Manuscript Psy01-A@1 · Psy01-A.tex · section · 94477fa3

Common latent task distribution

Definitions and exact pruning operators › Common latent task distribution · lines 250–280

Let $Y$ be uniform on ${0, 1, 2, 3}$ , and let

X = μ_{Y} + ν Z, Z \sim N (0, I_{2}), ν = 0.8,

where

μ_{0} = (- 3, - 3), μ_{1} = (3, 3), μ_{2} = (- 3, 3), μ_{3} = (3, - 3) .

Define the fixed-point-free permutation

π (0) = 2, π (1) = 3, π (2) = 0, π (3) = 1.

Thus $π^{2}$ is the identity and $π (y) \neq = y$ .

The visual task has label $Y$ , whereas the voice task has label $π (Y)$ . Their input distributions are identical. For theoretical comparisons it is useful to evaluate both rules on the same latent sample $(X, Y)$ , even though the supplied implementation generates separate visual and voice test datasets.

The cues are

c_{v} = (1, 0), c_{a} = (0, 1), c_{α} = α c_{v} + (1 - α) c_{a} .

Displayed formulas in this passage

unlabeled · line 253
$X = μ_{Y} + ν Z, Z \sim N (0, I_{2}), ν = 0.8,$
unlabeled · line 258
$μ_{0} = (- 3, - 3), μ_{1} = (3, 3), μ_{2} = (- 3, 3), μ_{3} = (3, - 3) .$
unlabeled · line 263
$π (0) = 2, π (1) = 3, π (2) = 0, π (3) = 1.$
unlabeled · line 276
$c_{v} = (1, 0), c_{a} = (0, 1), c_{α} = α c_{v} + (1 - α) c_{a} .$