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Posts tagged with functional analysis

A fun exercise/problem/puzzle introducing function space.

hi-res




Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

What I dislike about the word #liminality http://t.co/uWCczGDiDj : it suggests the ∩ is small or temporary.
— isomorphismes (@isomorphisms)
July 7, 2013

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the [0,1] of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

By contrast ℝ→ℝ functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.
But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.
So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into
one type,
the other,
or “something in between”.
Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".

Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

https://upload.wikimedia.org/wikipedia/commons/3/33/Anna_P.jpg

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the [0,1] of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

image

By contrast ℝ→ℝ functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

In harmonic analysis and PDE, one often wants to place a function ƒ:ℝᵈ→ℂ on some domain (let’s take a Euclidean space ℝᵈ for simplicity) in one or more function spaces in order to quantify its “size”….  [T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.  …  For function spaces X on Euclidean space, two such exponents are the regularity s of the space, and the integrability p of the space.  …  …        —Terence Tao  Hat tip: @AnalysisFact

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.

But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.

So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into

  • one type,
  • the other,
  • or “something in between”.

Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".


hi-res







In harmonic analysis and PDE, one often wants to place a function ƒ:ℝᵈ→ℂ on some domain (let’s take a Euclidean space ℝᵈ for simplicity) in one or more function spaces in order to quantify its “size”….
[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.
…
For function spaces X on Euclidean space, two such exponents are the regularity s of the space, and the integrability p of the space.
…
…



—Terence Tao
Hat tip: @AnalysisFact

In harmonic analysis and PDE, one often wants to place a function ƒ:ℝᵈ→ℂ on some domain (let’s take a Euclidean space ℝᵈ for simplicity) in one or more function spaces in order to quantify its “size”….

[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.

For function spaces X on Euclidean space, two such exponents are the regularity s of the space, and the integrability p of the space.

—Terence Tao

Hat tip: @AnalysisFact


hi-res




Given a time-series of one security’s price-train P[t], a low-frequency trader’s job (forgetting trading costs) is to find a step function S[t] to convolve against price changes P[t]

image

with the proviso that the other side to the trade exists.

S[t] represents the bet size long or short the security in question. The trader’s profit at any point in time τ is then given by the above definite integral.

 
  • I haven’t seen anyone talk this way about the problem, perhaps because I don’t read enough or because it’s not a useful idea. But … it was a cool thought, representing a >0 amount of cogitation.
  • This came to mind while reading a discussion of “Monkey Style Trading” on NuclearPhynance. My guess is that monkey style is a Brownian ratchet and as such should do no useful work.
  • If I were doing a paper investigating the public-welfare consequences of trading, this is how I’d think about the problem.

    Each hedge fund / central bank / significant player is reduced to a conditional response strategy, chosen from the set of all step functions uniformly less than a liquidity constraint. This endogenously coughs up the trading volume which really should be fed back into the conditional strategies.
  • Does this viewpoint lead to new risk metrics?
  • Should be mechanical to expand to multiple securities. Would anything interesting come from that?

I wouldn’t usually think that multiplication of functions has anything to do with trading. Maybe some theorems can do a bit of heavy lifting here; maybe not.

It at least feels like an antidote to two wrongful axiomatic habits. For economists who look for real value, logic, and Information Transmission, it says The market does whatever it wants, and the best response is a response to whatever that is. For financial engineering graduates who spent too long chanting the mantraμ dt + σ dBt" this is just another way of emphasising: you can’t control anything except your bet size.

UPDATE: Thanks to an anonymous commenter for a correction.




The word ‘space’ has acquired several meanings, which is what you would expect of such a sexy, primitive, metaphorically rich, eminently repurposeable concept.

  1. Outer space, of course, is where cosmonauts, Hubble telescopes, television satellites, and aliens reside. It’s ℝ³, or something like that.
    image 
  2. Grammatical spaces keep words apart. The space bar got a little more exercise than the backspace key while I was writing this list.
    image 
  3. Non-printable area (space) is also free from ink or electronic text in newspapers: ad space. Would you like to buy one?
    image 
  4. Closely related is the negative area in sculpture, architecture, and other visual arts.

