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!
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Then Who?
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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
- 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?
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
+
=
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:
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put some goo in a jar surrounded by magnets.
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Turn on the radio.
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Watch it dance.
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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!)
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Watch magnetic changes under pulses that excite Hydrogen, Nitrogen, Carbonyl gives 3 axes.
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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.
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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)
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FFT needs more measurements to find all 3 peaks.
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Chiseling can retain statistically significant with data 1/2 to even 1/8 the data of the FFT.
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Twice as fast means...
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Half price.
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More scans.
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Finer features at same price.
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Why?
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FFT assumes frequency bins like keys on a piano.
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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