James.Wilson@ColoState.Edu.

Professor of Mathematics, Colorado State University

Patterns in Data & the Algebra That Finds It

Cluster Patterns in Tensors  Data

WhO Me?

Couldn't Be!

  • Then Who?

  • Maybe you?

Tensors as AreA &volume

Volume basics

Volume: west window bays x east window bays x floors \[\text{Vol}(4,5,3)  = 4\text{bays} \times 5\text{bays}\times 3\text{floors} = 60\text{rooms}.\]

Lessons:

  • Volumes are distribute products
  • They need multiplication tables ("tensors")

Oops! Miss counted, but easy update: \[\text{Vol}(4,5+{\color{red}1},3)=\text{Vol}(4, 5, 3)+{\color{red}\text{Vol}(4,1,3)}=60+12=72\]

Need it in Cubic feet? Easy.

Vol(4,6,3) = 160 \(\text{ft}^4/\text{bay}^2~\text{floor}\) \(\times 4 \text{bays} \times 6\text{bays} \times 3 \text{floors}\)

Area...easy

Area...

Less easy

\[\text{Area}\left(\begin{bmatrix} a\\ c\end{bmatrix}, \begin{bmatrix} b\\ d\end{bmatrix}\right) = (a+b)(c+d)-2bc-2\frac{1}{2}ac-2\frac{1}{2}bd=ad-bc= \begin{bmatrix} a & b\end{bmatrix}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} c\\ d \end{bmatrix}\]

Adjust the Multiplication Table "tensor"

Indirect Area...Even Harder

Area/Volume wrapped onto surfaces, measure inputs by proxy \[Jac(x,y)=\begin{bmatrix}\partial_x & \partial_y\end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} x\\ y\end{bmatrix}\]

Proportionate measures add strangely \[p_1\boxplus p_2=\frac{V_1}{V_3}p_1+\frac{V_2}{V_3}p_2\neq p_2\boxplus p_1\] E.g. adding gas pressure or supply/demand.

New uses? Make up new measurements

New uses? Sales Volume?

Slightly more honest

Scores from from VSChartz.com in 2021

Histogram Visualization

New uses? interactions probabilities and quantum information

I could keep going....

...And I have!

 

HTTPS://TheTensor.Space (Community)

HTTPS://PleaseDistribute.Org. (me)

  • Goal: Locate nutrition of fruit salad.
  • Problem: Data table has no fruit salad row.

Solution: compute the combination of rows specified by the recipe.

Galway --- Enfield

   18:00 ----   20:30

   19:00 ----   21:30

             Galway --- Enfield

    0.5 *  18:00 ----   20:30

 + 0.5 *  19:00 ----   21:30

-------------------------------------

               18:30 ---- 21:00

 

  • Goal: Arrive in Enfield at 21:00
  • Problem: Data only has 20:30 and 21:30

Solution: Use the recipe

1/2 of bus 1  and 1/2 bus 2!

A table of data is a tensor (for me) only when it is informative to generate new data as combinations of the rows, columns, pages, etc.

Consequence: You can change coordinates!

Back to Patterns

What are the GAME markets?

Data table --> Multiplication table

Use algebra to factor

Ok... Now Factor This?

\begin{array}{c|ccc|} * & f_1 & f_2 & f_3 \\ \hline e_1 & 0.2g_1+0.9g_2-0.4g_3 & 0.2g_1+0.9g_2-0.4g_3 & -1.0g_1+0.4g_2-0.1g_3\\ e_1 & -9.1g_1+8.9g_2+0.7g_3 & -2.0g_1+0.1g_2-0.1g_3 & 0.0g_1+1.8g_2-0.4g_3 \\ \hline & & \end{array}

Or this...

Yes, even movies can be tensors...

Finding Good Algebra in Real Life

How do you study something curved?

 

How do you study something curved?

 

With lines tangent to it!

 

Product Rule:

\[D(x*y)=D(x)*y+x*D(y)\]

 

We can do this on ANY tensor, and notice D is linear (exponent 1)

If we have products of different sizes or content, you independent coordinates \[H(x*y) = F(x)*y + x * G(y).\]

G

+

=

\begin{aligned} -12 D(\text{Vol}(x,y,z)) & = 4\text{Vol}(Ax, y, z) -2 \text{Vol}(x,By, z) + 13\text{Vol}(x,y,Cz)\\ 1 D(\text{Vol}(x,y,z)) & = 0\text{Vol}(Ax, y, z) +1 \text{Vol}(x,By, z) + 5\text{Vol}(x,y,Cz)\\ \end{aligned}

Want a higher valence product, like volume, video,  Cumulants & Moments? Want to weight some axis or combine variations?

