08-Norms of Vectors and Matrices.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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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，所以这个大麻省理工学院我只是来自一个
okay so this big MIT I just came from a

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教师很棒的教师安迪低
faculty terrific faculty member Andy low

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在斯隆学校和我必须
in in the Sloan School and I have to

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告诉你他告诉我们的是什么，然后我有
tell you what he told us and then I had

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在他解释之前我不得不离开
I had to leave before he could explain

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为什么这是真的，但这就像是一个
why it's true but this is like an

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我不想要的惊人事实
amazing fact which I don't want to

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忘了所以在这里你去这就是一切
forget so here you go this is everything

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将在该板所以它是很
will be on that board so it's it's it's

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关于我们或其他人的观察
an observation about us or other people

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也许不是我们，但假设你有一个
maybe not us but so suppose you have a

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有偏见的硬币也许你不是人
biased coin maybe you don't the people

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玩这个游戏不知道，但它
playing this game don't know but it's

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75％的人可能生产25％的头部
75% likely to produce heads 25% likely

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产生尾巴然后玩家有
to produce tails and then the player has

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猜到一个接一个的翻转
to guess for one flip after another

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头或尾，如果你得到一美元
heads or tails and you get a dollar if

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如果你是的话，你支付一美元是对的
you're right you pay a dollar if you're

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错了，所以你只想得到尽可能多的
wrong so you just want to get as many

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从这枚硬币中做出正确的选择
right choices as possible from this coin

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继续翻转所以你应该怎么做
flip that continues so what should you

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做得好，我想我希望我们会这样做
do well what I guess I hope we would do

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我们不知道是什么
is we would not know what the

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概率是如此我们猜测
probabilities were so we would guess

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也许是第一次尾巴的头
maybe heads the first time tails the

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第二次前往第三次
second time heads to the third time and

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等等，但实际结果却是
so on but the actual result would be

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大多是头
mostly heads

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所以我们会在某些方面学到这一点
so we would learn at some point that

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或许也许不会那么快我们
maybe maybe not quite as soon as that we

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最终会知道我们应该这样做
would eventually learn that we should

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继续猜测正确然后我们
keep guessing heads right and then we

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那将是我们的最佳选择
would be that would be our optimal

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战略猜测头所有的时间，但
strategy to guess heads all the time but

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人们实际上做了什么，他们确实开始
what do people actually do they do start

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像这样，然后他们
like this same way and then they're

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开始学习头脑更多
beginning to learn that heads is more

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很常见，也许他们的头脑比
common so maybe they do more heads than

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尾巴，但有时尾巴是正确的
tails but sometimes tails is right and

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然后过一小会儿，他们也许看到
then after a little while they maybe see

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是的，或者他们很好
that it's yeah or they're just well

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也许他们不算他们只是
maybe they're not counting they're just

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像普通人一样经营
like operating like ordinary people

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普通人实际上做了什么
and what do ordinary people actually do

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从长远来看，你会说猜
in the long run you would say guess

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每次都对，但他们没有
heads every time right but they don't in

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从长远来看，人们也许是动物
the long run people and maybe animals

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无论猜测是四分之三
and whatever guess heads three-quarters

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的时间，并在尾部四分之一
of the time and tails one quarter at a

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他们的时间并不令人难以置信
time isn't that unbelievable they're

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四分之一的时间猜测尾巴
guessing tails a quarter of the time

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当赔率永远不会改变的时候
when the odds are never changing anyway

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这就是经济学家的观点
that's that's something that economists

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和其他人必须解释和
and and other people have to explain and

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如果我能再住一小时
if I had been able to stay another hour

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我可以告诉你这个解释
I could tell you about the explanation

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但无论如何，那就是我看到的
but anyway that's that's I I see I've

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写在我没有的板上
written that on a board that I have no

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埋葬的方式，所以它会在那里
way to bury so it's gonna be there and

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它不是1806年的主题，而是
it's not the subject of 1806 five but

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00:04:06,070 --> 00:04:10,354
它有点令人惊奇，所以有
it's kind of amazing right so there's

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良好的数学问题无处不没关系我可以
good math problems everywhere okay can I

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只要离开你我所知道的和
just leave you with what I know and and

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如果我了解更多，我会回到那里
if I learn more I'll come back to that

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问题没关系
question okay

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请注意这种方式
please turn attention this way right

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Norm在规范上有点说几句
Norm's a little a few words on norms

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就像那个应该是你的一句话
just like that should be a word in your

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语言，所以你应该知道它是什么
language and so you should know what it

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意思是你应该知道他们是少数
mean and you should know they're a few

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重要的规范再一个规范是一个
of the important norms again a norm is a

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测量矢量或大小的方法
way to measure the size of a vector or

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矩阵的大小或a的大小
the size of a matrix or the size of a

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张紧我们拥有的任何东西或功能
tensor whatever we have or a function

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非常重要的是规范
very important the norm would be a

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00:04:56,260 --> 00:04:59,685
功能就像正弦X从零到PI
function like sine X from zero to PI

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那个函数的大小是多少？
what would be the size of that function

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00:05:02,300 --> 00:05:05,175
好吧，如果它是两个正弦x的大小
well if it was two sine x the size would

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00:05:05,180 --> 00:05:08,205
两倍，所以规范应该
be twice as much so the norm should

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反映，但昨天或
reflect that but so yesterday or

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星期三我告诉你那里有P.
Wednesday I told you the so there P

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等于1实际无穷大然后
equals to 1 actually infinity and then

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我要用一个零的标准
I'm gonna put in the zero norm with a

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问号，因为你会看到它
question mark because you'll see it's

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它有问题，但让我公正
that it has a problem but let me just

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从上次回忆起这些是P
recall from last time so these are the P

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等于2是通常的平方和
equal to two is the usual sum of squares

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平方根通常长度的向量P.
square root usual length of a vector P

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这是一个非常重要的规范
equal one is this very important norm so

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我们称之为l1规范，我们会看到
a we call that the l1 norm and we'll see

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很多我真正提到过的
a lot of that that really I mentioned

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它起着非常重要的作用
that it plays a very significant part

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00:06:01,820 --> 00:06:04,275
现在在压缩传感中它确实是一个
now in compress sensing it really was a

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信号处理中的重磅炸弹
bombshell in signal processing to

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发现和在其他领域
discover and in other fields to to

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发现有些东西确实有效
discover that some things really work

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00:06:13,670 --> 00:06:17,355
在l1规范中做得最好
well work best in the l1 norm the

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最大规范有自然的作用
maximum norm has a natural part to play

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我们会看到它或它的矩阵模拟
and we'll see that or its matrix analog

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00:06:25,730 --> 00:06:30,435
所以我没有提到l0规范
so I didn't mention the l0 norm all this

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LP业务因此LP规范
LP business so the LP norm

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00:06:35,450 --> 00:06:44,625
或者对于任何P，你是否获得了力量
or for any P is you take the piece power

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00:06:44,630 --> 00:06:49,135
在这里，P是2和
to the piece power up here P was 2 and

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你拿这块根也好吧
your take the piece root so maybe I

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应该把它写在P的那个上面
should write it to the one over P that's

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然后那样
then that way

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以片断力量和片断为基础我们
taking piece powers and piece roots we

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得到两个V的标准有一个因素
do get the norm of two V has a factor

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两个比较V的标准，所以P等于
two compared to the norm of V so P equal

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到2你看到我们做对了
to 2 you see that we've got it right

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那里的P等于你在这里看到的
there P equal one you see it here

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00:07:13,140 --> 00:07:16,225
因为它只是它的总和
because it's just just sum of the

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00:07:16,230 --> 00:07:19,375
绝对值人们无穷，如果我要是
absolute values people infinity if I if

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00:07:19,380 --> 00:07:23,035
我向上和向上移动P它会选择
I move P up and up and up it will pick

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你知道，因为我增加P无论哪个
out you know as I increase P whichever

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一个是最大的将要采取
one is biggest is going to just take

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00:07:28,410 --> 00:07:30,955
结束，这就是为什么你达到最大规范
over and that's why you get to max norm

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00:07:30,960 --> 00:07:35,155
现在，我正在使用零标准
now there's zero norm where I'm using

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00:07:35,160 --> 00:07:37,915
你会看到这个词不正确
that word like improperly as you'll see

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00:07:37,920 --> 00:07:42,475
因为如果是这样的话，那是什么呢？
because if so what is the 0 norm it's

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00:07:42,480 --> 00:07:55,065
所以它是非零的数量
the so so it's the number of non-zeros

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00:07:55,070 --> 00:08:05,275
它的非零组件就是它
of nonzero components its it it's the

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00:08:05,280 --> 00:08:07,345
你想知道有关的事
thing that you'd like to know about in

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稀疏性的问题确实存在
question of sparsity does the is there

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00:08:10,680 --> 00:08:13,705
只有一个非零组件或那里11
just one nonzero component or there 11

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00:08:13,710 --> 00:08:17,635
你有什么样的可能吗？
are there 101 like what that you might

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00:08:17,640 --> 00:08:21,925
想要最小化，因为稀疏
want to minimize that because sparse

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00:08:21,930 --> 00:08:24,355
向量和稀疏矩阵很多
vectors and sparse matrices are much

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00:08:24,360 --> 00:08:26,785
你得到了很好的计算速度更快
faster to compute with you've got good

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00:08:26,790 --> 00:08:29,335
东西，但现在我声称这不是一个常态
stuff but now i claim that's not a norm

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00:08:29,340 --> 00:08:32,565
在0非零分量的数目
in the number of 0 nonzero components

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00:08:32,570 --> 00:08:36,474
因为什么是对如何做规范
because what's the norm of to how does

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00:08:36,479 --> 00:08:39,865
2V的标准与规范比较
the norm of 2v compare with the norm of

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00:08:39,870 --> 00:08:44,335
伏0规范这将是相同的到V
V the 0 norm it would be the same to V

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00:08:44,340 --> 00:08:46,865
具有相同数量的非零值
has the same number of non-zeros as

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00:08:46,870 --> 00:08:52,265
所以它违反了规范，所以我
so it violates the rule for a norm so I

