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text:
\section{Reduction of rational maps}
\label{sec:reduc}

Given a polynomial $f(z)\in {\mathcal O}_{\overline{K}}[z]$, denote by
$\overline{f}(z)\in \overline{k}[z]$ the polynomial formed by
reducing all coefficients of $f$ modulo ${\mathcal M}_{\overline{K}}$.

\begin{defin}
\label{def:goodred}
Let $\phi(z)\in K(z)$ be a rational function
of degree $d\geq 1$.  Write $\phi=f/g$ with
$f,g\in{\mathcal O}_K[z]$ and with at least one coefficient
of $f$ or $g$ having absolute value $1$.
Let $\overline{\phi}:=\overline{f}/\overline{g}$.
We say that $\phi$ has {\em good reduction}
if $\deg \overline{\phi} = \deg \phi$.
Otherwise,
we say that $\phi$ has \emph{bad reduction}.

We say that $\phi$ has \emph{potentially good reduction}
if there is some $h\in\PGL(2,\overline{K})$ such that
$h\circ\phi\circ h^{-1}\in\overline{K}(z)$ has good reduction.
\end{defin}

There is some inconsistency in the literature as to whether
``bad reduction'' of a dynamical system $\phi$ should mean
the lack of good reduction, or, more restrictively, the lack
of good reduction even after a $K$-rational change of coordinates.
For example, if $a\in K$ is a uniformizer, then the map
$\phi(z) = az^2$ has bad reduction as written, but its
$K$-rational conjugate $a\phi(a^{-1}z)=z^2$ has good reduction.
Meanwhile, the map $\psi(z)=az^3$ also has bad reduction,
and in fact it can be shown to have bad reduction even after
any $K$-rational coordinate change.  However, $\psi$ has
potentially good reduction, because
$a^{1/2}\psi(a^{-1/2}z) = z^3$ has good reduction.

Given $\phi\in K(z)$, it is easy to check that polynomials
$f,g\in{\mathcal O}_K[z]$ exist as specified in Definition~\ref{def:goodred},
and that the reduction type of $\phi$ does not change if
two other such polynomials
$\tilde{f},\tilde{g}\in{\mathcal O}_K[z]$ are chosen in their stead.
Note that $\deg\overline{\phi}\leq \deg\phi$, but
the degree could drop if the formerly relatively
prime $f,g\in {\mathcal O}_K[z]$ gain a common factor when
they are reduced, or if
$\max\{\deg\overline{f},\deg\overline{g}\}$
is strictly smaller than $\max\{\deg f,\deg g\}$.

The reduction map $\red:{\mathcal O}_{\overline{K}}\to \overline{k}$
induces a map $\red:\PP^1(\overline{K})\to\PP^1(\overline{k})$, which coincides with the
original reduction map on ${\mathcal O}_{\overline{K}}$ and maps
$\PP^1(\overline{K})\smallsetminus{\mathcal O}_{\overline{K}}$ to the point
$\overline{\infty}\in\PP^1(\overline{k})$.  It is easy to check that
a rational function $\phi\in\overline{K}(z)$ has good reduction
if and only if it respects the reduction map, i.e., if
$$\overline{\phi}\big(\bar{x}\big) = \overline{\phi(x)}
\quad\text{for all}\quad x\in\PP^1(\overline{K}).$$
As a result, the composition of two maps of good reduction
again has good reduction.
In addition, among maps of degree~1, i.e., in $\PGL(2,\overline{K})$,
the good reduction maps are precisely the elements of
$\PGL(2,{\mathcal O}_{\overline{K}})$.

\begin{lemma}
\label{lem:multabs}
Let $\psi\in\overline{K}(z)$ have good reduction, and write $\psi=f/g$
with $f,g\in {\mathcal O}_{\overline{K}}[z]$ as in Definition~\ref{def:goodred}.
Suppose that $0$ is a fixed point of $\psi$,
and let $\lambda$ be the associated multiplier.
Then $f(0)=0$, $|g(0)|_v = 1$, and $|\lambda|_v = |f'(0)|_v$.
\end{lemma}

\begin{proof}
Since $f$ and $g$ are relatively prime, we have
$f(0)=0$ and $g(0)\neq 0$.  However, because
$\bar{f},\bar{g}\in{\mathcal O}_{\overline{K}}[z]$ are also relatively prime,
we must in fact have $|g(0)|_v=1$.  Thus,
\[
|\lambda|_v = \Big|\frac{f'(0)g(0) - f(0)g'(0)}{g(0)^2}\Big|_v
= \frac{|f'(0) g(0)|_v}{|g(0)|_v^2} = |f'(0)|_v .
\qedhere
\]
\end{proof}

\section{Proof of the Theorem}
\label{sec:proof}

The following lemmas are well known in the more general setting
of dynamics on a non-archimedean disk, but their proofs in our
context are very short and simple, and so we include
them for the convenience of the reader.

\begin{lemma}
\label{lem:norepgood}
Let $\psi\in \overline{K}(z)$ be a rational function
of good reduction, and let $x\in\PP^1(\overline{K})$ be a fixed point.
Then $x$ is indifferent or attracting.
\end{lemma}

\begin{proof}
There is some degree one map $h\in\PGL(2,{\mathcal O}_{\overline{K}})$
of good reduction such that $h(x)=0$.  Since conjugating by $h$
preserves good reduction and multipliers, we may assume without
loss that $x=0$.
Let $\lambda:=\psi'(0)$ be the associated multiplier.

Write $\psi=f/g$ as in Definition~\ref{def:goodred}.
Then $f'\in{\mathcal O}_{\overline{K}}[z]$, since $f\in{\mathcal O}_{\overline{K}}[z]$.
Thus, by Lemma~\ref{lem:multabs}, we have
$|\lambda|_v = |f'(0)|_v\leq 1$.
\end{proof}

\begin{lemma}
\label{lem:attrgood}
Let $\psi\in\overline{K}(z)$ be a rational function
of good reduction, and let $x\in\PP^1(\overline{K})$ be an attracting fixed point
of $\psi$.
Then for any other fixed point $y\in\PP^1(\overline{K})$ of $\psi$,
the reductions $\red(x),\red(y)\in\PP^1(\overline{k})$ are distinct.
\end{lemma}

\begin{proof}
By a $\PGL(2,{\mathcal O}_{\overline{K}})$ change of coordinates, we may assume that
$x=0$.  Write $\psi=f/g$ as in Definition~\ref{def:goodred},
and let $\lambda:=\psi'(0)$ be the associated multiplier.
By Lemma~\ref{lem:multabs}, we have $f(0)=0$,
$|g(0)|_v=1$, and $|f'(0)|_v=|\lambda|_v<1$.  That is,
$f(z) = a_1 z + \cdots + a_d z^d$ with $|a_i|_v\leq 1$
and $|a_1|_v<1$, whereas
$g(z) = b_0 + b_1 z + \cdots + b_d z^d$ with $|b_i|_v\leq 1$
and $|b_0|_v=1$.

For any $y\in\PP^1(\overline{K})$ with $\red(y)=\red(0)$, we have $|y|_v<1$.  Thus,
if $y\neq 0$, we have
$$|\psi(y)|_v
= |y|_v\cdot \frac{|a_1 + \cdots + a_d y^{d-1}|_v}{|b_0 + \cdots + b_d y^d|_v}
= |y|_v\cdot \frac{|a_1 + \cdots + a_d y^{d-1}|_v}{|b_0|_v}
< |y|_v,$$
and hence $y$ cannot be fixed.
\end{proof}

\begin{lemma}
\label{lem:indiffgood}
Let $\psi\in\overline{K}(z)$ be a rational function
of good reduction and degree $d\geq 2$,
and let $x\in\PP^1(\overline{K})$ be an indifferent fixed point
of $\psi$.  Then $\psi^{-1}(x)\smallsetminus\{x\}$
is nonempty, and $\red(x)\neq \red(y)$
for any $y\in\psi^{-1}(x)\smallsetminus\{x\}$.
\end{lemma}

\begin{proof}
By a $\PGL(2,{\mathcal O}_{\overline{K}})$ change of coordinates, we may assume that
$x=0$.  Write $\psi=f/g$ as in Definition~\ref{def:goodred},
and let $\lambda:=\psi'(0)$ be the associated multiplier.
By Lemma~\ref{lem:multabs}, we have $f(0)=0$,
$|g(0)|_v=1$, and $|f'(0)|_v=|\lambda|_v=1$.  Thus,
$f(z) = a_1 z + \cdots + a_d z^d$ with $|a_i|_v\leq 1$
and $|a_1|_v=1$.
It is immediate that $f$, and hence $\psi$, has no roots in
${\mathcal M}_{\overline{K}}\smallsetminus\{0\}$.

Meanwhile, the hypothesis on $\psi'(0)$ also implies that
$0$ is not a critical point.
Since $d\geq 2$, $\psi^{-1}(0)\smallsetminus\{0\}$ is nonempty.
Given $y\in\psi^{-1}(0)\smallsetminus\{0\}$,
the previous paragraph shows that $y\not\in{\mathcal M}_{\overline{K}}$,
and hence $\red(y)\neq\red(0)$.
\end{proof}

\begin{remark}
Lemmas~\ref{lem:attrgood} and~\ref{lem:indiffgood} are special
cases more general statements about fixed points of a map
$\phi\in K(z)$ of good reduction.  Specifically, if $x$ is
an attracting fixed point (or more generally, periodic point)
of such a map, then the residue class of $x$ contains no other
periodic points, although it almost certainly contains many
other \emph{preperiodic} points that ultimately map to $x$
itself.  Similarly, if $x$ is an indifferent periodic point
of a map of good reduction, then the residue class of
$x$ maps bijectively onto itself, and hence the only preperiodic
points in the residue class are in fact periodic.
(In fact, there are usually infinitely many periodic points in such a
residue class.)

Furthermore, these facts about preperiodic points near nonrepelling periodic
points were generalized by Rivera-Letelier in his thesis \cite{Riv1}
to maps of bad reduction, by replacing residue
classes by an appropriate notion of Fatou components.
However, we will only need the far simpler statements of
Lemmas~\ref{lem:attrgood} and~\ref{lem:indiffgood} here.
\end{remark}

\begin{proof}[Proof of the Theorem]
For each $i=1,\ldots,d+1$, let $\lambda_i\in\overline{K}$ be the
multiplier of the fixed point $x_i$.
If $\phi$ has potentially good reduction, then there is some
$h\in\PGL(2,\overline{K})$ such that $h\circ\phi\circ h^{-1}$
has good reduction.  The fixed points of
$h\circ\phi\circ h^{-1}$ have the same multipliers as those
of $\phi$, and therefore, by Lemma~\ref{lem:norepgood},
we have $|\lambda_i|_v \leq 1$ for all $i$, proving the
first statement of the Theorem.
We now consider two cases.

