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Register a new userAn approximation algorithm for approximation rank
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Added by
Troy Lee
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.
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The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound was only established when the number of orthonormal factors is one. To this end, by further exploring the multilinearity and orthogonality of the problem, we introduce a modified approximation algorithm. Approximation lower bound is established, either in deterministic or expected sense, no matter how many orthonormal factors there are. In addition, a major feature of the new algorithm is its flexibility to allow either deterministic or randomized procedures to solve a key step of each latent orthonormal factor involved in the algorithm. This feature can reduce the computation of large SVDs, making the algorithm more efficient. Some numerical studies are provided to validate the usefulness of the proposed algorithm.
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are textsc{Low GF(2)-Rank Approximation}, textsc{Low Boolean-Rank Approximation}, and vario
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to $textit{edit}$ a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then $textit{lift}$ the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, ($ell$-)Dominating Set, Edge ($ell$-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of several important graph classes (in some cases, also approximating the target parameter of the family). For bounded degeneracy, we obtain a bicriteria $(4,4)$-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria $(O(log^{1.5} n), O(sqrt{log w}))$-approximation, and for bounded pathwidth, we obtain a bicriteria $(O(log^{1.5} n), O(sqrt{log w} cdot log n))$-approximation. For treedepth $2$ (also related to bounded expansion), we obtain a $4$-approximation. We also prove complementary hardness-of-approximation results assuming $mathrm{P} eq mathrm{NP}$: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor ($2$ assuming UGC).
We study a combinatorial problem called Minimum Maximal Matching, where we are asked to find in a general graph the smallest that can not be extended. We show that this problem is hard to approximate with a constant smaller than 2, assuming the Unique Games Conjecture. As a corollary we show, that Minimum Maximal Matching in bipartite graphs is hard to approximate with constant smaller than $frac{4}{3}$, with the same assumption. With a stronger variant of the Unique Games Conjecture --- that is Small Set Expansion Hypothesis --- we are able to improve the hardness result up to the factor of $frac{3}{2}$.
In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values, and then minimize the resulting approximation subject to the linear constraints. The accuracy of the approximation is controlled via a scaling parameter $delta$, where a smaller $delta$ corresponds to a more accurate fitting. The consequent optimization problem for any finite $delta$ is nonconvex. Therefore, in order to decrease the risk of ending up in local minima, a series of optimizations is performed, starting with optimizing a rough approximation (a large $delta$) and followed by successively optimizing finer approximations of the rank with smaller $delta$s. To solve the optimization problem for any $delta > 0$, it is converted to a new program in which the cost is a function of two auxiliary positive semidefinete variables. The paper shows that this new program is concave and applies a majorize-minimize technique to solve it which, in turn, leads to a few convex optimization iterations. This optimization scheme is also equivalent to a reweighted Nuclear Norm Minimization (NNM), where weighting update depends on the used approximating function. For any $delta > 0$, we derive a necessary and sufficient condition for the exact recovery which are weaker than those corresponding to NNM. On the numerical side, the proposed algorithm is compared to NNM and a reweighted NNM in solving affine rank minimization and matrix completion problems showing its considerable and consistent superiority in terms of success rate, especially, when the number of measurements decreases toward the lower-bound for the unique representation.
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