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00:08:52,270 --> 00:08:57,065
想一想这些规范和所有的豌豆
think with these norms and all the peas

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00:08:57,070 --> 00:09:02,495
实际上是数学之间
in between so actually the the math

147
00:09:02,500 --> 00:09:06,005
论文充满了让P介于1之间
papers are full of let P be between 1

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00:09:06,010 --> 00:09:08,315
和无限，因为那是
and infinity and because that's the

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00:09:08,320 --> 00:09:10,475
在那里，你做的范围内有一个适当的标准为
range where you do have a proper norm as

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00:09:10,480 --> 00:09:14,435
我们会看到，我认为好的事情
we will see I think the good thing to do

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00:09:14,440 --> 00:09:17,735
这些规范是有一个画面中
with these norms is to have a picture in

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00:09:17,740 --> 00:09:20,675
你的思想规范的几何是真实的
your mind the geometry of a norm is real

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00:09:20,680 --> 00:09:23,255
是好的所以我要建议的图片
is good so the picture I'm gonna suggest

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是绘制所有的矢量让我们说在2d
is plot all the vectors let's say in 2d

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00:09:28,170 --> 00:09:34,055
所以二维空间或两个所以我
so two dimensional space or two so I

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00:09:34,060 --> 00:09:37,475
想要绘制他们所拥有的向量
want to plot then the vectors they have

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00:09:37,480 --> 00:09:43,145
V在这些不同的规范中等于1
V equal one in these different norms so

158
00:09:43,150 --> 00:09:48,205
让我问你这是什么，这里是2d空间
let me ask you what so here's 2d space

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00:09:48,210 --> 00:09:51,695
r2现在我想绘制所有的
r2 and now I want to plot all the

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00:09:51,700 --> 00:09:55,865
具有普通l2的向量
vectors that have the ordinary l2

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长度等于一个，这样又有什么
lengths equal one so what does that

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图片看起来像我只是想一张照片
picture look like I just think a picture

163
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真的是值得的东西它是一个圆
is really worth something it's a circle

164
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谢谢它是一个圆圈，这是一个圆圈
thanks it's a circle it's a circle this

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00:10:07,420 --> 00:10:10,175
圆是这是当然的方程式
circle is that's the equation of course

166
00:10:10,180 --> 00:10:17,965
v1平方加v2平方1以便我
v1 squared plus v2 squared 1 so that I

167
00:10:17,970 --> 00:10:22,175
会称之为单位球
would call that the unit ball for the

168
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现在规范或其他任何东西都可以
norm or whatever is is a circle okay now

169
00:10:27,160 --> 00:10:30,365
现在这里更有趣了
now here comes more interesting what

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关于在l1所以再次告诉我如何
about in the l1 so again tell me how to

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绘制所有具有V1加V点
plot all the points that have V 1 plus V

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2等于1什么是边界
2 equal 1 what what what's the boundary

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现在看起来就像是让我们一样
gonna look like now it's gonna be let's

174
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看清楚我可以把一定数量的
see well I can put in a certain number

175
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那里有一个点和那个点
of points there one and there one and

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那里是负一号那里
there at minus one and there

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这个就是那个
this one there was that would be the

178
00:11:00,860 --> 00:11:03,195
将反映向量10的向量
vector that would reflect the vector 1 0

179
00:11:03,200 --> 00:11:06,615
这将反映向量0
and this would reflect the vector 0

180
00:11:06,620 --> 00:11:12,435
减去1所以是OK所以这些都是像
minus 1 so yeah ok so those are like

181
00:11:12,440 --> 00:11:15,915
四点易于绘图容易看到
four points easy to plot easy to see

182
00:11:15,920 --> 00:11:19,574
l1规范，但是什么是完整的
that the l1 norm but what's the full

183
00:11:19,579 --> 00:11:23,805
那是什么呢？剩下的边界是什么
what so what's the rest of the boundary

184
00:11:23,810 --> 00:11:28,245
这是一颗钻石，它是钻石
here it's a diamond good it's a diamond

185
00:11:28,250 --> 00:11:35,565
这是我们所拥有的线性设定为等于一起来
it's we have linear set equal to one up

186
00:11:35,570 --> 00:11:37,695
这里是正象限
here in the positive quadrant

187
00:11:37,700 --> 00:11:40,815
它只是V1加上V2等于1
it's just V 1 plus V 2 equal 1 in the

188
00:11:40,820 --> 00:11:45,584
这个图是一条直线所以
graph of that is a straight line so all

189
00:11:45,589 --> 00:11:48,375
这些家伙这是V的所有要点
these guys this is all the points with V

190
00:11:48,380 --> 00:11:52,514
1加上V2等于1并且在这里和
1 plus V 2 equal 1 and over here and

191
00:11:52,519 --> 00:11:56,625
在这里和这里所以单位球
over here and over here so the unit ball

192
00:11:56,630 --> 00:12:02,745
在l1规范中是钻石，那是一个
in the l1 norm is a diamond and that's a

193
00:12:02,750 --> 00:12:05,834
它反映在一个非常重要的图片
very important picture it reflects in a

194
00:12:05,839 --> 00:12:08,985
简单的东西很简单
simple very simple way something

195
00:12:08,990 --> 00:12:11,024
重要的是l1规范和
important about the l1 norm and the

196
00:12:11,029 --> 00:12:14,805
因为它刚刚爆炸了他
reason it's just exploded him in

197
00:12:14,810 --> 00:12:17,685
重要的是让我继续下去
importance let me continue though what

198
00:12:17,690 --> 00:12:22,355
关于Max或V无穷大的最大范数
about the max norm the Max or V infinity

199
00:12:22,360 --> 00:12:26,385
又是什么让我策划这些家伙
what so again let me plot these guys

200
00:12:26,390 --> 00:12:29,715
这些家伙肯定会参与其中
these guys are certainly gonna be in it

201
00:12:29,720 --> 00:12:32,715
他们再次归零
again they're zero

202
00:12:32,720 --> 00:12:35,894
你知道加减我和加号
you know the plus or minus I and plus or

203
00:12:35,899 --> 00:12:39,975
减去那些是好朋友的J是什么
minus J that are good friends what's the

204
00:12:39,980 --> 00:12:42,885
边界的其余部分现在看起来像现在
rest of the boundary look like now now

205
00:12:42,890 --> 00:12:53,475
这就是这意味着什么是最大的
this is this means max so what what are

206
00:12:53,480 --> 00:12:58,454
你看到它的其余部分
the rest of the points you see it does

207
00:12:58,459 --> 00:13:00,135
稍微想一想，然后你就明白了
take a little thought but then you get

208
00:13:00,140 --> 00:13:04,065
它，你不要忘记它，所以
it and you don't for forget it ok so

209
00:13:04,070 --> 00:13:10,155
什么是我正在寻找的东西
what's what's up I'm looking so suppose

210
00:13:10,160 --> 00:13:12,865
最大值是V1
the maximum is V one

211
00:13:12,870 --> 00:13:16,375
我想你要去II认为这是
I think you're gonna go I I think it's

212
00:13:16,380 --> 00:13:22,435
看起来像那样的外出-一个
gonna look like that out - out - one

213
00:13:22,440 --> 00:13:29,575
零和零到一，在这里
zero and up to zero one and up here the

214
00:13:29,580 --> 00:13:35,005
矢量将是一个点
vector would be one one point for

215
00:13:35,010 --> 00:13:38,155
所以最大的东西就是一个
something so the maximum would be one is

216
00:13:38,160 --> 00:13:42,834
那真是太好了，不知怎的
that okay so so so somehow what really

217
00:13:42,839 --> 00:13:45,955
看到你更改这个数字P你
sees as you change this number P you

218
00:13:45,960 --> 00:13:50,515
从一个人或者daimond开始吧
start with people one or daimond and it

219
00:13:50,520 --> 00:13:54,115
有点膨胀成为P的一个圆圈
kind of swells out to be a circle at P

220
00:13:54,120 --> 00:13:57,955
等于2然后它保持
equal to 2 and then it kind of keeps

221
00:13:57,960 --> 00:14:01,435
膨胀为P等于平方
swelling to be a square at P equal

222
00:14:01,440 --> 00:14:04,435
无限，这是一个有趣的
infinity that's a that's a interesting

223
00:14:04,440 --> 00:14:11,455
事情，是的，现在是什么问题
thing and yeah now what's the problem

224
00:14:11,460 --> 00:14:25,105
0标准点在哪里
with the 0 norm where are the points

225
00:14:25,110 --> 00:14:31,785
一个非零所以所以我正在密谋
with one non zero so so so I'm plotting

226
00:14:31,790 --> 00:14:36,355
球在哪里在哪里
ball where are the where the where are

227
00:14:36,360 --> 00:14:39,055
这个东西中的向量有一个
the vectors in this thing that have one

228
00:14:39,060 --> 00:14:42,565
非零不是不为零非零使
non-zero not not zero nonzero so that

229
00:14:42,570 --> 00:14:48,685
这不包括在内但我也是如此
that's not included and but I but so

230
00:14:48,690 --> 00:14:49,704
我有什么
what do I have

231
00:14:49,709 --> 00:14:53,035
我不允许向量三分之一
I'm not allowed the vector one-third

232
00:14:53,040 --> 00:14:55,075
三分之二因为有两个非
two-thirds because that has two non

233
00:14:55,080 --> 00:14:58,465
零，那么只有点数在哪里
zeros so where are the points with only

234
00:14:58,470 --> 00:15:03,535
斧头上有一个非零的是啊
one nonzero yeah on the axes yeah that

235
00:15:03,540 --> 00:15:06,985
告诉你，他可以在那里
does tell you so he can be there and

236
00:15:06,990 --> 00:15:11,785
有没有好男人，当然这样的
there okay without guy and of course so

237
00:15:11,790 --> 00:15:14,575
只是一直走出去，完全是这样
is just keep going out so it totally

238
00:15:14,580 --> 00:15:21,445
违反你所知道的也许是重点
violates the you know so maybe the point

239
00:15:21,450 --> 00:15:25,615
我应该对这些数字做些什么
that I should make about these figures

240
00:15:25,620 --> 00:15:28,525
所以就像我去的时候发生的那样
so so like what's happening when I go