\textbf{Case 1}.
If $|\lambda_i|_v=1$ for some $i$, then the associated fixed point
$x_i$ is not a critical point (since $\lambda_i\neq 0$), and hence
$\phi^{-1}(x_i)\smallsetminus\{x_i\}$ is nonempty.  Pick
$y_1\in \phi^{-1}(x_i)\smallsetminus\{x_i\}$, and pick
$y_2\in \phi^{-1}(y_1)$.  Then $y_1\neq x_i$ by definition,
and hence $y_2\neq y_1$ also.  Moreover, $y_2\neq x_i$, because
if $y_2=x_i$, then taking $\phi$ of both gives $y_1=x_i$.
Let $h$ be the unique map in $\PGL(2,\overline{K})$ with
$h(x_i)=0$, $h(y_1)=1$, and $h(y_2)=\infty$,
and let $\psi:=h\circ\phi\circ h^{-1}$
If $\psi$ has good reduction, then $\phi$ has
potentially good reduction, and we are done.

Conversely, if $\phi$ has potentially good reduction,
there is some $\tilde{h}\in\PGL(2,\overline{K})$ such that
$\tilde{\psi}:=\tilde{h} \circ\phi\circ\tilde{h}^{-1}$
has good reduction.
By Lemma~\ref{lem:indiffgood}, the indifferent fixed point
$\tilde{h}(x_i)$ of $\tilde{\psi}$ and its preimage
$\tilde{h}(y_1)$ (under $\tilde{\psi}$) have different reductions
in $\PP^1(\overline{k})$.  Since $\tilde{\psi}$ respects reduction,
we must have $\bar{\tilde{\psi}}(\overline{\tilde{h}(y_2)})
= \overline{\tilde{h}(y_1)}$, and for the same reasons as in the previous
paragraph, the reductions
$\tilde{h}(x_i)$, $\tilde{h}(y_1)$, and $\tilde{h}(y_2)$
must all be distinct points in $\PP^1(\overline{k})$.  Thus, there is
some map $r\in\PGL(2,{\mathcal O}_{\overline{K}})$ such that
$$r\big(\tilde{h}(x_i)\big) = 0,
\quad
r\big(\tilde{h}(y_1)\big) = 1,
\quad\text{and}\quad
r\big(\tilde{h}(y_2)\big)=\infty.$$
Then we must have $r\circ \tilde{h}=h$,
and $\psi=r\circ\tilde{\psi}\circ r^{-1}$.
Since $\tilde{\psi}$, $r$, and $r^{-1}$ all have good
reduction, it follows that $\psi$ also has good reduction.

\textbf{Case 2}.
The remaining case is that no $x_i$ is repelling or indifferent,
and hence they are all attracting.  That is,
$|\lambda_i|_v<1$ for all $i$.  Then no $\lambda_i$
is $1$, and hence the $d+1$ fixed points all
have multiplicity $1$ and therefore are distinct.
In particular, $x_1$, $x_2$, and $x_3$ are all distinct.
Let $h$ be the unique map in $\PGL(2,\overline{K})$ with
$h(x_1)=0$, $h(x_2)=1$, and $h(x_3)=\infty$,
and let $\psi:=h\circ\phi\circ h^{-1}$.
If $\psi$ has good reduction, then $\phi$ has
potentially good reduction, and we are done.

Conversely, if $\phi$ has potentially good reduction,
there is some $\tilde{h}\in\PGL(2,\overline{K})$ such that
$\tilde{\psi}:=\tilde{h} \circ\phi\circ\tilde{h}^{-1}$
has good reduction.
By Lemma~\ref{lem:attrgood}, the attracting fixed points
$\tilde{h}(x_1)$, $\tilde{h}(x_2)$, and $\tilde{h}(x_3)$
of $\tilde{\psi}$ all have different reductions
in $\PP^1(\overline{k})$.  Thus, there is
some map $r\in\PGL(2,{\mathcal O}_{\overline{K}})$ such that
$$r\big(\tilde{h}(x_1)\big) = 0,
\quad
r\big(\tilde{h}(x_2)\big) = 1,
\quad\text{and}\quad
r\big(\tilde{h}(x_3)\big)=\infty.$$
Then we must have $r\circ \tilde{h}=h$,
and $\psi=r\circ\tilde{\psi}\circ r^{-1}$.
Since $\tilde{\psi}$, $r$, and $r^{-1}$ all have good
reduction, it follows that $\psi$ also has good reduction.
\end{proof}

{\bf Acknowledgements.}
The author thanks Xander Faber and Joseph Silverman for their
helpful suggestions.
The author also
gratefully acknowledges the support of NSF grant DMS-1201341.

redpajama_set_name: RedPajamaArXiv


text:
/**
 * @file
 * This file contains main point for Martlet
 */

redpajama_set_name: RedPajamaGithub


Years after Carly Cardellino quit her university sorority, she couldn't leave one piece of it: a teal star as well as a red heart outlined in black, the group's symbol, which she had actually spent $50 to have actually tattooed on her left foot during her student year.
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" The skin is a bit lighter," stated Ms. Cardellino, who associates that partly to her compulsive use sun block in the area. (Tattoos can't be subjected to sun while they're being treated.) "But if I revealed you I had a tattoo there, you 'd resemble, 'I don't think you.'".
Yet it has actually been commonly hailed as the very first significant improvement in tattoo removal in Twenty Years. That's because, compared to the old lasers, which worked merely in billionths of seconds, medical professionals state the picosecond both cuts therapy time in fifty percent as well as can remove shades of ink (including reds, blues as well as greens) that formerly hardly budged. A little study published in the journal Dermatology revealed that two-thirds of tattoos with blue as well as green pigments virtually vanished after a couple of therapies with a picosecond laser.
Advocates prefer to explain the distinction in how carefully the new lasers smash ink as the difference between pebbles and sand. (Researchers are already at work on a femtosecond laser, which would pulse at a quadrillionth of a second, which in this example, most likely would pulverize the ink to silt.).
" It's a truly substantial advancement," stated Dr. Roy Geronemus, a dermatology teacher at NYU Langone Medical Center, who has actually worked with lasers since 1983 and carried out several of the first picosecond studies. (Dr. Geronemus gets on the clinical advisory board of the company that makes the laser.).
As with all aesthetic therapies, there is some component of marketing hype. Tattoo removal is a roughly $75-million-a-year company– mostly accommodating young experts that believe tattoos are preventing their rise, mothers that determine the art no longer fits their picture and also tattoo fanatics who just want to redecorate.
There is likewise a random sample of heartbreak and also hopes rushed. Dr. Bruce Baker, a dermatologist in Manhattan, has actually twice gotten rid of the same lady's name from the very same male's buttocks. You link the dots.
However the variety of treatments done in the United States has actually declined sharply in recent years, to 33,363 in 2014 from 58,429 in 2012, one of the most current year for which the American Society for Aesthetic Cosmetic surgery has numbers.
Some doctors connect the drop to the brand-new laser's capacity to reduce therapies per person (each treatment counts as a procedure, in the culture's bookkeeping), but Dr. James Grotting, the culture's head of state, put a less delighted spin on it.
" The numbers are decreasing because tattoo-removal treatments haven't provided naturally good results," Dr. Grotting wrote in an email. He called tattoos "still an unsolved issue" because of the still-present opportunity of scarring and the large variant in how various shades and types of ink react to lasers. Pink ink, for instance, frequently contains iron oxide, which means it could turn black under a laser, a less-than-desirable outcome, specifically if one has, for example, tattooed pink lip liner.
No one challenges that having a tattoo eliminated is much more pricey and time-consuming compared to having one put on in the first place, and the cost is seldom, if ever, covered by clinical insurance policy. Only one lady spoke with reported a tattoo coming close to "cleaning," as it's called, in a treatment or two, and also hers was a little brownish Zodiac indicator on her appropriate ring finger. (Price and also initial outcomes vary extensively by the size of the tattoo and also its place– leg as well as feet tattoos are slower to go away, likely due to the fact that blood flow is less.).
After a touch of the laser, the skin crusts instantly. Most doctors offer numbing cream as well as lidocaine shots, which implies that by the time the laser hits (with an ominous-sounding breeze), the worst component is over.
Mr. Schratter happily spent five hours under the needle having a redwood tree tattooed from his ideal knee to his groin last year, he is afraid shots. "Ironic, I know," he stated. For his first consultation, his anxiousness slowed down the shot of the 10 lidocaine shots he needed throughout a two-and-a-half-hour challenge. Real laser time: seven minutes.
Mr. Schratter, that has 7 other tattoos and strategies to change the redwood with another thing, was much more unwinded for his second consultation, which took only about 45 mins. Still, he joked, "deforesting is hard.".
Lots of states don't mandate that a medical professional execute laser treatments. You might desire one.
Fundamentally, laser tattoo removal is touching a wand to a spot of skin. The treatment is managed by states, a number of which enable laser drivers to function after just 16 hrs of hands-on training.
Consider this, published in boldface kind, from the website of the laser business Astanza: "Entrepreneurs with no clinical history in over 45 states operate highly profitable laser tattoo removal businesses on a daily basis.".
The company takes place to explain that even if a doctor is required to serve as medical supervisor, "this role usually requires just periodic check-ins.".
New york city State calls for that a doctor manage laser therapies, yet especially adds that this "shall not be interpreted as necessarily requiring the physical existence of the overseeing doctor at the time and also location where such services are done." New Jacket is more stringent: physicians only.
Physicians normally bill at the very least two times as high as laser clinics, so why would you spend for an M.D.?
" You can teach a monkey to press a button," said Dr. S. Tyler Hollmig, assistant teacher of dermatologic surgical treatment at Stanford University. "It's judgment. That's why you want a physician.".
Every doctor has a tale of an individual who can be found in after a botched removal treatment. Treating a tattoo properly consists of comprehending the biology of the skin around it. There is no harmony among tattoo ink.
And also an individual may want a person who can evaluate the color of the skin (darker skin is harder), the background of the tattoo (has it been treated prior to?) and also its age (older tattoos are less complicated, since the body gradually eliminates the ink, which could have faded anyway).
Tattoos should go to least six months old to be removed, due to the fact that the inflammatory feedback has to stop, or elimination treatment could make it worse.
As well as it could be less painful to head to a doctor. Also if your state doesn't think about running a laser to be the practice of medicine, administering a shot of lidocaine typically is.


text:
Q: What can I use to get the paths of all parent directories for a given path I'm looking for a tool that I can use from a bash script that can give me the list of parent directories for a given path.
e.g. given the input foo/bar/moocow I would like to get out:
foo
foo/bar
foo/bar/moocow

It would also be great if I could pipe in multiple paths, and get unique results back, e.g.:
toolimlookingfor << EOF
dir1/file1
dir1/file2
foo/bar/moocow
EOF

Output:
dir1
dir1/file1
dir1/file2
foo
foo/bar
foo/bar/moocow

dirname is close to what I'm looking for, but it only gives the immediate parent. I'm looking for the path it's self and all parents.

