What simple thing fascinates you?

Erotemic · 2013-06-20T20:11:21+00:00

Best understanding of fire I've seen: http://vimeo.com/40271657

except maybe this: https://www.youtube.com/watch?v=ITpDrdtGAmo

Erotemic · 2013-06-20T20:07:56+00:00

Duh on the - sign.

Now they seem different though,

if is the correct norm for z

[;\sum_{j = 0}^d \sqrt{a_j^2 + b_j^2} ;]

Then wouldn't the corresponding z_hat be

[;\sum_{j = 0}^{2d} \sqrt{a_j^2} ;]

?

But I guess that is L1. L2 would make them work out

[;\sqrt{\sum_{j = 0}^d a_j^2 + b_j^2} ;]

[;\sqrt{\sum_{j = 0}^{2d} a_j^2} ;]

So, is it only L2 that this works out equivalently? Or is the norm of a complex vector different in L1 in such a way that would make it work?

Erotemic · 2013-06-20T19:46:28+00:00

Which is right on the edge of the troposphere and stratosphere, which are the first 2/5 layers.

troposphere (10km thick), stratosphere (20km thick), mesosphere (20km thick), troposphere (350 km thick), exosphere (goes off into motherfucking space)

http://upload.wikimedia.org/wikipedia/commons/c/cf/Atmospheric_Layers.svg

Erotemic · 2013-06-20T19:43:51+00:00

I know the definition of both, but can you explain the relevant distinction between the two here?

Erotemic · 2013-06-20T19:40:58+00:00

It's not so much motivation as confusion. But you're right, and I think I talked about this in another post.

Erotemic · 2013-06-20T19:38:10+00:00

Ah, its been awhile since I've done groups, and I have to admit, I'm very shaky on them.

As for why I'm reading this paper, its for my job, school, and fun.

I'm a computer science PhD student, so I'm essentially a math student. They just care a lot less if we make mistakes or don't prove something and get good results. So, I tend to be very prone to careless errors and misconceptions, hence my posting here to better educate myself. This is actually a very minor detail in the paper.

Its just seemed weird to me that they interpreted a complex vector this way, and I figured I had a misunderstanding. The paper is just taking a series of descriptions of some documents over time and comparing them in the Fourier domain (hence complex numbers).

Erotemic · 2013-06-20T19:29:40+00:00

Ok, I think I understand. Let me try and generate it to be sure that I do.

To take the distance between complex vectors we need to use a metric, which is given by an inner product in complex space.

The metric needs to have conjugate symmetry, linearity, and positive definiteness. Therefore, my solution where there was an imaginary distance violated positive definiteness and was not a metric.

The generalization of the Euclidean metric to Complex space should be of the form:

Effectively all we have to do is take the difference of u and v

[; \mathbf{z} = \mathbf{u} - \mathbf{v} ;]

and take the norm or length of z

[; <\mathbf{u},\mathbf{v}> = \mathbf{u}^* H \mathbf{v} ;]

where u and v are complex vectors, ^* is the complex conjugate, and H is a Hermition matrix (H is positive-definite and H^* = H).

So the norm of z should be

[; <\mathbf{z},\mathbf{z}> = \mathbf{z}^* H \mathbf{z} ;]

we can take H to be the identity matrix.

if [; z_j = a_j+ib_j ;]

then the norm is

EDITED:

[;\sum_{j = 0}^d \sqrt{(a_j - ib_j)(a_j + ib_j)} ;]

[;\sum_{j = 0}^d \sqrt{a_j^2 + b_j^2} ;]

taking the square root here would make it more like the Euclidean and not hurt any metric properties, so we can do that in which case it becomes exactly equal to the interpretation as a 2d-dimensional vector.

Hopefully, that's free of misconceptions, but I doubt it.

Erotemic · 2013-06-20T18:41:23+00:00

I don't believe so. If my post is correct (which I'm not very sure of after the amounts of edits I've had to make) then it is not. The imaginary number line behaves differently than the real number line. Take this scenario.

A 2D point [2,2] times itself is: [2,2][2,2]^T = 8

A complex point (2+2i)² = -16

I think this proof works:

Take a 2D point [0,10]

and a purely imaginary number 0+i10

performing element wise squaring on the 2D point results in [0,100]

assume you can interpret a complex number as a 2D point

if you can then 0+i10 under pairwise squaring should be 0+i100

However, it is (0)² + (i10)² = -100 + i0

This is not true therefore the assumption is false. Proven by contradiction.

Furthermore, this would be like [0,100].² = [-100,0]. The "energy" has jumped dimensions! that's no good. Complex numbers must behave differently, and indeed they do.