241
00:15:28,530 --> 00:15:31,375
降到零我真的很满意
down to zero I'm served it really that

242
00:15:31,380 --> 00:15:33,175
手指应该是在另一端向右
finger should be on the other end right

243
00:15:33,180 --> 00:15:37,075
哦，不要拍这个家伙在中间
oh no shoot this guy's in the middle

244
00:15:37,080 --> 00:15:41,815
这是一个画得很厉害的手指l2是善良的
this is a badly drawn finger l2 is kind

245
00:15:41,820 --> 00:15:46,495
一个中心人l1是一端el
of a center guy l1 is the at one end el

246
00:15:46,500 --> 00:15:48,715
在另一端和这一个的无限性
infinities at the other end and this one

247
00:15:48,720 --> 00:15:53,845
已离开，左边那里
has gone off and the left there the the

248
00:15:53,850 --> 00:15:56,865
钻石有没有发生在这里的事情
diamond has yeah what's happened here as

249
00:15:56,870 --> 00:16:00,715
一个人走向零没有
that one goes down towards zero none of

250
00:16:00,720 --> 00:16:04,345
这些都会好起来的这些这些这些
these will be okay these these these

251
00:16:04,350 --> 00:16:09,235
球将或这些集将失去
balls will or these sets will lose

252
00:16:09,240 --> 00:16:14,245
重量是啊所以他们总是会有这些
weight yeah so they'll always have these

253
00:16:14,250 --> 00:16:17,635
指出，但他们会像这样
points in but they'll be like this and

254
00:16:17,640 --> 00:16:20,215
然后像这样，然后终于在
then like this and then finally in the

255
00:16:20,220 --> 00:16:23,065
在不可接受的限度但没有
in the unacceptable limit but none of

256
00:16:23,070 --> 00:16:25,885
这些，这是没有任何好处的用户可以在本
those this was not any good either this

257
00:16:25,890 --> 00:16:31,285
这是P等于一半让我们说排序
this was for P equal half let's say sort

258
00:16:31,290 --> 00:16:34,465
呃那是P等于发生的那个
of uh that's a P equal to happen that's

259
00:16:34,470 --> 00:16:38,115
不是一个好的规范是的所以也许是事情
not a good norm yeah so maybe the thing

260
00:16:38,120 --> 00:16:42,595
你有这个圈子的属性
you so there's a property of the circle

261
00:16:42,600 --> 00:16:47,215
钻石和广场是一个
the diamond and the square which is a

262
00:16:47,220 --> 00:16:50,095
那些那些很好的数学属性
nice math property of those of those

263
00:16:50,100 --> 00:16:55,915
三套，不属于此
three sets and is not possessed by this

264
00:16:55,920 --> 00:17:00,295
因为这件事减肥我输了
as this thing loses weight I lose the

265
00:17:00,300 --> 00:17:02,185
财产当然那是
property and then of course that's

266
00:17:02,190 --> 00:17:04,165
完全迷失在那里你知道吗？
totally lost over there do you know what

267
00:17:04,170 --> 00:17:06,655
那个属性将是它的本质
that property would be what what it's

268
00:17:06,660 --> 00:17:10,345
为你凹凹凸我会
for you concave convex convex I would

269
00:17:10,350 --> 00:17:21,105
说凸是真正的规范有凸
say convex this is true norm has convex

270
00:17:21,110 --> 00:17:24,865
单球也许可能是我正在拿球
uni-ball maybe maybe for ball I'm taking

271
00:17:24,870 --> 00:17:30,925
所有这些是啊所以我允许
all of these yeah so I'm allowing the

272
00:17:30,930 --> 00:17:33,805
这些形状的内部所以这不是一个
insides of these shapes so this is not a

273
00:17:33,810 --> 00:17:37,345
凸集，设置我不应该
convex set that set which I shouldn't

274
00:17:37,350 --> 00:17:47,595
所以不是不凸会是这样一个与
so not not convex would be this one with

275
00:17:47,600 --> 00:17:51,375
哦，这反映了事实哦
and that reflects the fact that the oh

276
00:17:51,380 --> 00:17:55,435
规范的规则被打破了
the rules for a norm are broken the

277
00:17:55,440 --> 00:17:57,534
三角不等式可能已被打破
triangle inequality is probably broken

278
00:17:57,539 --> 00:18:03,654
那是的，我觉得有点像
and that yep I think that's sort of like

279
00:18:03,659 --> 00:18:13,374
值得记住，然后再一个规范
worth remembering and then one more norm

280
00:18:13,379 --> 00:18:21,404
这是很自然的思考
that's natural to think about is so s as

281
00:18:21,409 --> 00:18:24,685
在广场问题中总是如此
in the Piazza question s does always

282
00:18:24,690 --> 00:18:28,855
代表1806年的对称矩阵
represent a symmetric matrix in 1806

283
00:18:28,860 --> 00:18:32,215
五，现在我要问我的规范是什么
five and now I'm gonna ask my norm is

284
00:18:32,220 --> 00:18:34,975
将要这样我会称之为s
going to be so I'm gonna call it the s

285
00:18:34,980 --> 00:18:38,754
规范，所以评估实际上它是一个
norm so asses up actually it's a

286
00:18:38,759 --> 00:18:41,874
正定对称矩阵F是
positive definite symmetric matrix F is

287
00:18:41,879 --> 00:18:44,215
一个正定的对称矩阵和
a positive definite symmetric matrix and

288
00:18:44,220 --> 00:18:45,235
我该怎么办
what do I do

289
00:18:45,240 --> 00:18:54,174
我会带V转置AVsV对不起好吧
I'll take V transpose AV s V sorry okay

290
00:18:54,179 --> 00:18:57,294
我们的能量是什么呢？
what's our word for that the energy

291
00:18:57,299 --> 00:18:59,965
这是在矢量V的能量和
that's the energy in the vector V and

292
00:18:59,970 --> 00:19:05,304
我会采取平方根，以便我这样做
I'll take the square root so that I so

293
00:19:05,309 --> 00:19:10,544
如果我，我现在有两个长度
that I now have the lengths of two if I

294
00:19:10,549 --> 00:19:14,245
从V2到V的双V然后我得到2
double V from V 2 to V then I get a 2

295
00:19:14,250 --> 00:19:16,105
这里和2在这里，当我拿这个
here and a 2 here and when I take this

296
00:19:16,110 --> 00:19:18,475
平方根我得到一个整体两个和
square root I get a overall two and

297
00:19:18,480 --> 00:19:21,235
这就是我想要的我想要的规范
that's what I want I want the norm to

298
00:19:21,240 --> 00:19:25,044
与两个或三个线性增长或
grow linearly with the two or three or

299
00:19:25,049 --> 00:19:27,654
无论我乘以什么，但是什么是
whatever I multiply by but what is the

300
00:19:27,659 --> 00:19:33,115
这个东西的形状，什么是
shape of this thing so what is the

301
00:19:33,120 --> 00:19:40,175
我会得到什么样的形状
what is the shape I'm gonna get a

302
00:19:40,180 --> 00:19:42,515
像那样的形状是什么形状
picture like that so what is the shape

303
00:19:42,520 --> 00:19:51,905
V转置的V等于或小于1
of V transpose s V equal one less or

304
00:19:51,910 --> 00:19:58,355
如果s是正对称的，则等于1
equal one if s is a positive symmetric

305
00:19:58,360 --> 00:20:04,415
正面的信件告诉我们，我声称
positive letters tell us I'm claiming

306
00:20:04,420 --> 00:20:07,325
我们什么时候得到一堆规范
that we get a bunch of norms when do we

307
00:20:07,330 --> 00:20:12,545
得到l2规范什么矩阵s会
get the l2 norm what what matrix s would

308
00:20:12,550 --> 00:20:15,965
这给了我们l2标准的身份
this give us the l2 norm the identity

309
00:20:15,970 --> 00:20:18,515
当然现在发生了什么事情发生，如果
certainly now what's going to happen if

310
00:20:18,520 --> 00:20:21,365
我使用了一些不同的矩阵
I use some different matrix s this

311
00:20:21,370 --> 00:20:25,955
圆圈会改变我的形状
circle is going to change shape I'm

312
00:20:25,960 --> 00:20:28,325
会有不同的规范取决于
gonna have a different norm depending on

313
00:20:28,330 --> 00:20:33,095
S和一个典型的情况下，将作为s等于
s and a typical case would be s equal to

314
00:20:33,100 --> 00:20:38,705
3这是一个正定的对称
3 that's a positive definite symmetric

315
00:20:38,710 --> 00:20:43,175
矩阵，和本，现在我会
matrix and this this now I would be

316
00:20:43,180 --> 00:20:47,015
绘制2V1平方加上的图形
drawing the graph of 2 V 1 squared plus

317
00:20:47,020 --> 00:20:49,385
3V2平方即能量
3 V 2 squared that would be the energy

318
00:20:49,390 --> 00:20:52,985
对等，我只想要你
right equal one and I just want you to

319
00:20:52,990 --> 00:20:56,945
告诉我这是什么形状，这是一个
tell me what shape that is so that's a

320
00:20:56,950 --> 00:20:59,915
你可以检查的完美的规范
perfectly good norm that you could check

321
00:20:59,920 --> 00:21:02,555
他们所有的财产都出来了
all its properties they all come out

322
00:21:02,560 --> 00:21:06,515
很容易，但我得到一个新的图片新
easily but I get a new picture a new

323
00:21:06,520 --> 00:21:09,815
你可以调整的规范
norm that's kind of adjustable you could

324
00:21:09,820 --> 00:21:12,095
说它是加权范数权重的意思
say it's a weighted norm weights mean

325
00:21:12,100 --> 00:21:15,035
你选择了一些
that you kind of have picked some

326
00:21:15,040 --> 00:21:17,045
数字有点适合
numbers sort of appropriate to the

327
00:21:17,050 --> 00:21:19,385
特别好的问题就是假设那些
particular problem well suppose those

328
00:21:19,390 --> 00:21:21,995
数字是2和3是什么形状
numbers are 2 and 3 what shape is the

