Isometric embedding probability distributions with tree transportation cost into $ell_1$
I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric.
Let $T = ([n], E)$ be an unweighted, undirected tree with root $r in [n]$, and shortest path metric $d$. Let $p, q in mathbf{R}^{n}$ be two probability distributions over the vertices $[n]$ (i.e. $p_i geq 0$, $sum_i p_i = 1$). Define the transportation distance between $p$ and $q$ as
$$
tau(p, q)= inf left{sum_{i,j=1}^{n} d(i, j) w_{ij} : w_{ij} geq 0, sum_{i=1}^n w_{ij} = q_j, sum_{j=1}^n w_{ij} = p_i right}.
$$
Let $Delta_n$ denote the space of probability distributions on $[n]$. Show that there is some integer $k$ such that the metric space $(Delta_n, tau)$ isometrically embeds into $(mathbf{R}^k, d_1)$, where
$$
d_1(x, y) := sum_1^k |x_i - y_i|
$$
for $x, y in mathbf{R}^k$.
I also don't quite know why the tree part is so relevant.
probability-distributions metric-spaces trees isometry
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
add a comment |
I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric.
Let $T = ([n], E)$ be an unweighted, undirected tree with root $r in [n]$, and shortest path metric $d$. Let $p, q in mathbf{R}^{n}$ be two probability distributions over the vertices $[n]$ (i.e. $p_i geq 0$, $sum_i p_i = 1$). Define the transportation distance between $p$ and $q$ as
$$
tau(p, q)= inf left{sum_{i,j=1}^{n} d(i, j) w_{ij} : w_{ij} geq 0, sum_{i=1}^n w_{ij} = q_j, sum_{j=1}^n w_{ij} = p_i right}.
$$
Let $Delta_n$ denote the space of probability distributions on $[n]$. Show that there is some integer $k$ such that the metric space $(Delta_n, tau)$ isometrically embeds into $(mathbf{R}^k, d_1)$, where
$$
d_1(x, y) := sum_1^k |x_i - y_i|
$$
for $x, y in mathbf{R}^k$.
I also don't quite know why the tree part is so relevant.
probability-distributions metric-spaces trees isometry
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
add a comment |
I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric.
Let $T = ([n], E)$ be an unweighted, undirected tree with root $r in [n]$, and shortest path metric $d$. Let $p, q in mathbf{R}^{n}$ be two probability distributions over the vertices $[n]$ (i.e. $p_i geq 0$, $sum_i p_i = 1$). Define the transportation distance between $p$ and $q$ as
$$
tau(p, q)= inf left{sum_{i,j=1}^{n} d(i, j) w_{ij} : w_{ij} geq 0, sum_{i=1}^n w_{ij} = q_j, sum_{j=1}^n w_{ij} = p_i right}.
$$
Let $Delta_n$ denote the space of probability distributions on $[n]$. Show that there is some integer $k$ such that the metric space $(Delta_n, tau)$ isometrically embeds into $(mathbf{R}^k, d_1)$, where
$$
d_1(x, y) := sum_1^k |x_i - y_i|
$$
for $x, y in mathbf{R}^k$.
I also don't quite know why the tree part is so relevant.
probability-distributions metric-spaces trees isometry
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric.
Let $T = ([n], E)$ be an unweighted, undirected tree with root $r in [n]$, and shortest path metric $d$. Let $p, q in mathbf{R}^{n}$ be two probability distributions over the vertices $[n]$ (i.e. $p_i geq 0$, $sum_i p_i = 1$). Define the transportation distance between $p$ and $q$ as
$$
tau(p, q)= inf left{sum_{i,j=1}^{n} d(i, j) w_{ij} : w_{ij} geq 0, sum_{i=1}^n w_{ij} = q_j, sum_{j=1}^n w_{ij} = p_i right}.
$$
Let $Delta_n$ denote the space of probability distributions on $[n]$. Show that there is some integer $k$ such that the metric space $(Delta_n, tau)$ isometrically embeds into $(mathbf{R}^k, d_1)$, where
$$
d_1(x, y) := sum_1^k |x_i - y_i|
$$
for $x, y in mathbf{R}^k$.
I also don't quite know why the tree part is so relevant.
probability-distributions metric-spaces trees isometry
probability-distributions metric-spaces trees isometry
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
edited Dec 5 '18 at 5:06
Alex Ravsky
39.2k32080
39.2k32080
asked Nov 19 '18 at 2:29
Drew Brady
649315
asked Nov 19 '18 at 2:29
Drew Brady
649315
asked Nov 19 '18 at 2:29
Drew Brady
649315
649315
add a comment |
add a comment |
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