩阵起来
it so this is made up of a matrix of all

459
00:28:58,110 --> 00:29:06,235
那些加上J的矩阵，所以它是12
ones plus a matrix of J's so that's 1 2

460
00:29:06,240 --> 00:29:11,755
多达n12多达n因为每
up to n 1 2 up to n because the every

461
00:29:11,760 --> 00:29:13,435
该矩阵的输入仅取决于
entry of that matrix just depends on the

462
00:29:13,440 --> 00:29:17,095
行索引然后这家伙取决于
row index and then this guy depends on

463
00:29:17,100 --> 00:29:20,575
既J和K，所以这是一个乘法
both J and K so this is a multiplication

464
00:29:20,580 --> 00:29:27,375
表右，所以这是n24到2Nn
table right so this is n 2 4 up to 2 N n

465
00:29:27,380 --> 00:29:35,395
2nnsquareok清楚所有的矩阵
2n n square ok clearly the matrix of all

466
00:29:35,400 --> 00:29:42,715
者是一个秩一点矩阵具有相同的
ones is a rank one matrix the same with

467
00:29:42,720 --> 00:29:44,935
这家伙列空间只是
this guy the column space is just of

468
00:29:44,940 --> 00:29:48,565
尺寸1，这个家伙发生了
dimension 1 and this guy happens that

469
00:29:48,570 --> 00:29:50,815
最后一个人也恰好是排名
the last guy also happens to be of rank

470
00:29:50,820 --> 00:29:54,405
1，因为我可以写在这个矩阵
1 because I can write this matrix in

471
00:29:54,410 --> 00:30:02,425
等级1形式是列向量Rho
Rank 1 form which is a column vector Rho

472
00:30:02,430 --> 00:30:08,165
矢量所以这个矩阵X的等级为3
vector so this matrix X is of rank 3

473
00:30:08,170 --> 00:30:15,245
我猜最多迈克3是怎么回事
I guess at most Mike 3 is what about

474
00:30:15,250 --> 00:30:20,705
肖肖尼现在好吧，当然这不是
Shoshone okay now of course this isn't

475
00:30:20,710 --> 00:30:23,765
得到数字低排名，所以让我们
got to numerical low rank yet so let's

476
00:30:23,770 --> 00:30:29,795
ReidReid知道，让自己在那里
get ourselves there so Reid Reid knew

477
00:30:29,800 --> 00:30:32,705
这对他说自己好不好
this and he said to himself okay well if

478
00:30:32,710 --> 00:30:36,335
如果X实际上我可以近似
I can approximate if X is actually

479
00:30:36,340 --> 00:30:38,945
来自抽样功能和我
coming from sampling a function and I

480
00:30:38,950 --> 00:30:41,075
通过多项式逼近该函数
approximate that function by polynomial

481
00:30:41,080 --> 00:30:43,925
然后我会让自己变得低级别
then I'm going to get myself a low-rank

482
00:30:43,930 --> 00:30:46,865
近似并得到一个约束
approximation and get a bound on the

483
00:30:46,870 --> 00:30:52,985
数字排名如此一般如此我
numerical rank so so in general if I

484
00:30:52,990 --> 00:30:55,735
给你一个两个变量的多项式
give you a polynomial of two variables

485
00:30:55,740 --> 00:30:59,915
可以写下来的度M
which can be written down its degree M

486
00:30:59,920 --> 00:31:05,495
在x和y中，让我们说保留这些
in both x and y let's say keep these

487
00:31:05,500 --> 00:31:09,655
索引远离矩阵索引I.
indexes away from the matrix index I

488
00:31:09,660 --> 00:31:13,865
给你这样的多项式，我走了
give you this such polynomial and I go

489
00:31:13,870 --> 00:31:19,645
离开，我取样并制作矩阵X.
away and I sample it and make a matrix X

490
00:31:19,650 --> 00:31:23,225
然后通过查看每个术语X.
then X by looking at each term

491
00:31:23,230 --> 00:31:26,065
就像我在那里一样，我们会有
individually like I did there we'll have

492
00:31:26,070 --> 00:31:30,905
数学上低等级与epsilon
low rank mathematically with epsilon

493
00:31:30,910 --> 00:31:34,805
等于零，我们至多M的平方
equals zero we have at most M Squared

494
00:31:34,810 --> 00:31:38,195
排名如果M是三或四或十
rank and if M is three or four or ten it

495
00:31:38,200 --> 00:31:40,835
因为这个X可能会很低
possibly could be low because this X

496
00:31:40,840 --> 00:31:44,855
可能是一个大矩阵好，所以我们
could be a large matrix okay so what we

497
00:31:44,860 --> 00:31:47,285
为希尔伯特矩阵所做的确定没问题
did for the Hilbert matrix was said okay

