Tomminn · HackerTrans

6 jaar geledendiscuss

The exterior product also exists in a different vector space to its factors.

6 jaar geledendiscuss

The type signature of the outer product is not correct. We are mapping two functions to a function in the same domain.

Convolution is neither a valid example of an inner product or an outer product. No basic geometric operation on vectors has the correct type signature and axioms for convolution to be interpreted as a generalized "x". What we'd be looking for is a billinear mapping of vectors to vectors, and complex number style multiplication in R^2, or cross product multiplication in R^3 or R^7 are the only real candidates in that category unless we start interpreting functions as matrices.

Tomminn

6 jaar geledendiscuss

The parents point is that when you do the primary school multiplication algorithm, you are actually performing a discrete convolution.

Tomminn

6 jaar geledendiscuss

I don't think you can really call it a generalized dot product, because it doesn't map to a scalar. The inner product is the well accepted definition of a generalized dot product, and convolution does not follow the axioms that an inner product must follow.

Tomminn

6 jaar geledendiscuss

This example is great, hope you don't mind if I spell it out how your impulse weight f(d) and impulse response functions g(d) are functions over d (each associated with 10^d):

  f(0) = 1      g(0) = 3      f*g(0) = 3 
  f(1) = 0      g(1) = 2      f*g(1) = 2 
  f(2) = 1                    f*g(2) = 3 
  f(3) = 0                    f*g(3) = 2
  f(4) = 0                    f*g(4) = 0
  f(5) = 1                    f*g(5) = 3 
                              f*g(6) = 2

You can see how when there is an impulse weight "x" in digit space entry d, the impulse response "3x" comes at d, and the impulse response "2x" comes at d+1. Note how f and g can be anything and it convolution (f * g) is still their multiplication (except for the obvious rub when f*g(d) is greater than 9, so you need to add a rule about that).

Tomminn

6 jaar geledendiscuss

Okay. This is decent. I do like the idea of "fancy multiplications" because I do think you should understand convolution as well as you understand multiplication.

But I still feel like this kinda obscures and confuses the origin of the reversal.*

If you don't understand the convolution formula instinctively, read this comment enough times till you do.

The point is this.

-We have a function originBangSound(t) that maps the effect at (t) of an impulse or "bang" coming from the origin (t=0).

-We have a function bangWeights(t), which measures the distribution of impulses or "bangs" over time.

-The question is: how do we get the total allBangSounds(t)?

Simple: We make every point in time the origin, and add all the results together. Let's call (tau) the current origin. The size of the bang at this origin is bangWeights(tau). The size of the sound at (t) which is coming from this origin is originBangSound(t- tau), since we care about the position of (t) relative to the current origin (tau). Adding them up leads to an integral in the continuous case.

  allBangSounds(t) = \integral bangWeights(tau)*originBangSound(t-tau) d(tau)

The point is this. Don't think of it as a flip. Think of (tau) as defining the origin point for a particular "bang".

Here's a nice sanity check: if (tau) is larger than (t), (or equivalently, (t-tau)<0) then do you expect to hear its bang? Ofcourse not. The bang hasn't happened yet. So unless its bang travels backward in time (which definitely does happen in spatial convolutions!) you ain't hearing it.

_____________________________

*Come to think of it, this is a very useful pun. When you think to yourself "what's the origin of the reversal again?", just remember, the moving the origin is the origin of the reversal.

Tomminn

7 jaar geledendiscuss

Sure, but the kinds of requests NVC codifies are ones where "you hurt my feelings via action X".

Tomminn

7 jaar geledendiscuss

I think non-violent communication is great when you actually have to ask someone to change something about the way they operate in the world.

If you want to improve your relationships though, make rule zero "minimize the extent to which you require other people in the world to change."

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