# On the Communication Complexity of Linear Algebraic Problems in the Message Passing Model

@inproceedings{Li2014OnTC, title={On the Communication Complexity of Linear Algebraic Problems in the Message Passing Model}, author={Yi Li and Xiaoming Sun and Chenguang Wang and David P. Woodruff}, booktitle={DISC}, year={2014} }

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized s-player communication complexity of these problems is at least s times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, we can…

## 17 Citations

### Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

- Computer Science, MathematicsSIAM J. Comput.
- 2016

This paper proves lower bounds on randomized multiparty communication complexity, both in the blackboard model and in the message-passing model, and introduces a new technique for proving such bounds, called symmetrization, which is natural, intuitive, and often easy to use.

### Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

- Computer Science, MathematicsICML
- 2015

This work shows that any deterministic algorithm solving the generalized matrix rank estimation problem must communicate Ω(n2) bits, which is order equivalent to transmitting the whole matrix, and proposes a randomized algorithm that communicates only e O(n) bits.

### The Effect of Range and Bandwidth on the Round Complexity in the Congested Clique Model

- Computer ScienceCOCOON
- 2016

The space between the unicast and broadcast congested clique models is very rich and interesting, and it is shown that the round complexity of the pairwise set-disjointness function \(\textsc {pwdisj}\) is completely sensitive to the range r.

### On the Multiparty Communication Complexity of Testing Triangle-Freeness

- Mathematics, Computer SciencePODC
- 2017

Evidence that the problem is hard on testing triangle-freeness is given by showing that finding an edge that participates in a triangle is hard, even when promised that the graph is far from triangle-free.

### A Rounds vs. Communication Tradeoff for Multi-Party Set Disjointness

- Computer Science2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

The trade-off between the number of interaction rounds the authors allow the players, and the total number of bits they must send to solve set disjointness is studied, and it is shown that if R rounds of interaction are allowed, the communication cost is Ω(nk^{1/R}/R^4), which is nearly tight.

### The Range of Topological Effects on Communication

- Computer ScienceICALP
- 2015

It is shown that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting the terminals, and for natural composed functions like Element-Distinctness ED and XOR the naive protocols suggested by their definition is optimal for general networks.

### Tight Bounds for the Subspace Sketch Problem with Applications

- Computer Science, Mathematics
- 2019

The results are optimal up to logarithmic factors, and show in particular that one cannot compress A to O(d) “directions” v1, vO(d), such that for any x, ‖Ax‖1 can be well-approximated from 〈v1, x〉, . . . , 〉vO (d), x》.

### Communication-Efficient Distributed Covariance Sketch, with Application to Distributed PCA

- Computer ScienceJ. Mach. Learn. Res.
- 2021

This paper proves an almost tight deterministic communication lower bound, then provides a new randomized algorithm with communication cost smaller than the deterministic lower bound and gives an improved distributed PCA algorithm for sparse input matrices, which uses the distributed sketching algorithm as a key building block.

### 1 0 Ju l 2 01 9 Tight Bounds for the Subspace Sketch Problem with Applications

- Computer Science, Mathematics
- 2019

The results are optimal up to logarithmic factors, and show in particular that one cannot compress A to O(d) “directions” v1, x, vO(d), such that for any x, ‖Ax‖1 can be well-approximated from 〈v1,x〉, . . . , 〉vO( d), x〉.

### Space lower bounds for linear prediction in the streaming model

- Computer Science, MathematicsCOLT
- 2019

It is shown that fundamental learning tasks, such as finding an approximate linear separator or linear regression, require memory at least \emph{quadratic} in the dimension, in a natural streaming setting, and that such problems cannot be solved by scalable memory-efficient streaming algorithms.

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