My life as a Ph.D. student, S01E07: Pondering life and others’ life

While my life has been not quite eventful for the past months, I do want to mention one thing I didn’t see coming.

Two professors came to ask me about my former Yao-class classmates who are applying to their open PhD positions. Incidentally both of them will be hiring their first batch of students, so that’s presumably an important decision for them.

And one of the two students in question also turned to me for help. That’s very important to them for sure!

I wondered why my classmates are applying for PhD now instead of at this time last year in the first place… Maybe they did a one-year Master’s program. Maybe they took a one-year gap. I really didn’t know. I should have known better about my classmates. That’s my bad.

Well, you can’t really blame me on that. In a place like Yao-class, everyone cares about themselves.

It feels strange. Just months ago you were still classmates. Now you kind of have a say on their future. I don’t even feel qualified to give my opinions on their performances that could be taken very seriously! But I said what I knew and what I thought. Even if I can’t help them, I hope that I at least didn’t hinder them.

Life is so strange and unpredictable after all.

When you have nothing to do, you think about life. Like what I did several days ago when I left my laptop in the lab at night and was sitting at my desk in my apartment, facing the wall, grabbing a Subway.

“If you are what you eat, then after years half of me would be made of Subway!”

Well, I didn’t think about sandwiches.

Why do PhD? That’s the one question every PhD student keep asking himself. It’s a hard one. Sometimes you also ask, why is everyone doing PhD?

Look at ourselves. We are Yao-class graduates. It’s good to be humble but we can just be bold and say that we are the top CS students from China and among the most intelligent ones of all Chinese students. We could probably achieve success easily in many fields with our talent. But most of us choose to do the one least cost-efficient thing. It seems that it has become some kind of obligation. If you are the best student, you do PhD.

Let’s not think too hard about it. For us is has been a long and lonely journey to get where we are, and let’s hope that the answer would become clear further down the road.

My life as a Ph.D. student, S01E06: Entering working mode

I haven’t written anything here for two months… That’s what happens when you have research to do and no interesting story to tell. Now that several months has passed since I arrived here, the feeling of freshness is completely gone, and its time for serious work.

I don’t know how an average student spend the very earliest stage of his academic career – before his first publication. It must take some unusual talent to give big results and publish first-author top conference papers outright! I guess many have to do some mundane task as part of a large project, then get his first publication as a credit for his work. Then I should be happy that I don’t have those mundane task to do!

I’m basically still exploring GANs. I’ve probably found a way to generate high-resolution images with a single model. It would be really exciting if I can beat the state-of-the-art, but I’ll still consider it to be a good experience even if I can’t. I’ve learned a lot about neural networks, and I got a lot of interesting ideas to try, which I won’t tell you before I publish 🙂

The life is kind of dull, though. This is probably the first time in my life when I could keep working hard for weeks, but also the first time when I sometimes feel mentally exhausted. I know that that’s gonna be the norm. The world is in turmoil, but as long as WWIII does not happen and no one is shooting at you, it won’t make your life any different. As long as progress in your work keeps you cheered up things would be fine, but if you got stuck in your work and at the same time does not have a life, it would be bleak…

The semester is over. But you don’t really have holidays as a PhD student. You always have one conference submission deadline after another, and for me next year’s SIGGRAPH would be the first one. We are not sure if we could do what we want yet, but I hope we could, of course.

A little bit of thought on solving Rubik’s Cubes and the possibility of doing that with neural networks

I’ve been thinking about solving combination puzzles with computer for a long time.

I believe that every CS student have learned how to solve the 15-puzzle with heuristic search in elementary AI courses, or at least to solve the simpler 8-puzzle with BFS in elementary algorithm courses. At first sight, Rubik’s Cube may seem to be a similar puzzle, just a bit bigger.