498
00:31:47,290 --> 00:31:49,835
好好看看那家伙看起来那个人
well look at that guy that guy looks

499
00:31:49,840 --> 00:31:52,355
就像它正在采样它看起来的功能
like it's sampling a function it looks

500
00:31:52,360 --> 00:31:54,365
就像它在X上对函数1进行采样一样
like it's sampling the function 1 over X

501
00:31:54,370 --> 00:31:58,285
加y减1，所以他对自己说
plus y minus 1 so he said to himself

502
00:31:58,290 --> 00:32:03,575
好吧，如果我看看希尔伯特，那就是X.
well that X if I look at the Hilbert

503
00:32:03,580 --> 00:32:08,735
矩阵然后是一个函数的抽样
matrix then that is sampling a function

504
00:32:08,740 --> 00:32:13,325
它恰好不是一个多项式它
it happens to not be a polynomial it

505
00:32:13,330 --> 00:32:16,745
碰巧是这个功能，但那是
happens to be this function but that's

506
00:32:16,750 --> 00:32:19,085
好的，因为采样多项式是
okay because sampling polynomials are

507
00:32:19,090 --> 00:32:21,175
整数整数给出
integer integers gives

508
00:32:21,180 --> 00:32:24,595
低排名可能正好采样顺畅
low rank exactly maybe sampling smooth

509
00:32:24,600 --> 00:32:27,955
像这样的函数函数可以
functions functions like this can be

510
00:32:27,960 --> 00:32:29,815
很好地近似于多项式和
well approximated by polynomials and

511
00:32:29,820 --> 00:32:32,235
因此具有较低的数值等级和
therefore have low numerical rank and

512
00:32:32,240 --> 00:32:34,585
这就是他在这种情况下所做的
that's that's what he did in this case

513
00:32:34,590 --> 00:32:38,185
所以这里他做得很好，他试图找到一个
so here's he did fine he tried to find a

514
00:32:38,190 --> 00:32:42,205
P是Fin的多项式近似
P a polynomial approximation to F in

515
00:32:42,210 --> 00:32:44,485
特别是他看着这个
particular he looked at exactly this

516
00:32:44,490 --> 00:32:51,625
那种近似所以他有一些
kind of approximation so he has some

517
00:32:51,630 --> 00:32:54,895
数字在这里，以便事情得到解决
numbers here so that things get resolved

518
00:32:54,900 --> 00:32:59,665
后来他试图找到一个这样做的P.
later he tried to find a P that did this

519
00:32:59,670 --> 00:33:09,385
那种近似然后他会
kind of approximation and then he would

520
00:33:09,390 --> 00:33:14,215
开发X的低秩近似
develop a low-rank approximation to X by

521
00:33:14,220 --> 00:33:18,775
抽样P所以他会说好的Y
sampling P so he would say okay well Y

522
00:33:18,780 --> 00:33:26,455
如果我让Y成为P的样本
if I let Y be a sampling of P then from

523
00:33:26,460 --> 00:33:27,775
事实上F是一个很好的近似值
the fact that F is a good approximation

524
00:33:27,780 --> 00:33:30,145
到PX是好的Y是好的
to P X is a good Y is a good

525
00:33:30,150 --> 00:33:33,805
近似于X，所以这有
approximation to X and so this has

526
00:33:33,810 --> 00:33:42,325
有限的等级你写下这个
finite rank you wrote down that this

527
00:33:42,330 --> 00:33:47,635
必须坚持，epsilon出现在这里
must hold and the epsilon comes out here

528
00:33:47,640 --> 00:33:49,825
因为我选择了这些因素
because I do these factors were chosen

529
00:33:49,830 --> 00:33:52,885
恰到好处地选择除以n
just right the divided by n was chosen

530
00:33:52,890 --> 00:33:55,335
所以epsilon就在那里出现了
so that the epsilon came out just there

531
00:33:55,340 --> 00:33:58,165
好吧这么多年了
okay so for many years that was kind of

532
00:33:58,170 --> 00:33:59,875
人们愿意的典型理由
the canonical reason that people would

533
00:33:59,880 --> 00:34:03,265
如果矩阵X是，那就好了
give that well if the matrix X is

534
00:34:03,270 --> 00:34:07,585
从平滑函数中采样然后我们
sampled from a smooth function then we

535
00:34:07,590 --> 00:34:08,845
可以通过a近似该函数
can approximate that function by a

536
00:34:08,850 --> 00:34:13,045
多项式并得到多项式秩
polynomial and get polynomial rank

537
00:34:13,050 --> 00:34:16,375
近似，因此矩阵
approximations and therefore the matrix

538
00:34:16,380 --> 00:34:22,435
X的数值排名很低
X will be of low numerical rank there's