 

"Chisels" are selections of equations and constraints on the derivations, e.g.:\[C=\begin{bmatrix} 12 & 4 & -2 & 13\\ -1 & 0 & 1 & 5 \end{bmatrix}\] together with \[A=A^{\dagger}, B=Diag(B), C=D\]

They control what patterns can be discovered.

And it's solutions are a Lie algebra

The main fact:\[A*B=0\Rightarrow A=0\text{  or  } B=0\]

  • \(D_0 (x*y) = D_1(x)*y+ x*D_2(y)\)
  • \(D_0 (x*y) - D_1(x)*y - x*D_2(y) = 0\)
  • \(5 (x*y) - 3x*y +2 x*y = 0\)
  • \((5- 3+2)(x*y) = 0\)
  • So \(x*y=0\)!  I.e. a pattern! \[\begin{array}{c|ccc|} * & \cdots & y & \cdots \\ \hline \vdots & ? & ? & ? \\ x & ? & 0 & ?\\ \vdots & ? & ? & ? \\ \hline \end{array}\]

If \(D_0(x*y)=5(x*y)\), \(D_1(x)=3x\) and \(D_2(y)=-2y\). [True for Eigenvectors]

First every universal chisel!

Theorem.  (Brooksbank-Kassabov-W.) Chisels detect cluster patterns in time \(O(n^{2})\).

  • Tucker (vector space) \[\begin{bmatrix} 1 & 0 & 0 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix},\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]
  • Centroids (commutative ring)\[\begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & -1 \end{bmatrix}\]
  • Adjoints (ring with involution) \[\begin{bmatrix} 1 & -1 & 0 \end{bmatrix}\]
  • Derivations (Lie algebra)\[\begin{bmatrix} 1 & 1 & -1 \end{bmatrix}\]
  • In 4+ valence there \(\infty\) types.

Liu-Maglione-W.

Now \(O(n^{1+1/d})\) on average.

Theorem.  Clusters you detect are essentially unique, because of the algebra.

  • (Folklore) Tucker (vector space) unique, greedy solve.
  • (W.08) Centroids (commutative ring) have unique patterns.
  • (W.08, Maehara-Murota 10) Adjoints (ring with involution) have conjugate patterns.
  • (BKW) Derivations (Lie algebra) unique over \(\mathbb{C}\)

Is it useful?

"Random" Basis Change

Blind source separation

Oulier Detection

USDA Study "What We Eat in America" (add sugars data) .

Compression? THeory says yes, Quality works, beats HOSVD, DCT, but Still Tied or worse than HEVC

Forget about looks; deeper structure

Mouse thiamine triphosphatase : pdb_00002jmu https://www.rcsb.org/3d-view/2JMU

Nuclear Magnetic Resonance NMR:

  • put some goo in a jar surrounded by magnets.  

  • Turn on the radio.  

  • Watch it dance.  

  • Decide what your protein molecule looks like.

Mouse thiamine triphosphatase : pdb_00002jmu https://www.rcsb.org/3d-view/2JMU

NMR Tensor?

(Experts: I don't know any better!  Teach me!)

  • Watch magnetic changes under pulses that excite Hydrogen, Nitrogen, Carbonyl gives 3 axes.

  • Look for coordinates in wave space to identify what "tugs" on what, i.e. is the hydrogen attached to the nitrogen or the Carbon, etc.

  • Peaks in the wave frequency are the key.

To the eye, the FFT looks far more organized than Dleto (not always the case just this example)

Look with instruments; Dleto is actually quite systematic.

Molecule shape tracks with how atoms move.

More Peaks = More structure (more dots to connect)

  • FFT needs more measurements to find all 3 peaks.

  • Chiseling can retain statistically significant with data 1/2 to even 1/8 the data of the FFT.

  • Twice as fast means...

    • Half price.

    • More scans.

    • Finer features at same price.

Why?

  • FFT assumes frequency bins like keys on a piano.

  • Need more frequencies? Build a bigger piano.

  • Chiseling scans the data you have and decides what bins to make.
  • It's tuning your instrument for the music instead of building a bigger one.

When Data tables => Multiplication Tables

  • Cluster patterns => Factoring, zero-divisor, algebra.
  • Algorithms to find patterns come from derived algebra.
  • Uniqueness claims follow from the algebra.
  • Algebraist can learn from data sciences and perhaps also give back.

Summary

Talks, Articles, and Books

PleaseDistribute.Org

Software:

TheTensor.Space (links to GitHub projects)

Tell me what I need to learn: James.Wilson@ColoState.Edu

Data-Algebra

By James Wilson

Data-Algebra

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