Yes, they are all “sequential move” type of puzzles where you try to restore a bunch of things to some initial configuration by performing some operation sequentially. But Rubik’s Cube is definitely much more difficult, both for humans and computers. Size is not what really matters – you can make yourself an arbitrarily large (n^2-1)-puzzle, and it will still be easier than Rubik’s Cube, just quadratically more boring and time consuming. Apparently there are deep group theoretic reasons. Actually I once read about why some of these puzzles are more difficult than others in a Scientific American article. But I think this can be explained by 15-puzzle being more local than Rubik’s Cube: in Rubik’s Cube, one move could affect 1/3 of all blocks where in 15-puzzle, one move affect only one piece.

The consequence is that in 15-puzzle you can see clearly which configuration is closer to being solved – you can see it more easily as a human, and you can design a heuristic more easily for a computer. In Rubik’s Cube, this is much more difficult. I haven’t even seen one example of a good search heuristic for Rubik’s Cube. Maybe people just does not bother finding one: by memorizing some 200 move sequences and practice people could solve a Rubik’s Cube under 10 seconds, and the total number of states of a Rubik’s Cube, 43,252,003,274,489,856,000 (I can memorize this number) and the God’s number, 20, is just small enough to make it tractable to solve with a (bidirectional) brute-force search.

But when you have a larger combination puzzle – a 4×4×4 or 5×5×5 or bigger cube or a megaminx, the number of possible states explodes and brute-force search becomes mostly useless.

Somehow people could still solve them by finding patterns. Someone have even designed an algorithm to solve a n×n×n in Θ(n^2/log n) moves – what a strange time complexity, a log in the denominator!

Actually, finding solutions to all those exotic puzzles is my favorite part in this Rubik’s Cube game. The 200 or so formulas may have been found by brute-force computer search, but one you have played with a Rubik’s Cube for some time and if you are clever enough, you can construct those formulas yourself for other puzzles. You start by just randomly – or maybe purposefully – turning the faces and see what happens. Then you suddenly find a short move sequence that does something interesting – say, rotating a small number of pieces while leaving their position unchanged, or only moving some corner pieces but does not affect edge pieces.

Then you get a bunch of formula, each one of them will keep a certain set of pieces unchanged, move a second set of pieces in some desired fashion, and move the remaining pieces in someway you don’t really care.

Then with what you have at hand, you define your strategy. Be it edge-first, corner-first, layer-first or whatever you may want to call it, usually you have several stages. In each stage, you want to keep unchanged a larger set of pieces that are already restored than the previous stages, and try to restore several more pieces to advance to the next stage.

You might not notice it but what you are doing is actually repeatedly reducing the group of possible states to one of its subgroups, until you get a trivial group consisting only of the solved state.

The process might not be optimal, but it works.

But it seems that we have not formulated this process well enough so that we can let computers do it. I’ve been thinking about it for a while. May be we human and computers could cooperate: we define the stages so that the quotient group between two stages is not so large so that we can brute-force search for a formula that brings us from one stage to the next. It would be so good if the computer can also learn to define the stages. But that part does involve some insight into the structure of the puzzle.

Yesterday when I was dealing with the GAN thing, I suddenly came up with the idea of solving Rubik’s Cubes with a neural network.

I think that’s quite natural. These days people seem to believe that neural networks can do everything. But after a quick google search I found that few people have considered the same thing.

So it’s time to have a try! Brute-force search definitely is ineffective for solving complex combination puzzles but maybe neural networks could do the magic. The idea cannot be simpler: encode the state of each piece into one-hot vectors and just build a network to predict from the input state the mext move towards solving the puzzle.

Then it comes to the encoding, the training data and the structure of the network. I’ve seen some people represent the state of a Rubik’s Cube by one-hot encoding the color of stickers at each position… that sounds ridiculous. I think one-hot encoding the position of each piece makes more sense. We have 20 movable pieces, and each of them have 24 positions, so the input would be a 480-dimensional 0-1 vector. Note that this is actually more than what you need if you choose to use stickers: 48 movable stickers, each with 6 possible colors, producing a 288-dimensional 0-1 vector. But still, I think the position of the pieces is more intrinsic.