539
00:34:22,440 --> 00:34:26,315
这个推理的问题
an issue with this reasoning

540
00:34:26,320 --> 00:34:28,565
特别是对于希尔伯特矩阵
especially for the Hilbert matrix that

541
00:34:28,570 --> 00:34:31,715
它实际上并没有那么好用
it doesn't actually work that well so

542
00:34:31,720 --> 00:34:36,245
例如，如果我把1000乘1000
for example if I take the 1000 by 1000

543
00:34:36,250 --> 00:34:40,825
希尔伯特矩阵，我看看它的排名
Hilbert matrix and I look at its rank

544
00:34:40,830 --> 00:34:42,875
好吧我已经告诉过你了
okay well I've already told you this is

545
00:34:42,880 --> 00:34:46,924
满级，你将获得一千个
full rank you'll get a thousand all the

546
00:34:46,929 --> 00:34:51,155
如果我看，奇异值是正的
singular values are positive if I look

547
00:34:51,160 --> 00:34:56,075
在这个1000的数字排名
at the numerical rank of this 1000 by

548
00:34:56,080 --> 00:34:58,595
1000希尔伯特矩阵，我计算它
1000 Hilbert matrix and I compute it I

549
00:34:58,600 --> 00:35:01,145
计算SVD，我看看有多少
compute the SVD and I look at how many

550
00:35:01,150 --> 00:35:06,695
在epsilon上面，epsilon是10到
are above epsilon where epsilon is 10 to

551
00:35:06,700 --> 00:35:10,595
减去15这意味着我可以
the minus 15 so that means I can

552
00:35:10,600 --> 00:35:12,815
由1000希尔伯特逼近1000
approximate the 1,000 by 1,000 Hilbert

553
00:35:12,820 --> 00:35:16,535
矩阵由秩28矩阵而且只给出
matrix by rank 28 matrix and only give

554
00:35:16,540 --> 00:35:20,525
一个精确到15的数字
up a digit that will be exact to 15

555
00:35:20,530 --> 00:35:25,295
这是一个巨大的数字，所以这是
digits which is a huge amount so this is

556
00:35:25,300 --> 00:35:30,865
我们从实践中得到的东西，但杂草
what we get from in practice but weeds

557
00:35:30,870 --> 00:35:41,675
这里的说法表明，军衔
argument here shows that the rank of

558
00:35:41,680 --> 00:35:45,515
这个矩阵的数值排名就是那个
this matrix the numerical rank is that

559
00:35:45,520 --> 00:35:50,885
最让它不会做了很好的工作在
most so it doesn't do a very good job on

560
00:35:50,890 --> 00:35:55,765
用于界定等级的希尔伯特矩阵
the Hilbert matrix for bounding the rank

561
00:35:55,770 --> 00:35:59,135
对，所以里德出现了这个
right so Reed comes along takes this

562
00:35:59,140 --> 00:36:01,115
函数他试图找到一个多项式
function he tries to find a polynomial

563
00:36:01,120 --> 00:36:02,195
这样做
that does this

564
00:36:02,200 --> 00:36:05,015
而10到15的他发现了
whereas 10 to the minus 15 he finds that

565
00:36:05,020 --> 00:36:07,415
他需要的项数，他
he needs the number of terms that he

566
00:36:07,420 --> 00:36:10,645
这个表达式的需求在这里
needs in this expression here is around

567
00:36:10,650 --> 00:36:15,935
719，因此这是排名
719 and therefore that's the rank that

568
00:36:15,940 --> 00:36:18,515
他在数字排名上受到了限制
he gets the bound on the numerical rank

569
00:36:18,520 --> 00:36:22,955
麻烦是719告诉我们的
the trouble is that 719 tells us that

570
00:36:22,960 --> 00:36:25,445
这个数字并不低
the numerical this is not of low

571
00:36:25,450 --> 00:36:29,255
数字排名，但我们知道它是如此
numerical rank but we know it it so it's

572
00:36:29,260 --> 00:36:34,325
这是令人不满意的原因所以有
it's unsatisfactory reason so there's

573
00:36:34,330 --> 00:36:37,085
有几个人试图上来
been several people trying to come up

574
00:36:37,090 --> 00:36:40,185
有更合适的理由
with more appropriate reasons that

575
00:36:40,190 --> 00:36:45,555
布莱恩在这里第二十八位，所以一位
Blayne the twenty-eighth here and so one

576
00:36:45,560 --> 00:36:49,695
我开始使用的原因是
reason that has I've started to use is

577
00:36:49,700 --> 00:36:52,755
另一个稍微不同的方式
that another slightly different way of

578
00:36:52,760 --> 00:36:56,715
看着要说的东西
looking at things which is to say the

579
00:36:56,720 --> 00:37:09,584
世界是学期现在西尔维斯特什么的
world is semester now Sylvester what

