Communication complexity
Rectangular matrix multiplication revisited
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On data structures and asymmetric communication complexity
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Fast rectangular matrix multiplication and applications
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The space complexity of approximating the frequency moments
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Finding Frequent Items in Data Streams
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Sampling lower bounds via information theory
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Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
An improved data stream summary: the count-min sketch and its applications
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Foundations and Trends® in Theoretical Computer Science
Improved Approximation Algorithms for Large Matrices via Random Projections
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Fast computation of low-rank matrix approximations
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Spectral norm of random matrices
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A sparse Johnson: Lindenstrauss transform
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Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
1-pass relative-error Lp-sampling with applications
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Lower bounds for sparse recovery
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Streaming algorithms for extent problems in high dimensions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Streaming graph computations with a helpful advisor
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
A Randomized Algorithm for Principal Component Analysis
SIAM Journal on Matrix Analysis and Applications
Theory of data stream computing: where to go
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A unified framework for approximating and clustering data
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Fast moment estimation in data streams in optimal space
Proceedings of the forty-third annual ACM symposium on Theory of computing
Subspace embeddings for the L1-norm with applications
Proceedings of the forty-third annual ACM symposium on Theory of computing
Multiplicative approximations of random walk transition probabilities
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Everywhere-tight information cost tradeoffs for augmented index
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Almost optimal explicit Johnson-Lindenstrauss families
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Compressed matrix multiplication
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Sparser Johnson-Lindenstrauss transforms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Optimal bounds for Johnson-Lindenstrauss transforms and streaming problems with sub-constant error
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Low rank matrix-valued chernoff bounds and approximate matrix multiplication
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Graph sketches: sparsification, spanners, and subgraphs
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Explicit Dimension Reduction and Its Applications
SIAM Journal on Computing
Streaming computations with a loquacious prover
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An Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Optimal Bounds for Johnson-Lindenstrauss Transforms and Streaming Problems with Subconstant Error
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Simple and deterministic matrix sketching
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Low rank approximation and regression in input sparsity time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Compressed matrix multiplication
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
Direct product via round-preserving compression
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an arbitrary order. We give the first lower bounds known for the space needed by the sketches, for a given estimation error ε. We sharpen prior upper bounds, with respect to combinations of space, failure probability, and number of passes. The sketch we use for matrix A is simply STA, where S is a sign matrix. Our results include the following upper and lower bounds on the bits of space needed for 1-pass algorithms. Here A is an n x d matrix, B is an n x d' matrix, and c := d+d'. These results are given for fixed failure probability; for failure probability δ0, the upper bounds require a factor of log(1/δ) more space. We assume the inputs have integer entries specified by O(log(nc)) bits, or O(log(nd)) bits. (Matrix Product) Output matrix C with F(ATB-C) ≤ ε F(A) F(B). We show that Θ(cε-2log(nc)) space is needed. (Linear Regression) For d'=1, so that B is a vector b, find x so that Ax-b ≤ (1+ε) minx' ∈ Reald Ax'-b. We show that Θ(d2ε-1 log(nd)) space is needed. (Rank-k Approximation) Find matrix tAk of rank no more than k, so that F(A-tAk) ≤ (1+ε) F{A-Ak}, where Ak is the best rank-k approximation to A. Our lower bound is Ω(kε-1(n+d)log(nd)) space, and we give a one-pass algorithm matching this when A is given row-wise or column-wise. For general updates, we give a one-pass algorithm needing [O(kε-2(n + d/ε2)log(nd))] space. We also give upper and lower bounds for algorithms using multiple passes, and a sketching analog of the CUR decomposition.