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paper modifications

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This commit is contained in:
Garry Boyer
2007-09-15 05:38:36 +00:00
parent 24fb0a30b3
commit dd4ee19a68
1 changed files with 41 additions and 42 deletions
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fastlib/u/garryb/nbr/paper/paper.tex
+41 -42
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@@ -29,7 +29,8 @@
\newcommand{\odothat}{\widehat{\odot}}
\newcommand{\prefsplit}[2]{#1 \succ #2}
\newcommand{\summary}{\hat{\sigma}}
\newcommand{\summary}{\delta}
%hat{\sigma}}
\DeclareMathOperator*{\map}{map}
\DeclareMathOperator*{\worst}{worst}
@@ -46,6 +47,7 @@
\DeclareMathOperator{\ATDISCRETION}{}
\newcommand{\fig}[1]{Figure~\ref{fig:#1}}
\newcommand{\eqn}[1]{Equation~\ref{eqn:#1}}
\newcommand{\Gnp}{\Psi}
\newcommand{\gnp}{\psi}
@@ -96,40 +98,42 @@
\newcommand{\canpruneglob}{C_{\!\letterglob}}
\newcommand{\deltaglob}{\summary_{\!\letterglob}}
\newcommand{\letterqr}{\rho}
\newcommand{\outqr}{\varrho}
\newcommand{\inqr}{\rho}
\newcommand{\letterqr}{v}
\newcommand{\outqr}{V}
\newcommand{\inqr}{v}
\newcommand{\Opqr}{\myOp{\letterqr}}
\newcommand{\opqr}{\myop{\letterqr}}
\newcommand{\fqr}{f_{\!\letterqr}}
\newcommand{\gqr}{g_{\!\letterqr}}
\newcommand{\letterqrv}{\vec{\rho}}
\newcommand{\letterqrv}{v}
%\newcommand{\outqrv}{\vec{\rho}}
\newcommand{\inqrv}{\vec{\rho}}
\newcommand{\inqrv}{v}
%\newcommand{\fqrv}{f_{\letterqrv}}
%\newcommand{\gqrv}{g_{\letterqrv}}
\newcommand{\deltaqrv}{\summary_{\!\letterqrv}}
\newcommand{\canpruneqrv}{C_{\!\letterqrv}}
\newcommand{\canpruneqrv}{C}%_{\!\letterqrv}}
\newcommand{\identqr}{0_{\!\letterqrv}}
\newcommand{\varqrv}{\letterqrv^{\:C\!}}
\newcommand{\varqrvparent}{\letterqrv^{\:P\!}}
\newcommand{\varqrv}{\tilde{\letterqrv}}
%\newcommand{\varqrv}{\letterqrv^{\:C\!}}
\newcommand{\varqrvparent}{\letterqrv^{P}}
\newcommand{\lettermu}{\mu}
\newcommand{\lettermu}{e}
%\newcommand{\inmu}{\mu}
\newcommand{\inmu}{\mu}
\newcommand{\Outopmu}{\widehat{\nameOp{\bigodot}{\lettermu}}}%\mathop{\widehat{\bigodot\nolimits}\!\scriptstyle{\mu}}}
\newcommand{\outopmu}{\:\widehat{\odot}_{\!\mu}\:}
\newcommand{\inmu}{e}
\newcommand{\Outopmu}{\nameOp{\bigodot}{\lettermu}}%\mathop{\widehat{\bigodot\nolimits}\!\scriptstyle{\mu}}}
\newcommand{\outopmu}{\:\odot_{\!\mu}\:}
\newcommand{\Opmu}{\myOp{\lettermu}}
\newcommand{\opmu}{\myop{\lettermu}}
\newcommand{\fmu}{f_{\!\lettermu}}
\newcommand{\fmuv}{\vec{f_{\!\lettermu}}}
\newcommand{\fmuv}{f_{\!\lettermu}}
\newcommand{\deltamu}{\summary_{\!\lettermu}}
\newcommand{\canprunemu}{C_{\!\lettermu}}
\newcommand{\canprunemu}{C}
\newcommand{\heurqr}{H}
\newcommand{\identmu}{0_{\lettermu}}
\newcommand{\varmuchild}{\lettermu^{\!C}}
\newcommand{\varmuparent}{\lettermu^{\!P}}
\newcommand{\varmuchild}{\tilde{\lettermu}}
%\newcommand{\varmuchild}{\lettermu^{\!C}}
\newcommand{\varmuparent}{\lettermu^{P}}
%\newcommand{\muparent}{\inmu_{\text{coarse}}}
%\newcommand{\muchild}{\inmu_{\text{children}}}
@@ -596,32 +600,27 @@ We next show, for query-reference problems, a set of generalized ``rules'' enume
%Nonetheless, this simple model leads to effective parallelization of problems such as two-point correlation\footnote{list more}.
