On the other hand, to split a space into tensor components, you need a basis $v_{ij}, i\in I, j\in J$ labeled by two indices, and then it's the tensor product of two spaces with the cardinality of $I$ and $J$. However, these spaces are not subspaces of the total space. The state space of a single particle is not a subspace of that of a two-particle system. It is a bit confusing that the physical notion of "subsystem" does not conform to the mathematical notion of subspace.
@enumaris I'm not familiar with it, but I don't imagine it would be too hard to pick up. I figure that working on a RL lib would get me up to speed with what you mean by sparse RL
@danielunderwood I also have another project I'm working on but my app coder ghosted me for some reason...that one is currently stalled...but that one involves collaborative filtering. It's also the only project I'm working on that I would hope actually makes money...but I haven't done a market study for it yet or anything like that...
pytorch is pretty cool...in many ways I think it's better than tensorflow
@ACuriousMind (I mean, you do have an isomorphism from $|\psi\rangle \mapsto |\psi\rangle \otimes |\phi\rangle$ where $|\phi\rangle$ is some particular state of a second particle.)
@Semiclassical Yes, but there is no canonical choice for that second state, in contrast to the direct sum, where you can canonically add the zero vector of the second space.
@enumaris I'll have to check it out. I like being able to do anything in tf, but it's a bit of a pain to use compared to keras. Collaborative filtering sounds interesting too since evidently it's used in recommender systems? Though pretty much all of ML is interesting to me at this point. I feel kind of like when I started physics and had no in-depth knowledge of any particular field
Trying to figure out some simple proofs of the following. Suppose I entangle two spin-s particles in a singlet state, and I measure their spin components $S_{1a}$, $S_{2b}$ along two perpendicular axes $a,b$. Then $\langle S_{1a}S_{2b}\rangle =0$.
The simplest proof is to rotate both systems by 180 degrees about the a-axis. This leaves a unchanged but maps $b\mapsto -b$. The rotational invariance of the singlet state then ensures $\langle S_{1a}S_{2b}\rangle =\langle S_{1a}S_{2(-b)}\rangle = -\langle S_{1a}S_{2b}\rangle =0$
That's a good proof, but (for the purposes of conversation with someone else) I'd prefer to have some others
Best I've come up with so far is to use rotational invariance to restrict to the case $\langle S_{1z}S_{2x}\rangle$ without loss of generality
And then argue that $S_{1z}S_{2x}$ sends the singlet state into the triplet state, so that $\langle 0 0 | (S_{1z}S_{2x}|00\rangle)=0$
But it'd be better if I could directly argue that $S_{1a}S_{2b}|00\rangle$ is a triplet state
(hmm, what I want to argue isn't that $S_{1a}S_{2b}|00\rangle$ is a triplet state, but that it contains nothing from the singlet state. so that's not great.)
I wonder how entanglement will shape the behaviour of bacterial like life?
In other news, the triple loophole closing experiment in this week's new scientist is an enjoyable read especially the section where their device keep get screwed by storms and the researchers continue to persists until successful
That's a very motivating take of the human virtues of scientists
What do they mean here exactly by crystal lattice? Do they just mean the potential function, which has flipped, due to the reversal of the magnetic field? And why do they mention vibrational energy?
I don’t really see understand the difference between the short- en long-time scales, except that I do understand that in the short-time scale scenario, we can think of our system as a paramagnet, without having to take into consideration other things. Could someone clarify? (if my question makes sense:d)
I think I'll get back into it this week or something
@danielunderwood always possible, but I've tried both signs in the obvious places and neither sign works so...
but in fact, I'm not 100% sure that theoretically that's not what should happen
because I'm implementing a very simplified version of policy gradient right now
without a lot of the modifications that are recommended to "make it work right"
all the reward signals are positive so I'm not sure the theoretical direction of the weights wouldn't always just be to increase
it could be that I need to use a baseline or something for the algorithm to be stable...but I didn't see that as a "hard requirement" in the RL book so I didn't implement it
@enumaris open AI is deep into video games, if you work in them more it could be a strategy to open the door more there. you can work on low level or high level, and low level is increasingly well understood, but deep learning is starting to run into inherent limitations at the low level. high(er) level is where the core innovation has to happen in intermediate future for meaningful advances... "determining reward signals" (vs "merely" assuming them!) is going to be a big deal...