A: A Python script using the standard pathlib module, working over arguments instead of input:
#! /usr/bin/env python3

import sys

from collections import OrderedDict
from pathlib import Path

paths = OrderedDict()  # to maintain ordering and skip duplicates

for arg in sys.argv[1:]:
    path = Path(arg)
    for subpath in reversed(path.parents):
        # add the parents
        paths[subpath.as_posix()] = subpath
    # add the path itself
    paths[path.as_posix()] = path

# we don't need '.' in the output
if '.' in paths:
    paths.pop('.')

print('\n'.join(paths.keys()))

(One could easily modify it to use stdin, by looping over sys.stdin instead.)
Example:
% ./foo.py dir1/file1 dir1/file2 foo/bar/moocow
dir1
dir1/file1
dir1/file2
foo
foo/bar
foo/bar/moocow

A: If you are okay with creating new directory in /tmp folder then, you can use find command, this method will also take care of print only unique directory names.
#!/bin/bash
if [[ $# -eq 0 ]]; then exit; fi
mkdir -p /tmp/printnames
for i
do
  mkdir -p /tmp/printnames/"$i"
done
cd /tmp/printnames
find *  -type d -print
rm -r /tmp/printnames

So you can create a temporary directory structure and then traverse though it using find.

A: This code will do it. It uses no temporary files. It works with white space, however newlines are ambiguous( change \n to \0 to fix). You should use \0 if it is to be used in automated processes. \0 is the only character that is guaranteed not to be in a Unix file name.
It just recursively uses dirnames. You can tweak it by removing/enabling some of the printfs. To make it ascend or descend, or to not show the root node (. or /). The showing of the root node was not in your example, but I think it is needed. Without it /a/b/c gives the same output as a/b/c.
#!/bin/bash

function dirnames {
    local input
    local parent
    input="$1"
    if [[ "$input" == "" ]]
    then
        true
    elif [[ "$input" == "." ||  "$input" == "/" ]]
    then
        printf '%s\n' "$input" #print the root node
    else
        #printf '%s\n' "$input"  #print node (descending)
        parent="$(dirname "$input")"
        dirnames "$parent"
        printf '%s\n' "$input" #print node (ascending)
    fi
}

for d in "$@"
do
    dirnames "$d"
done

Example usage
%↳ ./dirnames a/b/c
.
a
a/b
a/b/c
%↳ ./dirnames /a/b/c
/
/a
/a/b
/a/b/c
%↳

To get it to work like this
%↳ ./dirnames a/b/c
.
a
b
c
%↳ ./dirnames /a/b/c
/
a
b
c
%↳

then save file as dirnames then run sed -r -i  '17 s/("[$]input")/"$(basename \1)"/' dirnames. Now you will have a script that only outputs the leafs at each level.

A: Given that you are going to treat paths as text, with the implicit assumptions that they won't contain newline characters and that they don't need to be canonicalized, you may pipe them to this script:
#!/bin/awk -f
BEGIN {
  FS="/"
}
{
  f = ""
  for (i=1; i<=NF; i++) {
    f = f (i>1 ? FS : "") $i
    if (! p[f]++ && f)
      print f
  }
}

Every input line is split on /. Fields are then concatenated, first to last, separated by a /, and each intermediate result (fragment) is saved as a value of the index of an associative array (paths). Each non-empty1 fragment that has not been encountered yet is printed.
Note that this will consider foo, /foo, foo/, /foo/ and even /foo// to be distinct paths and it will print out all of them separately.
1 The first one in absolute paths, which in AWK have an empty field before the first /

A: if a trailing slash (or other separator) is fine for you, like in my case, the following solution was the easiest and most transparent one for me:
pth='one/two/three'
sep='/'

# split string into array by separator
arr=($(echo $pth | tr $sep '\n'))

# loop over array and append
sub=''
for index in ${!arr[@]}
do
    sub+="${arr[index]}$sep"
    printf "$index \t ${arr[index]} \t $sub \n"
done

Output:
0        one     one/
1        two     one/two/
2        three   one/two/three/

A: If you don't mind a sorted output, you could obtain a good result with a pipeline using a sed script plus sort like this:
$ sed -nE 's%/+(/[^/]|$)%\1%g;:loop;\%(^[^/]+|/[^/]*)$%{p;s///};tloop' | LC_ALL=C sort -u

The above Sed script goes to the extent of coalescing adjacent / characters (the very first s command up to the first ; does that) but apart from that it has no knowledge of relative paths etc.
Note that the script is POSIX compliant although on some systems you might need to type it in multiline, for instance like this:
$ sed -nE 's%/+(/[^/]|$)%\1%g
:loop
\%(^[^/]+|/[^/]*)$%{
p;s///
}
tloop
' | LC_ALL=C sort -u

Note also that it cannot handle newlines in filenames. However, on enhanced tools that support -z option you can just add it to both sed and sort to obtain nul-delimited I/O.
I also had fun in making a script capable of full canonicalization, including any arbitrary . and .. occurrences as well as unqualified paths like dir1/file1 prepended by a leading ./. However, it does not resolve symlinks as it does not access the filesystem at all.
I used Awk for its useful builtin machineries to split and rebuild strings.
#!/usr/bin/awk -f

BEGIN { FS="/+"; OFS="/" }
{
    for (nfields=dotdot=0; NF; NF--) {
        if (path[nfields]=$NF) {
            if ($NF=="..") dotdot++
            else if ($NF!=".") {
                if (dotdot) dotdot--
                else nfields++
            }
        }
    }
    if (nfields in path && !path[nfields])
        $1=""
    else {
        while (dotdot--) path[nfields++]=".."
        path[nfields++]="."
    }
    while (nfields--) {
        $(NF+1) = path[nfields]
        if (!seen[$0]++) print $0
    }
}

You may handle newlines in filenames by invoking it with -v RS='\0' -v ORS='\0' on Awk implementations (such as GNU Awk) that support a nul RS.
HTH

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text:
Reagan Revolution
Let's Keep the Reagan Revolution Alive
Defining the Reagan Revolution
Key Reagan Speeches
Reagan Bookshelf
American Energy Independence
Congressional and Government Reform
Constitutional Supremacy and American Exceptionalism
Benghazi Terror Attack
Missile Defense and New Strategic Arms Reduction Treaty START
Free Trade and Globalization
Brexit: Britain Votes to Leave European Union
Immigration and Border Security
Assimilation not Multiculturalism
Interior Enforcement
Protections for Private Property
Kelo: An Examination
Agenda 21 Earth Summit
Climate Change (UNFCCC)
Law of the Sea Treaty (LOST)
Model UN and UNA-USA
Issues Library
Shopping Opportunities
Let's Keep the Reagan Revolution Alive!
Peace Through Strength
Individual Freedom
An "Informed" Patriotism
10/27/64, "A Time For Choosing", Televised Campaign Address for Goldwater Presidential Campaign
8/19/76, Ronald Reagan's Remarks at the Republican National Convention, Kansas City, MO
2/6/77, The New Republican Party, Conservative Political Action Conference, Washington, DC
1/20/81, First Inaugural Address, U.S. Capitol
4/28/81, Address on the Program for Economic Recovery, Joint Session of Congress
6/8/82, Address to the British Parliament, Royal Gallery at the Palace of Westminster, London
3/8/83, Address to the National Association of Evangelicals ("Evil Empire" Speech), Sheraton Twin Towers Hotel, Orlando, FL
3/23/83, Address to the Nation on National Security ("Star Wars" SDI Speech), Oval Office
6/6/84, Address at the U.S. Ranger Monument Commemorating the 40th Anniversary of D-Day, Pointe Du Hoc, Normandy, France
8/23/84, Presidential Nomination Acceptance Address, Republican National Convention, Dallas, Texas
1/21/85, Second Inaugural Address, U.S. Capitol
1/28/86, Address to the Nation on the Explosion of the Space Shuttle "Challenger", Oval Office
7/3/86, Address at the Statue of Liberty Centennial Celebration, Governors Island, New York Harbor
9/26/86, Remarks at the Swearing-in Ceremony for William H. Rehnquist as Chief Justice and Antonin Scalia as Associate Justice of the Supreme Court of the United States, East Room of the White House
6/12/87, Address at the Brandenburg Gate (Berlin Wall), Berlin
7/3/87, Fourth of July Address on America's Economic Bill of Rights, Jefferson Memorial, Washington DC
5/31/88, Address at Moscow State University, Moscow, USSR
1/11/89, Farewell Address to the Nation, Oval Office
11/5/94, Farewell Letter to the American People
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Come read along on our rather logical fairy tale journey!
Also, make sure you don't miss reading the excerpt below!!
I think it could be loads of fun.
This looks like a fun fairy tale- the math angle sounds interesting!
It isn't the cleanest, but I did get sucked into the story and the humor was great.