329
00:21:22,000 --> 00:21:27,005
单位球在这个标准中它是椭圆形的
unit ball in this s norm it's in ellipse

330
00:21:27,010 --> 00:21:30,215
对，它是一个椭圆，我猜它
right it's an ellipse and I guess it

331
00:21:30,220 --> 00:21:33,665
实际上将是更大的数字3
will actually be so the larger number 3

332
00:21:33,670 --> 00:21:37,185
好吧意味着你不能走得那么远
well will mean you can't go as far as

333
00:21:37,190 --> 00:21:39,735
一个较小的数字-我认为它会
a smaller number - I think it would

334
00:21:39,740 --> 00:21:43,755
可能是像这样的椭圆
probably be an ellipse like this and the

335
00:21:43,760 --> 00:21:46,815
椭圆的轴长度
axes axis lengths of the ellipse would

336
00:21:46,820 --> 00:21:49,185
可能与它有关-
probably have something to do with it -

337
00:21:49,190 --> 00:21:53,055
在三个没关系，所以这就是现在你
in the three okay so that's just now you

338
00:21:53,060 --> 00:21:56,745
真的知道所有的矢量规范
know really all the vector norms that

339
00:21:56,750 --> 00:22:02,475
有点自然而然地使用这些
are sort of natural to use these come up

340
00:22:02,480 --> 00:22:05,505
以自然的方式，他们就像我们说的一样
in a natural way they're all as we said

341
00:22:05,510 --> 00:22:07,935
身份矩阵让我们回归
the identity matrix brings us back to

342
00:22:07,940 --> 00:22:12,095
这两种规范都是如此
the two norm so these are all sort of

343
00:22:12,100 --> 00:22:14,465
两种规范的变化
variations on the two norm

344
00:22:14,470 --> 00:22:19,755
这些是P的变化，从1开始
these are variations as P runs from 1 up

345
00:22:19,760 --> 00:22:24,225
2到无穷大，不允许
to 2 onto infinity and is not allowed to

346
00:22:24,230 --> 00:22:31,095
低于1，那是规范然后
go below 1 okay that's norms and then

347
00:22:31,100 --> 00:22:33,525
也许你实际上可以从这个看
maybe you can actually see from this

348
00:22:33,530 --> 00:22:39,275
图片在这里，我们有点喜欢
picture here's our like somewhat hokey

349
00:22:39,280 --> 00:22:44,225
为什么它是该规范的想法
idea of why it is that this norm

350
00:22:44,230 --> 00:22:48,075
最小化那里或最小化
minimizing the so minimizing there or in

351
00:22:48,080 --> 00:22:53,295
这个规范所以我在这里的意思是什么
this norm so what do I mean by that here

352
00:22:53,300 --> 00:22:59,135
将是一个典型的问题最小化
would be a typical problem minimize

353
00:22:59,140 --> 00:23:05,775
受到斧头等于B的l1哦对不起
subject to ax equal B the l1 oh sorry

354
00:23:05,780 --> 00:23:10,935
我正在使用X现在是l1的一个标准
I'm using X now yeah one norm of the l1

355
00:23:10,940 --> 00:23:18,855
X的规范，这将是一个重要的
norm of X so that would be an important

356
00:23:18,860 --> 00:23:21,345
问题实际上它有一个名字的人
problem actually it has a name people

357
00:23:21,350 --> 00:23:23,505
花了很多时间思考一个
have spent a lot of time thinking of a

358
00:23:23,510 --> 00:23:25,965
快速解决它的方式几乎就像
fast way to solve it it's almost like

359
00:23:25,970 --> 00:23:28,185
最小二乘法什么会什么会令
least squares what would what would make

360
00:23:28,190 --> 00:23:31,365
它更像是这些正方形
it more like these squares would be

361
00:23:31,370 --> 00:23:38,385
改变两个耶耶我能不能
change that to two yeah yeah can I just

362
00:23:38,390 --> 00:23:41,975
有点像草图而不是一个大的
sort of like sketch without making a big

363
00:23:41,980 --> 00:23:45,855
这里论证L之间的区别
argument here the difference between L

364
00:23:45,860 --> 00:23:47,475
等于1或
equal 1 or

365
00:23:47,480 --> 00:23:51,435
到这里，是的，我只是画一个
to here yeah just I'll just draw a

366
00:23:51,440 --> 00:23:54,435
图片现在擦除这个椭圆但你
picture now erase this ellipse but you

367
00:23:54,440 --> 00:24:00,524
不会忘记好的，所以这是我们的
won't forget okay so this this is our

368
00:24:00,529 --> 00:24:04,185
问题l与l1它有一个着名的名字
problem l with l1 it has a famous name

369
00:24:04,190 --> 00:24:05,924
基础追求
basis pursuit

370
00:24:05,929 --> 00:24:08,445
在工作的人都很有名
well famous to people who work in

371
00:24:08,450 --> 00:24:12,284
对于l2的优化它具有重要意义
optimization for l2 it has an important

372
00:24:12,289 --> 00:24:14,325
好吧，它有点像这些
name well it's sort of like these

373
00:24:14,330 --> 00:24:18,644
方块岭回归各种各样的
squares Ridge regression all sorts of

374
00:24:18,649 --> 00:24:22,034
这就像一个漂亮的模特
the this is like a beautiful model

375
00:24:22,039 --> 00:24:24,524
所有解决方案中的问题
problem among all solutions to ax

376
00:24:24,529 --> 00:24:28,985
假设这只是一个等式
suppose this is just one equation like

377
00:24:28,990 --> 00:24:34,365
c1x1加c2x2等于右侧
c1 x1 plus c2 x2 equals some right side

378
00:24:34,370 --> 00:24:39,644
B所以X的约束表明了这一点
B so the X's the constraint says that

379
00:24:39,649 --> 00:24:42,105
向量X必须是在一条线上
the vectors X have to be on a line

380
00:24:42,110 --> 00:24:45,195
假设这是一个图表
there's suppose that's a graph of that

381
00:24:45,200 --> 00:24:50,865
所有这些X中的哪一行
line so as among all these X's which one

382
00:24:50,870 --> 00:24:54,404
是的我从来没有意识到我是什么
yeah I never realizing what is what I'm

383
00:24:54,409 --> 00:24:57,794
要说是聪明的我的意思是好的
gonna say is gonna be smart okay I mean

384
00:24:57,799 --> 00:25:01,005
没有它会是不错的不会是
not it's gonna be nice not gonna be

385
00:25:01,010 --> 00:25:01,725
难
difficult

386
00:25:01,730 --> 00:25:05,985
让我们做，我们知道最好的L2这样一个
let's do the one we know best l2 so

387
00:25:05,990 --> 00:25:08,355
这是一条线让我放过的图片
here's a picture of the line let me let

388
00:25:08,360 --> 00:25:10,995
我让它更倾斜一点
me make it a little more tilted so so

389
00:25:11,000 --> 00:25:18,784
你是的是的，是的，就像二三
you yeah yeah yeah like like two three

390
00:25:18,789 --> 00:25:22,875
好吧，这是XY平面，这里是X.
okay where this is the XY plane here's X

391
00:25:22,880 --> 00:25:26,115
1这里有X2这对是
1 there's X 2 here are the pair of the

392
00:25:26,120 --> 00:25:29,205
满足我的条件的点
points that satisfy my condition which

393
00:25:29,210 --> 00:25:32,705
一个最小化该线上的哪个点
one minimizes which point on that line

394
00:25:32,710 --> 00:25:37,695
最小化具有最小的l2范数
minimizes has the smallest l2 norm which

395
00:25:37,700 --> 00:25:41,595
线上的点具有最小的l2
point on the line has the smallest l2

396
00:25:41,600 --> 00:25:47,174
规范是的，你正在画右边
norm yeah you're drawing the right

397
00:25:47,179 --> 00:25:50,024
用你的双手数字最小的l2
figure with your hands the smallest l2

398
00:25:50,029 --> 00:25:53,534
要记住的规范L是多远的
norm L to remember is just how far out

399
00:25:53,539 --> 00:25:54,215
你走
you go

400
00:25:54,220 --> 00:25:56,185
就是这样
it's it's

401
00:25:56,190 --> 00:25:58,355
在这里循环所以无所谓
circular here so it doesn't matter what

402
00:25:58,360 --> 00:26:00,695
方向他们都给予同样的l
direction they all giving the same l

403
00:26:00,700 --> 00:26:03,035
2-norm它只是我们有多远
2-norm it's just how far so we're

404
00:26:03,040 --> 00:26:04,895
找对的最近点
looking for the closest point on the

405
00:26:04,900 --> 00:26:08,135
因为我们不想去任何一条线
line because we don't want to go any

406
00:26:08,140 --> 00:26:10,554
我们想要进入最小距离
further we want to go a minimum distance

407
00:26:10,559 --> 00:26:17,674
有两个现在那么如果是这样哪来的点
with two now so if so where's the point

408
00:26:17,679 --> 00:26:19,625
至少距离是啊它只是显示
at minimum distance yeah it's just show

409
00:26:19,630 --> 00:26:22,054
我再一次用手或者再次
me again once more with hands or

410
00:26:22,059 --> 00:26:27,145
无论我不想要什么
whatever it'll be that I didn't want

411
00:26:27,150 --> 00:26:30,544
那里有45度角，我要擦掉
45-degree angles there I'm gonna erase

412
00:26:30,549 --> 00:26:34,235
再次，真正这个时候我会
it again and really this time I'm gonna

413
00:26:34,240 --> 00:26:41,375
获得不是45的角度
get angles that are not 45 all right

414
00:26:41,380 --> 00:26:45,365
辉煌得到它，这是我的路线
brilliant got it okay that's my line

415
00:26:45,370 --> 00:26:47,705
好的，最近的点是什么
okay and what's the nearest point in the

416
00:26:47,710 --> 00:26:48,395
l2规范
l2 norm

417
00:26:48,400 --> 00:26:51,125
这里是在L2赢家权
here's the winner in l2 right the

418
00:26:51,130 --> 00:26:54,635
最近点每个人都看到了
nearest point everybody sees that

419
00:26:54,640 --> 00:26:57,185
这就是你所知道的
picture so that's the that's you know

420
00:26:57,190 --> 00:27:02,785
这是一个基本的图片为
that's a basic picture for for

421
00:27:02,790 --> 00:27:05,285
用约束最小化某事
minimizing something with a constraint