\subsection{Query-reference instrinsic pruning}
A query-reference problem computes for each query $q$,
A query-reference problem computes for each query a \defterm{query result} $\outqr(q, \kdroot{R})$, expressed
\begin{eqnarray}
\outqr(q, R) &=& \gqr(q, \inqr(q, R)),
\outqr(q, R) &=& \gqr(q, \inqrv(\{q\}, R)),
\\
\inqr(q, R) &=& \Opqr_{r \in R} \fqr(q, r).
\inqrv(\{q\}, R) &=& \Opqr_{r \in R} \fqr(q, r).
\label{eqn:qrdef}
\end{eqnarray}
\noindent where $R$ is initially $\kdroot{R}$.
In addition to the classic \nbody\ force calculation problem, this encompasses all nearest-neighbors, k-nearest-neighbors classification, nonparametric Bayes classification, kernel density estimation, affinity propagation, and more.
Although each query is independent, speedup is achievable by considering queries {\it en masse}; that is, the contribution of a set of references might be shown to have an exact value for an entire distant set of queries.
A \defterm{mass result} $\inqrv(Q, R)$ is defined if it is acceptable to treat
\[
\forall q \in Q,~~ \inqr(q, R) \gets \inqrv(Q, R)
\]
\noindent
within the context of the entire computation.
In addition to the classic \nbody\ force calculation problem, this encompasses all nearest-neighbors, k-nearest-neighbors classification, nonparametric Bayes classification, kernel density estimation, affinity propagation, and more.
Although each query is independent, speedup is achievable by considering queries {\it en masse}; that is, the contribution of a set of references might be shown to have an exact value for an entire set of queries.
A \defterm{mass result} $\inqrv(Q, R)$ may exist if it is acceptable to use that same value for each query given a reference node.
Otherwise, $\inqrv(Q,R)$ is undefined.
Expressed in tree notation,
Hierarchically, we express,
\begin{equation}
\text{if prune occurs for } \kdparent{Q} \supset Q \text{, then } \inqrv(Q, R) = \inqrv(\kdparent{Q}, R).
\text{if } \kdparent{Q} \supset Q \text{ and }\inqrv(\kdparent{Q}, R)\text{ is defined, } \inqrv(Q, R) \gets \inqrv(\kdparent{Q}, R).
\label{eqn:qrvparent}
\end{equation}
\noindent
Note that $\inqrv$ must be defined for singleton queries, and a single-tree algorithm conceptually treates $\inqrv$ in only this way.
Recall that $\inqrv$ is defined for singleton queries, and a single-tree algorithm conceptually treates $\inqrv$ in only this way.
It is important to note that although a dual-tree algorithm recursively descends the query tree, there is no explicit data dependency within the query tree, except the ability to incorporate mass results pruned from a parent.
Our mathematical model leaves to the implementation how the query set is divided.
Both types of algorithms, though, perform divide and conquer with the reference tree,
@@ -636,7 +635,7 @@ Both types of algorithms, though, perform divide and conquer with the reference
\end{equation}
\noindent
with statistics $\outstat(Q)$ and $\outstat(R)$, an intrinsic prune indicator function $\canpruneqrv$, and intrinsic prune value $\deltaqrv$.
with node statistics $\outstat(Q)$ and $\outstat(R)$, an intrinsic prune indicator function $\canpruneqrv$, and intrinsic prune value $\deltaqrv$.
Statistics are frequently built from commutative, associative operators, and may also be built bottom-up,
\begin{eqnarray}
\outstat(X) &=& \gstat(\instat(X)),
@@ -663,7 +662,7 @@ Range count, the query-reference analog to two-point correlation, is fully defin
\noindent
The two-point correlation is then the sum of $\outqr$ for all queries.
Equations \ref{eqn:qrdef} through \ref{eqn:defstat} thus express the data flow of any query-reference problem that prunes only intrinsically using arbitrary commutative, associative statistics.
Equations \ref{eqn:qrdef} through \ref{eqn:defstat} thus express the data flow of any query-reference problem, and allows intrinsic prunes.