The training data would be obtained by scrambling the cube. We train the network to bring a state to its previous state in the scramble sequence. The scramble sequence should be generated from a random walk: with a fixed probability (say 0.05 which is the reciprocal of 20, the God’s number) we come back to the solved state, otherwise we choose randomly from 12 possible moves. This makes sense since it ensures that each state in the state space is visited exponentially more often from its nearest path to the solved state.

Then we come to the fun part – the structure. We could just use some plain fully-connected layers. But remember that there is symmetry in the Rubik’s Cube, as I mentioned in a previous post. So if we rotate the input state, the same rotation should apply to the prediction as well. This can be enforced through weight sharing. Imagine that all the neurons, including the input, are divided in to 24 classes, corresponding to 24 elements in the chiral octahedral symmetry group O. Then if there is a link between a neuron a a neuron b and under some spatial rotation neuron a becomes neuron c and neuron b becomes neuron d, then the weight between a and b and that between c and d should be identical! Now we have an interesting 24-fold weight sharing inside the network! This still looks like a fully connected network, yet it is strangely entangled in itself.

While I haven’t tried this yet, I’m expecting the network to do more than merely predicting the next step. By examining for each neuron which input state cause it to have maximum activation, we should be able to see what pattern the network is using to find the next move – what should be kept and what should be moved, which should help us find the stages in the human solving strategy.

玩玩我刚训练好的GAN……

吐个槽先……我感觉我的英文写作比中文好,但写英文还是挺费脑子的,写中文就可以很随便……毕竟中文是母语……

这是我第一次训练神经网络(跑别人写的代码不算),一周都在琢磨怎么把GAN训练出来,今天不想费脑子了,而且好像正好关于GAN的中文资料也很少,就用中文写了

Generative Adversarial Network,生成式对抗网络,是一种生成模型(废话)。基于神经网络的生成模型很常见,但主要是在序列形式的数据上训练RNN,比如生成文本啥的。GAN采取了不同的思路:对抗训练。两个神经网络:一个生成网络,学习从随机噪声向量产生与训练数据相似的样本,和一个判别网络,学习判别一个样本是来自训练数据还是来自生成网络。对抗的目标是,生成网络提高判别误差,判别网络降低判别误差。

对抗训练也不算是新思路了,比如让两个AI对弈或者对话啥的很久以前就有,但用于生成模型算是个比较新的用法。

训练过程也很简单:对生成网络和判别网络的训练交替进行,每一回合,首先从训练数据sample一个batch,标记为真,然后使用生成网络生成一个batch,标记为假,用这两个batch训练判别网络;再使用生成网络生成一个batch,标记为真,让判别网络进行判别,然后把误差反向传播到生成网络进行训练。

虽然思路很简单,但实际操作的时候还是比较tricky的,很难训练好。原因有几个:首先,生成网络的质量无法量化(使用判别误差显然是不行的),只能靠主观判断,而凭主观判断是难以看出训练到底进行到什么程度、有没有卡住的,也就导致很难找到合适learning rate。

其次,两个网络的训练速度是很不同的。为了确保两个网络共同进步,要使它们保持能力相当。原paper认为判别网络应当多训练。但我的实测结果是在使用相同的learning rate时判别网络对生成网络基本上是吊打,不知道是什么情况。如果尝试保持每一回合结束后的判别误差约等于random guess的话,每个回合大概要训练生成网络20次以上……我还没有做实验,不过感觉可能是和输入噪声变量的数量有关。

不过上面两点都是次要的。GAN最常见的失败模式是生成网络对所有输入给出相同的输出。这个很好理解:每一个回合,生成网络的最优策略当然是找到当前判别网络表现最差的那一个样本,然后对所有输入都给出那个输出。不过接下来判别网络马上就会发现这个样本是假的。然后下一个回合生成网络又会找到另一个判别网络表现最差的样本,就这样这个最差样本在样本空间里变来变去,两个网络捉迷藏……