422
00:27:05,290 --> 00:27:08,515
这是根本问题
which is the fundamental problem of

423
00:27:08,520 --> 00:27:10,895
神经网络的优化
optimization of neural nets of

424
00:27:10,900 --> 00:27:15,595
我生活中的一切都很好
everything really of life well I'm

425
00:27:15,600 --> 00:27:19,924
哲学这样好吗这是一个的，但
philosophical so okay that's that's but

426
00:27:19,929 --> 00:27:22,025
当然，这总是意味着问题
of course it always mean the question

427
00:27:22,030 --> 00:27:24,125
永远是，也许这在生活中是真实的
always is and maybe it's true in life to

428
00:27:24,130 --> 00:27:28,895
你使用哪种规范好吧现在算了
which norm are you using okay now count

429
00:27:28,900 --> 00:27:30,544
并与自己展示那是
and show with yourself that was the

430
00:27:30,549 --> 00:27:35,345
最小的l2是最短的
minimum in l2 that's the shortest

431
00:27:35,350 --> 00:27:37,895
距离意味着我们的距离
distance where a distance means what we

432
00:27:37,900 --> 00:27:40,775
通常认为它是有意义但现在
usually think of it as meaning but now

433
00:27:40,780 --> 00:27:44,825
让我们来看看指出的l1规范
let's go for the l1 norm which point on

434
00:27:44,830 --> 00:27:50,164
这条线现在具有最小的l1范数
the line has the smallest l1 norm so now

435
00:27:50,169 --> 00:27:53,825
如果是的话，我打算加两个
I'm going to add the two so if this is

436
00:27:53,830 --> 00:27:58,315
有些人认为这是零点
some point a this is some point a zero

437
00:27:58,320 --> 00:28:03,575
这是零点B的权利
and this is some point zero B right

438
00:28:03,580 --> 00:28:04,115
那里
there

439
00:28:04,120 --> 00:28:05,945
所以这两点显然是
so those two points are obviously

440
00:28:05,950 --> 00:28:08,405
重要而且我们可以想到这一点
important and that point we could figure

441
00:28:08,410 --> 00:28:09,635
出的公式
out the formula for

442
00:28:09,640 --> 00:28:13,235
因为我们知道这是什么
because it's we we know what the

443
00:28:13,240 --> 00:28:15,965
几何，但我只是把这两个
geometry is but I've just put those two

444
00:28:15,970 --> 00:28:19,294
如此点我得到零B是的
points in so did I get a zero B yeah

445
00:28:19,299 --> 00:28:23,435
这是一个零井，所以让我问一下
that's a zero well so let me just ask

446
00:28:23,440 --> 00:28:24,245
你那个问题
you that question

447
00:28:24,250 --> 00:28:26,225
那条线上的哪一点最小
what point on that line has the smallest

448
00:28:26,230 --> 00:28:32,785
l1范数具有最小的l1范数
l1 norm which has the smallest l1 norm

449
00:28:32,790 --> 00:28:34,895
有人说它只是说了一点
somebody said it just say it a little

450
00:28:34,900 --> 00:28:36,455
大声，这样你就错了
louder so that you're wrong

451
00:28:36,460 --> 00:28:43,205
磁带永远零B这一点是
tape forever zero B this point that's

452
00:28:43,210 --> 00:28:44,635
获胜者，冠军
the winner

453
00:28:44,640 --> 00:28:50,405
这是l1冠军，这就是
this is the l1 winner and this was the

454
00:28:50,410 --> 00:28:56,735
l2获胜者并注意我说的话
l2 the winner and notice what I said

455
00:28:56,740 --> 00:28:58,655
早些时候，我没有看到它的到来
earlier and I didn't see it coming but

456
00:28:58,660 --> 00:29:00,815
现在我意识到这是一个手指放入
now I realize this is a finger to put in

457
00:29:00,820 --> 00:29:07,715
获胜者最多的笔记
the notes the winner has the most zeroes

458
00:29:07,720 --> 00:29:10,745
它是稀疏的向量，它有一个零
it's the sparse vector it's got a zero

459
00:29:10,750 --> 00:29:12,875
它没有两个组件
well out of two components it didn't

460
00:29:12,880 --> 00:29:15,455
有很大的自由，但它有零
have much freedom but it has a zero

461
00:29:15,460 --> 00:29:16,235
零件
component

462
00:29:16,240 --> 00:29:24,065
它坐落在上上它的轴
it sits on the on the on the axes it's

463
00:29:24,070 --> 00:29:26,075
轴上的东西有
the things on the axes that have the

464
00:29:26,080 --> 00:29:29,735
最小数量的组件，所以是的
smallest number of components so so yes

465
00:29:29,740 --> 00:29:34,294
所以这是l2中的图片
so this is the picture in l2 in in two

466
00:29:34,299 --> 00:29:36,875
抱歉，在二维图像，使
sorry the picture in two dimensions so

467
00:29:36,880 --> 00:29:40,385
我在2d，你可以看到
I'm in 2d and you can see that the

468
00:29:40,390 --> 00:29:43,565
获胜者是零组件是啊是啊
winner is has a zero component yeah yeah

469
00:29:43,570 --> 00:29:49,565
并且这是一个延伸到的事实
and and that's a fact that extends into

470
00:29:49,570 --> 00:29:52,294
更高的尺寸，这使得
higher dimensions too and that makes the

471
00:29:52,299 --> 00:29:55,924
l1规范特别，因为我说是的
l1 norm special as I've said yeah is

472
00:29:55,929 --> 00:29:58,294
还有更多关于这个例子的说法
there more to say about that example

473
00:29:58,299 --> 00:30:04,804
这是一个简单的第二个问题
that's for a simple 2d question that

474
00:30:04,809 --> 00:30:07,924
真的说明了l1
really makes the point that the l1

475
00:30:07,929 --> 00:30:10,145
胜利者在那里它不是你
winner is there it's not further you

476
00:30:10,150 --> 00:30:11,765
不要再上去行权
don't go further up the line right

477
00:30:11,770 --> 00:30:14,435
因为因为那么你，那么你
because because then you're then you

478
00:30:14,440 --> 00:30:17,705
甚至在各方面都很糟糕
have even that's bad in all ways when

479
00:30:17,710 --> 00:30:20,164
你进一步上升，你正在添加
you go up further you're you're adding

480
00:30:20,169 --> 00:30:22,185
有些非零
some nonzero

481
00:30:22,190 --> 00:30:24,315
第一个组件，你正在增加
first component and you're increasing

482
00:30:24,320 --> 00:30:27,075
非零超过第二个组成部分
the non zero over second component so

483
00:30:27,080 --> 00:30:29,385
你的想法不错
your that's a bad idea

484
00:30:29,390 --> 00:30:31,815
这是一个坏主意，这是这个
that's a bad idea this is this is the

485
00:30:31,820 --> 00:30:37,335
赢家，在某种程度上是这里的图片
winner and in a way here's the picture

486
00:30:37,340 --> 00:30:41,535
是的哦是的我应该准备这些
yeah oh yeah I should prepare these

487
00:30:41,540 --> 00:30:43,905
讲座，但这一个都出来了
lectures but this one's coming out all

488
00:30:43,910 --> 00:30:46,725
无论如何，所以这样的图片
right anyway so this so the picture

489
00:30:46,730 --> 00:30:50,175
那里有最近的球
there is the nearest ball it's at that

490
00:30:50,180 --> 00:30:54,674
点，你能看到什么呢？
point and what is it can you see that so

491
00:30:54,679 --> 00:30:57,975
该星是外循环，这是
that star is outside the circle this is

492
00:30:57,980 --> 00:31:01,545
l1冠军，所以这是最近的
the l1 winner so and that's the nearest

493
00:31:01,550 --> 00:31:05,225
那就是那个爆发了
that's the that's is the blow-up of the

494
00:31:05,230 --> 00:31:09,375
l1规范直到达到这一点为止
l1 norm until it hits that's the point

495
00:31:09,380 --> 00:31:16,424
在l1规范命中你看到它只是
where the l1 norm hit do you see it just

496
00:31:16,429 --> 00:31:18,855
给它一点想到的是
give it a little thought that that

497
00:31:18,860 --> 00:31:21,615
另一种几何方式来看
another another geometric way to see the

498
00:31:21,620 --> 00:31:24,195
你开始回答这个问题
answer to this problem is you started

499
00:31:24,200 --> 00:31:26,745
原点然后你炸毁了一个规范，直到
the origin then you blow up a norm until

500
00:31:26,750 --> 00:31:30,315
你得到一点就行了
you get a point on the line that

501
00:31:30,320 --> 00:31:32,685
满足你的约束，因为
satisfies your constraint and because

502
00:31:32,690 --> 00:31:35,205
当它击中时，你正在炸毁一个常态
you were blowing up a norm when it hits

503
00:31:35,210 --> 00:31:38,085
首先，这是最小的爆炸
first that's the smallest blow-up

504
00:31:38,090 --> 00:31:38,804
可能
possible

505
00:31:38,809 --> 00:31:41,145
这是这家伙的那个薄荷
that's the mint that's the guy that

506
00:31:41,150 --> 00:31:44,265
最小化你只是考虑一下
minimizes ya just think about that

507
00:31:44,270 --> 00:31:47,565
图片，我会画得更好
picture and I'll draw it better

508
00:31:47,570 --> 00:31:55,245
某处-那是矢量规范和
somewhere - well that's vector norms and

509
00:31:55,250 --> 00:31:58,725
然后我介绍了一些矩阵规范和
then I introduced some matrix norms and

510
00:31:58,730 --> 00:32:06,395
我只想说那些话
let me just say a word about those

511
00:32:06,400 --> 00:32:09,755
关于矩阵规范的好话，所以
okay word about matrix norm so so the

512
00:32:09,760 --> 00:32:17,375
矩阵规范是如此如此现在我有一个
matrix norms were the so so now I have a

