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Bj,k = jkBc, (4.32) for some matrices 2 Fp×m, 2 Fp×n, and constants Ac 2 F s p × r m , Bc 2 F s p × t n satisfying Ac Bc 6= 0. Consequently, we have dK 0 @ 8< : 0 @ X j0,k0 j0k0aij0k0 1 A 0 @ X j00,k00 j00k00bij00k00 1 AAc Bc 9= ; i2K 1 A = AB (4.33) for all possible values of , , and K. Fixing the value i, we can view each subtensor aijk as a vector of length pm, and each subtensor bijk as a vector of length pn. For brevity, we denote each such vector by ai and bi respectively. Similarly, we can also view matrices and as vectors of length pm and pn, and we denote these vectors by and . Furthermore, we can define dot products within these vector spaces following the conventions. Using these notations, (4.33) can be written as dK {( · ai) ( · bi)Ac Bc}i2K = AB. (4.34) Given the above definitions, we now prove that within each subset K of size K linear, the vectors {ai}i2K span the space Fpm. Essentially, we need to prove that for any such given subset K, there does not exist a nonzero 2 Fp×m such that the corresponding vector 2 Fpm satisfies · ai = 0 for all i 2 K. Assume the opposite that such an exists, so that · ai is always 0, then the LHS of (4.34) becomes a fixed value. On the other hand, since is nonzero, we can always find different values of such that  is variable. Recalling (4.31) and (4.32), the RHS of (4.34) cannot be fixed if  is variable, which results in a contradiction. Now we use this conclusion to prove (4.29). For any fixed K with size K linear, let B be a subset of indices in K such that {ai}i2B form a basis. Recall that we are considering the case where K linear < N, meaning that we can find a worker ˜k 62 K. For convenience, we define K+ = K [ {˜k}, and K− , K+\B. Obviously, B = pm, and K− = K+−B = K linear +1−pm. Hence, it suffices to prove that K− pn, which only requires that {bi}i2K− forms a basis of Fpn. Equivalently, we only need to prove that any 2 Fp×n such that its vectorized version 2 Fpn satisfies · bi = 0 for any i 2 K− must be zero. For brevity, we let B denotes the subspace that contains all values of satisfying this condition. To prove this statement, we first construct a list of matrices as follows, denoted by { i}i2B. Recall that {ai}i2B forms a basis. We can find a matrix i 2 Fp×m for each i 2 B such that their vectorized 42
Object Description
Title  Coded computing: a transformative framework for resilient, secure, private, and communication efficient large scale distributed computing 
Author  Yu, Qian 
Author email  qyu880@usc.edu;qianyu0929@gmail.com 
Degree  Doctor of Philosophy 
Document type  Dissertation 
Degree program  Electrical Engineering 
School  Viterbi School of Engineering 
Date defended/completed  20200324 
Date submitted  20200804 
Date approved  20200805 
Restricted until  20200805 
Date published  20200805 
Advisor (committee chair)  Avestimehr, Salman 
Advisor (committee member) 
Luo, HaiPeng Ortega, Antonio Soltanolkotabi, Mahdi 
Abstract  Modern computing applications often require handling massive amounts of data in a distributed setting, where significant issues on resiliency, security, or privacy could arise. This dissertation presents new computing designs and optimality proofs, that address these issues through coding and informationtheoretic approaches. ❧ The first part of this thesis focuses on a standard setup, where the computation is carried out using a set of worker nodes, each can store and process a fraction of the input dataset. The goal is to find computation schemes for providing the optimal resiliency against stragglers given the computation task, the number of workers, and the functions computed by the workers. The resiliency is measured in terms of the recovery threshold, defined as the minimum number of workers to wait for in order to compute the final result. We propose optimal solutions for broad classes of computation tasks, from basic building blocks such as matrix multiplication (entangled polynomial codes), Fourier transform (coded FFT), and convolution (polynomial code), to general functions such as multivariate polynomial evaluation (Lagrange coded computing). We develop optimal computing strategies by introducing a general coding framework called “polynomial coded computing”, to exploit the algebraic structure of the computation task and create computation redundancy in a novel coded form across workers. Polynomial coded computing allows for orderwise improvements over the state of the arts and significantly generalizes classical codingtheoretic results to go beyond linear computations. The encoding and decoding process of polynomial coded computing designs can be mapped to polynomial evaluation and interpolation, which can be computed efficiently. ❧ Then we show that polynomial coded computing can be extended to provide unified frameworks that also enable security and privacy in the computation. We present the optimal designs for three important problems: distributed matrix multiplication, multivariate polynomial evaluation, and gradienttype computation. We prove their optimality by developing informationtheoretic and linearalgebraic converse bounding techniques. ❧ Finally, we consider the problem of coding for communication reduction. In the context of distributed computation, we focus on a MapReducetype framework, where the workers need to shuffle their intermediate results to finish the computation. We aim to understand how to optimally exploit extra computing power to reduce communication, i.e., to establish a fundamental tradeoff between computation and communication. We prove a lower bound on the needed communication load for general allocation of the task assignments, by introducing a novel informationtheoretic converse bounding approach. The presented lower bound exactly matches the inverseproportional coding gain achieved by coded distributed computing schemes, completely characterizing the optimal computationcommunication tradeoff. The proposed converse bounding approach strictly improves conventional cutset bounds and can be widely applied to prove exact optimally results for more general settings, as well as more classical communication problems. We also investigate a problem called coded caching, where a single server is connected to multiple users in a cache network through a shared bottleneck link. Each user has an isolated memory that can be used to prefetch content. Then the server needs to deliver users’ demands efficiently in a following delivery phase. We propose caching and delivery designs that improve the stateoftheart schemes under both centralized and decentralized settings, for both peak and average communication rates. Moreover, by developing informationtheoretic bounds, we prove the proposed designs are exactly optimal among all schemes that use uncoded prefetching, and optimal within a factor of 2.00884 among schemes with coded prefetching. 