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Tools & Resources / Publications & References / Publications / A high-throughput drug screening strategy for detecting rhodopsin P23H mutant rescue and degradation.
A high-throughput drug screening strategy for detecting rhodopsin P23H mutant rescue and degradation.
Authors: Chen Y, Tang H, Seibel W, Papoian R, Li X, Lambert NA, Palczewski K.
Publisher/Year: Invest Ophthalmol Vis Sci. 2015 Apr;56(4):2553-67.
Inherent instability of the P23H mutant opsin accounts for approximately 10% of autosomal dominant retinitis pigmentosa cases. Our purpose was to develop an overall set of reliable screening strategies to assess if either stabilization or enhanced degradation of mutant rhodopsin could rescue rod photoreceptors expressing this mutant protein. These strategies promise to reveal active compounds and clarify molecular mechanisms of biologically important processes, such as inhibition of target degradation or enhanced target folding.
Cell-based bioluminescence reporter assays were developed and validated for high-throughput screening (HTS) of compounds that promote either stabilization or degradation of P23H mutant opsin. Such assays were further complemented by immunoblotting and image-based analyses.
Two stabilization assays of P23H mutant opsin were developed and validated, one based on β-galactosidase complementarity and a second assay involving bioluminescence resonance energy transfer (BRET) technology. Moreover, two additional assays evaluating mutant protein degradation also were employed, one based on the disappearance of luminescence and another employing the ALPHA immunoassay. Imaging of cells revealed the cellular localization of mutant rhodopsin, whereas immunoblots identified changes in the aggregation and glycosylation of P23H mutant opsin.
Our findings indicate that these initial HTS and following assays can identify active therapeutic compounds, even for difficult targets such as mutant rhodopsin. The assays are readily scalable and their function has been proven with model compounds. High-throughput screening, supported by automated imaging and classic immunoassays, can further characterize multiple steps and pathways in the biosynthesis and degradation of this essential visual system protein.


text:
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ERA Home About ERA Past Editorial Board
About Electronic Research Announcements
This journal is currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences
Journal overview:
This electronic-only journal was published by the American Mathematical Society from 1995 to 2007 and is currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences. As published by the AMS, a research announcement was designed to communicate its contents to a broad mathematical audience.
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Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.
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According to a recent Nielsen's study published in February of this year, the average American spends just over 34 hours each month on a mobile device. The continued increase in usage of mobile devices has led many businesses to offer public Wi-Fi to customers. But, what do you need to know before hopping on a public Wi-Fi network?
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Geology Chert Strata overlying The Great Limestone (Ref: Geology of the Country around Barnard Castle. Memoirs of the Geological Survey of Great Britain. 1976. HMSO.
Record Name Aspen Grove below Hugill Force, Gilmonby CP.
Record Description Cloned grove of approximately 15 aspen trees including multi stem trees and fallen trees, on chert strata on the eastern side of the steep sided ravine immediately below Hugill Force.
Common / Notable Species Rowan the only other tree species on the chert strata here. This is in contrast to the species-rich woodland in the Lower Ravine which is on the underlying Great Limestone.
Image 1 Description Aspen Grove on chert cliff strata at top of the ravine below Hugill Force.
Image 2 Description Aspen Grove on chert cliff strata at top of the ravine below Hugill Force.
Image 3 Description The aspens, detail with fallen tree.
Image 4 Description Hugill Force with rowans the only tree on chert strata.


text:
Ervín Kováč (16 June 1911 – 30 October 1972) was a footballer who played international football for both Czechoslovakia and Slovakia. He played as a defender for Slovan Bratislava.

References

1911 births
1972 deaths
Czechoslovak footballers
Czechoslovakia international footballers
Slovak footballers
Slovakia international footballers
Dual internationalists (football)
ŠK Slovan Bratislava players
Association football defenders

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text:
\section{Introduction}

Out-of-Distribution (OoD) detection is an important machine learning
problem that finds instances that do not belong to training classes~\cite{liang2018enhancing}.
Deep learning models tend to make incorrect predictions
with high confidence for instances from unseen
classes~\cite{szegedy2014intriguing,moosavi2017universal}.
By OoD detection, we can safely deploy machine learning models
in the open world, where new classes can emerge.
OoD detection can also be beneficial for
ensuring the quality of the collected data,
and finding instances with unusual behavior for data analyists~\cite{yu2019unsupervised}.
Many neural network-based OoD detection methods have been proposed~\cite{morningstar2021density,ren2019likelihood,devries2018learning,winkens2020contrastive,liu2020energy,chen2020robust,xiao2020likelihood,feng2021improving}.
However, these methods require a large number of data for training.
In some real-world applications, enough data might be unavailable
in target tasks.
For example, it is important to detect manufacturing failures
that are not categorized in existing failure classes in each factory
for improving productivity,
but many data are not accumulated in newly operated factories.

We propose a simple yet effective method
for improving the OoD detection performance on unseen target tasks
by meta-learning from data in tasks different from the target tasks.
Even when enough data are unavailable for a newly operated factory,
many data for different factories would be available.
For the meta-training data,
instances with class labels from various tasks are assumed to be given,
where classes are different across tasks.
For a target task,
a small number of instances with class labels
are given as in-distribution (ID) data.
We want to identify whether test instances
are ID or OoD for the target task.

With the proposed method,
OoD scores are calculated by density estimation in a latent space.
Density estimation has been used for OoD detection~\cite{bishop1994novelty}.
However, ID data can have low likelihoods
in extremely high dimensions, and density estimation
in the original high dimensional space sometimes fails
to detect OoD~\cite{choi2018waic,nalisnick2018deep,hendrycks2018deep}.
To avoid such problems,
the proposed method embeds instances
into a latent space by a neural network that is shared among all tasks.

Meta-learning is formulated as a bilevel optimization problem.
In the inner optimization problem,
task-specific parameters are adapted to the given task-specific data.
In the outer optimization problem,
common parameters shared across all tasks
are meta-learned to improve the expected test performance
when the task-specific parameters
adapted in the inner optimization are used.
With the proposed method,
the inner optimization problem
corresponds to fitting a density model
to the task-specific data in a latent space.
We use a Gaussian mixture model (GMM) for the density model.
The GMM parameters,
which are the class probability, mean, and covariance for each class,
are adapted in a closed form by maximizing the likelihood.
In the outer optimization problem,
we meta-learns the neural network
such that the OoD detection performance improves
when the density is estimated with the adapted GMM parameters.
By incorporating the closed form solution of the inner optimization problem
into the outer optimization problem,
we can solve the bilevel optimization problem efficiently
by a stochastic gradient descent method.

For each meta-learning step,
instances in the meta-training data
are randomly selected for ID,
and instances in classes different from the ID instances
are randomly selected for OoD.
By generating ID and OoD data in this way,
we can evaluate the OoD detection performance using the meta-training data,
which is required for the objective function
in the outer optimization problem.
The performance is evaluated by the area under the ROC curve (AUC),
which is often used in
the existing literature~\cite{liang2018enhancing,wang2019out,yu2019unsupervised}.
Although data in target tasks are labeled with their classes,
only ID instances are given as the task-specific data.
Therefore, the inner optimization needs to be
performed without OoD instances.
The proposed method enables this by
defining the inner optimization as density estimation using ID instances.
Figure~\ref{fig:model} illustrates
a meta-learning step of the proposed method.

\begin{figure}[t!]
  \centering
  \includegraphics[width=21em]{images/model.png}
  \caption{Meta-learning step for the proposed method. For the meta-training data, we are given multiple tasks with different classes. 1) Randomly sample support instances and ID query instances from a task,
    and sample OoD query instances from another task.
    2) Embed sampled instances by a neural network in a latent space.
    3) Estimate task-specific Gaussian mixture model parameters based on the maximum likelihood.
    4) Calculate the density of the query instances using the estimated GMM parameters.
    5) Evaluate the OoD score based on AUC.
    6) Backpropagate the AUC to update the neural network parameters.}
  \label{fig:model}
\end{figure}

The main contributions of this paper are as follows:
\begin{enumerate}
\item We propose a meta-learning method for OoD detection that uses labeled data in various tasks to improve the performance on unseen target tasks that consist of unlabeled data.
\item The proposed method enables an efficient bilevel optimization by modeling the inner optimization with the maximum likelihood of a Gaussian mixture model in a latent space, which gives a closed-form solution for the adaptation to each task.
\item We confirm that the proposed method achieves better performance than existing methods.
\end{enumerate}
The remainder of this paper is organized as follows.
In Section~\ref{sec:related},
we briefly review related work.
In Section~\ref{sec:proposed},
we formulate our problem,
and present our meta-learning method for OoD detection.
In Section~\ref{sec:experiments},
we demonstrate that the proposed method achieves better performance than existing meta-learning and OoD detection methods.
Finally, we present concluding remarks and a discussion of future work
in Section~\ref{sec:conclusion}.

\section{Related work}
\label{sec:related}

Many meta-learning methods have been proposed~\cite{schmidhuber:1987:srl,bengio1991learning,finn2017model,vinyals2016matching,snell2017prototypical,lee2019meta}.
Meta-learning methods learn how to learn from a small amount of
labeled data in various tasks,
and use the learned knowledge in unseen tasks.
However, most of these methods are not designed for OoD detection,
and are inapplicable to our problem.
A meta-learning method for OoD detection
based on model-agnostic meta-learning (MAML)~\cite{finn2017model}
has been proposed~\cite{jeong2020ood}.
For the inner optimization,
MAML requires costly back-propagation through iterative gradient descent steps.
On the other hand, the proposed method achieves an efficient inner optimization
in a closed form based on density estimation with GMMs.
The closed form inner optimization for meta-learning has been successfully used
in various methods~\cite{bertinetto2018meta,iwata2020few,snell2017prototypical,fortuin2019deep,harrison2018meta}.
However, they define the inner optimization problem as
regression or classification, but not as density estimation.
An OoD detection method based on Mahalanobis distance
assumes a GMM in a latent space~\cite{lee2018simple}.
However, it uses a covariance shared among all classes,
which has lower expressive power than class-specific covariances.
In addition,
it requires many training data since it is not a meta-learning method.
Anomaly detection is related to OoD detection.
Anomaly detection methods usually do not use
class labels~\cite{chalapathy2018anomaly,kumagai2019transfer,akcay2018ganomaly,zenati2018efficient,schlegl2017unsupervised},
where class labels indicate categories that ID instances belong to
and they are different from ID/OoD labels.
On the other hand, the proposed method uses class labels
for effectively estimating the density.
The better performance of the proposed method
compared with these existing methods are demonstrated in our experiments
in Section~\ref{sec:experiments}.

\section{Proposed method}
\label{sec:proposed}

\subsection{Problem formulation}

In a meta-training phase,
meta-training data
$\mathcal{D}=\{\{(\vec{x}_{tn},y_{tn})\}_{n=1}^{N_{t}}\}_{t=1}^{T}$ are given,
where $T$ is the number of tasks,
$N_{t}$ is the number of labeled instances in the $t$th task,
$\vec{x}_{tn}\in\mathcal{X}$ is the feature of the $n$th instance,
$y_{tn}\in\{c_{t1},\dots,c_{tK_{t}}\}$ is its class label,
$c_{tk}$ is the $k$th class,
and $K_{t}$ is the number of classes.
In a meta-test phase,
labeled support set $\mathcal{S}=\{(\vec{x}_{n},y_{n})\}_{n=1}^{N_{\mathrm{S}}}$
is given in a target task that are different from the training tasks,
where $y_{n}\in\{c_{1},\cdots,c_{K}\}$,
$K$ is the number of classes in the support set,
and the classes in the target task do not overlap with
those in the training tasks.
Our aim is to identify
whether unlabeled instances $\{\vec{x}\}$
belong to the classes
in the support set $\{c_{1},\cdots,c_{K}\}$ (in-distribution)
or not (out-of-distribution).
We assume that feature space $\mathcal{X}$ is the same across all tasks.