580
00:37:09,589 --> 00:37:10,905
这意味着什么是这个词
does that mean what is the word

581
00:37:10,910 --> 00:37:13,125
在这种情况下，西尔维斯特意味着它意味着
Sylvester mean in this case it means

582
00:37:13,130 --> 00:37:15,614
矩阵满足某种类型
that the matrices satisfy a certain type

583
00:37:15,619 --> 00:37:17,864
等式称为西尔维斯特
of equation called the Sylvester

584
00:37:17,869 --> 00:37:22,655
等式，所以原因是真的
equation and so the reason is really

585
00:37:22,660 --> 00:37:27,395
许多这些矩阵满足a
many of these matrices satisfy a

586
00:37:27,400 --> 00:37:29,834
西尔维斯特方程式并且需要
Sylvester equation and that takes the

587
00:37:29,839 --> 00:37:45,165
对于一些B和C来说形式还可以，所以X是
form okay for some a B and C so X is

588
00:37:45,170 --> 00:37:47,235
您想要展示的感兴趣的矩阵
your matrix of interest you want to show

589
00:37:47,240 --> 00:37:51,555
X的数值低等级和任务
X's of numerical low rank and the task

590
00:37:51,560 --> 00:37:55,025
在手边就是找到一个B和C这样的
at hand is to find an a B and C so that

591
00:37:55,030 --> 00:38:00,225
对于方程好吗X满足
X satisfies that equation okay for

592
00:38:00,230 --> 00:38:02,354
例如两个矩阵的所有矩阵
example all the matrices the two

593
00:38:02,359 --> 00:38:05,045
我已经登上了矩阵
matrices I've had on the board

594
00:38:05,050 --> 00:38:09,735
满足semestre方程西尔威斯特
satisfy a semestre equation a sylvester

595
00:38:09,740 --> 00:38:11,805
矩阵方程有aaB和a
matrix equation there is an a a B and a

596
00:38:11,810 --> 00:38:15,185
例如，他们这样做的C.
C for which they do this for example

597
00:38:15,190 --> 00:38:18,165
记住我们的希尔伯特矩阵
remember the Hilbert matrix which we

598
00:38:18,170 --> 00:38:21,515
有那里我会再次把它写下来
have there I'll write it down again

599
00:38:21,520 --> 00:38:27,015
满足这些条目所以我们所有
satisfies has these entries so all we

600
00:38:27,020 --> 00:38:29,084
需要做的是试图找出一个
need to do is to try to figure out an a

601
00:38:29,089 --> 00:38:32,625
和B然后是C，这样我们就可以做到
and B and then a C so that we can make

602
00:38:32,630 --> 00:38:34,635
它适合西尔维斯特方程式有很多
it fit a Sylvester equation there's many

603
00:38:34,640 --> 00:38:36,975
这样做的不同方式
different ways of doing this the one

604
00:38:36,980 --> 00:38:40,695
我喜欢的是以下内容
that I like is the following

605
00:38:40,700 --> 00:38:43,814
如果我把一个半在这里和三面翻
what if I put a half here and three over

606
00:38:43,819 --> 00:38:49,555
到这里一直到n减1/2
to here all the way down to n minus 1/2

607
00:38:49,560 --> 00:38:54,335
次此矩阵和我，所以这是
times this matrix and I so this is

608
00:38:54,340 --> 00:38:58,685
将此矩阵的顶部乘以1/2
timesing the top of this matrix by 1/2

609
00:38:58,690 --> 00:39:01,595
然后3超过2，然后5在2
and then 3 over 2 and then 5 over 2

610
00:39:01,600 --> 00:39:03,515
所以我们基本上把这个矩阵计时
so we're basically timesing that matrix

611
00:39:03,520 --> 00:39:07,415
该矩阵的每个条目由J减1/2
each entry of this matrix by J minus 1/2

612
00:39:07,420 --> 00:39:11,975
然后我做一些事情上如果有合适的
and then I do something on the right if

613
00:39:11,980 --> 00:39:13,145
在这里，我被允许做，因为
here which I'm allowed to do because

614
00:39:13,150 --> 00:39:15,185
我有B自由，我选择了这个
I've got the B freedom and I choose this

615
00:39:15,190 --> 00:39:23,795
是一样的了减号然后
to be the same up to a minus sign then

616
00:39:23,800 --> 00:39:25,925
当你想到它是什么时
when you think about this what is it

617
00:39:25,930 --> 00:39:26,255
干
doing

618
00:39:26,260 --> 00:39:32,075
这是JK条目的时间，这是由j
it's timesing the JK entry this is by j

619
00:39:32,080 --> 00:39:34,415
减去1/2，这就是这样做
minus 1/2 that's what this is doing and

620
00:39:34,420 --> 00:39:37,175
什么是它与JK的时间
what's this doing its timesing the JK