513
00:32:17,380 --> 00:32:24,455
矩阵a和我想定义那些相同的
matrix a and I want to define those same

514
00:32:24,460 --> 00:32:28,865
对于矩阵而言这又是三个规范
three norms again for a matrix and this

515
00:32:28,870 --> 00:32:35,345
是两个常态，两个标准是什么
was two norm and what was the two norm

516
00:32:35,350 --> 00:32:44,645
一个矩阵很好，它是西格玛一个转身
of a matrix well it was Sigma one turned

517
00:32:44,650 --> 00:32:47,765
是这样的，没有定义它
out to be so what that doesn't define it

518
00:32:47,770 --> 00:32:50,375
或者它可以找到它只是说好吧
or it could to find it just say okay the

519
00:32:50,380 --> 00:32:52,985
最大的奇异值是两个常态
largest singular value is the two norm

520
00:32:52,990 --> 00:32:55,685
矩阵，但实际上它来自
of the matrix but actually it comes from

521
00:32:55,690 --> 00:32:58,505
在某个地方所以我想谈谈这个
somewhere so I want to speak about this

522
00:32:58,510 --> 00:33:06,125
首先是墓葬规范，所以它就是这样
one first the tomb norm so so so it's

523
00:33:06,130 --> 00:33:09,005
矩阵的两个范数和一种方法
the two norm of a matrix and one way to

524
00:33:09,010 --> 00:33:18,845
看矩阵的两个范数就是
see the two norm of a matrix is to

525
00:33:18,850 --> 00:33:23,795
连接到我的2个矢量范数
connected to the 2 norm of vectors I'd

526
00:33:23,800 --> 00:33:26,075
喜欢连接矩阵的两个范数
like to connect the two norm of matrices

527
00:33:26,080 --> 00:33:31,205
向量的两个范数以及如何应用
to the two norm of vectors and how shall

528
00:33:31,210 --> 00:33:34,345
我这样做，我想我会去看看
I do that I think I'm going to look at

529
00:33:34,350 --> 00:33:42,535
AX的2范数超过X的两个范数
the 2 norm of AX over the two norm of X

530
00:33:42,540 --> 00:33:46,295
所以对我来说这个比例是
so in a way to me that that ratio is

531
00:33:46,300 --> 00:33:50,765
比如爆炸因子，如果a是7次
like the blow up factor if a was 7 times

532
00:33:50,770 --> 00:33:53,945
如果一个人的身份是一个简单的案例
the identity to take an easy case if a

533
00:33:53,950 --> 00:33:56,225
是7倍的身份会是什么那
is 7 times the identity what would that

534
00:33:56,230 --> 00:34:05,225
比如是7如果是a如果a是7我这个
ratio be yeah 7 if a is if a is 7 I this

535
00:34:05,230 --> 00:34:08,225
这将是7X，这将是X和
this will be 7 X and this will be X and

536
00:34:08,230 --> 00:34:12,035
规范因素7出来了
norms the factor 7 comes out so that

537
00:34:12,040 --> 00:34:16,075
比率将是7好
ratio will be 7 okay

538
00:34:16,080 --> 00:34:20,395
对我而言，规范就是这样的
for me the norm is take that's the

539
00:34:20,400 --> 00:34:25,935
爆炸因素，所以这里是我的
blow-up factor when so here here's my

540
00:34:25,940 --> 00:34:29,605
这是我现在的矩阵规范的想法
here's the idea of a matrix norm now I'm

541
00:34:29,610 --> 00:34:34,465
从矢量做矩阵矩阵范数
doing matrix matrix norm from vector

542
00:34:34,470 --> 00:34:46,315
规范和答案将是最大的
norm and the answer will be maximum the

543
00:34:46,320 --> 00:34:49,735
这个比率的最大值我称之为比率
maximum of this ratio I call that ratio

544
00:34:49,740 --> 00:34:51,505
爆炸矢量只是一个
of the blow up vector that's just a

545
00:34:51,510 --> 00:34:57,015
化妆品名称是所有X的最大值
made-up name the maximum over all X's

546
00:34:57,020 --> 00:35:00,355
所有我想看看哪个矢量得到
all the I look to see which vector gets

547
00:35:00,360 --> 00:35:05,755
炸得最多，这是常态
blown up the most and that is the norm

548
00:35:05,760 --> 00:35:08,305
矩阵
of the matrix

549
00:35:08,310 --> 00:35:11,905
我知道我已经接受了规范
I know I've I've settled on norms of

550
00:35:11,910 --> 00:35:13,825
这是在楼上完成的
vectors that's that's done upstairs

551
00:35:13,830 --> 00:35:17,065
现在我正在寻找规范
there now I'm looking at norms of

552
00:35:17,070 --> 00:35:20,455
矩阵，这是获得一个的方法
matrices and this is one way to get a

553
00:35:20,460 --> 00:35:23,335
矩阵的良好规范来了
good norm of a matrix that kind of comes

554
00:35:23,340 --> 00:35:25,225
从这两个规范，所以会有
from the two norm so there would be

555
00:35:25,230 --> 00:35:27,655
来自的矩阵的其他规范
other norms for matrices coming from

556
00:35:27,660 --> 00:35:30,955
其他矢量规范和我们没有的规范
other vector norms and those we haven't

557
00:35:30,960 --> 00:35:34,165
看到但这两个规范是非常的
seen but the two norm is this is a very

558
00:35:34,170 --> 00:35:36,955
重要的是什么是最大的
important one so what is the maximum

559
00:35:36,960 --> 00:35:42,445
矩阵的这个比率的值
value of this of that ratio for a matrix

560
00:35:42,450 --> 00:35:48,595
声称它是Sigma1是什么的
a the claim is that it's Sigma 1 what a

561
00:35:48,600 --> 00:35:54,435
那里的大平等现在可以看出原因
big equality there now can we see why is

562
00:35:54,440 --> 00:35:57,595
为什么Sigma1能够解决这个问题
why is Sigma 1 the answer to this

563
00:35:57,600 --> 00:36:04,675
问题我可以看到一对夫妇的方式来
problem I can see a couple of ways to

564
00:36:04,680 --> 00:36:05,995
考虑一下，但那是非常的
think about that but that's a very

565
00:36:06,000 --> 00:36:11,775
事实上这是一个重要的事实
important fact in fact this is a way to

566
00:36:11,780 --> 00:36:15,475
发现西格玛1没有全部
discover what Sigma 1 is without all the

567
00:36:15,480 --> 00:36:18,145
其他西格玛如果我寻找X那个
other Sigma's if I look for the X that

568
00:36:18,150 --> 00:36:20,095
有最大的爆炸因素和
has the biggest blow-up factor and by

569
00:36:20,100 --> 00:36:23,065
会是其中X的方式，其中X会
the way which X will it be which X will

570
00:36:23,070 --> 00:36:26,935
在这里赢得最大的竞争，并成为
win the max competition here and be

571
00:36:26,940 --> 00:36:28,824
Sigma的时间是1倍
Sigma 1 times as long

572
00:36:28,829 --> 00:36:34,645
Raja的比例为Sigma1
Raja's as the the ratio will be Sigma 1

573
00:36:34,650 --> 00:36:37,044
什么是西格玛1这是什么东西
that'll be Sigma 1 when is this thing

574
00:36:37,049 --> 00:36:39,895
西格玛1倍大，对于其
Sigma 1 times as large as that for which

575
00:36:39,900 --> 00:36:45,804
如果X是一个特征向量，则X为零
X naught for an eigenvector if X was an

576
00:36:45,809 --> 00:36:49,574
特征向量是什么比例
eigenvector what would that ratio be

577
00:36:49,579 --> 00:36:54,925
lambda但是如果a不是对称的
lambda but if a is not a symmetric

578
00:36:54,930 --> 00:36:56,125
矩阵
matrix

579
00:36:56,130 --> 00:36:59,064
也许特征向量不会告诉你
maybe the eigenvectors don't tell you

580
00:36:59,069 --> 00:37:03,084
服务员准确的方式去做什么
the the waiter the exact way to go what

581
00:37:03,089 --> 00:37:05,334
那么你现在猜的是什么向量视图
so what vector view would you now guess

582
00:37:05,339 --> 00:37:09,714
它不是一个特征向量，它是一个单数
it's not an eigenvector it is a singular

583
00:37:09,719 --> 00:37:11,905
矢量和哪些奇异向量
vector and which singular vectors are

584
00:37:11,910 --> 00:37:16,014
可能会是你v1
probably going to be you v1

585
00:37:16,019 --> 00:37:18,534
是啊V一个有意义的赢家所以
yeah V one makes sense winner so the

586
00:37:18,539 --> 00:37:22,584
本次比赛的获胜者是X等于V.
winner of this competition is X equal V

587
00:37:22,589 --> 00:37:29,024
1第一个右边的单数
1 the first the first right singular

588
00:37:29,029 --> 00:37:34,254
矢量，我们最好能够检查
vector and we better be able to check

589
00:37:34,259 --> 00:37:39,385
这又是一个最大化问题
that so again this maximization problem

590
00:37:39,390 --> 00:37:44,484
答案告诉你的是
the answer tells you is in terms of the

591
00:37:44,489 --> 00:37:47,875
奇异向量，这是一种寻找方式
singular vectors so that's a way to find

592
00:37:47,880 --> 00:37:49,854
这第一个奇异向量没有
this first singular vector without

593
00:37:49,859 --> 00:37:53,635
找到他们所有，让我们只是插入
finding them all and let's just plug in

594
00:37:53,640 --> 00:37:57,114
他们向量的第一件事就是看
the first thing of their vector and see

595
00:37:57,119 --> 00:38:00,954
这个比例是看到它看到的
that the ratio is see what it see that

596
00:38:00,959 --> 00:38:03,984
这个比例是西格玛1所以现在让我插上
the ratio is Sigma 1 so now let me plug

597
00:38:03,989 --> 00:38:09,135
它所以我有什么我想要一个V1
it in so so what do I have I want a V 1

598
00:38:09,140 --> 00:38:15,264
超过V1的长度好吧，我希望
over length of V 1 okay and I'm hoping

599
00:38:15,269 --> 00:38:18,834
得到答案的是什么回合
to get that answer well what's the bout

