統計解析R たぶんpart3くらい (587ﾚｽ)
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302: デフォルトの名無しさん [] 2019/09/21(土) 00:43:04.47 ID:MLPk5U2Q(1/4) AAS
## Zigzag on Rails

[ブログ](http://blog.mirkoklukas.com/finite-sample-expressivity/)の内容を
実装してみる。

``` {r}
zigzag_on_rails = function (xi, yi) {
js = order (xi); xi = xi [js]; yi = yi [js]; y0 = yi [1];
ai = diff (c (0, diff (yi) / diff (xi), 0));
f = function (x) {
purrr::reduce (.init = y0, .x = seq_along (xi), .f = function (out, j) {
out + ai [j] * prelu (x - xi [j]);
});
};
big_data $ add (big_data, xi = xi, yi = yi, y0 = y0, ai = ai, zigzag = f);
};

dog_data = with (dog_data, {
yi = c (2, 0, - 1, 1); rails = zigzag_on_rails (xi, yi);
x = seq (min (xi) - 1, max (xi) + 1, len = 1e+3);
plot (x, rails $ zigzag (x), type = 'l');
points (xi, yi); add (dog_data, yi = yi);
});
```

観測値からの外延の仕方がブログとは異なると思うが、こんな感じじゃないかと
思う。[区分線形関数](https://en.wikipedia.org/wiki/Piecewise_linear_function)
を式にすると、自然とReLUの和が現れる。折れ線の変化`ai`を正と負の成分に
分けると、それぞれが凸 on Railsに乗る。
続く

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303: デフォルトの名無しさん [] 2019/09/21(土) 00:44:03.23 ID:MLPk5U2Q(2/4) AAS
``` {r}
convex_on_rails = function (xi, ai) {
js = order (xi); xi = xi [js]; ai = ai [js]; bi = cumsum (ai);
ci = cumsum (ai * xi); f = function (j, x) bi [j] * x - ci [j];
d_f = function (j, x) ai [j] * (x - xi [j]);
i_f = function (j, x) (x + ci [j]) / bi [j];
convex = function (x) { js = seq_along (xi);
purrr::reduce (.init = 0, .x = js, .f = function (out, j) {
pmax (out, f (j, x)); }); };
resnet = function (x) { js = rev (seq_along (xi));
purrr::reduce (.init = f (js [1], x), .x = js, .f = function (out, j) {
out - plure (d_f (j, i_f (j, out))); }); };
list (convex = convex, resnet = resnet);
};

resnet_on_rails = function (xi, yi) {
out = zigzag_on_rails (xi, yi); with (out, {
xi = out $ xi; ai = out $ ai; js = which (ai > 0);
fp = convex_on_rails (xi [js], ai [js]); js = which (ai < 0);
fm = convex_on_rails (xi [js], - ai [js]);
convex = function (x) y0 + fp $ convex (x) - fm $ convex (x);
resnet = function (x) y0 + fp $ resnet (x) - fm $ resnet (x);
add (out, convex = convex, resnet = resnet);
});
};
```

続く

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304: デフォルトの名無しさん [] 2019/09/21(土) 00:44:40.99 ID:MLPk5U2Q(3/4) AAS
``` {r}
with (dog_data, {
rails = resnet_on_rails (xi, yi);
x = seq (min (xi) - 1, max (xi) + 1, len = 1e+3);
plot (x, rails $ resnet (x), type = 'l');
points (xi, yi);
doit = function (xi, yi) {
rails = resnet_on_rails (xi, yi);
x = seq (min (xi) - 1, max (xi) + 1, len = 1e+3);
equal (rails $ zigzag (xi), rails $ convex (xi));
equal (rails $ zigzag (xi), rails $ resnet (xi));
};
doit (xi, yi);
n = 10;
doit (c (rnorm (n, 0, 1), rnorm (n, 2, 3)), c (rnorm (n, 0, 2), rnorm (n, 0, 1)));
});
```

区分線形関数は二本のResNet with ReLUの
[アフィン写像](https://en.wikipedia.org/wiki/Affine_transformation)で
書けたことになる。
おしまい

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305: デフォルトの名無しさん [] 2019/09/21(土) 00:50:33.16 ID:MLPk5U2Q(4/4) AAS
と思ったけどバグを見つけた
誤:
equal (rails $ zigzag (xi), rails $ convex (xi));
equal (rails $ zigzag (xi), rails $ resnet (xi));
正:
equal (rails $ zigzag (x), rails $ convex (x));
equal (rails $ zigzag (x), rails $ resnet (x));

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