621
00:39:37,180 --> 00:39:41,965
由K减去1/2，所以这是总的来说
entry by K minus 1/2 so this is in total

622
00:39:41,970 --> 00:39:46,235
将JK输入加到J加K减去
timesing the JK entry by J plus K minus

623
00:39:46,240 --> 00:39:49,985
1/2减1/2这是减1所以这个
1/2 minus 1/2 which is minus 1 so this

624
00:39:49,990 --> 00:39:52,925
是J加K的JK入场时间
is timesing the JK entry by J plus K

625
00:39:52,930 --> 00:39:56,345
减1，所以它击败了分母
minus 1 so it knocks out the denominator

626
00:39:56,350 --> 00:40:01,385
我们从这个等式得到的是一个
and what we get from this equation is a

627
00:40:01,390 --> 00:40:12,845
一堆那些在这种情况下a和B.
bunch of ones so in this case a and B

628
00:40:12,850 --> 00:40:15,575
是对角线，C是所有的矩阵
are diagonal and C is the matrix of all

629
00:40:15,580 --> 00:40:19,705
好吧，我们也可以为范德曼做这件事
ones ok we can also do this for vandeman

630
00:40:19,710 --> 00:40:25,205
所以vanaman，你会记得看起来像
so vanaman you'll remember looks like

631
00:40:25,210 --> 00:40:34,705
这个，然后在这里我们有这个人
this and then over here we have this guy

632
00:40:34,710 --> 00:40:38,705
以多项式出现的矩阵
the matrix that appears with polynomial

633
00:40:38,710 --> 00:40:42,935
插好了，所以如果我想
interpolation ok so if I think about

634
00:40:42,940 --> 00:40:46,205
我也可以想出一个B和
this I can also come up with an a B and

635
00:40:46,210 --> 00:40:50,765
C，例如这里有效
C and for example here's one that works

636
00:40:50,770 --> 00:40:58,865
我可以把X贴在对角线上
I can stick the X's on the diagonal

637
00:40:58,870 --> 00:41:01,755
所以如果你想象那是什么
so this is if you imagine what that

638
00:41:01,760 --> 00:41:03,975
左边的矩阵正在做它
matrix on the left is doing it's

639
00:41:03,980 --> 00:41:08,505
将每列乘以向量Xok
timesing each column by the vector X ok

640
00:41:08,510 --> 00:41:10,005
所以这个矩阵的第一列
so the first column of this matrix

641
00:41:10,010 --> 00:41:13,455
X成为第二个向量X.
becomes X the vector X the second

642
00:41:13,460 --> 00:41:16,695
成为矢量x平方在哪里
becomes the vector x squared where

643
00:41:16,700 --> 00:41:18,975
平方是明智的，然后是
squared is done entry wise and then the

644
00:41:18,980 --> 00:41:21,285
第三个条目现在是X立方体，当我们
third entry is now X cubed and when we

645
00:41:21,290 --> 00:41:24,885
到最后是X到nok所以
get to the last is X to the n ok so

646
00:41:24,890 --> 00:41:27,195
这就像每列乘以
that's like multiplied each column by

647
00:41:27,200 --> 00:41:31,575
矢量X所以如果我想尝试来
the vector X so if I want to try to come

648
00:41:31,580 --> 00:41:33,705
了一个矩阵，还剩下些什么是
up with a matrix so what's left is of

649
00:41:33,710 --> 00:41:37,095
低级别就像我能做的那样
low rank is like of this form what I can

650
00:41:37,100 --> 00:41:41,955
做的是移动列，所以我注意到了
do is shift the columns so I've noticed

651
00:41:41,960 --> 00:41:44,175
这个产品在这里这个对角线
that this product here this diagonal

652
00:41:44,180 --> 00:41:46,875
矩阵已经使第一列X如此
matrix has made the first column X so if

653
00:41:46,880 --> 00:41:48,285
我想杀掉那个专栏
I want to kill off that column I can

654
00:41:48,290 --> 00:41:51,105
取第二列并将其置换为
take the second column and permute it to

655
00:41:51,110 --> 00:41:52,905
第一栏我可以拿第三栏
the first column I can take the third

656
00:41:52,910 --> 00:41:55,095
列并将其置换到第二个
column and permute it to the second the

657
00:41:55,100 --> 00:41:57,045
最后一栏推算将其置换为
last column imputed permute it to the

658
00:41:57,050 --> 00:41:59,235
倒数第二列，这将是
penultimate column here and that will

659
00:41:59,240 --> 00:42:01,005
实际上杀掉了我的很多东西
actually kill off a lot of what I've

660
00:42:01,010 --> 00:42:03,945
在这个矩阵就在这里创建，以便让
created in this matrix right here so let