\subsection{Model}

In this subsection,
we present our model to output
task-specific OoD score $u(\vec{x}\mid\mathcal{S};\bm{\Phi})$
of unlabeled instance $\vec{x}$
given supposet set
$\mathcal{S}=\{(\vec{x}_{n},y_{n})\}_{n=1}^{N_{\mathrm{S}}}$,
where $\bm{\Phi}$ is the set of the common model parameters shared across tasks.

The proposed model
uses the following negative log density in a latent space
for the OoD score,
\begin{equation}
  u(\vec{x}\mid\mathcal{S};\bm{\Phi}) = -\log p\left(f(\vec{x};\bm{\phi})\mid\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})\right),
  \label{eq:u}
\end{equation}
where $p$ is the probabilisty density function in the latent space,
$f:\mathcal{X}\rightarrow\mathbb{R}^{D}$
is a neural network that maps an instance from
the original space to the $D$-dimensional latent space,
$\bm{\phi}\in\bm{\Phi}$ is the neural network parameters
that are shared across tasks,
and
$\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})$ is the set of
the task-specific parameters
of the probability density function adapted to support set $\mathcal{S}$.
The negative log likelihood is an natural OoD score
since instances with the low probability density can be seen OoD,
and it has been used for OoD detection~\cite{bishop1994novelty}.
With neural network $f$,
we can find a latent space that is appropriate for detecting OoD
even when the density estimation in the original space
does not perform well on OoD detection.

The probability density function parameters $\bm{\Theta}$
is adapted to support set $\mathcal{S}$ by maximizing the log likelihood,
\begin{align}
  \hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})
  &=\mathop{\rm arg~max}\limits_{\bm{\Theta}}\log p(\mathcal{S}\mid\bm{\Theta};\bm{\Phi})
  \nonumber\\
  &=\mathop{\rm arg~max}\limits_{\bm{\Theta}}\sum_{(\vec{x},y)\in\mathcal{S}}\log p(\vec{x}\mid y,\bm{\Theta};\bm{\Phi}).
  \label{eq:ml}
\end{align}
The proposed model assumes a Gaussian mixture model (GMM)
in the latent space,
\begin{align}
  \lefteqn{p\left(f(\vec{x};\bm{\phi})\mid\bm{\Theta}\right)
    = \sum_{k=1}^{K}\gamma_{k}\mathcal{N}\left(f(\vec{x};\bm{\phi})\mid
    \bm{\mu}_{k},\bm{\Sigma}_{k}\right)}
    \nonumber\\
    &=\sum_{k=1}^{K}\gamma_{k}(2\pi)^{-\frac{D}{2}}\vert\bm{\Sigma}_{k}\vert^{-\frac{1}{2}}
    \exp\left(-\frac{1}{2}(f(\vec{x};\bm{\phi})-\bm{\mu}_{k})^{\top}\bm{\Sigma}_{k}^{-1}(f(\vec{x};\bm{\phi})-\bm{\mu}_{k})\right),
    \label{eq:gmm}
\end{align}
where $\gamma_{k}\in [0,1]$ is the class probability of the $k$th class,
$\mathcal{N}(\cdot\mid\bm{\mu},\bm{\Sigma})$ is the Gaussian distribution
with mean $\bm{\mu}\in\mathbb{R}^{D}$ and covariance $\bm{\Sigma}\in\mathbb{R}^{D\times D}$,
and $\bm{\Theta}=\{\gamma_{k},\bm{\mu}_{k},\bm{\Sigma}_{k}\}_{k=1}^{K}$
is the set of the task-specific GMM parameters.
Since the class label is given for each support instance,
we can obtain the adapted parameters in a closed form using the GMM
by solving Eq.~(\ref{eq:ml}) as follows,
\begin{align}
  \hat{\bm{\gamma}}_{k}(\mathcal{S})
  =\frac{1}{\mid\mathcal{S}_{k}\mid}, \quad
  \hat{\bm{\mu}}_{k}(\mathcal{S})=\frac{1}{\vert\mathcal{S}_{k}\vert}\sum_{\vec{x}\in\mathcal{S}_{k}}f(\vec{x};\bm{\phi}),\nonumber
\end{align}
\begin{align}
  \hat{\bm{\Sigma}}_{k}(\mathcal{S})
    =\frac{1}{\vert\mathcal{S}_{k}\vert}
  \left(\sum_{\vec{x}\in\mathcal{S}_{k}}
  (f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k}(\mathcal{S}))
  (f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k}(\mathcal{S}))^{\top}
  +\beta\vec{I}\right),
  \label{eq:gmm_params}
\end{align}
where $\mathcal{S}_{k}=\{\vec{x}\mid(\vec{x},y)\in\mathcal{S},y=k\}$ is the set of instances with label $k$ in the support set, $\vert\mathcal{S}_{k}\vert$ is its size, $\beta\in\mathbb{R}_{>0}$ is the parameter,
and $\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})=\{\hat{\gamma}_{k}(\mathcal{S}),\hat{\bm{\mu}}_{k}(\mathcal{S}),\hat{\bm{\Sigma}}_{k}(\mathcal{S})\}_{k=1}^{K}$.
By parameter $\beta$, the covariance is regularized,
and is guaranteed to be positive definite,
which is necessary to compute the inverse of the covariance stably.
The inverse is required to calculate
the likelihood in Eq.~(\ref{eq:gmm}).

\subsection{Meta-training}

The common parameters to be meta-learned
are $\bm{\Phi}=\{\bm{\phi},\beta\}$, where $\bm{\phi}$ is
the set of the neural network parameters, and $\beta$ is the
regularization parameter for covariance estimation.
Let $\mathcal{Q}_{\mathrm{I}}=\{\vec{x}^{\mathrm{I}}_{n}\}_{n=1}^{N_{\mathrm{I}}}$ be a query set of instances
that are ID of the support set,
and $\mathcal{Q}_{\mathrm{O}}=\{\vec{x}_{n}^{\mathrm{O}}\}_{n=1}^{N_{\mathrm{O}}}$
be a query set of instances
that are OoD of the support set.
Each meta-training task contains support set $\mathcal{S}$ and
query set $\mathcal{Q}=\{\mathcal{Q}_{\mathrm{I}},\mathcal{Q}_{\mathrm{O}}\}$.
We want to improve the expected AUC over tasks,
\begin{align}

  \hat{\bm{\Phi}}=
  \mathop{\rm arg~max}\limits_{\bm{\Phi}}\mathbb{E}_{t\in\{1,\cdots,T\}}\left[\mathbb{E}_{\mathcal{S},\mathcal{Q}\subset\mathcal{D}_{t}}\left[\mathrm{AUC}\right(\mathcal{Q}\mid\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi}),\bm{\Phi}\left)\right]\right],
  \label{eq:expected_auc}
\end{align}
where $\mathbb{E}$ represents the expectation,
and
$\mathrm{AUC}(\mathcal{Q}\mid\bm{\Theta},\bm{\Phi})$
is the AUC on query set $\mathcal{Q}$ calculated using
task-specific parameters $\bm{\Theta}$
and common parameters $\bm{\Phi}$,
and $\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})$
is a task-specific parameters adapted to support set $\mathcal{S}$
by maximizing the log likelihood in Eq.~(\ref{eq:ml}).
It is a bilevel optimization problem where
Eq.~(\ref{eq:ml}) is the inner problem,
and Eq.~(\ref{eq:expected_auc}) is the outer problem.
Since the inner problem is solved in a closed form
as shown in Eq.~(\ref{eq:gmm_params}),
the bilevel optimization can be efficiently solved
by incorporating the inner solution into the outer problem.
The objective functions of the inner and outer problems
are the likelihood and AUC, respectively, and they are different.
It is because
while only ID data are given in the inner problem,
which is the same setting with the meta-test phase,
both of the ID and OoD data are given in the outer problem.

The AUC is calculated by the ratio that the OoD score of
the OoD query instances is higher than that of the ID query instances,
\begin{align}
  \mathrm{AUC}(\mathcal{Q}\mid\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi}),\bm{\Phi})=
    \frac{1}{N_{\mathrm{O}}N_{\mathrm{I}}}
  \sum_{\vec{x}^{\mathrm{O}}\in\mathcal{Q}_{\mathrm{O}}}
  \sum_{\vec{x}^{\mathrm{I}}\in\mathcal{Q}_{\mathrm{I}}}
  I\left(u(\vec{x}^{\mathrm{O}}\mid\mathcal{S};\bm{\Phi})>u(\vec{x}^{\mathrm{I}}\mid\mathcal{S};\bm{\Phi})\right),
\end{align}
where $I$ is the indicator function, i.e., $I(A)=1$ is $A$ is true,
and $I(A)=0$ otherwise.
To make the AUC differentiable,
we use the following smooth approximation of the AUC,
\begin{align}
  \widetilde{\mathrm{AUC}}(\mathcal{Q}\mid\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi}),\bm{\Phi})=
    \frac{1}{N_{\mathrm{O}}N_{\mathrm{I}}}
  \sum_{\vec{x}^{\mathrm{O}}\in\mathcal{Q}_{\mathrm{O}}}
  \sum_{\vec{x}^{\mathrm{I}}\in\mathcal{Q}_{\mathrm{I}}}
  \sigma\left(u(\vec{x}^{\mathrm{O}}\mid\mathcal{S};\bm{\Phi})-u(\vec{x}^{\mathrm{I}}\mid\mathcal{S};\bm{\Phi})\right),
  \label{eq:smooth_auc}
\end{align}
where $\sigma(x)=\frac{1}{1+\exp(-x)}$ is the sigmoid function
which is used for the smooth approximation of the indicator function~\cite{iwata2019supervised}.

Algorithm~\ref{alg:train} shows the meta-learning procedures of the
proposed model.
The expectation in Eq.~(\ref{eq:expected_auc}) is calculated
by the Monte Carlo method,
where support and query sets are randomly sampled
from the meta-training data in Lines 3--6.

