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Equation 30 · Part 4 · The Bend an Elevator Cannot Fake

Symbol B^top

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,
B⊤B^\top

What this part means

BtB^top is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

BtB^top 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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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