  5. Or in music. Don’t forget to “play” the notes you don’t play, Thelonious!
  6. Or the space you need to give someone in a relationship, if you want to allow them to be themselves whilst also being with you.
  7. Space on my hard drive to store an exact digital replica of all my vinyl? This kind of space also applies to human memory capacity, computer RAM, and other electronic pulsings which seem rather more time-based than spatial & static.
    image 
  8. Businessmen refer to competitive neighbourhoods: the online payments space; the self-help books category; the $99-and-under motel space; and so on.
    image
    image
     
  9. Space as distinct from time. Although cosmologists will tell you that spacetime is a pseudo-Riemannian manifold which looks locally like ℝ⁴, a geographer or ecologist will tell you that locally space looks like ℝ² (since we live solely on the surface of the Earth).
    imageimageimageimageimageimage
    I believe the ℝ² view is also taken by programmers who geotag things (flickr photos, twitter tweets, 4square updates): second basement = 85th floor and canopy = rainforest floor as far as that’s concerned.

    Both perspectives are valid. They’re just different ways of modelling “the world” with tuples. Is it surprising that cold, rigid, soulless mathematics allows for different, contradictory viewpoints? Time is like space in the grand scheme of things, but for life on Earth time-averages and space-averages are very different.
    Europe, upside-down. 
  10. Parameter space. The first graphs one learns in school plot input x versus output ƒ(x).
    image
    But another kind of plot — like a solid liquid gas diagram
    image
    — plots input a versus input b, with the area coloured or labelled by output ƒ(x). (In the case of matter’s phases, the codomain of ƒ is the set {solid, liquid, gas, plasma} rather than the familiar .)

    • When I push this lever, what happens? What about when I push that one?
    • There are connections to Fourier spectrum.
  11. Phase space. Paths, orbits, and trajectories taken through other spaces. Like the string of (x₍ᵤ₎,y₍ᵤ₎,z₍ᵤ₎)-coordinates that a water rocket takes across the lawn. Or the path of temperature (temp₍ᵤ₎) during a year in Bloomington.
    image
    image
    image
    image

    Or the trajectory of the dynamical system (your feelings₍ᵤ₎, your partner’s feelings₍ᵤ₎) representing your marriage.
    image

    Roger Penrose uses the example of the configuration space of a belt to explain that phases can happen on non-trivial manifolds. (A belt can take on as many configurations as a string, plus it can be twisted into a Moebius band, but if it’s twisted twice that’s the same as twisted zero times.) 
    image 

[Sorry, I don’t have a Unicode character for subscript t, so I used u to represent the time-indexing of path variables. Maybe that’s better anyway, because time isn’t the only possible index.]

  1. Personal space. I forgot personal space. Excuse me; pardon me.



  2. All of the spaces above are like an existing nothing. The space between your arm and your chest, the space where I draw—all of these are conceptually “empty” but impinge on and interact with the rest of reality.

    All of those senses of the word are completely nothing alike to how mathematicians use the word. Mathematicians mean “stuff plus structure to the stuff” which is not at all like the other spaces.


    Abstract spaces.
    These are best understood as ordered tuples, i.e. “Things plus the relationships and desired interpretation of those things.” The space—more like “the entire logical universe I’m going to be talking about here”—is supposed to contain EVERYTHING you need, in order to work with any of the parts. So for example to use a division sign ÷, the space must include numbers like and . (Or you could just do without the ÷ sign. You can make a ring that’s not a division ring; look it up.)

    • A Banach space is made up of vectors (things that can be added together), is complete (there are enough things that infinite limit sequences make sense), with a notion of distance (norm), but not necessarily angle. Also two things can be 0 distance away from each other without being the same thing. (That’s unlike points in Euclidean space: (2,5,2) is the only thing 0 away from (2,5,2)).
    • A group is complete in the sense that everything you need to do the operation is included. (But not complete in the way that Banach space is complete with respect to sequences converging. Geez, this terminology is overloaded with meanings!)
      image 
    • A vector space is complete in the same way that a group is. In the abstract sense. Again, a vector is “anything that can be added together”. The vectors’ space completely brings together all the possible sums of any combination of summands.

      For example, in a 2-space, if you had (1,0) and (0,1) in the space, you would need (1,1) so that the vector space could be complete. (You would also need other stuff.)

      And if the vector space had a and b, it would need to contain a+b — whatever that is taken to mean — as well as a+b+b+(a+b)+a and so on. In jargon, “closed under addition”.
    • A topological space (confusingly, sometimes called “a topology”) is made up of things, bundled together with the necessary overlap, intersection, union, superset, subset concepts so that “connectedness” makes sense.
      image 
    • A Hilbert space has everything a Banach space does, plus the notion of "angle". (Defining an inner product is as good as defining an angle, because you can infer angle from inner multiplication.) ℂ⁷ is a hilbert space, but the pair ({0, 1, 2}, + mod 2) is not.
      image
       imageimageimageimageimageimageimageimageimageimage
    • Euclidean space is a flat, rigid, stick-straight, all-joins-square Hilbert space.
    • To recap that: vector space  Banach space  Hilbert space, where the  symbol means “is less structured than”.