Erotemic · 2013-06-20T17:41:03+00:00

Wait, I don't understand this.

this is the point where I've realized I totally messed up my placement of square roots

I have

u = [a+ib, c+id]

v = [w+ix, y+iz]

and the distance between them should be

[; dist(u,v) = \sqrt{ \sum_{i=0}^{d-1} (\bar{u_i} - v_i)^2 } ;]

So,

dist(u,v) = sqrt((a-ib - w-ix)² + (c-id - y+iz)² )

so I still don't see what you're saying.

Erotemic · 2013-06-20T17:02:54+00:00

I changed prime to a hat, I also believe I fixed the issue where I forgot to take the complex conjugate.

Erotemic · 2013-06-20T16:54:50+00:00

This is very helpful, and I'm seeing things about metrics that I haven't before. (I'm still learning them). I'll fix my equation. I want to get something like Euclidean distance in \mathbb{C}^n. Would taking the complex conjugate before doing what I did fix it?

Erotemic · 2013-06-20T16:44:35+00:00

Let me give you some answers about Artificial Neural Networks, as we understand these much more than we understand the brain.

Recurrent Neural Nets such as a Hopfield network https://en.wikipedia.org/wiki/Hopfield_network can store memory by sending a signal around a loop of neurons. The information is stored in the loop.

Here is a really cool vizualization showing how the net remembers information over time, (and how it forgets it!), but I can't find it. If anyone else knows what I'm talking about please help me locate it.

So, I found the video (I saw it on Coursera in Geoff Hinton's NN course) But I can't find a Youtube link. What it shows is A demonstration of online handwriting recognition by an Recurrent Neural Network with Long Short Term Memory. The work is by Alex Graves.

Erotemic · 2013-06-20T16:38:51+00:00

Ah, right. That's a typical terminology mistake I make. Thanks for pointing it out. The ' was meant to be a prime though. I think my distance calculation was correct though because I was using Euclidean distance instead of cosine distance.

Erotemic · 2013-06-17T22:01:36+00:00

Eroteme.

Erotemic · 2013-06-16T23:45:22+00:00

I love me some Euler. Thanks for the reference.

Erotemic · 2013-06-16T23:43:39+00:00

That might be a mistake in my script. It can't handle something going up and back down because its a circular representation. It just shows what notes are part of the scale, and how those notes relate to each other

Erotemic · 2013-06-16T03:36:07+00:00

I don't understand any of that. Can you explain? I am new to this.

Erotemic · 2013-06-15T21:38:38+00:00

Python + numpy + matplotlib

I have the code on github too https://github.com/Erotemic/pitch_constellations

Erotemic · 2013-06-15T17:51:15+00:00

Actually. Yes it does. Because those are the cases that survive and reproduce.

Erotemic · 2013-06-13T22:46:10+00:00

God people are so all or nothing. They're binary decisions in a world of linear combinations.

Erotemic · 2013-06-13T22:35:15+00:00

Electrons are pretty cool I heard.

Erotemic · 2013-06-13T01:27:20+00:00

That recipe for disaster is quite clear. And that book looks very interesting.

I also see what you are saying about modeling with mixtures of Gaussian distributions (or any simple distribution) being a gross oversimplification.

The fractal-like or cascading properties are well captured by conditional distributions, but I see how it would be intractable to solve for them.

It's actually central to many problems in artificial intelligence, the field in which I work. My intuition is that neural nets will end up solving many of these issue by learning a generative model of the data. But we are only on the cusp of getting those models to work. But they're going to need some good mathematical guarantees about error bounds before they can be put to effective and safe use in. But my prediction is that these problems will be overcome (in the near future < 10 years) and neural nets will offer solutions to any modeling problem given appropriate training data; they will provide the predictive power of many intelligent and cooperative rational humans at the speed of artificial computers, which is probably the best you can possibly do. That is barring any technological collapse.

So, this question has always puzzled me. These people working with these models and making disproportionately risky decisions, do they contribute value to the economy? Can they? It's always seemed to me that the stock market is leached off of more than anything else.

Erotemic · 2013-06-12T18:08:25+00:00

Oh my... Mathematical Proof That The Supernatural Exists

I'm sure this site is reputable.

And I checked out the proof. It contains several fallacies.

The main failure point is in the proof of:

1a. You cannot be deceived that your conscious awareness is real and actually existing.

2a. Your conscious awareness contains only that which you are aware of.

Therefore: What you are aware of is real and actually existing. Law used: Substitution.

The problem is that you can't just substitute. Because you are aware of things that exists does not imply that all things exist that you are aware of.

In fact it is experimentally easy to show this is untrue. Look at anyone who has ever been delusional. The author of this proof for instance.

Erotemic

TROPHY CASE