The time complexity to adapt GMM parameters in Eq.~(\ref{eq:gmm_params}) is
$O(KN^{2}D)$, where
$K$ is the number of classes,
$N$ is the number of support instances per class,
and $D$ is the latent space dimensionality.
The complexity to calculate the OoD score is $O(KD^{3})$,
where the complexity of the inverse of the covariance matrix is
the cubic of the matrix size.
When the latent space dimensionality is larger than the number of instances,
$D>N$, we can reduce the complexity to $O(KN^{3})$
using the Woodbury formula~\cite{petersen2008matrix}.
The complexity to calculate the smooth AUC is $O(N_{\mathrm{O}}N_{\mathrm{I}})$
since it is a summation over $N_{\mathrm{O}}$ OoD query instances
and $N_{\mathrm{I}}$ ID query instances.

\begin{algorithm}[t!]
  \caption{Meta-training procedure of our model.}
  \label{alg:train}
  \begin{algorithmic}[1]
    \renewcommand{\algorithmicrequire}{\textbf{Input:}}
    \renewcommand{\algorithmicensure}{\textbf{Output:}}
    \REQUIRE{Meta-training data $\mathcal{D}$, support set size $N_{\mathrm{S}}$, ID query set size $N_{\mathrm{I}}$, OoD query set size $N_{\mathrm{O}}$}
    \ENSURE{Trained common parameters $\bm{\Phi}$}
    \STATE Initialize common parameters $\bm{\Phi}$.
    \WHILE{End condition is satisfied}
    \STATE Randomly select task index $t$ from $\{1,\cdots,T\}$.
    \STATE Generate support set $\mathcal{S}\subset\mathcal{D}_{t}$ by randomly sampling $N_{\mathrm{S}}$ instances from the selected task.
    \STATE Generate ID query set $\mathcal{Q}_{\mathrm{I}}$ by randomly sampling $N_{\mathrm{I}}$ instances from the selected task, $\mathcal{Q}_{\mathrm{I}}\subset\mathcal{D}_{t}\setminus\mathcal{S}$ where the ID query set and support set do not share the instances, $\mathcal{Q}_{\mathrm{I}}\cap\mathcal{S}=\phi$.
    \STATE Generate OoD query set $\mathcal{Q}_{\mathrm{O}}$ by randomly sampling $N_{\mathrm{O}}$ instances from the unselected tasks, $\{1,\cdots,T\}\setminus t$.
    \STATE Adapt GMM parameters $\hat{\bm{\Theta}}(\mathcal{S};\bm{\Phi})$ to
    the support set by maximizing the likelihood with Eq.~(\ref{eq:gmm_params}).
   \STATE Calculate the OoD score by Eq.~(\ref{eq:u}) using the adapted GMM parameters.
    \STATE Calculate the smooth AUC by Eq.~(\ref{eq:smooth_auc}).
    \STATE Update common parameters $\bm{\Phi}$ using the smooth AUC and its gradient by a stochastic gradient method.
    \ENDWHILE
  \end{algorithmic}
\end{algorithm}

\subsection{Relation to existing methods}

A OoD detection method
based on Mahalanobis distance~\cite{lee2018simple}
assumes a GMM with a full covariance matrix shared by all classes,
\begin{align}
  p_{\mathrm{M}}\left(f(\vec{x};\bm{\phi})\mid\bm{\Theta}\right)
    = \sum_{k=1}^{K}\gamma_{k}\mathcal{N}\left(f(\vec{x};\bm{\phi})\mid
    \bm{\mu}_{k},\bm{\Sigma}\right).
    \label{eq:gmm_maha}
\end{align}
The negative OoD score is calculated by the minimum of
the Mahalanobis distance over classes,
\begin{align}
  u_{\mathrm{M}}(\vec{x}\mid\mathcal{S};\bm{\Phi})
  &= -\min_{k}[(f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k})^{\top}\hat{\bm{\Sigma}}^{-1}(f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k})]
  \nonumber\\
  &=-\max_{k}\log\mathcal{N}(f(\vec{x};\bm{\phi})\mid\hat{\bm{\mu}}_{k},\hat{\bm{\Sigma}})
  \nonumber\\
  &\approx-\log\sum_{k}\mathcal{N}(f(\vec{x};\bm{\phi})\mid\hat{\bm{\mu}}_{k},\hat{\bm{\Sigma}}),
\end{align}
where we used the fact that the LogSumExp is a smooth approximation
of the maximum, $\log\sum_{k}\exp(\cdot)\approx\max_{k}(\cdot)$.
Therefore, when class probability $\gamma_{k}$ are the same for all classes,
the Mahalanobis distance-based method can be seen that
it uses log likelihood in a latent space as the OoD score
as with the proposed method.

The prototypical network is a meta-learning method
for classification~\cite{snell2017prototypical},
and it assumes
the following
Gaussian mixture model with a spherical covariance in a latent space,
\begin{align}
  p_{\mathrm{P}}\left(f(\vec{x};\bm{\phi})\mid\bm{\Theta}\right)
    = \sum_{k=1}^{K}\gamma_{k}\mathcal{N}\left(f(\vec{x};\bm{\phi})\mid
    \bm{\mu}_{k},\tau\vec{I}\right),
    \label{eq:gmm_proto}
\end{align}
with uniform class probability $\gamma_{k}=\frac{1}{K}$.
Mean vectors $\{\bm{\mu}_{k}\}_{k=1}^{K}$ are obtained
by maximizing the log likelihood of the support set.
The proposed model can express more complicated distributions
by adapting a full covariance matrix for each class.
When parameter $\beta$ in Eq.~(\ref{eq:ml}) is large,
the proposed model becomes similar to a GMM with
a fixed spherical covariance as with the prototypical network.
By meta-learning $\beta$ such that the OoD detection performance is improved,
the proposed method can estimate the density flexibly
while avoiding overfitting.
Different from the proposed method,
the objective function of the outer optimization problem of
the prototypical network is the classification cross-entropy loss,
where the posterior class probability is calculated by
\begin{align}
  p(k\mid\vec{x})=\frac{\exp(-\frac{1}{2\tau}\parallel f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k}(\mathcal{S})\parallel^{2})}
  {\sum_{k'=1}^{K}\exp(-\frac{1}{2\tau}\parallel f(\vec{x};\bm{\phi})-\hat{\bm{\mu}}_{k'}(\mathcal{S})\parallel^{2})}.
  \label{eq:proto}
\end{align}

With classifier-based OoD detection methods~\cite{liang2018enhancing},
the negative OoD score is calculated by the maximum of the posterior class probability,
\begin{align}
  u_{\mathrm{P}}(\vec{x}\mid\mathcal{S};\bm{\Phi})&=-\max_{k} p(k\mid\vec{x})
  \propto -\max_{k} \mathcal{N}(f(\vec{x};\bm{\phi})\mid\hat{\bm{\mu}}_{k},\tau\vec{I}),
\end{align}
which is also related to the density estimation in a latent space.

Temperature scaling is used for OoD detection~\cite{liang2018enhancing},
where $\tau$ is
the temperature in Eq.~(\ref{eq:proto}),
and the temperature scaling controls the covariance.
Since the proposed method adapts the covariance for each class
to the support set,
the proposed method is related to the temperature scaling
that adapt to each task.

\section{Experiments}
\label{sec:experiments}

\subsection{Data}

We evaluated the proposed method
with the following six datasets:
Omniglot, Miniimagenet, CIFAR10, SVHN, Patent, and Dmoz.

The Omniglot dataset~\cite{lake2015human} consisted of
hand-written images of 964 characters from 50 alphabets (real and fictional).
There were 20 images for each character category.
Each image was represented by gray-scale with 28 $\times$ 28 pixels.
The number of instances, attributes, and categories were
19280, 784, and 964, respectively.

The Miniimagenet dataset
consisted of images from 100 categories~\cite{vinyals2016matching}.
Each image was represented by RGB with 84 $\times$ 84 pixels.
The number of instances, attributes, and categories were
60000, 21168, and 100, respectively.

The CIFAR10 dataset consisted of images from
10 categories~\cite{krizhevsky2009learning}.
Each image was represented by RGB with 32 $\times$ 32 pixels.
The number of instances, attributes, and categories were
60000, 3072, and 10, respectively.

The SVHN dataset consisted of digit images obtained from
house numbers in Google street view~\cite{netzer2011reading}.
Each image was represented by RGB with 32 $\times$ 32 pixels.
The number of instances, attributes, and categories were
73257, 3072, and 10, respectively.

The Patent dataset consisted of patents published
in Japan from January to March in 2004,
which were categorized by International Patent Classification.
Each patent was represented by bag-of-words.
We omitted words that occurred in fewer than 100 patents,
omitted patents with fewer than 100 unique words, and
omitted categories with fewer than ten patents.
The number of instances, attributes, and categories were
5714, 3201, and 285, respectively.

The Dmoz dataset consisted of webpages crawled in 2006
from the Open Directory Project~\cite{lorenzetti2019dmoz,lorenzetti2009semi}
\footnote{The data were obtained from \url{https://data.mendeley.com/datasets/9mpgz8z257/1}.}.
Each webpage was categorized in a web directory,
and represented by bag-of-words.
We omitted words that occurred in fewer than 300 webpages,
omitted webpages with fewer than 300 unique words,
and omitted categories with fewer than ten webpages.
The number of instances, attributes, and categories were
15159, 17659, and 354, respectively.

For each of the Omniglot, Miniimagenet, Patent, and Dmoz datasets,
we randomly used 60\% of the categories for training,
20\% for validation,
and the remaining categories for testing.
For each of the CIFAR10 and SVHN datasets,
since their number of classes was small,
the Miniimagenet dataset was used for the training and validation,
where we rescaled images in the Miniimagenet dataset
to 32 $\times$ 32 pixels.
For the test of the CIFAR10 dataset,
CIFAR10 images were used for the ID,
and SVHN images were used for the OoD.
For the test of the SVHN dataset,
SVHN images were used for the ID,
and CIFAR10 images were used for the OoD.
It is known that when a neural network-based density model
is trained on the CIFAR10 images,
the likelihood of the SVHN images is likely to be higher than that of
the CIFAR10 images~\cite{nalisnick2018deep}.
For all datasets, we generated 64 tasks for each validation and test data.
For each task in the validation and test data,
we first randomly selected five categories for ID,
and one category for OoD.
Then, we randomly selected five support instances
from each of the ID categories,
and five query instances from each of the ID and OoD categories.
We performed five experiments with different data splits
for each dataset.

\subsection{Comparing methods}

We compared our proposed method (Ours) with
ablation of the proposed method (Ours-C),
three types of prototypical networks~\cite{snell2017prototypical}
(Proto-C, Proto-A, and Proto-O),
three types of model-agnostic meta-learning~\cite{finn2017model}
(MAML-C, MAML-A, and MAML-O),
OoD-MAML~\cite{jeong2020ood},
learnable class boundary networks~\cite{wang2019out} (LCBO),
Mahalanobis distance-based method~\cite{lee2018simple} (Mahalanobis),
deep support vector data description (SVDD)~\cite{ruff2018deep},
density estimation with a single Gaussian model (Gaussian),
and kernel density estimation (KDE) in a latent space.
All methods use neural networks
to map instances from the original space to a latent space.
`C' in the method names indicates that
the method is meta-learned
by minimizing the expected test classification cross-entropy loss,
`A' indicates that
the method is meta-learned
by maximizing the AUC of OoD detection,
and `O' indicates that
the method is based on
out-of-distribution detector for neural networks
(ODIN)~\cite{liang2018enhancing}.

