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

Level set — where it appears

The set of places at which a function takes a value at or below a stated level. Its shape is what an optimisation is really searching: a connected sheet means the near-optimal answers are one family, and a set of disconnected pieces means they are several.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The set of aspects within a stated distance of the best, for Robinson over Japan. Each row takes every point of a 36 × 19 × 24 grid in the three aspect parameters that scores within (1 + t) of the best, joins neighbouring points, and identifies the pieces the exact degeneracy relates. At t = 3 it is one connected piece spanning 170° of pole; by t = 1 it has broken into 14 pieces; and by t = 0.3 the largest of them spans 11°. So it is not one valley and it is not one basin — it is a sheet that fractures.

The shape of the valley

An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that fractures into fourteen pieces once the threshold tightens.

choosing · Choosing Where Robinson's valley breaks, against how large the region is. The threshold at which the set of near-optimal aspects stops being one connected piece, for square regions of growing size at 38° north. It falls from 2.13 at 6° to 0.27 at 30°, a factor of 8.0. The previous rung measured this at one region and quoted "about twice the optimum"; that value belongs to a small region, and the prediction that it should fall as the region grows is what this tests.

Where the valley breaks in two

The previous rung found a near-optimal set that is one connected sheet at a loose threshold and fourteen basins at a tight one, and explained the transition without testing it. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.13 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.

choosing · Choosing 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.

AspectAspect searchBasinExponentFracture thresholdLocal minimumOptimisationOptimisation landscapeReproducibilityShortfallDegeneracyIdentifiability

All concepts