\begin{algorithm}
\label{subalg}
 

\begin{description}


\item[{\sc Input:}]
 Three integers $m$, $n$, and $r$ satisfying $0<2r \le n\le m$;
an $m\times n$ matrix $M$, given explicitly or implicitly.


\item[{\sc Initialization:}]
\begin{enumerate}
\item Generate a pair of  independent abridged   SRHT  matrices  $F$ and $H$  of length 3 and sizes $2r\times m$ and $n\times r$, respectively 
 (see Appendix \ref{spreprmlt});

\item Compute the sketches $FM$ and $MH$.
\end{enumerate}

\item[{\sc Computations:}]
\begin{enumerate}
\item
Compute matrix $Q$ as the Q factor of the thin QR factorization\footnote{Recall  the 
uniqueness of the thin QR factorization of an $m\times k$ matrix 
where  an $k\times k$ upper triangular factor
$R$ has positive diagonal entries
\cite[Thm. 5.3.2]{GL13}.} of $EH$.
\item
Compute the matrices $U$ and $T$ of the thin QR factorization $UT$ of $FQ$. 
\item
Compute and output the matrix $\Delta = QT^+U^T(FM)$, an LRA of $M$.
\end{enumerate}

\end{description}
\end{algorithm}

\begin{algorithm}
\begin{description}
\item[{\sc Input:}]
Three integers $m$, $n$, and $r$ satisfying $0<2r \le n\le m$;
an $m\times n$ matrix $M$, given explicitly or implicitly.
\item[{\sc Initialization:}]
\begin{enumerate}
\item Generate a pair of independent abridged SRHT matrices $F$ and $H$ of length 3 and sizes $2r\times m$ and $n\times r$, respectively
(see Appendix \ref{spreprmlt});
\item Compute the sketches $FM$ and $MH$.
\end{enumerate}
\item[{\sc Computations:}]
\begin{enumerate}
\item
Compute matrix $Q$ as the Q factor of the thin QR factorization\footnote{Recall the
uniqueness of the thin QR factorization of an $m\times k$ matrix
where an $k\times k$ upper triangular factor
$R$ has positive diagonal entries
\cite[Thm. 5.3.2]{GL13}.} of $EH$.
\item
Compute the matrices $U$ and $T$ of the thin QR factorization $UT$ of $FQ$.
\item
Compute and output the matrix $\Delta = QT^+U^T(FM)$, an LRA of $M$.
\end{enumerate}
\end{description}
\end{algorithm}
{
    "title": "superfast escalators for near-optimal low rank approximation of a matrix",
    "id": "1906.04223",
    "abstract": "a superfast (aka sublinear cost) algorithm only accesses a small fraction of all entries of an input matrix. we seek such algorithms for low rank approximation (lra) of a matrix, but for some matrix families any such algorithm at best halves the large error of the trivial approximation by the matrix filled with 0s. nevertheless, our simple superfast escalating algorithms compute near optimal lras of a new large class of real-world matrices according to our analysis and numerical tests.",
    "categories": "math.na cs.na",
    "doi": "",
    "created": "2019-06-10",
    "updated": "2024-01-06",
    "authors": [
        "soo go",
        "qi luan",
        "victor y. pan"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/1906.04223"
}
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