661
00:42:03,950 --> 00:42:07,595
我写下来这是一种抑扬
me write that down this is a circumflex

662
00:42:07,600 --> 00:42:16,305
这样做就是排列
this does that permutation this does

663
00:42:16,310 --> 00:42:18,135
那个排列我放了一个减1
that permutation I've put a minus 1

664
00:42:18,140 --> 00:42:19,725
我可以在那里放任何号码
there I could have put any number there

665
00:42:19,730 --> 00:42:22,725
它没有任何区别但是这个
it doesn't make any difference but this

666
00:42:22,730 --> 00:42:24,795
是一个非常好的工作
is the one that works out really nicely

667
00:42:24,800 --> 00:42:28,635
现在这个0出了很多东西，因为
now this 0 is out lots of things because

668
00:42:28,640 --> 00:42:30,495
我做乘法的方式
of the way I've done the multiplication

669
00:42:30,500 --> 00:42:32,805
通过X和列的周向移动
by X and the circum shift of the columns

670
00:42:32,810 --> 00:42:36,155
所以第一列是0因为
and so the first column is 0 because

671
00:42:36,160 --> 00:42:39,555
此第一列是X此第一列
this first column is X this first column

672
00:42:39,560 --> 00:42:43,695
是X所以我的X减X这个专栏
is X so I've got X minus X this column

673
00:42:43,700 --> 00:42:46,605
是x平方减去x平方所以我得到0
was x squared minus x squared so I got 0

674
00:42:46,610 --> 00:42:50,115
我只是保持沿下去，直到那
and I just keep going along until that

675
00:42:50,120 --> 00:42:52,335
最后一列的最后一列是
last column that last column is a

676
00:42:52,340 --> 00:42:54,225
问题，因为这是最后一列
problem because the last column of this

677
00:42:54,230 --> 00:42:57,765
家伙是X到N，而我没有X
guy is X to the N whereas I don't have X

678
00:42:57,770 --> 00:43:00,255
到N和V所以有一些数字
to the N and V so there's some numbers

679
00:43:00,260 --> 00:43:05,055
好的
here ok

680
00:43:05,060 --> 00:43:07,975
你会发现两种情况都看得出来
you'll notice that see in both cases

681
00:43:07,980 --> 00:43:11,305
恰好是这些中的低秩矩阵
happens to be a low rank matrix in these

682
00:43:11,310 --> 00:43:14,995
这种情况发生的情况是等级1和
cases that happens to be of Rank 1 and

683
00:43:15,000 --> 00:43:19,225
所以人们想知道可能是它
so people were wondering maybe it's

684
00:43:19,230 --> 00:43:20,845
是与满足这些
something to do with satisfying these

685
00:43:20,850 --> 00:43:24,535
制造这些的方程式
kind of equations that makes these

686
00:43:24,540 --> 00:43:27,045
在实践中出现的矩阵
matrices that appear in practice

687
00:43:27,050 --> 00:43:31,315
数字上的低排名和经过很多次
numerically of low rank and after a lot

688
00:43:31,320 --> 00:43:34,945
在这方面的工作人们已经出现了
of work in this area people have come up

689
00:43:34,950 --> 00:43:40,615
有一个证明这个的界限
with a bound that demonstrates that this

690
00:43:40,620 --> 00:43:43,075
这些类型的方程是关键
these kind of equations are key to

691
00:43:43,080 --> 00:43:47,415
理解数值低等级
understanding the numerical low rank so

692
00:43:47,420 --> 00:43:57,025
如果X满足像学期一样的学期方程
if X satisfies a semester equation like

693
00:43:57,030 --> 00:44:05,875
这和一个是正常的
this and a is normal

694
00:44:05,880 --> 00:44:07,825
B是正常的我真的不想
B is normal I don't really want to

695
00:44:07,830 --> 00:44:11,505
专注于那两个条件
concentrate on that those two conditions

696
00:44:11,510 --> 00:44:19,315
这比人们有点学术
it's a little bit academic then people

697
00:44:19,320 --> 00:44:21,295
在单数上发现了一个界限
have found a bound on the singular

698
00:44:21,300 --> 00:44:24,115
任何满足此要求的矩阵的值
values of any matrix that satisfies this

699
00:44:24,120 --> 00:44:29,665
他们发现了这种表达方式
kind of expression and they found this

700
00:44:29,670 --> 00:44:45,835
更充分和约束确定所以这里的等级
fuller and bound ok so here the rank of

701
00:44:45,840 --> 00:44:50,455
C是关闭，因此去那里所以在我们的
C is off so that goes there so in our

702
00:44:50,460 --> 00:44:53,635
我们有R的两个例子就是1
cases the two examples we have R is 1 so

703
00:44:53,640 --> 00:44:56,935
我们可以忘记这个讨厌的家伙
we can forget about our this nasty guy

