JEE coaching and high school learning

One reason I’m not as good at machine learning as I can possibly be is because I suck at linear algebra. I totally completely suck at it. Seven years of usage of R has meant that at least I no longer get spooked out by the very sight of vectors or matrices, and I understand the concept of matrix multiplication (an operator rotating a vector), but I just don’t get linear algebra.

For example, when I see terms such as “singular value decomposition” I almost faint. Multiple repeated attempts at learning the concept have utterly failed. Don’t even get me started on the more complicated stuff – and machine learning is full of them.

My inability to understand linear algebra runs deep, and it’s mainly due to a complete inability to imagine vectors and matrices and matrix operations. As far back as I remember, I have hated matrices and have tried to run away from it.

For a long time, I had placed the blame for this on IIT Madras, whose mathematics department in its infinite wisdom had decided to get its brilliant Graph Theory expert to teach us matrices. Thinking back, though, I remember going in to MA102 (Vectors, Matrices and Differential Equations) already spooked. The rot had set in even earlier – in school.

The problem with class 11 in my school (a fairly high-profile school which was full of studmax characters) was that most people harboured ambitions of going to IIT, and had consequently enrolled themselves in formal coaching “factories”. As a result, these worthies always came to maths, physics and chemistry classes “ahead” of people like me who didn’t go for such classes (I’d decided to chill for a year after a rather hectic class 10 when I’d been under immense pressure to get my school a “centum”).

Because a large majority of the class already knew what was to be taught, teachers had an incentive to slack. Also the fact that most students were studmax had meant that people preferred to mug on their own rather than display their ignorance in class. And so jai happened.

I remember the class when vectors and matrices were introduced (it was in class 11). While I don’t remember too many details, I do remember that a vocal majority already knew about “dot product” and “cross product”. It was similar a few days later when the vocal majority knew matrix multiplication.

And so these concepts were glossed over, and lacking a grounding in fundamentals, I somehow never “got” the concept.

In my year (2000), CBSE decided to change format for its maths examination – everyone had to attempt “Part A” (worth 70 marks) and then had a choice between “Part B” (vectors, matrices, etc.) and “Part C” (introductory statistics). Most science students were expected to opt for Part B (Part C had been introduced for the benefit of commerce students studying maths since they had little to gain from reading about vectors). For me and one other guy from my class, though, it was a rather obvious choice to do Part C.

I remember the invigilator (who was from another school) being positively surprised during my board exam when I mentioned that I was going to attempt Part C instead of Part B. He muttered something to the extent of “isn’t that for commerce students?” but to his credit permitted us to do the paper in whatever way we wanted (I fail to remember why I had to mention to him I was doing Part C – maybe I needed log tables to do that).

Seventeen odd years down the line, I continue to suck at linear algebra and be stud at statistics. And it is all down to the way the two subjects were introduced to me in school (JEE statistics wasn’t up to the same standard as Part C so the school teachers did a great job of teaching that).

Maths, machine learning, brute force and elegance

Back when I was at the International Maths Olympiad Training Camp in Mumbai in 1999, the biggest insult one could hurl at a peer was to describe the latter’s solution to a problem as being a “brute force solution”. Brute force solutions, which were often ungainly, laboured and unintuitive were supposed to be the last resort, to be used only if one were thoroughly unable to implement an “elegant solution” to the problem.

Mathematicians love and value elegance. While they might be comfortable with esoteric formulae and the Greek alphabet, they are always on the lookout for solutions that are, at least to the trained eye, intuitive to perceive and understand. Among other things, it is the belief that it is much easier to get an intuitive understanding for an elegant solution.

When all the parts of the solution seem to fit so well into each other, with no loose ends, it is far easier to accept the solution as being correct (even if you don’t understand it fully). Brute force solutions, on the other hand, inevitably leave loose ends and appreciating them can be a fairly massive task, even to trained mathematicians.

In the conventional view, though, non-mathematicians don’t have much fondness for elegance. A solution is a solution, and a problem solved is a problem solved.

With the coming of big data and increased computational power, however, the tables are getting turned. In this case, the more mathematical people, who are more likely to appreciate “machine learning” algorithms recommend “leaving it to the system” – to unleash the brute force of computational power at the problem so that the “best model” can be found, and later implemented.