这大概也是原paper认为判别网络应该多训练的原因:生成网络不可能一步就使得所有输入都输出那个最差样本,如果判别网络多训练,及时发现生成网络的输出变化趋势并将这个样本判别为假,让这个最差样本在样本空间里跑得比较快,生成网络就追不上,也就不会对所有输入都给出相同输出了。

同样道理,learning rate不能太大。

但实际操作的时候,还是总是会发生这种生成网络collapse的情况……

我是在MNIST手写数字数据集上尝试训练的。网络结构什么的……这么简单的数据集其实随便什么结构都好吧……判别网络输入接两层卷积,接一层全连接,接输出,使用batch bormalization。生成网络正好反过来。

在调整网络结构、调整learning rate、调整两个网络的训练速度比均无果之后,我想了个不太优美的办法。判别网络每次只判别一个样本。如果可以一次判别整个batch呢?虽然每一个样本都像是真的,但整个batch长得一样,明显就是假的嘛……

所以怎样让判别网络在判别单个样本的时候可以参照整个样本的信息?一个方法是,求整个batch的平均值,然后将每个样本与平均值的差值作为一个额外的channel。如果整个batch很相似,这个channel的值(的绝对值)会比较小,否则会比较大。

加了这个trick之后,还真的就训练出来了……看一下训练过程,可以发现生成网络还是会有将所有输入收敛到同一输出的趋势,但马上就要collapse的时候,判别网络习得了真batch和假batch的统计差异,然后再过几个回合,生成网络的输出就发散了。

看一下结果。以下是生成网络生成的数字:

211100

以下是真实数据:

true

还不错吧。不过也有一些明显的缺点:数字分布不均匀(1明显太多而8明显太少),以及有一些四不像的输出。再看看训练过程中输出的演化:

vis_x3

有一些很快就稳定了,还有一些跳来跳去,大概是位于输出不同数字的输入区域的边界上。

再看看其他好玩的事情,比如输入的128个标准正态分布随机变量都是干啥的。事实证明,不太好玩……由于128个太多了,而MNIST数据集很简单,没那么多的变化自由度,所以大部分变量只做了一点微不足道的工作,即使能看出来也难以解释到底是个啥作用……举个例子,以下每一组输出中只有一个输入从-2变化到2,其余变量固定:

var_vis_41

以下每组是随机两个输入之间的线性插值:

1d_vis_5

以下是四个输入之间的双线性插值:

2d_vis_8

针对在不同数字生成区域边界上存在的四不像输出,有可以改进的办法:由于MNIST是有class label的,可以做有监督学习:将class label转换成one-hot vector放在输入的种子里,确保不同数字生成区域分离,然后将判别网络改成判别真假+分类、生成网络的目标改成增加判别误差、减小分类误差,可以期望产生更好的输出。目前还在尝试中,结果如下

40000

看起来起码数字分布不均匀的问题是解决了……

这一周算是学习,接下来要把GAN投入实战来做本lab的一个项目了。

不过我自己是有其他打算的……GAN这么6的东西,赶紧拿来随机生成萌妹子啊!

想要高清无码大图是不太可能,而且看看一些在ImageNet之类的数据集上训练的结果可以知道,花花草草还行,想让GAN生成动物啥的结果会很猎奇……

但我想生成个眼睛啥的总还是可以做的吧,收集几万张图片也不是问题,有空试试。

My life as a Ph.D. student, S01E05: Still working on my first neural network…

I was planning to write a post after getting everything done but it took longer than what I like…

We are working on some generative models now and for the last week I was learning how to train a Generative Adversarial Network. GANs are super cool, but also notoriously difficult to train! They probably aren’t the best choice for a machine learning newbie like me.

But doing something challenging is fun! Using something that works out of the box is just too lame. And if it already works well, it likely isn’t worth studying.