704
00:44:56,940 --> 00:45:06,545
这里叫做Ghazala关税号
here is called Ghazala tariff number

705
00:45:06,550 --> 00:45:11,405
e是一个集，其中包含的特征值
e is a set that contains the eigenvalues

706
00:45:11,410 --> 00:45:16,865
a和F是包含的集合
of a and F is the set that contains the

707
00:45:16,870 --> 00:45:23,915
B的特征值好吧现在看起来像
eigenvalues of B okay now it looks like

708
00:45:23,920 --> 00:45:26,615
我们通过获得绝对没有
we have gained absolutely nothing by

709
00:45:26,620 --> 00:45:29,405
这是因为我刚刚告诉你
this down because I've just told you

710
00:45:29,410 --> 00:45:31,235
奇异价值受佐拉关税的约束
singular values are bound by Zola tariff

711
00:45:31,240 --> 00:45:34,175
数字，这并不意味着什么
numbers that doesn't mean anything to

712
00:45:34,180 --> 00:45:37,505
这对我来说意味着什么，但是
anyone it means a little bit to me but

713
00:45:37,510 --> 00:45:41,705
没那么多，这个关键的约束
not that much so the key to this bound

714
00:45:41,710 --> 00:45:45,065
这是非常有用的原因是这样的
the reason this is useful is that so

715
00:45:45,070 --> 00:45:48,275
很多人都弄清楚了这些
many people have worked out what these

716
00:45:48,280 --> 00:45:52,295
佐拉关税数字实际上意味着没关系
zola tariff numbers actually mean okay

717
00:45:52,300 --> 00:45:56,585
因此，这些都是工作的两个关键人
so these are two key people that worked

718
00:45:56,590 --> 00:45:59,975
这个界限意味着什么，我们有
out what this bound means and we have

719
00:45:59,980 --> 00:46:02,855
获得了很多，因为人们已经
gained a lot because people have been

720
00:46:02,860 --> 00:46:05,315
研究这个数字就像一个
studying this number this is like a

721
00:46:05,320 --> 00:46:09,395
1870年人们关心的数字
number that people cared about from 1870

722
00:46:09,400 --> 00:46:12,035
直到现在和人们
onwards to the present day and people

723
00:46:12,040 --> 00:46:14,165
我非常好地研究了这个数字
have studied this number extremely well

724
00:46:14,170 --> 00:46:17,495
所以我们通过转动获得了一些东西
so we've gained something by turning it

725
00:46:17,500 --> 00:46:20,075
成为一个更抽象的问题的人
into a more abstract problem that people

726
00:46:20,080 --> 00:46:23,375
已经考虑过以前和现在我们
have thought about previously and now we

727
00:46:23,380 --> 00:46:26,015
可以去关于佐拉关税的文献
can go to the literature on zola tariff

728
00:46:26,020 --> 00:46:28,055
数字无论他们是什么并发现
numbers whatever they are and discover

729
00:46:28,060 --> 00:46:31,505
这个或整个工作的整个文献
this whole literature of work on this or

730
00:46:31,510 --> 00:46:34,355
关税号码和关键部分我会
a tariff number and the key part I'll

731
00:46:34,360 --> 00:46:42,575
告诉你关键是集E
just tell you the key is that the sets E

732
00:46:42,580 --> 00:46:53,084
和F分开
and F are separated

733
00:46:53,089 --> 00:46:56,604
例如，在希尔伯特矩阵中
so for example in the Hilbert matrix the

734
00:46:56,609 --> 00:46:58,794
a的本征值可以从中读出
eigen values of a can be read off the

735
00:46:58,799 --> 00:47:07,814
对角线它们之间是什么
diagonal what are they they are between

736
00:47:07,819 --> 00:47:13,705
减去1/2和n减去1/2并且
minus 1/2 and n minus 1/2 and the

737
00:47:13,710 --> 00:47:19,225
b的特征值位于集合中
eigenvalues of b lie in the set minus

738
00:47:19,230 --> 00:47:25,135
1/2减去n加1和关键原因
1/2 minus n plus 1 and the key reason

739
00:47:25,140 --> 00:47:28,465
为什么希尔伯特矩阵很低
why the hilbert matrix is of low

740
00:47:28,470 --> 00:47:30,895
数字排名是这些事实
numerical rank is the fact that these

741
00:47:30,900 --> 00:47:33,804
两套分离并且使
two sets are separated and that makes

742
00:47:33,809 --> 00:47:35,844
这是所有关税数量变小
this is all a tariff number gets small

743
00:47:35,849 --> 00:47:39,534
你可能会非常快速地使用k
extremely quickly with k now you might

744
00:47:39,539 --> 00:47:41,655
不知道为什么有一个问号
wonder why there's a question mark on