The methods based on prototypical networks
(Proto-C, Proto-A, and Proto-O)
assume a Gaussian mixture model
with a unit spherical covariance in a latent space,
and the class probability is calculated based o the Bayes rule.
The methods based on model-agnostic meta-learning~\cite{finn2017model}
(MAML-C, MAML-A, and MAML-O)
model the class probability by a softmaxed linear projection
from the latent space.
With the methods based on ODIN (Proto-O and MAML-O),
the temperature of the class probability is meta-learned,
and small perturbations are added to the input
to the direction that maximizes the maximum class probability.
The neural network is meta-learned
by minimizing the expected test classification cross-entropy loss.
With prototypical network-based and MAML-based methods,
the negative OoD score is calculated by
the maximum of the class probability,
$\max_{k}p(k\mid\vec{x})$.
With OoD-MAML,
fake OoD instances are generated with the framework of MAML.
With learnable class boundary networks (LCBO),
the OoD score is modeled by
a neural network that takes the mean for each class and
the instance in the latent space as input.
With Mahalanobis,
the negative OoD score is calculated by
the minimum Mahalanobis distance to the class mean in the latent space.
SVDD, Gaussian, and KDE do not use class label information
in the support set.
With SVDD~\cite{ruff2018deep},
the OoD score is calculated by the distance from
the mean of the support instances in the latent space.
With Gaussian and KDE,
the OoD score is calculated by the likelihood in the latent space.
Except for Ours-C, Proto-C, MAML-C, Proto-O, and MAML-O,
the neural network is meta-learned
by maximizing the AUC of OoD detection.

\subsection{Setting}

In image datasets (Omniglot, Miniimage, CIFAR10, and SVHN),
we used a four-layered convolutional neural network of
filter size 32, kernel size three, and padding size one for $f$
with all methods.
In text datasets (Patent and Dmoz),
we used a three-layered feed-forward neural network
with 256 hidden and output units for $f$.
The instances in a latent space
were normalized by dividing by the standard deviation
of the support set.
For the activation function,
we used rectified linear unit $\mathrm{ReLU}(x)=\max(0,x)$.
We optimized using Adam~\cite{kingma2014adam}
with learning rate $10^{-3}$,
and dropout rate $10^{-1}$~\cite{srivastava2014dropout}.
For MAML-based method, we used three inner epochs.
The validation data were used for early stopping,
for which the maximum number of training epochs was 5,000.
Our implementation was based on PyTorch~\cite{paszke2017automatic}.

\subsection{Results}

\begin{table}[t]
  \centering
  \caption{Average test AUC for OoD detection and its standard error. Values in bold are not statistically significantly different at the 5\% level from the best performing method in each column according to a paired t-test.}
  \label{tab:auc}
  {\tabcolsep=0.4em
    \begin{tabular}{lrrrrrr}
      \hline
      & Omniglot & Miniimagenet & CIFAR10 & SVHN & Patent & Dmoz \\
      \hline
      Ours &{\bf 0.992$\pm$0.001} &{\bf 0.673$\pm$0.005} &0.619$\pm$0.017 &{\bf 0.989$\pm$0.001} &{\bf 0.873$\pm$0.006} &{\bf 0.828$\pm$0.005} \\
 Ours-C & 0.990$\pm$0.001 & 0.621$\pm$0.006 & 0.471$\pm$0.034 & {\bf 0.989$\pm$0.000} & 0.824$\pm$0.009 & 0.785$\pm$0.006 \\
 Proto-C & 0.888$\pm$0.007 & 0.577$\pm$0.006 & 0.565$\pm$0.012 & 0.369$\pm$0.008 & 0.737$\pm$0.009 & 0.705$\pm$0.006 \\
 Proto-A & 0.980$\pm$0.002 & 0.185$\pm$0.014 & 0.351$\pm$0.022 & 0.274$\pm$0.034 & 0.827$\pm$0.010 & 0.783$\pm$0.006 \\
 Proto-O & 0.768$\pm$0.012 & 0.567$\pm$0.005 & {\bf 0.799$\pm$0.010} & 0.199$\pm$0.013 & 0.163$\pm$0.012 & 0.072$\pm$0.005 \\
 MAML-C & 0.935$\pm$0.004 & 0.592$\pm$0.003 & 0.605$\pm$0.007 & 0.350$\pm$0.013 & 0.623$\pm$0.010 & 0.563$\pm$0.005 \\
 MAML-A & 0.521$\pm$0.009 & 0.498$\pm$0.006 & 0.636$\pm$0.022 & 0.309$\pm$0.040 & 0.487$\pm$0.014 & 0.464$\pm$0.005 \\
 MAML-O & 0.498$\pm$0.011 & 0.504$\pm$0.005 & 0.634$\pm$0.023 & 0.466$\pm$0.010 & 0.511$\pm$0.008 & 0.003$\pm$0.001 \\
 OoD-MAML & 0.982$\pm$0.003 & {\bf 0.672$\pm$0.003} & 0.669$\pm$0.025 & 0.656$\pm$0.024 & 0.810$\pm$0.006 & 0.767$\pm$0.007 \\
 LCBO & 0.559$\pm$0.005 & 0.522$\pm$0.007 & 0.481$\pm$0.012 & 0.541$\pm$0.032 & 0.573$\pm$0.008 & 0.564$\pm$0.006 \\
 Mahalanobis & 0.989$\pm$0.002 & 0.520$\pm$0.006 & 0.623$\pm$0.011 & {\bf 0.988$\pm$0.001} & {\bf 0.875$\pm$0.007} & {\bf 0.826$\pm$0.007} \\
 SVDD & 0.832$\pm$0.003 & 0.528$\pm$0.003 & 0.462$\pm$0.033 & {\bf 0.718$\pm$0.050} & 0.693$\pm$0.009 & 0.619$\pm$0.005 \\
 Gauss & 0.989$\pm$0.000 & 0.539$\pm$0.002 & 0.626$\pm$0.011 & 0.984$\pm$0.002 & {\bf 0.878$\pm$0.007} & {\bf 0.828$\pm$0.007} \\
 KDE & 0.858$\pm$0.006 & 0.511$\pm$0.015 & 0.421$\pm$0.029 & 0.706$\pm$0.032 & 0.733$\pm$0.008 & 0.655$\pm$0.016 \\
\hline
\end{tabular}}

\end{table}

\begin{figure*}[t!]
  \centering
  {\tabcolsep=-0.8em
    \begin{tabular}{ccc}
      \includegraphics[width=16em]{images/tsne0.png}&
      \includegraphics[width=16em]{images/tsne2.png}&
      \includegraphics[width=16em]{images/tsne5.png}\\
  \end{tabular}}
  \caption{Examples of density estimation in a two-dimensional latent space by the proposed method on the Omniglot dataset. The green triangles are support instances, blue crosses are ID query instances, and red circles are OoD query instances. The contours represent the log likelihood, i.e., negative OoD score,
    where the darker color indicates the higher log likelihood.}
  \label{fig:vis}
\end{figure*}

Table~\ref{tab:auc} shows the test AUC.
Our proposed method achieved high AUC with all datasets.
With the CIFAR10 dataset, the AUC by the proposed method
was lower than that by Proto-O.
However, the AUC by Proto-O with the SVHN dataset was very low.
It indicates that Proto-O estimated that the CIFAR10 images were ID and
the SVHN images were OoD by meta-learning with the Miniimagenet dataset.
By meta-learning a neural network for density estimation in a latent space,
the proposed method achieves a relatively high AUC on
both of the CIFAR10 and SVHN datasets.

The proposed method meta-trained by maximizing
the OoD detection performance (Ours)
was better than that by minimizing the classification loss (Ours-C).
This result indicates that the
directly maximizing the OoD detection performance is important.

The prototypical network-based methods performed worse
than the proposed method.
This result demonstrates the effectiveness of density estimation by
full covariance GMMs in a latent space.
The MAML-based methods adapt the whole neural network to the support set.
On the other hand, the proposed method adapts only a GMM in a latent space to the support set.
The better performance of the proposed method than the MAML-based methods
indicates that the GMM in a latent space is effective for OoD detection
when the number of instances in the support set is small.
OoD-MAML improved the AUC compared with the other MAML-based methods
by generating fake OoD instances.
However, since it is difficult to generate
a wide variety of OoD instances for each task,
it was worse than the proposed method.
The proposed method detects OoD by density estimation
using only ID instances without OoD instances,
and it does not need to generate fake OoD instances.

With LCBO, the OoD score is calculated by a neural network
that takes the support set as input.
Therefore, the OoD score of the support set is not necessarily high.
In contrast, with the proposed method,
the density-based OoD score of the support set is high
since we maximize the likelihood of the support set.
Therefore, the proposed method can flexibly adapt to a given support set
even when it does not appear in the meta-training dataset,
and it led to better performance of the proposed method.
While Mahalanobis achieved good performance,
it was worse than the proposed method on the Omiglot and Miniimagenet datasets.
This result indicates the effectiveness to model
class-specific full covariances in GMMs.
The AUC by the methods that do not use class label information
was lower than the proposed method.
This result shows the usefulness of the class label information
for OoD detection.

