1,518 research outputs found

    Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms

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    Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to np/2n^{\lfloor p/2 \rfloor} for a pp-th order tensor in Rnp\mathbb{R}^{n^p}. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as n3/2/polylognn^{3/2}/\textrm{polylog} n. We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds, which can be useful in other settings

    Generalizing Amdahl’s Law for Power and Energy

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    Extending Amdahl\u27s law to identify optimal power-performance configurations requires considering the interactive effects of power, performance, and parallel overhead

    Towards a better approximation for sparsest cut?

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    We give a new (1+ϵ)(1+\epsilon)-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size n/rn/r expand by a factor lognlogr\sqrt{\log n\log r} bigger, for some small rr; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-rr Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time 2O(r)poly(n)2^{O(r)} \mathrm{poly}(n). We also show similar approximation algorithms in graphs with genus gg with an analogous local expansion condition. This is the first algorithm we know of that achieves (1+ϵ)(1+\epsilon)-approximation on such general family of graphs