600
00:38:18,839 --> 00:38:21,234
这里的分母是V的长度
of the denominator here the length of V

601
00:38:21,239 --> 00:38:26,125
1是什么没什么大不了的还有那一个
1 is what no big deal there that's one

602
00:38:26,130 --> 00:38:29,185
什么是顶部的长度
what what's the length of the of the top

603
00:38:29,190 --> 00:38:30,875
一个如此
one so so

604
00:38:30,880 --> 00:38:35,315
是V1那么什么是V1，如果v1是
is a v1 now what is a v1 if v1 is the

605
00:38:35,320 --> 00:38:38,465
第一个奇异的第一个右奇异
first singular first right singular

606
00:38:38,470 --> 00:38:45,385
向量然后v1是Sigma的1倍u1
vector then a v1 is Sigma 1 times u1

607
00:38:45,390 --> 00:38:49,895
记住奇异的矢量交易了
remember the singular vector deals were

608
00:38:49,900 --> 00:38:58,195
AV等于SigmauAVk等于sigmakUK
AV equals Sigma u AV k equals sigma k UK

609
00:38:58,200 --> 00:39:00,485
记住，所以他们不是
remember that so they're not

610
00:39:00,490 --> 00:39:03,275
特征向量是它们的奇异向量
eigenvectors their singular vectors so

611
00:39:03,280 --> 00:39:09,125
这就是v1的长度
that's that is a v1 is the length of

612
00:39:09,130 --> 00:39:13,835
西格玛1U1和它是由1和分
sigma 1 u 1 and it's divided by 1 and of

613
00:39:13,840 --> 00:39:17,705
当然你也是一个单位向量，所以我
course u 1 is also a unit vector so I

614
00:39:17,710 --> 00:39:24,035
只是得到信号好，这是另一个
just get signal okay so that's another

615
00:39:24,040 --> 00:39:27,485
这样说，你可以找到西格玛1
way to say that you can find Sigma 1 by

616
00:39:27,490 --> 00:39:31,145
解决这个最大的问题，你得到
solving this maximum problem and you get

617
00:39:31,150 --> 00:39:36,065
西格玛1确定，我可以得到其他
that Sigma 1 ok and I could get other

618
00:39:36,070 --> 00:39:40,835
规范矩阵规范通过最大化
norms matrix norms by by maximizing that

619
00:39:40,840 --> 00:39:44,285
在那个向量范数I中炸毁因子
blow up factor in that vector norm I

620
00:39:44,290 --> 00:39:48,965
现在不会这样做只是为了保持控制
won't do that now just to keep control

621
00:39:48,970 --> 00:39:51,935
什么，我们现在得到了什么什么的
of what we've got now what were the what

622
00:39:51,940 --> 00:39:54,575
是下一个进入的矩阵规范
was the next matrix norm that came in

623
00:39:54,580 --> 00:39:57,545
最后的时间非常非常重要的一个
last time very very important one for

624
00:39:57,550 --> 00:40:01,385
以某种方式深度学习和神经网络
deep learning and neural nets somehow

625
00:40:01,390 --> 00:40:04,135
它比这家伙更简单一点，
it's a little simpler than this guy and

626
00:40:04,140 --> 00:40:07,655
什么是矩阵规范
what's the what was that matrix norm

627
00:40:07,660 --> 00:40:12,985
什么字母的名字在这里
what letter whose name goes here

628
00:40:12,990 --> 00:40:16,505
Frobenius所以Frobenius和F的首都F
Frobenius so capital F for Frobenius and

629
00:40:16,510 --> 00:40:19,535
这一点，那是什么那是
that and what was that that was the

630
00:40:19,540 --> 00:40:22,475
平方根的总和
square root of the sum of the all the

631
00:40:22,480 --> 00:40:31,805
加起来为所有人添加所有的IJ方块
add up add all the a IJ squares for all

632
00:40:31,810 --> 00:40:34,745
在矩阵上然后取一个正方形
over the matrix and then takes a square

633
00:40:34,750 --> 00:40:38,105
根，然后有人问了一个好
root and then somebody asked a good

634
00:40:38,110 --> 00:40:41,585
周三课后问题是什么
question after class on Wednesday what

635
00:40:41,590 --> 00:40:44,605
这与Sigma的关系有关
has that got to do with the Sigma's

636
00:40:44,610 --> 00:40:49,135
因为我失望的是这些
because I my disappoint was that these

637
00:40:49,140 --> 00:40:53,395
宿舍是随身携带的人
dorms are the guys that go with the

638
00:40:53,400 --> 00:40:55,045
Sigma的做法很好
Sigma's that does that have nice

639
00:40:55,050 --> 00:40:57,535
西格玛的公式就在这里
formulas for the Sigma's and here it is

640
00:40:57,540 --> 00:41:00,175
它是总和的平方根
it's the square root of the sum of the

641
00:41:00,180 --> 00:41:08,155
所有信号的正方形是对不起让
squares of all the signals yes sorry let

642
00:41:08,160 --> 00:41:16,765
我再写Frobenius但是这个
me write Frobenius again but this

643
00:41:16,770 --> 00:41:18,805
用F表示现在很漂亮
notation with an F is now pretty

644
00:41:18,810 --> 00:41:22,675
标准，我们应该能够看到
standard and we should be able to see

645
00:41:22,680 --> 00:41:26,065
为什么这个数字与那个数字相同
why that number is the same as that

646
00:41:26,070 --> 00:41:40,045
号码是的，我可以给你一个
number yeah I could I could give you a

647
00:41:40,050 --> 00:41:41,665
理由还是我不能把它放在上面
reason or I couldn't put it on the

648
00:41:41,670 --> 00:41:47,935
问题集是啊我认为它更好
problem set yeah I think it better on

649
00:41:47,940 --> 00:41:50,455
问题集首先是因为
the problem set because first of all

650
00:41:50,460 --> 00:41:52,165
我脱身马上和
I'll get off the hook right away and

651
00:41:52,170 --> 00:41:57,205
其次是中间的这种联系
secondly this connection between in

652
00:41:57,210 --> 00:41:59,485
ProbieNia所以它就像一个美丽的事实
Probie Nia so it's like a beautiful fact

653
00:41:59,490 --> 00:42:01,735
关于Frobenius规范，你加起来所有
about Frobenius norm that you add up all

654
00:42:01,740 --> 00:42:05,035
SigmaSquared是他们的M倍
the Sigma Squared's as M times n of them

655
00:42:05,040 --> 00:42:08,755
因为它是一个全部填充的矩阵或你
because it's a all filled matrix or you

656
00:42:08,760 --> 00:42:12,115
加起来以另一种方式说它是我们
add up so another way to say it is we

657
00:42:12,120 --> 00:42:14,275
还没写下今天的SVD
haven't written down the SVD today's a

658
00:42:14,280 --> 00:42:21,685
谁你的西格玛V转置和重点
whoyou Sigma V transpose and the point

659
00:42:21,690 --> 00:42:26,905
实际上就是弗罗贝尼乌斯的规范
is that for the Frobenius norm actually

660
00:42:26,910 --> 00:42:30,775
对于所有这些规范，我可以改变你
for all these norms I can change you it

661
00:42:30,780 --> 00:42:33,235
不改变常态，所以我可以做
doesn't change the norm so I can make

662
00:42:33,240 --> 00:42:36,445
你们认同它就是我们所知道的
you the identity it's you as we all know

663
00:42:36,450 --> 00:42:38,815
是一个正交矩阵，我是什么
is an orthogonal matrix and what I'm

664
00:42:38,820 --> 00:42:41,575
说是正交矩阵，你没有
saying is orthogonal matrix you doesn't

665
00:42:41,580 --> 00:42:44,445
改变任何这些特定的规范
change any of these particular norms so

666
00:42:44,450 --> 00:42:47,185
假设这里的身份相同
suppose it was the identity same here

667
00:42:47,190 --> 00:42:48,955
这可能是没有的身份
that could be the identity without

668
00:42:48,960 --> 00:42:52,405
改变规范，所以我们到了
changing the norm so we're down to the

669
00:42:52,410 --> 00:42:55,705
什么是弗罗贝尼乌斯的规范
norm of the the Frobenius so what's the

670
00:42:55,710 --> 00:42:57,015
弗罗贝尼乌斯规范
Frobenius norm

671
00:42:57,020 --> 00:43:02,145
那个家伙是什么ProbieGnaeus
of that guy what's the Probie Gnaeus

672
00:43:02,150 --> 00:43:06,795
那个对角矩阵的范数很好
norm of that diagonal matrix well you're

673
00:43:06,800 --> 00:43:08,864
应该把所有的正方形加起来
supposed to add up the squares of all

674
00:43:08,869 --> 00:43:11,385
矩阵中的数字和做什么
the numbers in the matrix and what do

675
00:43:11,390 --> 00:43:16,695
你让你做对了所以这就是原因
you get you get that right so that's why

676
00:43:16,700 --> 00:43:19,335
这与此相同，因为
this is the same as this because the

677
00:43:19,340 --> 00:43:21,945
正交人在那里正交
orthogonal guy there on the orthogonal

678
00:43:21,950 --> 00:43:23,775
那个家伙在常态上没有任何区别
guy there make no difference in the norm

679
00:43:23,780 --> 00:43:28,995
但这需要检查是的，但是
but that takes checking right yeah but

680
00:43:29,000 --> 00:43:30,525
这是另一种了解原因的方法
that's another way to see why the

681
00:43:30,530 --> 00:43:33,045
证明规范给出了这个然后
provenience norm gives this and then

682
00:43:33,050 --> 00:43:37,985
最后这是核规范和
finally this was the nuclear norm and

683
00:43:37,990 --> 00:43:41,145
实际上就在我的午餐前
actually just before the my lunch

684
00:43:41,150 --> 00:43:43,815
讲授的大概的主题
lecture on the subject of probably about

685
00:43:43,820 --> 00:43:48,555
午餐讲座是一个学习早晨
a learning morning the lunch lecture was

686
00:43:48,560 --> 00:43:51,765
关于这种人类行为的疯狂方式
about this crazy way that humans behave