Here I’d like to talk about what I’ve learned so far. Let’s see the result first. I’m getting something like this from a GAN trained on MNIST dataset:

27000

It is generating something. Not exactly digits but do resemble some hand written script…

I found that there are some practical issues not addressed in the DCGAN paper. It suggested using batch normalization. Batchnorm behaves differently when training and when evaluating. When training. it uses the mean and variance of the training batch. When evaluating, it uses the statistics of all the examples it has seen. It is logical to use evaluating mode for the generator when training the discriminator and vice versa.

In the original GAN paper, in each discriminator training round, the “true batch” from real data and the “false batch” from generated examples are fed into the discriminator separately. It is logical to make the distribution of each batch the same, so true and false examples should be mixed in the same batch.

To avoid making the wrong decision for the previous two issues I tried both ways round. But the suggestions given in the DCGAN paper still didn’t quite work out for me. I’ve followed them as much as I can. I can’t do the fractional-strided convolution for the generator network because that kind of convolution is not readily available in torch… But otherwise I did exactly what was advised. But still, the generator network keeps collapsing every input to a single output. According to what I read, this indeed is the most observed failure mode of GAN.

Then I came up with a not-so-elegant trick to solve this problem. The generator collapses because it thinks that output is the single best image to trick the discriminator into mistaking it for a real example from the dataset and learns to produce that output from any input. But then the discriminator quickly learns that that particular example is fake. Then the generator looks for the next output that can trick the discriminator…

This happens because the discriminator looks at a single example at a time. If we can somehow let the discriminator reject a batch if every example in the batch looks the same, then the generator should not be tempted to collapse every input into the same output anymore! How do we do this? My solution is, for each batch, take their mean. Then for each example in the batch, take its difference from the mean and concatenate the result with itself by adding them as additional channels. It is expected that if the batch is similar then the intensity in the additional channel would be small and the discriminator would be able to learn it.

But then there is a conflict with batchnorm. Since we want the discriminator to tell the difference between a true batch and a false bath, we cannot mix them in the same batch! But on the other hand, the difference of intensity between a true batch and a false batch is what makes the additional channels useful, with batchnorm, the differences are wiped out! To prevent this, we cannot let batchnorm normalize the batches individually and have to mix them!

So do we not use batchnorm? But without batchnorm, the training becomes very unstable.

The solution is to mix the two batches after their additional channels have been calculated separately.

This method actually works! the result is not perfect, but at least the generator does not collapse anymore! In the training process, you can see that at some point, the generator almost collapses but then the output soon starts to diverge.

At this point I’m not quite sure how to get the rest of things right. Tuning training parameters, or go model shopping?

And then there is one last mystery. The GAN paper says we should train the discriminator more. But in reality the discriminator constantly beat the crap out of the generator even if I train the generator 10 times as hard! Is this normal?

Actually the difficulty of balancing the training of the generator and the discriminator is another major factor that makes GAN hard to train, along with generator collapsing. The third factor probably is that since there are two networks competing, there is no single loss function to measure how well the training is going.

I’ll probably write something more elaborate if I do become better at training GANs.

And what is the one thing that I want to generate with a GAN?

Image of anime style eyes!

My life as a Ph.D. student, S01E04: Game on! (Well, not really)

USC is renowned for its School of Cinematic Arts. That’s at best remotely related to my work, though.

But only after I came did I discover that they have equally competitive game design programs.

I once played Cloud when I was in middle school. A beautiful game. And then, a while ago, I found that this game was made in USC and the creator is my compatriot and an alumnus of SJTU! Indie game development definitely was a rare thing back then in China, and arguably still is now.

I’m taking CSCI-522 “Game Engine Development” this semester. I did plan to take a graphics course, but my boss is not teaching this semester, and the other graphics course really is just elementary, so I opted to try something different.

I’m not planning to become a game developer. (Well, strictly speaking that possibility is not ruled out yet.) You don’t need a phd to make games!  Actually I doubt whether such courses will make you a “game developer”. My classmates are almost exclusively Master’s students. I don’t know whether they are MSCS or MFA, but either ways, being an expert in game engine is probably only gonna get you in to a big game company to work on their engines. You are still a programmer, just working with some different kind of program. You won’t be the next, say, Notch or Jonathan Blow. Does not sound cool.

