اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMatrix multiplication algorithms from group orbits
201
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n=2,3,4,5, recovering Strassens algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassens algorithm; the same proof works for all n, to yield a decomposition with $n^3 - n + 1$ terms.
قيم البحث
اقرأ أيضاً
We develop a notion of {em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $ntimes n$ matrix multip
lication over an arbitrary field is at least $2n^2-n+1$. While inner rank does not provide improvements to currently known lower bounds, we argue that this notion merits further study.
While algebrisation constitutes a powerful technique in the design and analysis of centralised algorithms, to date there have been hardly any applications of algebraic techniques in the context of distributed graph algorithms. This work is a case stu
dy that demonstrates the potential of algebrisation in the distributed context. We will focus on distributed graph algorithms in the congested clique model; the graph problems that we will consider include, e.g., the triangle detection problem and the all-pairs shortest path problem (APSP). There is plenty of prior work on combinatorial algorithms in the congested clique model: for example, Dolev et al. (DISC 2012) gave an algorithm for triangle detection with a running time of $tilde O(n^{1/3})$, and Nanongkai (STOC 2014) gave an approximation algorithm for APSP with a running time of $tilde O(n^{1/2})$. In this work, we will use algebraic techniques -- in particular, algorithms based on fast matrix multiplication -- to solve both triangle detection and the unweighted APSP in time $O(n^{0.15715})$; for weighted APSP, we give a $(1+o(1))$-approximation with this running time, as well as an exact $tilde O(n^{1/3})$ solution.
We propose COSMA: a parallel matrix-matrix multiplication algorithm that is near communication-optimal for all combinations of matrix dimensions, processor counts, and memory sizes. The key idea behind COSMA is to derive an optimal (up to a factor of
0.03% for 10MB of fast memory) sequential schedule and then parallelize it, preserving I/O optimality. To achieve this, we use the red-blue pebble game to precisely model MMM dependencies and derive a constructive and tight sequential and parallel I/O lower bound proofs. Compared to 2D or 3D algorithms, which fix processor decomposition upfront and then map it to the matrix dimensions, it reduces communication volume by up to $sqrt{3}$ times. COSMA outperforms the established ScaLAPACK, CARMA, and CTF algorithms in all scenarios up to 12.8x (2.2x on average), achieving up to 88% of Piz Daints peak performance. Our work does not require any hand tuning and is maintained as an open source implementation.
Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (suc
h as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group. In particular, we determine that for cryo-EM with noise variance $sigma^2$ and uniform viewing directions, the number of samples required scales as $sigma^6$. We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.
This PhD thesis summarizes research works on the design of exact algorithms that provide a worst-case (time or space) guarantee for NP-hard scheduling problems. Both theoretical and practical aspects are considered with three main results reported. T
he first one is about a Dynamic Programming algorithm which solves the F3Cmax problem in O*(3^n) time and space. The algorithm is easily generalized to other flowshop problems and single machine scheduling problems. The second contribution is about a search tree method called Branch & Merge which solves the 1||SumTi problem with the time complexity converging to O*(2^n) and in polynomial space. Our third contribution aims to improve the practical efficiency of exact search tree algorithms solving scheduling problems. First we realized that a better way to implement the idea of Branch & Merge is to use a technique called Memorization. By the finding of a new algorithmic paradox and the implementation of a memory cleaning strategy, the method succeeded to solve instances with 300 more jobs with respect to the state-of-the-art algorithm for the 1||SumTi problem. Then the treatment is extended to another three problems 1|ri|SumCi, 1|dtilde|SumwiCi and F2||SumCi. The results of the four problems all together show the power of the Memorization paradigm when applied on sequencing problems. We name it Branch & Memorize to promote a systematic consideration of Memorization as an essential building block in branching algorithms like Branch and Bound. The method can surely also be used to solve other problems, which are not necessarily scheduling problems.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
