Once you’re comfortable with 2-arrays and 2-matrices, you…

October 15, 2011

(This article was first published on Isomorphismes, and kindly contributed to R-bloggers)

Once you’re comfortable with 2-arrays and 2-matrices, you can move up a dimension or two, to 4-arrays or 4-tensors.

You can move up to a 3-array / 3-tensor just by imagining a matrix which “extends back into the blackboard”. Like a 5 × 5 matrix. With another 5 × 5 matrix behind it. And another 5 × 5 matrix behind that with 25 more entries. Etc.

The other way is to imagine “Tables of tables of tables of tables … of tables of tables of tables.” This imagination technique is infinitely extensible.

\begin{bmatrix}  \begin{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} & \begin{bmatrix} e & f \\ g & h \end{bmatrix} \\ \\ \begin{bmatrix} j & k \\ l & m \end{bmatrix} & \begin{bmatrix} n & o \\ p & q \end{bmatrix} \end{bmatrix} & \begin{bmatrix} \begin{bmatrix} r & s \\ t & u \end{bmatrix} & \begin{bmatrix} v & w \\ x & y \end{bmatrix} \\ \\ \begin{bmatrix} z & a' \\ b' & c' \end{bmatrix} & \begin{bmatrix} d' & e' \\ f' & g' \end{bmatrix} \end{bmatrix} \\ \\ \begin{bmatrix} \begin{bmatrix} h' & j' \\ k' & l' \end{bmatrix} & \begin{bmatrix} m' & n' \\ o' & p' \end{bmatrix} \\ \\ \begin{bmatrix} q' & r' \\ s' & t' \end{bmatrix} & \begin{bmatrix} u' & v' \\ w' & x' \end{bmatrix} \end{bmatrix} & \begin{bmatrix} \begin{bmatrix} y' & z' \\ a'' & b'' \end{bmatrix} & \begin{bmatrix} c'' & d'' \\ e'' & f'' \end{bmatrix} \\ \\ \begin{bmatrix} g'' & h'' \\ j'' & k'' \end{bmatrix} & \begin{bmatrix} l'' & m'' \\ n'' & o'' \end{bmatrix} \end{bmatrix} \end{bmatrix}

If that looks complicated, it’s just because simple recursion can produce convoluted outputs. Reading the LaTeX (alt text) is definitely harder than writing it was. (I just cut & paste \begin{bmatrix} stuff \end{bmatrix} inside other \begin{bmatrix} … \end{bmatrix}.)

(The technical difference between an array and a tensor: an array is a block which holds data. A tensor is a block of numbers which (linearly) transform matrices / vectors / tensors. Array = noun. Tensor = verb.)

As the last picture — the most important one — demonstrates, a 4-array can be filled with completely plain, ordinary, pedestrian information like age, weight, height.

Inside each of the yellow or blue boxes in the earlier pictures, is a datum. What calls for the high-dimensional array is the structure and inter-relationships of the infos. Age, height, sex, and weight each belongs_to a particular person, in an object-oriented sense. And one can marginalise, in a statistical sense, over any of those variables — consider all the ages of the people surveyed, for example.

One last takeaway:

  • Normal, pedestrian, run-of-the-mill, everyday descriptions of things = high-dimensional arrays of varying data types.

Normal people speak about and conceive of information which fits high-D arrays all the time. “Attached” (in the fibre sense) to any person you know is a huge database of facts. Not to mention data-intensive visual information like parameterisations of the surface of their face, which we naturally process in an Augenblick.

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