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Equation 30 · Part 6 · The Bend an Elevator Cannot Fake

Symbol y

r:=[I−B(B⊤C−1B)+B⊤C−1]y,r := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y,
yy

What this part means

y is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

y is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

A tempting shortcut suggests itself immediately: pick any closed loop of clocks and sum the edge readings around it. If the sum fails to return to zero, something has broken the idea that a single scalar potential — curved or otherwise — assigns each clock one honest number. Formally, define the fitted residual r:=[I−B(B⊤C−1B)+B⊤C−1]yr := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y. the part of y that weighted least squares could not explain with any node potential at all. It is tempting to read a nonzero r as curvature declaring itself. That reading is wrong, and the reason is worth deriving rather than waving at.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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