But being cool is one thing, actually getting something to work is another thing! I barely know anything about the game development community but I guess in a big company you are guaranteed a good income while making a living as a indie game developer is much harder. If you think you had a great idea about game design but people don’t buy it, as often is the case, you’d probably rather learn to draw cute girls than designing games! At least you know someone’s gonna buy it if there is waifu! (No I’m joking)

But all those sort of things are none of my business. I’m taking this course to gain experience in working on giant complex projects. I heard that CSCI-522 is one of the more demanding courses offered by the compute science department. We are given a game engine that can barely do the basic rendering, animation, scripting, etc. and we will work on it so that in the end it will actually be something workable. Not being bold enough to read the source code of GCC or Linux kernel, this is the most complicated thing I’ve ever get my hands on.

In game engines efficiency is everything. Optimization is everything. That means you will work with a lower level language, and you won’t even want to use standard libraries. You don’t want generic solutions. You want something that works best for this particular program. So you do everything from scratch in C++: from memory management, to data structures, to runtime type information, to event system, and more.

And then, on top of that, you figure out how to abstract out the difference between hardware platforms and graphic APIs. Yes, we are gonna make it work on Windows and PSVita and iPad and more.

Then you do all the computation – physics and linear algebra and computational geometry, move objects around, pass control events around, gather draw calls and send them down the graphics pipeline.

Around that, you also need a script system and some sort of level editor. That really is a lot.

And while doing that, you use all kind of tricks to reduce coding work, do profiling, speed up compilation and help debugging.

Yes. Engineering. But as a graphics researcher, those are very useful skills if you want your research to be more than a paper and become something useful.

Thus far the technical part of the course has not gone beyond my existing knowledge, but I did learn some little tricks.

Here is one pre-class quiz question: you want to count the number of calls to certain functions. Suppose you want to implement several functions that take two integer arguments named a and b and return one integer. Define a macro called PROFILED_FUNC and a class called Tracker, such that

PROFILED_FUNC(func)
{
    //do some computation
    //return some value
}

defines such a function func, and when you call Tracker::stat(), it prints out the number of times each such function has been called.

The macro would be something like

#define PROFILED_FUNC(func)                            \
int func(int a, int b) {                               \
    static int callCount = 0;                          \
    if (callCount == 0) {                              \
        Tracker::register(#func, &callCount);          \
    }                                                  \
    callCount++;                                       \
    return _unprofiled_ ## func(a, b);                 \
}                                                      \
int _unprofiled_ ## func(int a, int b)

the # and ## are really alien to me! This made me believe more that I’ll never master C++.

And another trick: if you have a huge class definition in a header, you won’t want to include every time you use the class, since it slows down compilation. Time is money. It is common knowledge that we can forward declare a class, but that way we can only use a pointer to that class and cannot dereference it.

Now suppose you want to allocate a object of that class. Then you need to know its size. How do you avoid including the header?

You forward declare a function that allocate the object and return the pointer and put the definition in the implementation of that class.

And then there are other tricks. Thanks god that we don’t have template meta-programming!

闲得蛋疼,试论为什么人照镜子的时候会左右反转而不会上下反转

昨天看到这样一个问题:为什么人照镜子的时候会左右反转而不会上下反转?答案当然不是“人的眼睛是横着长的”……

之前我就见过一个类似的版本,然后直接看了答案——其实人照镜子的时候既不会左右反转也不会上下反转,而是会前后反转。当时我觉得“啊,好有道理”,然后就没有多想。不过昨天再次看到这个问题,觉得之前的答案并不完整:确实,人照镜子的时候是前后反转,但为什么大家普遍认为是左右反转呢?我决定以我的理解尝试解释一下这个问题。

