arXiv subjects

Quentin F. Stout

Publications and source records attributed to Quentin F. Stout.

9 recordsLinked to original sources

arXiv2024

Fine-Grained Computation in 3-Space: Matrix Multiplication and Graph Problems

Obeying constraints imposed by classical physics, we give optimal fine-grained algorithms for matrix multiplication and problems involving graphs and mazes, where all calculations are done in 3-dimensional space. We assume that whatever the technology is, a bit requires a minimum volume and communication travels at a bounded speed. These imply that multiplying $n \times n$ matrices takes $\Omega(n^{2/3})$ time, and we show that this can be achieved by a fine-grained 3-d mesh of $n^2$ processors. While the constants are impractically large, this is asymptotically faster than parallel implementations of Strassen's algorithm, while the lower bound shows that some claims about parallelizing faster serial algorithms are impossible in 3-space. If the matrices are not over a ring then multiplication can be done in $\Theta(n^{3/4})$ time by expanding to a mesh larger than the input. In 2-d (such as the surface of a chip) this approach is useless and $\Theta(n)$ systolic algorithms are optimal even when the matrices are over a ring. Similarly, for path and maze problems there are approaches useful in 3-d but not 2-d.

cs.DS↗

arXiv2023

$L_p$ Isotonic Regression Algorithms Using an $L_0$ Approach

Significant advances in flow algorithms have changed the relative performance of various approaches to algorithms for $L_p$ isotonic regression. We show a simple plug-in method to systematically incorporate such advances, and advances in determining violator dags, with no assumptions about the algorithms' structures. The method is based on the standard algorithm for $L_0$ (Hamming distance) isotonic regression (by finding anti-chains in a violator dag), coupled with partitioning based on binary $L_1$ isotonic regression. For several important classes of graphs the algorithms are already faster (in O-notation) than previously published ones, close to or at the lower bound, and significantly faster than those implemented in statistical packages. We consider exact and approximate results for $L_p$ regressions, $p=0$ and $1 \leq p < \infty$, and a variety of orderings.

cs.DS↗

arXiv2023

Best $L_p$ Isotonic Regressions, $p \in \{0, 1, \infty\}$

Given a real-valued weighted function $f$ on a finite dag, the $L_p$ isotonic regression of $f$, $p \in [0,\infty]$, is unique except when $p \in [0,1] \cup \{\infty\}$. We are interested in determining a ``best'' isotonic regression for $p \in \{0, 1, \infty\}$, where by best we mean a regression satisfying stronger properties than merely having minimal norm. One approach is to use strict $L_p$ regression, which is the limit of the best $L_q$ approximation as $q$ approaches $p$, and another is lex regression, which is based on lexical ordering of regression errors. For $L_\infty$ the strict and lex regressions are unique and the same. For $L_1$, strict $q \scriptstyle\searrow 1$ is unique, but we show that $q \scriptstyle\nearrow 1$ may not be, and even when it is unique the two limits may not be the same. For $L_0$, in general neither of the strict and lex regressions are unique, nor do they always have the same set of optimal regressions, but by expanding the objectives of $L_p$ optimization to $p < 0$ we show $p{ \scriptstyle \nearrow} 0$ is the same as lex regression. We also give algorithms for computing the best $L_p$ isotonic regression in certain situations.

cs.DM↗

arXiv2022

$L_0$ Isotonic Regression With Secondary Objectives

We provide algorithms for isotonic regression minimizing $L_0$ error (Hamming distance). This is also known as monotonic relabeling, and is applicable when labels have a linear ordering but not necessarily a metric. There may be exponentially many optimal relabelings, so we look at secondary criteria to determine which are best. For arbitrary ordinal labels the criterion is maximizing the number of labels which are only changed to an adjacent label (and recursively apply this). For real-valued labels we minimize the $L_p$ error. For linearly ordered sets we also give algorithms which minimize the sum of the $L_p$ and weighted $L_0$ errors, a form of penalized (regularized) regression. We also examine $L_0$ isotonic regression on multidimensional coordinate-wise orderings. Previous algorithms took $Θ(n^3)$ time, but we reduce this to $o(n^{3/2})$.

cs.DS↗

arXiv2021

Generating Artificial Core Users for Interpretable Condensed Data

Recent work has shown that in a dataset of user ratings on items there exists a group of Core Users who hold most of the information necessary for recommendation. This set of Core Users can be as small as 20 percent of the users. Core Users can be used to make predictions for out-of-sample users without much additional work. Since Core Users substantially shrink a ratings dataset without much loss of information, they can be used to improve recommendation efficiency. We propose a method, combining latent factor models, ensemble boosting and K-means clustering, to generate a small set of Artificial Core Users (ACUs) from real Core User data. Our ACUs have dense rating information, and improve the recommendation performance of real Core Users while remaining interpretable.

cs.IR↗

arXiv2017

L infinity Isotonic Regression for Linear, Multidimensional, and Tree Orders

Algorithms are given for determining $L_\infty$ isotonic regression of weighted data. For a linear order, grid in multidimensional space, or tree, of $n$ vertices, optimal algorithms are given, taking $Θ(n)$ time. These improve upon previous algorithms by a factor of $Ω(\log n)$. For vertices at arbitrary positions in $d$-dimensional space a $Θ(n \log^{d-1} n)$ algorithm employs iterative sorting to yield the functionality of a multidimensional structure while using only $Θ(n)$ space. The algorithms utilize a new non-constructive feasibility test on a rendezvous graph, with bounded error envelopes at each vertex.

cs.DS↗

arXiv2015

An Algorithm for $L_\infty$ Approximation by Step Functions

An algorithm is given for determining an optimal $b$-step approximation of weighted data, where the error is measured with respect to the $L_\infty$ norm. For data presorted by the independent variable the algorithm takes $Θ(n + \log n \cdot b(1+\log n/b))$ time and $Θ(n)$ space. This is $Θ(n \log n)$ in the worst case and $Θ(n)$ when $b = O(n/\log n \log\log n)$. A minor change determines an optimal reduced isotonic regression in the same time and space bounds, and the algorithm also solves the $k$-center problem for 1-dimensional weighted data.

cs.DS↗

arXiv2015

Optimal component labeling algorithms for mesh-connected computers and VLSI

Given an undirected graph $G$ of $n$ weighted edges, stored one edge per processor in a square mesh of $n$ processors, we show how to determine the connected components and a minimal spanning forest in $Θ(\sqrt{n})$ time. More generally, we show how to solve these problems in $Θ(n^{1/d})$ time when the mesh is a $d$-dimensional cube, where the implied constants depend upon $d$.

cs.DS↗

arXiv2014

Optimal Reduced Isotonic Regression

Isotonic regression is a shape-constrained nonparametric regression in which the regression is an increasing step function. For $n$ data points, the number of steps in the isotonic regression may be as large as $n$. As a result, standard isotonic regression has been criticized as overfitting the data or making the representation too complicated. So-called "reduced" isotonic regression constrains the outcome to be a specified number of steps $b$, $b \leq n$. However, because the previous algorithms for finding the reduced $L_2$ regression took $Θ(n+bm^2)$ time, where $m$ is the number of steps of the unconstrained isotonic regression, researchers felt that the algorithms were too slow and instead used approximations. Other researchers had results that were approximations because they used a greedy top-down approach. Here we give an algorithm to find an exact solution in $Θ(n+bm)$ time, and a simpler algorithm taking $Θ(n+b m \log m)$ time. These algorithms also determine optimal $k$-means clustering of weighted 1-dimensional data.

stat.CO↗