For what it’s worth, my second year of PhD was relatively miserable compared to my first and third.
ignore the prestige games, knuckle down and do the best work you can do- it’s more impressive if you succeed at a lower rank institution anyways, imo. And if you want that signalling power you can always plan to go for a postdoc.
Wishing you the best of luck, it can be a lonely path.
I was considering signing up today, based on this I won’t.
I have had a great experience with gemini-cli, which may interest you: it only requires a google account to setup and you get a decent chunk of gemini 2.5 pro usage for free!
Will have to try this! Apparently Claude Enterprise includes everything in Pro except Claude Code, which is why I couldn't sign up. Just a warning in case you fall in the same position!
This is not quite right; you are actually losing information about each of the dimensions and your mental model of reducing the dimensionality by one is misleading.
Consider [1,0] and [x,x]
Normalised we get [1,0] and [sqrt(.5),sqrt(.5)] — clearly something has changed because the first vector is now larger in dimension zero than the second, despite starting off as an arbitrary value, x, which could have been smaller than 1. As such we have lost information about x’s magnitude which we cannot recover from just the normalized vector.
Well, depends. For some models (especially two tower style models that use a dot product), you're definitely right and it makes a huge difference. In my very limited experience with LLM embeddings, it doesn't seem to make a difference.
Magnitude is not a dimension, it’s information about each value that is lost when you normalize it. To prove this normalize any vector and then try to de-normalize it again.
Magnitude is a dimension. Any 2-dimensional vector can be explicitly transformed into the polar (r, theta) coordinate system where one of the dimensions is magnitude. Any 3-dimensional vector can be transformed into the spherical (r, theta, phi) coordinate where one of the dimensions is magnitude. This is high school mathematics. (Okay I concede that maybe the spherical coordinate system isn't exactly high school material, then just think about longitude, latitude, and distance from the center.)
There's something wrong with the picture here but I can't put my finger on it because my mathematical background here is too old. The space of k dimension vectors all normalized isn't a vector space itself. It's well-behaved in many ways but you lose the 0 vector (may not be relevant). Addition isn't defined anymore, and if you try to keep it inside by normalization post addition, distribution becomes weird. I have no idea what this transformation means for word2vec and friends.
But the intuitive notion is that if you take all 3D and flatten it / expand it to be just the surface of the 3D sphere, then paste yourself onto it Flatland style, it's not the same as if you were to Flatland yourself into the 2D plane. The obvious thing is that triangles won't sum to 180, but also parallel lines will intersect, and all sorts of differing strange things will happen.
I mean, it might still work in practice, but it's obviously different from some method of dimensionality reduction because you're changing the curvature of the space.
The space of all normalized k-dimensional vector is just a unit k-sphere. You can deal with it directly, or you can use the standard inverse stereographic projection to map every point (except for one) onto a plane.
> triangles won't sum to 180
Exactly. Spherical triangles have the sum of their interior angles exceed 180 degrees.
> parallel lines will intersect
Yes because parallel "lines" are really great circles on the sphere.
So is it actually the case that normalizing down and then mapping to the k-1 plane yields a useful (for this purpose) k-1 space? Something feels wrong about the whole thing but I must just have broken intuition.
My favorites are the ones from 2024 London mayoral election[1]:
> The Binface manifesto called for the abolition of VAR [video assisted referee] (presumably in football matches) and promised to force Thames Water managers to "take a dip in the Thames... see how they like it", in reference to the recent sewage discharge controversy; also to "build at least one affordable house", referring to the housing crisis in London.
My real favorite part is that some of these are obviously nonsensical, but some of them are actually reasonable sounding ideas... I'm not sure I would personally vote for free broadband for everyone, but that is absolutely legitimate platform that I could see somebody actually running on.
I'm not British, but I'm a fan a silly humour. One message I like is:
"I would make an absolutely cast-iron firm commitment to build the 100bn Trident weapon system ... but then I would make an equally firm private commitment after that public commitment not to build.
Because they are secret submarines and no one will ever know...
Win-win. A hundred billion quid. Shove it in the health service"
This was achieved before the Uxbridge by-election. Which is pretty good going for a pub [1] built at the same time as the first European settlement in the interior of the continental US [2].
The modern Lords is composed of a mix of Life Peers, who were sent there in their lifetime by the Monarch, generally in fact as a result of the government of the day (ie elected politicians) choosing them†, and a fixed number (from the pool) of Hereditary Peers, who have inherited this honour typically (but not always) from their father and so on, perhaps for centuries.
Historically all the Peers could sit, today a fixed number of Hereditary Peers are chosen, the rest get the same title etc. but have no role in Parliament. An election is held (internally) to decide who gets to do this, on the one hand it is paid (a few hundred pounds for each day you're there, so real money albeit you wouldn't get rich) but you're expected to actually do something useful, which if you are accustomed to just sitting on your backside getting rich off the labour of others will be a nasty surprise.
So, even if Binface were actually a peer (which he is not) he wouldn't necessarily be in the Lords today, and actually if he was in the Lords that means he'd need to quit to become an MP as it's not legal to be both - historically it wasn't even possible to quit but somebody in the Lords really wanted to be an elected politician in the 20th century so the rules were changed to allow them to stop being a Lord -- interestingly the law actually doesn't destroy the peerage, if you're a hereditary peer and you want to be an MP so you give that up the peerage exists anyway when you die and it gets inherited as normal.
† This is as self-serving and corrupt in practice as you'd expect. For every famously charitable sports person and beloved actor honoured, expect a career politician looking to retire, a party donor and some dodgy business guy who in a century everybody will know was a crook or a rapist or both... But in principle they could just send the nice lady who taught a generation of kids to read, that bloke who won six Olympic medals and somebody who was born with no legs and yet single handedly saved all the kids in a burning orphanage, so there's that.
Not quite. Scottish peers (to be precise, members of the Peerage of Scotland, which isn't quite the same thing) and Irish peers (again, more precisely, members of the Peerage of Ireland) elected representatives from their number to sit in the Lords, in much the same way as the hereditary peers do today.
Scottish peers got the universal right to sit in the Lords in 1963.
The right of Irish peers to sit in the Lords, if elected, survived Irish independence in 1922; however, the office tasked with overseeing their election, that of the Lord Chancellor of Ireland, was abolished with independence, so their numbers gradually dwindled: the last Irish peer to sit in the Lords died in 1961.
> a fixed number (from the pool) of Hereditary Peers, who have inherited this honour typically (but not always) from their father and so on, perhaps for centuries.
That's so incredibly embarrassing and anachronistic.