\subsection{Query-Reference Extrinsic Prunes}
@@ -788,7 +787,7 @@ Although further discussion of the merits is warranted, it is beyond the scope o
\[
\begin{array}[t]{l}
\\ \text{Input:}\left(
\begin{array}[c]{l}\kdroot{Q}, \kdroot{R}, \gqr, \opqr, \fqr, \deltaqrv, \\ \heurqr, \canpruneqrv, \canprunemu, \outopmu, \opmu, \fmuv, \deltamu\end{array}\right)
\begin{array}[c]{l}\kdroot{Q}, \kdroot{R}, \gqr, \opqr, \fqr, \deltaqrv, \\ \heurqr, \canprunemu, \outopmu, \opmu, \fmuv, \deltamu\end{array}\right)
\X \text{for all nodes } Q \in \kdroot{Q}\text{, } \varmuchild(Q) \gets \text{identity of }\Opmu
\X \text{for all nodes } Q \in \kdroot{Q}\text{, } \varqrv(Q) \gets \text{identity of }\Opqr%\identqr
\X \text{dfe}(\kdroot{Q}, \kdroot{R}, \text{identity of }\Opmu)
@@ -806,14 +805,14 @@ Although further discussion of the merits is warranted, it is beyond the scope o
\\ &\psty\!\!\!\!\opmu\!\!\!\!& \psty\deltamu(\outstat(Q), \outstat(R)) & \!\!\!\text{\com{node pair's $\lettermu$, Eqn~\ref{eqn:mudelta}}}
\\ &\psty\!\!\!\!\opmu\!\!\!\!& \psty\varmuparent & \!\!\!\text{\com{unvisited $\lettermu$, Eqn~\ref{eqn:muparent}}}
\end{array}
\x \text{if } \exists q \exists r ~ Q = \{q\}\text{ and }R = \{r\}\text{,}
\x \text{if } Q = \{q\}\text{ and }R = \{r\}\text{ singletons,}
\xx \text{\com{leaf-leaf interaction, Eqns \ref{eqn:qrdef}, \ref{eqn:qrvcompose}}}
\xx \varqrv(\{q\}) \gets \varqrv(\{Q\}) \opqr \fqr(q, r)
\x \text{else if } \canpruneqrv(\outstat(Q), \outstat(R))\text{ or }\canprunemu(\outstat(Q), \outstat(R), \lettermu)\text{,}
\xx \varqrv(\{q\}) \gets \varqrv(\{q\}) \opqr \fqr(q, r)
\x \text{else if } \canprunemu(\outstat(Q), \outstat(R), \lettermu)\text{,}
\xx \text{\com{prune node-node interaction, Eqns \ref{eqn:qrvprune}, \ref{eqn:qrvcompose}}}
\xx \varqrv(Q) \gets \varqrv(Q) \opqr \deltaqrv(\outstat(Q), \outstat(R))
\x \text{else if } |Q| \geq |R|\text{,}
\xx \text{\com{independently explore query children}}
\xx \text{\com{explore both query children}}
\xx \text{for } Q' \in \{\kdleft{Q}, \kdright{Q}\}\text{,}
\xxx \text{\com{apply pruning information, Eqns \ref{eqn:qrvparent}, \ref{eqn:qrvcompose}}}
\xxx \varqrv(Q') \gets \varqrv(Q') \opqr \varqrv(Q)
@@ -822,8 +821,8 @@ Although further discussion of the merits is warranted, it is beyond the scope o
\xx \varqrv(Q) \gets \text{identity of }\Opqr
\xx \text{\com{recompute high-quality $\lettermu$ bottom-up, Eqn~\ref{eqn:muchild}}}
\xx \!\!\!\begin{array}{lll}
\psty \varmuchild(Q) &\psty\!\!\gets\!\!&\psty (\varmuchild(\kdleft{Q}) \opmu \fmuv(\kdleft{Q}, \varqrv(\kdleft{Q})))
\\ &\psty\!\!\outopmu\!\!&\psty (\varmuchild(\kdright{Q}) \opmu \fmuv(\kdright{Q}, \varqrv(\kdright{Q})))
\psty \varmuchild(Q) &\psty\!\!\gets\!\!&\psty (\varmuchild(\kdleft{Q}) \opmu \fmuv(\outstat(\kdleft{Q}), \varqrv(\kdleft{Q})))
\\ &\psty\!\!\outopmu\!\!&\psty (\varmuchild(\kdright{Q}) \opmu \fmuv(\outstat(\kdright{Q}), \varqrv(\kdright{Q})))
\end{array}
\x \text{else}
\xx \text{\com{explore reference children in heuristic order}}
@@ -836,8 +835,8 @@ Although further discussion of the merits is warranted, it is beyond the scope o
\X \text{function fixup}(Q, \varqrvparent)\text{,}
\x \text{\com{propagate mass results to leaves}}
\x \varqrv(Q) \gets \varqrv(Q) \opqr \varqrvparent
\x \text{if } \exists q ~ Q = \{q\}\text{,}
\xx \outqr(q, \kdroot{R}) \gets \gqr(q, \varqrv(Q))
\x \text{if } Q = \{q\}\text{ singleton,}
\xx \outqr(q, \kdroot{R}) \gets \gqr(q, \varqrv(\{q\}))
\x \text{else,}
\xx \text{fixup}(\kdleft{Q}, \varqrv(Q))
\xx \text{fixup}(\kdright{Q}, \varqrv(Q))