Lab · c19.s03.L1
Bend the space
A hidden layer changes the coordinates; the output's straight line in the new coordinates is the bent boundary in the old ones.
Question: How does a second layer bend the boundary?
Frame 1 of 1: One line, two edgesThe output unit reads -h1 + h2 - 0.5, so it answers 1 exactly on the upper-left side of the straight line h2 = h1 + 0.5 in the hidden square; there the hidden pair is (0, 1). Carried back to the input plane, that one line is the boundary: two edges, x1 + x2 = 0.5 and x1 + x2 = 1.5, with the answer-1 band between them. Both hidden units read the same sum, so the two edges are parallel and the boundary has no corner. The probe at (0.30, 0.30) has sum 0.60, hidden pair (0, 1), output score 0.50 and answer 1.
One line, two edges. The output unit reads -h1 + h2 - 0.5, so it answers 1 exactly on the upper-left side of the straight line h2 = h1 + 0.5 in the hidden square; there the hidden pair is (0, 1). Carried back to the input plane, that one line is the boundary: two edges, x1 + x2 = 0.5 and x1 + x2 = 1.5, with the answer-1 band between them. Both hidden units read the same sum, so the two edges are parallel and the boundary has no corner. The probe at (0.30, 0.30) has sum 0.60, hidden pair (0, 1), output score 0.50 and answer 1.
Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.
Text description of this lab
Two panels. On the left, four points at the corners of a square, two gold and two navy, with two parallel diagonal lines cutting the square; the band between them is shaded russet. On the right, the same four points become three points at the corners of a unit square, and one straight russet line separates the gold point from the two navy points. A movable probe shows where any input lands in both panels.
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Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.