Figure~\ref{fig:vis} shows a visualization of density estimation in a two-dimensional latent space by the proposed method on the Omniglot dataset.
We additionally used a three-layered feed-forward neural network
for the two-dimensional embedding after the convolutional neural network.
By the neural network, ID instances were embedded in a high-density area,
and OoD instances were embedded in a low-density area.
By the GMMs, the task-specific density function was modeled flexibly
in the latent space depending on the support set.

\begin{figure}[t!]
  \centering
  \includegraphics[width=19em]{images/r2.png}
  \caption{Average test AUC for OoD detection with different numbers of meta-training classes by the proposed method on the Omniglot dataset. The bar shows standard error.}
  \label{fig:auc_class}
\end{figure}

\begin{figure}[t!]
  \centering
  \includegraphics[width=19em]{images/nhidden_r0.png}
  \caption{Average test AUC for OoD detection with different latent space dimensionality by the proposed method on the Omniglot dataset. The bar shows standard error.}
  \label{fig:auc_dim}
\end{figure}

\begin{table}[t]
  \centering
  \caption{Average test classification accuracy of class labels and its standard error. Values in bold are not statistically significantly different at the 5\% level from the best performing method in each column according to a paired t-test. Since LCBO, SVDD, Gauss, and KDE did not output the class label, we omit them.}
  \label{tab:acc}
  {\tabcolsep=0.4em
    \begin{tabular}{lrrrrrr}
      \hline
      & Omniglot & Miniimagenet & CIFAR10 & SVHN & Patent & Dmoz \\
      \hline
Ours &{\bf 0.995$\pm$0.000} &0.555$\pm$0.006 &{\bf 0.497$\pm$0.006} &{\bf 0.276$\pm$0.006} &0.880$\pm$0.003 &0.816$\pm$0.005 \\
 Ours-C & {\bf 0.994$\pm$0.001} & {\bf 0.602$\pm$0.005} & {\bf 0.505$\pm$0.007} & {\bf 0.269$\pm$0.004} & 0.868$\pm$0.004 & {\bf 0.822$\pm$0.001} \\
 Proto-C & {\bf 0.994$\pm$0.000} & {\bf 0.590$\pm$0.004} & {\bf 0.489$\pm$0.010} & {\bf 0.270$\pm$0.004} & 0.887$\pm$0.002 & 0.823$\pm$0.005 \\
 Proto-A & {\bf 0.995$\pm$0.000} & 0.293$\pm$0.012 & 0.267$\pm$0.009 & 0.198$\pm$0.002 & 0.879$\pm$0.001 & {\bf 0.825$\pm$0.004} \\
 Proto-O & 0.991$\pm$0.001 & {\bf 0.589$\pm$0.006} & 0.475$\pm$0.009 & 0.251$\pm$0.008 & 0.883$\pm$0.002 & 0.821$\pm$0.003 \\
 MAML-C & 0.990$\pm$0.001 & 0.576$\pm$0.006 & 0.489$\pm$0.006 & 0.239$\pm$0.003 & 0.754$\pm$0.005 & 0.692$\pm$0.007 \\
 MAML-A & 0.197$\pm$0.003 & 0.200$\pm$0.001 & 0.198$\pm$0.002 & 0.201$\pm$0.001 & 0.205$\pm$0.004 & 0.208$\pm$0.004 \\
 MAML-O & 0.678$\pm$0.071 & 0.522$\pm$0.008 & 0.433$\pm$0.020 & 0.219$\pm$0.004 & 0.239$\pm$0.016 & 0.658$\pm$0.020 \\
 OoD-MAML & 0.988$\pm$0.002 & 0.563$\pm$0.010 & 0.443$\pm$0.007 & 0.223$\pm$0.004 & 0.765$\pm$0.006 & 0.724$\pm$0.008 \\
 Mahalanobis & {\bf 0.994$\pm$0.001} & 0.319$\pm$0.005 & 0.491$\pm$0.007 & {\bf 0.275$\pm$0.006} & {\bf 0.905$\pm$0.002} & {\bf 0.839$\pm$0.006} \\
 \hline
\end{tabular}}
\end{table}

Figure~\ref{fig:auc_class} shows the test AUC with different numbers
of meta-training classes by the proposed method on the Omniglot dataset.
The test AUC increased as the number of meta-training classes increased.
This result indicates that the proposed method can increase the performance
by collecting meta-training data from many classes.

Figure~\ref{fig:auc_dim} shows the test AUC with different latent space
dimensionality by the proposed method on the Omniglot dataset.
When the latent space dimensionality was small, the test AUC was low.
It is because in a low-dimensional space representing complicated instances is difficult, and the density estimated with GMMs does not have enough expressive power.

Table~\ref{tab:acc} shows the test classification accuracy of class labels.
Since the proposed method (Ours)
is not meta-learned to improve the classification
performance, it was worse than the proposed method that is meta-learned
by minimizing the classification loss (Ours-C).
Since we focus on improving OoD detection performance in this paper,
we use the AUC on OoD detection as the objective function.
When we also want to improve classification performance,
we can add the classification cross-entropy loss in the objective function.
The classification accuracy by Ours-C was almost the same with
that by Proto-C.
In contrast, the AUC for OoD detection by Ours was better than Proto-A.
This result indicates that
although
GMMs with a common spherical covariance are sufficient for classification,
GMMs with full covariance for each class are needed for OoD detection.

Table~\ref{tab:time}
shows the average computational time for meta-training
and meta-testing
with a GTX 1080Ti GPU. The computational
time for the MAML-based methods was much longer than the other methods.
It was because MAML-based methods required
iterative gradient descent steps for the inner optimization.
Also, the ODIN-based methods took a long time since
they required to perturb the input for the query instances.

\begin{table}[t!]
  \centering
  \caption{Average meta-training and meta-testing computational time in seconds on the Omniglot dataset.}
  \label{tab:time}
\begin{tabular}{lrr}
\hline
& Train & Test \\
\hline
Ours &16619.6 & 0.416 \\
 Ours-C & 15230.5 & 0.329 \\
 Proto-C & 13381.7 & 0.325 \\
 Proto-A & 12846.6 & 0.345 \\
 Proto-O & 25299.8 &0.827 \\
 MAML-C & 57677.4 & 0.960 \\
 MAML-A & 55057.0 & 1.191 \\
 MAML-O & 177137.1 & 1.727 \\
 OoD-MAML & 47581.7 & 0.928 \\
 LCBO & 10673.2 & 0.198 \\
 Mahalanobis & 15169.2 & 0.283 \\
 SVDD & 10370.6 & 0.259 \\
 Gauss & 15056.3 & 0.294 \\
 KDE & 11922.2& 0.299 \\
\hline
\end{tabular}
\end{table}

\section{Conclusion}
\label{sec:conclusion}

We proposed a meta-learning method for OoD detection.
Our proposed method trains a neural network such that
density-based OoD scores perform well when
a Gaussian mixture model in the latent space is adapted
to given in-distribution data for each task.
Experiments on six datasets confirmed
that our proposed method had better OoD detection performance
than existing methods did.
For future work,
we will use other density estimation methods in a latent space
in our framework.
Also, we want to improve the performance by additionally using
OoD techniques such as input perturbations
and the use of representations in multiple layers~\cite{lee2018simple}.


text:
/ Portrait - NPG 5488; Edward Montagu, 1st Earl of Sandwich
Edward Montagu, 1st Earl of Sandwich
1 of 20 portraits matching 'Edward Montagu, 1st Earl of Sandwich'
by Sir Peter Lely
oil on canvas, circa 1655-1659
30 in. x 25 in. (762 mm x 635 mm)
Primary Collection
NPG 5488
Edward Montagu, 1st Earl of Sandwich (1625-1672), Admiral. Sitter in 14 portraits.
Artistback to top
Sir Peter Lely (1618-1680), Portrait painter. Artist associated with 842 portraits, Sitter in 19 portraits.
Linked publicationsback to top
Ollard, Richard, Character Sketches: Samuel Pepys and His Circle, 2000, p. 14
Ollard, Richard, Pepys and his Contemporaries, 2015, p. 18
Saywell, David; Simon, Jacob, Complete Illustrated Catalogue, 2004, p. 547
Simon, Jacob, The Art of the Picture Frame: Artists, Patrons and the Framing of Portraits in Britain, 1997 (accompanying the exhibition at the National Portrait Gallery from 8 November 1996 - 9 February 1997), p. 91, 154 Read entry
Carved and gilt oak with slight chamfer at back, mason's mitre, the back frame replaced, the gilding distressed showing the gesso. 3 3⁄ 4 to 5 inches wide, 6 inches at top cresting.
This portrait of the Admiral, Edward Montagu, created Earl of Sandwich in 1660, has an early Sunderland frame very similar to that on NPG 5461, but on a different scale.
Linked displays and exhibitionsback to top
Restoration Lives: Samuel Pepys and His Circle (31 March 2003 - 26 October 2003)
Secretary of State, John Thurloe, implements a highly efficient intelligence service and thwarts plans for a series of royalist uprisings which produced only Penruddock's revolt.
Following ineffectual royalist riots, Lord Protector, Oliver Cromwell, appoints nineteen Major-generals to manage regional government and prevent future challenges to the protectorate.
Publication of the controversial work De corpore, by philosopher, Thomas Hobbes, prompts mathematician, John Wallis to scornfully refute the work in Elenchus geometriae Hobbianae, starting a bitter, long-running polemical dispute between the two men.
General Robert Venables and Admiral William Penn lead an expedition to the Caribbean to threaten Spanish trade routes and weaken Catholic influence in the New World. An integral part of Cromwell's foreign policy to curb Spanish power, the campaign, Cromwell's 'western design', fails leading to war in Europe.


Marcon Heritage completed the fit-out works at the new Museum and Interpretive Centre for the Railway Preservation Society of Ireland over a 6 week period. The new visitor centre cost £3m and has transformed the RPSI's base into a modern tourist attraction.
Marcon was responsible for the manufacture and installation of all specialist joinery, display cases, integration of audio visual hardware, interactives and installation of graphics within the new museum space.
Working closely with leading interpretive designers GBDM, Marcon turned the carriage workshop and locomotive sheds into a fully functioning, living museum.


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text:
England vs India: ECB announced Test squad for first Test, Adil Rashid is recalled
England and Wales Cricket Board has declared England team for the first Test against India. To beat team India with his weapon of spin, leg-spinner Adil Rashid was recalled from the retirement and included in the 13-man squad for England's first match of the series.
England and Wales Cricket Board (ECB) announced the team for the first Test match on Thursday. Host England has given Essex's fast bowler Jamie Porter a chance for the first time in the team.
The 25-year-old Porter played 61 first-class matches. Porter had taken 75 wickets in his first county championship in 2017. He has been included in the team as a replaced of injured Chris Woakes.
The ECB has also included Moeen Ali in addition to Rashid to strengthen his spin department. Moeen picked up 19 wickets in his first Test series against India in 2014. Moeen, who has played 50 Test matches for England, played the last Test match against New Zealand in March.
The selection committee chaired by the new national selector Ed Smith decided to hand over the red ball to Rashid after almost two years, who retired from Test after Indian tour in December 2016.
Rashid can put Indian batsmen in trouble with his leg spin. Rashid was the man of the match against India in the third ODI, taking three wickets.
The England team for the first Test match is as follows…
Joe Root (captain), Moeen Ali, James Anderson, Jonny Bairstow (wicketkeeper). Stuart Broad. Jos Buttler. Alastair Cook. Sam Curran. Keaton Jennings. David Malan. Jamie Porter. Adil Rashid. Ben Stokes.
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