Keyword  coding theory; information theory; distributed computing; security; privacy; matrix multiplication; caching 
Language  English 
Part of collection  University of Southern California dissertations and theses 
Publisher (of the original version)  University of Southern California 
Place of publication (of the original version)  Los Angeles, California 
Publisher (of the digital version)  University of Southern California. Libraries 
Provenance  Electronically uploaded by the author 
Type  texts 
Legacy record ID  uscthesesm 
Contributing entity  University of Southern California 
Rights  Yu, Qian 
Physical access  The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository email address given. 
Repository name  University of Southern California Digital Library 
Repository address  USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 900897002, USA 
Repository email  cisadmin@lib.usc.edu 
Filename  etdYuQian8883.pdf 
Archival file  Volume13/etdYuQian8883.pdf 
Description
Title  Page 57 
Full text  Bj,k = jkBc, (4.32) for some matrices 2 Fp×m, 2 Fp×n, and constants Ac 2 F s p × r m , Bc 2 F s p × t n satisfying Ac Bc 6= 0. Consequently, we have dK 0 @ 8< : 0 @ X j0,k0 j0k0aij0k0 1 A 0 @ X j00,k00 j00k00bij00k00 1 AAc Bc 9= ; i2K 1 A = AB (4.33) for all possible values of , , and K. Fixing the value i, we can view each subtensor aijk as a vector of length pm, and each subtensor bijk as a vector of length pn. For brevity, we denote each such vector by ai and bi respectively. Similarly, we can also view matrices and as vectors of length pm and pn, and we denote these vectors by and . Furthermore, we can define dot products within these vector spaces following the conventions. Using these notations, (4.33) can be written as dK {( · ai) ( · bi)Ac Bc}i2K = AB. (4.34) Given the above definitions, we now prove that within each subset K of size K linear, the vectors {ai}i2K span the space Fpm. Essentially, we need to prove that for any such given subset K, there does not exist a nonzero 2 Fp×m such that the corresponding vector 2 Fpm satisfies · ai = 0 for all i 2 K. Assume the opposite that such an exists, so that · ai is always 0, then the LHS of (4.34) becomes a fixed value. On the other hand, since is nonzero, we can always find different values of such that  is variable. Recalling (4.31) and (4.32), the RHS of (4.34) cannot be fixed if  is variable, which results in a contradiction. Now we use this conclusion to prove (4.29). For any fixed K with size K linear, let B be a subset of indices in K such that {ai}i2B form a basis. Recall that we are considering the case where K linear < N, meaning that we can find a worker ˜k 62 K. For convenience, we define K+ = K [ {˜k}, and K− , K+\B. Obviously, B = pm, and K− = K+−B = K linear +1−pm. Hence, it suffices to prove that K− pn, which only requires that {bi}i2K− forms a basis of Fpn. Equivalently, we only need to prove that any 2 Fp×n such that its vectorized version 2 Fpn satisfies · bi = 0 for any i 2 K− must be zero. For brevity, we let B denotes the subspace that contains all values of satisfying this condition. To prove this statement, we first construct a list of matrices as follows, denoted by { i}i2B. Recall that {ai}i2B forms a basis. We can find a matrix i 2 Fp×m for each i 2 B such that their vectorized 42 