      Topological spaces can be even more unstructured than a vector space. Wikipedia explains all of the T0 T1 ⊰ T2 ⊰ T2.5  T3  T3.5  T4 ⊰ T5 ⊰ T6 progression which was thoroughly explored during the 20th century. (Those spaces differ in how separated “neighbours” are taken to be.)

I don’t mean to imply that these spaces can only be thought of as tuples: ({things}, operations). There are categorical ways to understand them which may be better. But don’t look at me; ask the ncatlab!

  1. Lastly, sometimes ‘space’ just means a collection of related things, without necessarily specifying, like above, the tools and viewpoints that we take to their relationships.
    • The space of all possible faces.
    • The space of all possible boyfriends.
    • The space of all possible songs.
    • The space of all possible sentences.
    • Qualia space, if you’re a theorist of consciousness.
    • The space of all possible romantic relationships.
    • The space of all possible computer programs of length 17239 bytes.
    • Whatever space politics occupies. (And we could debate about that.)
    • (consumption, leisure, utility) space
    • The space of all possible strategy pairs.
    • The space of all possible wealth distributions that sum to W.
    • The space of all bounded functions.
    • The space of all 8×8 matrices over the field ℤ₁₁.
    • The space of all polynomials.
    • The space of all continuous functions from [0,1] → [0,1].
    • The space of all square integrable functions.
    • The space of all bounded linear operators.
    • The space of all possible models of ______.
    • The space of all legal configurations of the Rubik’s cube.

(Some of these may be assumed to come packaged with a particular set of interpretations as in the previous ol:li.)




Vectors, concretely, are arrows, with a head and a tail. If two arrows share a tail, then you can measure the angle between them. The length of the arrow represents the magnitude of the vector.

The modern abstract view is much more interesting but let’s start at the beginning.

Force vectors

Originally vectors were conceived as a force applied at a point.

As in, “That lawn ain’t mowing itself, boy. Now you git over there and apply a continuous stream of vectors to that lawnmower, before I apply a high-magnitude vector to your bee-hind!”

Thanks Galileo, totally gonna get you back, man

The Galilean idea of splitting a point into its x-coordinate, y-coordinate and z-coordinate works with vectors as well. “Apply a force that totals 5 foot-pounds / second² in the x direction and 2 foot-pounds / second² in the y direction”, for instance.

Therefore, both points and vectors benefit from adding more dimensions to Galileo’s “coordinate system”. Add a w dimension, a q dimension, a ξ dimension — and it’s up to you to determine what those things can mean.

If a vector can be described as (5, 2, 0), then why not make a vector that’s (5, 2, 0, 1.1, 2.2, 19, 0, 0, 0, 3)? And so on.

4th Dimension Plus

So that’s how you get to 4-D vectors, 13-D vectors, and 11,929-D vectors. But the really interesting stuff comes from considering -dimensional vectors. That opens up functional space, and sooooo many things in life are functions.

(Interesting stuff also happens when you make vectors out of things that are not traditionally conceived to be “numbers”. Another post.)

Abstractions

In the most general sense, vectors are things that can be added together. The modern, abstract view includes as vectors:

Things you can do with vectors

Given two vectors, you should be able to take their outer product or their inner product.

The inner product allows you to measure the angle between two vectors. If the inner product makes sense, then the space you are playing in has geometry. (Not all spaces have geometry — some just have topology.)

And — this is weird — if the concept of angle applies, then the concept of length applies as well. Don’t ask me why; the symbols just work that way.

Magnitude

But the “length” of a song (one of my for-instances above) would not be something like 2:43. The magnitude of a song vector would be the total amount of energy in the sound wave | compression wave.

\| \text{song} \| = \int \text{compression wave}

What is the angle between two songs, two spike-trains, two security prices? What is the angle between two heartbeats? It’s the correlation between them.

Linear Algebra

Also, you can do linear algebra on vectors — provided they’re coming out of the same point. Some might say that the ability to do linear algebra on something is what makes a vector.

That can mean different things in different spaces — like maybe you’re superposing wave-forms, or maybe you’re converting bitmap images to JPEG. Or maybe you’re Photoshopping an existing JPEG. Oh, man, Photoshop is so math-y.

shearing the mona lisa

Shearing the mona lisa (linear algebra on an image — from the Wikipedia page on eigenvectors, one of which is the red arrow)