PGM-class and MRF parameter learning

I’m taking Stanford CS 228 (a.k.a. pgm-class) on Coursera. The class is great, I guess it provides close to the maximum one can do under the constraints of remoteness and bulkness. The thing I miss is theoretical problems, which were taken aside from the on-line version because they could not be graded automatically.

There is an important thing about graphical models I fully realized only recently (partly due to the class). This thing should be articulated clearly in every introductory course, but is often just mentioned, probably because lecturers consider it obvious. The thing is there is no probabilistic meaning of MRF potentials whatsoever. The partition function is there not only for amenity: in contrast to Bayesian networks, there is no general way to assign potentials of an undirected graphical model to avoid normalization. The loops make it impossible. The implication is one should not assign potentials by estimating frequencies of assignments to factors (possibly conditioned on features) like I did earlier. This is quite a bad heuristic because it is susceptible to overcounting. Let me give an example.

For the third week programming assignment we needed to implement a Markov network for handwriting OCR. The unary and pairwise potentials are somewhat obvious, but there was also a task to add ternary factors. The accuracy of the pairwise model is 26%. Mihaly Barasz tried to add ternary factors with values proportional to trigram frequencies in English, which decreased performance to 21% (link for those who have access). After removing pairwise factors, the performance rose to 38%. Why has the joint model failed? The reason is overcounting evidence: different factor types enforce the same co-occurrences, thus creating bias towards more frequent assignments, and this shows it can be significant. Therefore, we should train models with cycles discriminatively. 

One more thought I’d like to share: graphical model design is similar to software engineering in the way that the crucial thing for the both is eliminating insignificant dependencies on the architecture design stage. 

When and why it is safe to cast floor/ceil result to integer

Happy new year, ladies and gents! I am back, and the following couple of posts are going to be programming-related.

The C/C++ standard library declares floor function such that it returns a floating-point number:

     double floor (      double x );
      float floor (       float x );  // C++ only

I used to be tortured by two questions:

1. Why would the floor function return anything other than integer?
2. What is the most correct way to cast the output to integer?

At last I figured out the answer to both questions. Note that the rest of the post applies to the ceil function as well.

For the second point, I knew the common idiom was just type-casting the output:

double a;

int intRes = int(floor(a));  // use static_cast if you don't like brevity

If you’ve heard anything about peculiarities of floating-point arithmetic, you might start worrying that floor might return not exact integer value but  $\lfloor a \rfloor - \epsilon$, so that type-casting is incorrect (assume $a$ is positive). However, this is not the case. Consider the following statement.

If $a$ is not integer and is representable as IEEE-754 floating-point number, than both $\lfloor a \rfloor$ and $\lceil a \rceil$ are representable within that domain. This means that for any float/double value floor can return a number that is integer and fits in float/double format.

The proof is easy but requires understanding of representation of floating-point numbers. Suppose $a = c \times b^q, c  \in [0.1, 1)$ has $k$ significant digits, i.e. $k$-th digit after the point in $c$ is not null, but all the further ones are. Since it is representable, $k$ is less then the maximum width of significand. Since $a$ is not integer, both $\lfloor a \rfloor$ and $\lceil a \rceil$ have significands with the number of significant digits less then $k$. Rounding however might increase the order of magnitude but only by one. So, the rounded number’s significand fits in $k$ digits, Q.E.D.

However, this does not mean one can always type-cast the output safely, since, for example, not every integer stored in double can be represented as 4-byte int (and this is the answer to the question #1). Double precision numbers have 52 digit significands, so any integer number up to $2^{52}$ can be stored exactly. For the int type, it is only $2^{31}$. So if there is possibility of overflow, check before the cast or use the int64 type.

On the other hand, a lot of int’s cannot be represented as floats (also true for int64 and double). Consider the following code:

std::cout << std::showbase << std::hex 

      << int(float(1 << 23)) << " " << int(float(1 << 24)) << std::endl

      << int(float(1 << 23) + 1) << " " << int(float(1 << 24) + 1) << std::endl;

The output is:

0x800000 0x1000000
0x800001 0x1000000

$2^{24}+1$ cannot be represented as float exactly (it is represented as $2^{24}$), so one should not use the float version of floor for numbers greater than few millions.

There are classes of numbers that are guaranteed to be represented exactly in floating point format. See also the comprehensive tutorial on floating-point arithmetic by David Goldberg.

LidarK 2.0 released

The second major release of GML LidarK is now available. It reflects our 3-year experience on 3D data processing. The description from the project page:

The LidarK library provides an open-source framework for processing multidimensional point data such as 3D LIDAR scans. It allows building a spatial index for performing fast search queries of different kinds. Although it is intended to be used for LIDAR scans, it can be helpful for a wide range of problems that require spatial data processing.