And in this case, it is the “half-blood mathematicians” like me, who are aware of complex algorithms but are unsure of letting the system take over stuff end-to-end, who bat for elegance – to look at data, massage it, analyse it and then find that one simple method or transformation that can throw immense light on the problem, effectively solving it!

The world moves in strange ways.

High dimension and low dimension data science

I’ve observed that there are two broad approaches that people take to getting information out of data. One approach is to simply throw a kitchen sink full of analytical techniques at the data. Without really trying to understand what the data looks like, and what the relationships may be, the analyst simply uses one method after another to try and get insight from the data. Along the way, a “model” will get built.

The other approach (which I’m partial to) involves understanding each variable, and relationship between variables as a first step to getting insight from the data. Here, too, a model might get built, but it will be conditional on the analyst’s view on what kind of a model might suit the data after looking at the data.

Considering that both these approaches are used by large numbers of analysts, it is highly likely that both are legitimate. Then what explains the fact that some analysts use one approach, and others use another? Having thought about it for a long time, I have a hypothesis – it depends on the kind of data being analysed. More precisely, it has to do with the dimensionality of the data.

The first approach (which one might classify as “machine learning”) works well when the data is of high dimensions – where the number of predictors that can be used for predictors is really large, of the order of thousands or larger. For example, even a seemingly low-resolution 32 by 32 pixel image, looked at as a data point, has 1024 dimensions (colour of the image at each of the 1024 pixels is a different dimension).

Moreover, in such situations, it is likely that the signal in the data doesn’t come from one, or, two, or a handful of predictors. In high dimension data science, the signal usually comes from complex interplay of data along various dimensions. And this kind of search is not something humans are fit for – it is best that the machines are left to “learn” the model by themselves, and so you get machine learning.

On the other hand, when the dimensionality of the dataset is low, it is possible (and “easy”) for an analyst to look at the interplay of factors in detail, and understand the data before going on to build the model. Doing so can help the analyst identify patterns in the data that may not be that apparent to a machine, and it is also likely that in such datasets, the signal  lies with data along a small number of dimensions, where relatively simple manipulation will suffice. The low dimensionality also means that complex machine learning techniques are unlikely to contribute much in such cases.

As you might expect, from an organisational perspective, the solution is quite simple – to deploy high-dimension data scientists on high-dimension problems, and likewise with low-dimension data scientists. Since this distinction between high-dimension and low-dimension data scientists isn’t very well known, it’s quite possible that the scientists might be made to work on a problem of dimensionality that is outside of their comfort zone.

So when you have low dimensional data scientists faced with a large number of dimensions of data, you will see them use brute force to try and find signals in bivariate relationships in the data – an approach that will never work since the signal lies in a more complex interplay of dimensions.

On the other hand, when you put high dimension data scientists on a low dimension problem, you will either see them missing out on associations that a human could easily find but a machine might find hard to find, or they might unnecessarily “reduce the problem to a known problem” by generating and importing large amounts of data in order to turn it into a high dimension problem!

PS: I wanted to tweet this today but forgot. Basically, you use logistic regression when you think the signal is an “or” of conditions on some of the underlying variables. On the other hand, if you think the signal is more likely to be an “and” condition of certain conditions, then you should use decision trees!

 

Dreaming on about machine learning

I don’t know if I’ve written about this before (that might explain how I crossed 2000 blogposts last year – multiple posts about the same thing), but anyway – I’m writing this listening to Aerosmith’s Dream On.

I don’t recall when the first time was that I heard the song, but I somehow decided that it sounded like Led Zeppelin. It was before 2006, so I had no access to services such as Shazam to search effectively. So for a long time I continued to believe it was by Led Zep, and kept going through their archives to locate the song.

And then in 2006, Pandora happened. It became my full time work time listening (bless those offshored offices with fast internet and US proxies). I would seed stations with songs I liked (back then there was no option to directly play songs you liked – you could only seed stations). I discovered plenty of awesome music that way.

And then one day I had put on a Led Zeppelin station and started work. The first song was by Led Zeppelin itself. And then came Dream On. And I figured it was a song by Aerosmith. While I chided myself for not having identified the band correctly, I was happy that I hadn’t been that wrong – given that Pandora uses machine learning on song patterns to identify similar songs, that Dream On had appeared in a LedZep playlist meant that I hadn’t been too far off identifying it with that band.

Ten years on, I’m not sure why I thought Dream On was by Led Zeppelin – I don’t see any similarities any more. But maybe the algorithms know better!