# Generalized Kernel Trick

If you are machine learner and are working on something related to kernel methods, I am sure most of you are familiar with the so-called kernel trick, which is very fundamentally important for most kernel-based learning machines. The equation below gives a formal definition of the kernel trick:

That is, the inner product between the feature map $\phi(x)$ and $\phi(y)$ can be written in term of some positive semidefinite function $k$. This allows one to replace the inner product with the kernel evaluation, and thereby does not need to compute $\phi(x)$ explicitly. Similar to the standard kernel trick, the generalized version can be written as

where $\mathcal{T}$ is an operator in $\mathcal{L}(\mathcal{H})$. Note that the generalized kernel trick reduces to the standard kernel trick when $\mathcal{T}=\mathcal{I}$ where $\mathcal{I}$ is the identity operator. Kadri et al. (2012) showed that this trick holds for any implicit mapping $\phi$ of a Mercer kernel given for self-adjoint operator $\mathcal{T}$. This is trick particularly useful when deriving the learning algorithm for structured output learning.

# Baryon Acoustic Oscillations

I have been working on quasar target selection problem for awhile. Essentially, this is a classification problem where one want to identify the objects in the sky as quasars or stars based on their flux measurement. The problem is easy for the low-redshift range because there is a clear separation between quasars and stella objects, but as for the medium- and high-redshift ranges, quasar target selection becomes more difficult. For z>2.2, objects must be targeted down to g=22 mag, where the photometric measurement uncertainty becomes substantial. Moreover, at z = 2.8, the quasar and stella loci cross in color space.

Despite the challenges of the problem itself, it is very important to me to understand why such a distant object is worth detected at all. So I did some researches and came up with a simple explanation.

Shortly after the Big Bang, the cosmic plasma composed of photons and baryons were excited by the initial perturbation. Initially, the pressure from the cosmic microwave background keeps the photon+baryon plasma from decoupling. This plasma acts like a sound wave that moves outward until the Universe becomes neutral at redshift 1000. As the Universe has cooled enough, the proton captures the electron to form neutral Hydrogen, which also decouple the photons from the baryons. Photons continue to stream away, leading to the dramatic acoustic oscillations seen in cosmic microwave background anisotropy data. The baryons, on the other hand, remain in place and leave the baryon peak stalled at about 150 comoving Mpc. This causes a small excess in number of pairs of galaxies separated by such distance. These features are often referred to as the baryon acoustic oscillations (BAO). BAO determine the rate of growth of cosmic structure with the overall expansion  of the universe. The observability of BAO will help cosmologists measure the expansion history of the universe and thereby a probe of cosmic dark energy.

In principle, BAO can also be observed in all forms of cosmic structure including the distribution of intergalactic medium as probed by the Lyman alpha forest (LAF). The LAF can be seen in the spectra of high redshift quasars. To detect BAO in the LAF, one may cross-correlate absorption spectra in widely separate quasar pairs. This has been previously impossible due to lack of sufficient data. Therefore, detection of sufficiently large number of high redshift quasars becomes substantially important.

After working in this direction for awhile, I have a feeling that machine learning in astronomy has not been explored much. There might be some open problems that one can tackle from machine learning point of view. I have also got this inspiration from the talk by David Hogg.

# Empirical Inference Journal Club

I have joined the Department of Empirical Inference at Max Planck Institute for Intelligent Systems for already one year. I've learnt and experienced a lot during the course of my PhD.

Apart from getting my own research projects done, I have always been enthusiastic about learning new things, broadening and deepening my knowledge. So I've been reading rigorously on many different topics ranging from econometrics to cosmology. Understanding the theoretical aspects of machine learning, I think, is very important, but understanding its role in real-world applications is even more important.

Reading lots of papers, of course, already gives me a big picture of where machine learning is in scientific communities. However, it lacks social context. I would also like to know what other people think about it.

So I have recently set up a journal club called the Empirical Inference Journal Club with a strong hope that it will provide such a platform for students and postdocs in the department to share their knowledge on some particular topics related to empirical inference. In the department, people have actually been organizing the reading groups on different topics, but to my knowledge they had the reading for a short period of time and then stop.

I commit to keeping this journal club running. Of course, I have to do some extra works, but I think it's worthwhile. After three-week of the journal club, things seem to go smoothly. I hope more people will join and contribute to the journal club.

We always have two options: accepting things the way they are or having enough courage to change them.

# Hello world!

After many attempts in starting up an academic blog, I have been successful.

Primarily, I will try to write regularly about my ongoing works and ideas that I have during the day. I hope it will be somewhat helpful both to myself and to other people who will be reading my blog.

Cheers,

Krikamol