The API has been enriched with various features in this release. Indeed, it became more consistent and logical. New ways to access data (i.e. various iterators) are implemented. One can now find k nearest neighbours for any point, not just for one that belongs to the index. Since the data structure is a container, we've decided to parametrize it with template parameter. This decision is controversive: one does not need to cast tuples any more, but the code became clumsier and less portable, unfortunately.

The C++/MATLAB code is licensed free of charge for academic use. You can download it here.

Visual Assist's Tip of the Century

Probably you know the Visual Assist plug-in for Microsoft Visual Studio, which makes C++ programming in the environment zillion times handier. The story is about the bootstrap tip that highlights the feature of restoring files. While the tip window is shown, there is a second dialog box painted in the window. Since it is centred w.r.t. the screen, you perceive it as a real dialog box. Moreover, it is the usual case when Visual Assist suggests you to load some files from backup while MSVS is loading, because it does not always shut down properly. Thus, you try to close it clicking Yes or No, but nothing happens! The box is still in its place! That was really annoying.

In the later versions the developers solved the problem. They just marked the box as the example. The nice lack and the nice solution.

OpenCV bindings

If you are a computer vision researcher or an engineer, you cannot miss OpenCV library. Even if you are not, it could be useful to you. For example, here is a funny application: camera shots a programmer's face when a merge fails. If you are not familiar with the library, I recommend you to look through the list of its features on Wikipedia.

OpenCV is written in C to be extremely portable (for example, to DSP). The fact is C is not very popular nowadays. The recent release 2.0 contains (besides the other decent stuff) also C++ and Python wrappers. What about the other languages?

There are a number of C# wrappers. The most known is EmguCV, which is reported to be the only C# wrapper that supports OpenCV 2.0 (actually, I don't now what it means, but I suppose the API should correspond the C++ interface). It is distributed under GPL or the "Commercial License with a small fee".

As for Java, JavaCV seems to be the only viable wrapper. It also contains wrappers for other popular libraries like FFmpeg. It is also distributed under GPL, but the author promised to discuss weakening it if needed.

OpenCV was being supported by Intel, but it became a FOSS project recently. They are also going to participate in Google Summer of Code. If they will succeed, you might try to apply. I think it is a nice experience to develop such a popular library and be paid for it. :)

One of the fields they want to develop is augmented reality support for Android operating system. When I get known that there is an AR API in Android, I decided to try it. So, this is going to be a good opportunity!

UPD (Aug 6, 2011). There appeared a Haskell (!) wrapper for OpenCV by Noam Lewis.

Also, OpenCV folks are developing the official Java wrapper. Looking forward to use it!

Hello, world!

Well, I've just started my technical/research blog.

Why?

I already have a personal blog, which is (I hope) used to be read by a number of my friends. In recent time I noticed that I had begun post a lot of technical stuff my friends were not interested in. Also, that blog is in Russian, which limits its audience. So, I preferred not to change the blog policy but create a new blog. It was also Boris Yangel's amazing blog that inspired me to create mine.

Computer blindness?

Well, yes. Folks who know me might already divine that the blog is primarily about computer vision, and they are certainly right. I chose such a fancy name because computer vision is somewhat disappointing. Let me explain.

Do you remember that story when Marvin Minsky at MIT asked his student to teach a computer understand the scene retrieved from a camera during the 1966 summer break? Unsurprisingly, the student failed. Moreover, the general task is far from being solved even now. That days computers were slow, and AI scientists thought that performing logical inference is way more complicated than just scene analysis. Now, we have Prolog and a pile of different verification systems (thanks to the university curriculum -- we used some of them), but a computer is not able to recognize even simple object categories on different classes of images (e.g. relatively robust face detection was done only in early 2000s by Viola and Jones).

Actually, not only Minsky was misled. If you are a vision researcher, when you tell people about your research, they are likely to reply: "Is that all you can? Man, it's simple!" Sure, it is simple for you to figure out that it is a cow on the meadow, not a horse, but try to explain it to computer! It is actually a problem in our lab: when vision guys try to defend a Masters/PhD thesis in front of the committee (which consists of folks who do research in computer hardware, system programming etc), and the committee is usually not impressed by the results, because they think it is not too difficult.

Okay, you've got the point. Computers are blind, and we should cope with that.

Next. Do you know, what's the difference between computer vision of 1980s and modern computer vision? In 80s, they used to solve particular problems, not general ones. They could implement quite a decent vision system that performed its task, but it could not be transferred to another domain. Nowadays, such approach is no more scientific, while it is still good for engineers. Today, computer vision science is about general tasks. It became possible because of extensive using of machine learning methods (a lot of them were developed in '90s). Vision is nothing without learning now. I hope you've got this point too: Vision + Learning = Forever Together. That's why my blog cannot ignore machine learning issues.

What else? Programming language is a tool, but it can be interesting per se. I am about to finish a 5-year university programme in Computer Science, so I have been being taught different language concepts and programming paradigms, and I find it interesting sometimes. That is another possible topic.

Finally,

Subscribe me, read me and comment me. Everybody is welcome! Let it roll!