because their explanation was on the level of information asymmetry...which suggests to me that they think the market has some sort of natural relaxation time to it's efficient value which would be shorter the more information is more easily broadcast
so in the internet era...I would expect the relaxation time to be quite short - hence the hourly trading
but maybe economic actors are not rational either
in which case the market might never relax to its efficient values
@enumaris "info asymmetry" is not a bad summary, but notice theres an analytical asymmetry. intelligent agents have different analytical capabilities (even given identical info).
or maybe not...a brownian motion is Markovian...it's not white noise......maybe there's some way to change that Markovian nature into some MDP for a RL agent :D
@ACuriousMind One bit Im still confused about is that the $H_\text{tot} = \bigoplus_{i = 1}^\infty \left( H_1^{\otimes i} \right)$ and not just $H_\text{tot} = H_1 \otimes H_2 \otimes ... H_n$? The former looks like how we write superpositions of kets (of two body system for instance)
@ACuriousMind ahhh right, I see :) so this is in particular for the space of all photon states, had we just had a two photons problem, the state space would have simply been the H_1 \otimes H_2 (the two body Hilbert space) I guess
@ACuriousMind from actual calculation pov, if I have two sets $\{1,3,5\}$ and $\{0,2,4\}$ then their direct sum would yield $\{0,1,2,3,4,5\}$ whereas their tensor product would yield $\{(1,0), (1,2), (1,4), (3,0), (3,2), (3,4), (5,0), (5,2), (5,4)\}$, right?
There is actually a certain (and some may say deeper) truth to your statement, in that the disjoint union of sets shares certain categorial properties with the direct sum of vector spaces and that the Cartesian product of sets shares categorial properties with the tensor product of vector spaces
But at this point that's just confusing, especially since the underlying set of the direct sum of vector spaces is the same as the Cartesian product of the underlying sets.
@user929304 But anyway, examples: The space of three dimensional vectors is the direct sum of the space of vectors in the x-y place (a 2d subspace) and the z-axis (a 1d subspace).
A geometric example for the tensor product eludes me, so it will have to be algebraic: The vectors space of square matrices over a vector space is isomorphic to the tensor product of that space with itself. For instance, the 2-by-2 matrices are the tensor product of $\mathbb{R}^2$ with itself. (The isomorphism is given by sending the tensor product of the i-th with the j-th basic vector to the elementary matrix with all zero except at the $(i,j)$-th position.
@user929304 Well, the tensor product of the plane and the line is just isomorphic to the plane. The tensor product is not very geometric, at least not to me
For instance, the monomials {1,x} generate a 2D vector space. The monomials {1,y,y^2} generate a 3D vector space. If you consider products of those generators, you get {1,x,y,xy ,y^2,xy^2} which generates a 6D vector space
@danielunderwood this attnGAN architecture is actually quite interesting. It's more sophisticated than the one I was originally envisioning since it stacks multiple GANs rather than just use one.
Actually it's not that hard to understand the high level of what the model does, the paper is relatively clear. Reproducing the paper thought would be a bit of a chore since it's kinda of complicated. They released a github repo with the source code though (pytorch backend) so if you're interested in the nitty gritty details you can look at the source code.
@enumaris You could throw the GBM inside some financial derivative and make your neural network or whatever you're working on come up with a hedging strategy (and confirm that it matches the analytical/Monte Carlo/PDE sol'n), which you can predict.
I think the EMH would say that the current price already reflects all these considerations and so there is no preference of one financial product over another
strictly speaking, if the EMH is true, one should be indifferent between all strategies...I think...
@enumaris believe it or not its not really much different than standard data science... a key difference is that results (or methods that is) are not "published" :P