687
00:43:51,770 --> 00:43:58,815
实际上不是我们，而是其他人
not us but other humans other actually

688
00:43:58,820 --> 00:44:06,854
好吧，从磁带中拿出来好吧
well take that out of the tape yeah okay

689
00:44:06,859 --> 00:44:09,075
无论如何那是那次讲座
anyway that was that lecture well before

690
00:44:09,080 --> 00:44:13,445
这是一个小时的讲座加
that was a lecture for an hour plus

691
00:44:13,450 --> 00:44:16,275
关于深刻学习的人
about deep learning by somebody who

692
00:44:16,280 --> 00:44:19,425
真的真的已经开始明白了
really really has begun to understand

693
00:44:19,430 --> 00:44:22,745
里面有什么东西发生了这是如何
what is happening inside how does that

694
00:44:22,750 --> 00:44:25,995
梯度下降优化算法
gradient descent optimization algorithm

695
00:44:26,000 --> 00:44:29,265
挑出什么什么它挑出来作为
pick out what what does it pick out as

696
00:44:29,270 --> 00:44:33,885
随着作为东西它学习它的
that as as the thing it learns it's

697
00:44:33,890 --> 00:44:37,275
会是你知道我们的目标
gonna be the you know our goal in in

698
00:44:37,280 --> 00:44:40,515
这个课程我们还没有，但他的
this course we're not there yet but his

699
00:44:40,520 --> 00:44:44,685
猜想是的，所以它是一个
conjecture is that that yeah so it's a

700
00:44:44,690 --> 00:44:46,215
猜想他没有我们的证据
conjecture he doesn't have our proof

701
00:44:46,220 --> 00:44:49,815
他有一些很好的案例证明
he's got proofs of some nice cases where

702
00:44:49,820 --> 00:44:51,705
事情通勤，但他没有得到
things commute but he hasn't got the

703
00:44:51,710 --> 00:44:54,945
整件事，但它非常棒
whole thing yet but it's pretty terrific

704
00:44:54,950 --> 00:45:00,835
工作所以这是cerebro教授
work so this this was professor cerebro

705
00:45:00,840 --> 00:45:04,615
在芝加哥所以他刚宣布了他的
in Chicago so he just announced his

706
00:45:04,620 --> 00:45:07,945
猜想和他猜想的是
conjecture and his conjecture is that in

707
00:45:07,950 --> 00:45:11,845
一个模型案例深入学习
a model case the deep learning that

708
00:45:11,850 --> 00:45:14,245
我们将通过渐变来学习
we'll learn about with a gradient

709
00:45:14,250 --> 00:45:16,465
我们将学会找到的血统
descent that we'll learn about to find

710
00:45:16,470 --> 00:45:20,185
最重要的是重点
the best weights that the point is that

711
00:45:20,190 --> 00:45:24,565
他们你知道典型的深度学习
they're you know typical deep learning

712
00:45:24,570 --> 00:45:27,685
问题这几天有更多的
problem these days there are many more

713
00:45:27,690 --> 00:45:31,525
方式比样本等都有很多
ways than samples and so there are a lot

714
00:45:31,530 --> 00:45:33,505
可能的最小值
of possible minima

715
00:45:33,510 --> 00:45:35,905
许多不同的重量给出相同的
many different weights give the same

716
00:45:35,910 --> 00:45:39,115
最小的损失，因为有这么多
minimum loss because there are so many

717
00:45:39,120 --> 00:45:42,265
权衡它的问题也是如此
weights it's the problem is like too

718
00:45:42,270 --> 00:45:45,055
许多变量，但事实证明是一个
many variables but it turns out to be a

719
00:45:45,060 --> 00:45:47,245
非常非常好的事情那部分
very very good thing that's part of the

720
00:45:47,250 --> 00:45:52,725
成功，他认为，在一个
success and the he believes that in a

721
00:45:52,730 --> 00:45:58,195
通过优化的模型情况
model situation that the optimization by

722
00:45:58,200 --> 00:46:04,165
梯度下降挑出权重
gradient descent picks out the weights

723
00:46:04,170 --> 00:46:07,135
这样可以最大限度地减少核规范
that minimize the nuclear norm so this

724
00:46:07,140 --> 00:46:10,785
将是很多权重的标准
would be a norm of a lot of weights and

725
00:46:10,790 --> 00:46:14,295
他认为这就是系统的用武之地
he thinks that's where the system goes

726
00:46:14,300 --> 00:46:17,425
会看到这个出现了
will see this this was comes up in

727
00:46:17,430 --> 00:46:19,705
我上次提到的压缩传感
compressed sensing as I mentioned last

728
00:46:19,710 --> 00:46:23,035
这个时间，但现在我必须记住
time so this but now I have to remember

729
00:46:23,040 --> 00:46:26,905
什么是定义做得好
what was what was the definition well do

730
00:46:26,910 --> 00:46:29,005
你还记得核规范是什么
you remember what the nuclear norm was

731
00:46:29,010 --> 00:46:32,545
这也是他经常使用的他用过的
it's also his he often used he used a

732
00:46:32,550 --> 00:46:35,395
我会把小明星而不是结束
little star instead of an end I'll put

733
00:46:35,400 --> 00:46:37,915
在其他人称之为的笔记中
that in the notes other people call it

734
00:46:37,920 --> 00:46:40,765
跟踪规范，但我想这个
the trace norm that's but I think this

735
00:46:40,770 --> 00:46:46,705
最后一种给它起一个名字符号
end kind of gives it a name notation you

736
00:46:46,710 --> 00:46:48,655
还记得让我们叫它
can remember so let's call it the

737
00:46:48,660 --> 00:46:50,395
核规范你还记得那是什么
nuclear norm do you remember what that

738
00:46:50,400 --> 00:46:52,665
一个是
one was

739
00:46:52,670 --> 00:46:55,665
是的，有人正在唱歌
it was yes somebody's singing it right

740
00:46:55,670 --> 00:46:59,065
添加Sigma的耶只是总和
add the Sigma's yeah just the sum of the

741
00:46:59,070 --> 00:47:07,555
像你这样的信号知道一个规范是的
signals like you know one norm yeah this

742
00:47:07,560 --> 00:47:12,235
是一种天生的l1类型的规范
is natural sort of l1 type of norm for

743
00:47:12,240 --> 00:47:13,705
矩阵
matrices

744
00:47:13,710 --> 00:47:17,034
这是你拥有了l1范西格玛
it's the l1 norm you have that Sigma

745
00:47:17,039 --> 00:47:19,044
矢量这将是l2的规范
vector this would be the l2 norm of the

746
00:47:19,049 --> 00:47:20,455
西格玛因子即L
Sigma factor that would be the L

747
00:47:20,460 --> 00:47:23,334
无穷大规范注意到矢量
infinity norm notice that the vector

748
00:47:23,339 --> 00:47:28,375
数字无穷大为1会发生变化
numbers infinity to 1 gets changed

749
00:47:28,380 --> 00:47:34,125
当你看矩阵时，大约2
around 2 when you look at the matrix

750
00:47:34,130 --> 00:47:39,024
那么这是一个令人兴奋的想法吧
well so that's an exciting idea and it

751
00:47:39,029 --> 00:47:43,554
还有待证明的是人
remains to be proved people are

752
00:47:43,559 --> 00:47:46,015
尝试看看它说的是它
experimenting to see does it say is it

753
00:47:46,020 --> 00:47:46,554
真正
true

754
00:47:46,559 --> 00:47:52,255
是的，所以这是一个未来是的好吧
yep so that's a future yes okay so

755
00:47:52,260 --> 00:47:55,405
那是今天我们谈论的规范
that's today we've talked about norms

756
00:47:55,410 --> 00:47:57,955
这部分笔记将是
and this section of the notes will be

757
00:47:57,960 --> 00:48:05,155
关于规范我们已经迈出了一大步
all about norms we've taken a big leap

758
00:48:05,160 --> 00:48:11,475
在评论深度学习和
in to comment about deep learning and

759
00:48:11,480 --> 00:48:14,364
这是我想说的话最多，
this is what I want to say the most and

760
00:48:14,369 --> 00:48:18,385
我说给每个班级我的教
I say it to every class I teach at the

761
00:48:18,390 --> 00:48:22,284
我学期开始了
near the start of the semester my

762
00:48:22,289 --> 00:48:25,465
感觉我的工作就是教
feeling about what my job is is to teach

763
00:48:25,470 --> 00:48:28,195
你的事情或加入你加入你
you things or to join you join with you

764
00:48:28,200 --> 00:48:30,235
学习今天发生的事情
in learning things as has happened today

765
00:48:30,240 --> 00:48:33,774
这不是你的评分，我不花任何钱
it's not to grade you I don't spend any

766
00:48:33,779 --> 00:48:38,094
时间就像失去睡眠你们都应该
time like like losing sleep y'all should

767
00:48:38,099 --> 00:48:40,824
那个人应该拿一点还是一点
should that person take up one point or

768
00:48:40,829 --> 00:48:45,114
把它变成epsilon惩罚
epsilon penalty for for turning it in

769
00:48:45,119 --> 00:48:47,145
几分钟迟到
for minutes late

770
00:48:47,150 --> 00:48:50,304
你好，我们有很多东西
hello with that right we've got a lot to

771
00:48:50,309 --> 00:48:54,235
做到这一点，无论如何我们会继续
do here and so anyway we'll get on with

772
00:48:54,240 --> 00:49:00,324
一份工作，所以家庭作业三上来，我们
a job so homework three coming up and we

773
00:49:00,329 --> 00:49:03,145
使用你已经拥有的笔记
be using the notes that you already have

774
00:49:03,150 --> 00:49:05,364
为那些人发布了四星
posted in stellar four for those

775
00:49:05,369 --> 00:49:08,875
部分八九等，并
sections eight and nine and so on and

776
00:49:08,880 --> 00:49:11,965
星期一我们会继续，好的，看到你
we'll keep going on Monday okay see you

777
00:49:11,970 --> 00:49:15,410
周一，周末愉快

778
00:49:11,970 --> 00:49:15,870
Monday and have a great weekend

779
00:49:15,880 --> 00:49:17,940
您
you