Cartographic Projection
every map is wrong, and by exactly how much
Concept

Aspect search — where it appears

A search over the three angles of a rotation for the aspect that scores best over a region. Its answer is a map rather than a set of parameters, because the objective is nearly flat along one direction and two searches that agree about the map need not agree about the angles.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as basin, fracture threshold, optimisation landscape — the same set of essays touches all of them, so they are one junction rather than several.

The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.60 at a coefficient of determination of 0.913. The previous rung's level-set measurement gave −1.33 and the argument predicts −1, so measuring the height directly moves the answer FURTHER from the prediction rather than towards it.

The height of the pass between two basins

The previous rung recorded a shortfall: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.

choosing · Choosing The basin has three widths, and they differ by a factor of 4.6. Two sections through the near-optimal basin of the Robinson aspect over Japan, drawn at one scale. Each ellipse is the set of aspects whose score is twice the optimum's, from the objective's own second derivative at the optimum: 32.1°, 16.0°, 6.9° along the three principal directions. A single number for "the width of the basin" is the cube root of their product, 15.3°, and it is not any of them.

The basins have widths as well as depths

The previous rung measured the height of the pass and recorded a shortfall: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.

choosing · Choosing The piece count rises and falls. The number of connected pieces of the near-optimal aspect set for robinson over japan, swept finely through the threshold rather than sampled once below it. It is one piece at a wide threshold, reaches 23 at 1.256, and returns to one as the set shrinks onto the single best aspect. The set first disconnects at 2.244, which is above the peak: the pieces keep multiplying after the first break. This is the sweep the rung below could not afford and it costs one grid, because every threshold reads the same 4992 evaluations.

The threshold is not a percolation

The rung below found the near-optimal aspect set breaking into twelve pieces rather than two, called the transition a percolation, and recorded that it had not measured the exponent. Swept finely, the piece count rises from one to twenty-three and falls back to one — and refining the grid by a factor of fifteen does not move the peak, while an uncorrelated field on the same lattice grows by a factor of twelve.

choosing · Choosing

Named alongside it

The objects these essays reach for when they reach for this one.

BasinExponentFracture thresholdLevel setOptimisation landscapeShortfallAspectAnisotropyDegeneracyDiscretisationHessianLocal minimum

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