我认为根本的问题在于人自身的对称性和对左右、前后、上下三组方向的理解方式的不同。

通常而言,上下是一个绝对概念,不依赖于人自身的姿势:在有重力的地方,重力的方向就是“下”,而重力的反方向就是“上”。而前后左右是相对概念,自己面向的方向是前,背对的方向是后。至于左右……为了避免引用宇称不守恒,不妨定义心脏所在的一侧为左(镜面人请自行把定义反过来)(没有心脏的人请自己看着办),对侧为右。因此人日常使用的“上下-左右-前后”坐标系既不是是世界坐标系也不是局部坐标系,而是二者的某种混合。由于人通常是直立的,这一世界坐标系中上、下的的定义方式与“头顶为上、脚底为下”的局部坐标系中的定义方式通常是一致的,因而不影响人描述其他事物相对于自身的方向。但当人处于其他姿势的时候就可能造成麻烦:比如人平躺的时候描述自己头顶所指的方向的物体,就不能仅仅使用上下左右前后。

照镜子的时候,人类判断镜中的像是否在某一方向上反转,也是采用相同的方式:对于上下,相对于世界坐标系进行判断;对于前后左右,相对于镜中的像的局部坐标系进行判断。如果镜子在头顶,问照镜子的人他在那个方向被反转了,很多人会同意镜中的像被上下反转了,因为它在世界坐标系中上下反转了。

但至此我们仍不能解释为什么事实上的前后反转会被解释为左右反转。显然左右反转并不是由于镜子置于我们的前方:如果镜子在我们的右侧,我们仍会判断为自己被左右反转。对于前述镜子置于上方的情况,如果追问其有没有被左右反转,我相信多数人会给出肯定的回答。事实上仅有上下反转,多数人却会解释为上下和左右同时反转,在两个方向反转,岂不是相当于并没有反转!问题显然在于前后和左右这两个局部坐标轴仍具有不同的性质。

日常生活中,我们总是可以将他人的局部坐标系经旋转平移与自己的局部坐标系重合。在描述某事物相对于他人的位置时,可以理解为先将其经同一旋转平移变换到自己的局部坐标系下然后进行判断。但在镜中,坐标系的手性会改变,理论上经过旋转平移是不能将镜中自己的像的局部坐标系变换为自己的局部坐标系的,而总是会在某一个方向被反转。因此前述“相对于镜中的局部坐标系进行判断”理论上是无法实现的。

但人体在左右方向具有对称性,因此如果我们在将镜中像的局部坐标系与自身的局部坐标系对齐的时候对齐前后轴而反转左右轴,我们的像仍然可以与自己重合,这就使得在实际上我们仍然可以“相对于镜中的局部坐标系”判断物体的位置关系。但这一过程中,左右是被反向的。

人体左右对称而前后不对称,导致了我们在将镜中的局部坐标系与自身的局部坐标系进行对齐的时候总是对其前后轴而反转左右轴,因此无论将镜子置于那个方向,我们镜中的像左右永远都会反转,而前后永远不会反转。至于上下则依镜子的方向而定,置于上下轴上则会上下反转,置于前后左右则不会上下反转。

那么假如人类不具有对称性,或者前后轴和左右轴均对称以至于无从选择变换局部坐标系的时候在哪一个方向方向对齐、在哪一个方向反转呢?

对于前一种情况,我认为总会有某一个方向,例如视觉器官所在的方向,会被认可为“正面”即“前方”,判断相对位置时会以这一方向对齐坐标系。对于后一种情况,大概根本不会存在“前后左右”这种相对方向的概念。

至此基本解决了为什么人照镜子的时候会左右反转而不会上下反转。但我认为我们在描述镜中其他物体的像的方向时仍有一些微妙之处。一些物体沿镜子排列时镜中的像在哪些轴被反向?垂直于镜子时呢?人观察的位置对结果有何影响?镜子的方向有何影响?物体自身的对称性又有何影响?如果物体表面上印有文字呢?人类对“上下左右前后”这一系列方向的认知,似乎远比我们自己意识到的复杂。