745
00:47:41,660 --> 00:47:47,755
铅笔的名字有非正式的诅咒
pencils name there is unofficial curse

746
00:47:47,760 --> 00:47:51,594
这已经持续了一段时间都
that's been going on for a while both

747
00:47:51,599 --> 00:47:54,205
这些人在工作时死亡
these men died while working on the

748
00:47:54,210 --> 00:47:56,995
纸牌问题他们都死在了
solitaire problem they both died at the

749
00:47:57,000 --> 00:48:02,004
31岁的一个人被一个人击中而死
age of 31 one died by being hit by a

750
00:48:02,009 --> 00:48:05,475
训练目前还不清楚他是否这样做
train it's unclear whether he does

751
00:48:05,480 --> 00:48:10,104
自杀或意外铅笔死亡
suicidal or is accidental pencil died at

752
00:48:10,109 --> 00:48:12,594
加拿大山区的年龄为31岁
the age of 31 in the Canadian mountains

753
00:48:12,599 --> 00:48:14,804
通过雪崩
by an avalanche

754
00:48:14,809 --> 00:48:19,495
我现在还不到31岁但是要去
I am currently not yet 31 but going to

755
00:48:19,500 --> 00:48:22,645
是31很快，我很害怕，我
be 31 very soon and I'm scared that I

756
00:48:22,650 --> 00:48:29,725
可以加入这个列表，但对于休伯特来说
may join this list ok but for the Hubert

757
00:48:29,730 --> 00:48:32,245
矩阵你从这个分析得到什么
matrix what you get from this analysis

758
00:48:32,250 --> 00:48:35,955
基于这两个人的工作是
based on these two people's work is

759
00:48:35,960 --> 00:48:39,955
数字排名和数据比比皆是
abound on the numerical rank and the

760
00:48:39,960 --> 00:48:43,495
你得到的等级是让我们说一个世界
rank that you get is let's say a world

761
00:48:43,500 --> 00:48:50,594
希尔伯特矩阵的记录界限是
record bound for the Hilbert matrix is

762
00:48:50,599 --> 00:48:56,364
34这是不太28尚未但是
34 which is not quite 28 not yet but

763
00:48:56,369 --> 00:49:02,065
它比2890更具描述性
it's far more descriptive of 28 than 790

764
00:49:02,070 --> 00:49:06,445
所以这种边界技术
and so this technique of bounding

765
00:49:06,450 --> 00:49:08,305
使用这些太阳能的奇异值
singular values by using these solar

766
00:49:08,310 --> 00:49:10,465
关税数字开始增加
tariff numbers is starting to gain

767
00:49:10,470 --> 00:49:13,375
人气，因为我们终于可以回答
popularity because we can finally answer

768
00:49:13,380 --> 00:49:16,375
对自己为什么有这么多低
to ourselves why there are so many low

769
00:49:16,380 --> 00:49:18,025
出现的等级矩阵
rank matrices that appear in

770
00:49:18,030 --> 00:49:23,595
计算数学，它都是基于
computational math and it's all based on

771
00:49:23,600 --> 00:49:29,395
231岁的人死了，如果你死了
231 year olds that died and so if you

772
00:49:29,400 --> 00:49:30,895
无所谓，你什么时候做
ever wonder when you're doing

773
00:49:30,900 --> 00:49:32,725
计算科学时排名较低
computational science when a lower rank

774
00:49:32,730 --> 00:49:35,275
出现，平滑性论证
appears and the smoothness argument does

775
00:49:35,280 --> 00:49:37,405
你不工作，你可能会认为
not work for you you might like to think

776
00:49:37,410 --> 00:49:41,245
关于左拉关税和诅咒还好吧
about Zola tariffs and the curse okay

777
00:49:41,250 --> 00:49:52,185
非常感谢你现在的工作方式
thank you very much how does it work now

778
00:49:52,190 --> 00:49:55,825
好的，如果我们，我很乐意提问
okay I'm happy to take questions if we

779
00:49:55,830 --> 00:50:01,095
如果您有任何疑问，请等一下
have a minute if you have any questions

780
00:50:01,100 --> 00:50:04,645
好吧，我得到了我31岁的聚光灯
well then I get a spotlight on I'm 31 in

781
00:50:04,650 --> 00:50:08,875
12月所以他们在31岁时去世了
December so they died at the age of 31

782
00:50:08,880 --> 00:50:12,535
所以你知道明年是可怕的一年
so you know next year is the scary year

783
00:50:12,540 --> 00:50:16,405
对我来说，我不会在任何地方开车
for me so I will be not driving anywhere

784
00:50:16,410 --> 00:50:21,620
不会离开我的房子，直到我32变

785
00:50:16,410 --> 00:50:32,120
not leaving my house until I become 32

786
00:50:32,130 --> 00:50:34,190
您
you