Lils using left inverse: Difference between revisions

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When <math>X</math> is a rectangular complexity pretransformer, such as the <math>ZL</math> that is used for <math>\text{lils-C}()</math>, there are alternative inverses available which will still effectively leverage the dual norm inequality in order to cap the maximum damage across all intervals. Specifically, any matrix <math>X^{-}</math> (that's <math>X</math> but with a superscript ''minus'', as opposed to the superscript ''plus'' that we use for the Moore-Penrose inverse) which satisfies <math>X^{-}X=I</math> will suffice. We call such a matrix a "left-inverse", because it cancels the original matrix out when it is left-multiplied by it.

There's also a thing called a right-inverse, which is the opposite. Tall matrices (more rows than columns) like <math>ZL</math> have left-inverses (when they are [[full-rank]]. Wide matrices (more columns than rows) like mappings <math>M</math> have right-inverses (again, when they are full-rank). Square matrices (same count of rows and columns) have matrices that are both left-inverses and right-inverses, or in other words, true inverses (again, when they are full-rank).

There's also a thing called a right-inverse, which is the opposite. Tall matrices (more rows than columns) like <math>ZL</math> have left-inverses (when they are [[full-rank]]). Wide matrices (more columns than rows) like mappings <math>M</math> have right-inverses (again, when they are full-rank). Square matrices (same count of rows and columns) have matrices that are both left-inverses and right-inverses, or in other words, true inverses (again, when they are full-rank).

Fortunately for us, a left-inverse is the kind we need! That's because left-multiplication is the direction in which we need canceling out of <math>X</math> to occur, because of the way it figures in the dual norm inequality, as shown here:

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 When <math>X</math> is a rectangular complexity pretransformer, such as the <math>ZL</math> that is used for <math>\text{lils-C}()</math>, there are alternative inverses available which will still effectively leverage the dual norm inequality in order to cap the maximum damage across all intervals. Specifically, any matrix <math>X^{-}</math> (that's <math>X</math> but with a superscript ''minus'', as opposed to the superscript ''plus'' that we use for the Moore-Penrose inverse) which satisfies <math>X^{-}X=I</math> will suffice. We call such a matrix a "left-inverse", because it cancels the original matrix out when it is left-multiplied by it.
-There's also a thing called a right-inverse, which is the opposite. Tall matrices (more rows than columns) like <math>ZL</math> have left-inverses (when they are [[full-rank]]. Wide matrices (more columns than rows) like mappings <math>M</math> have right-inverses (again, when they are full-rank). Square matrices (same count of rows and columns) have matrices that are both left-inverses and right-inverses, or in other words, true inverses (again, when they are full-rank).
+There's also a thing called a right-inverse, which is the opposite. Tall matrices (more rows than columns) like <math>ZL</math> have left-inverses (when they are [[full-rank]]). Wide matrices (more columns than rows) like mappings <math>M</math> have right-inverses (again, when they are full-rank). Square matrices (same count of rows and columns) have matrices that are both left-inverses and right-inverses, or in other words, true inverses (again, when they are full-rank).
 Fortunately for us, a left-inverse is the kind we need! That's because left-multiplication is the direction in which we need canceling out of <math>X</math> to occur, because of the way it figures in the dual norm inequality, as shown here: