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We propose practical algorithms for entrywise $ell_p$-norm low-rank approximation, for $p = 1$ or $p = infty$. The proposed framework, which is non-convex and gradient-based, is easy to implement and typically attains better approximations, faster, than state of the art. From a theoretical standpoint, we show that the proposed scheme can attain $(1 + varepsilon)$-OPT approximations. Our algorithms are not hyperparameter-free: they achieve the desiderata only assuming algorithms hyperparameters are known a priori---or are at least approximable. I.e., our theory indicates what problem quantities need to be known, in order to get a good solution within polynomial time, and does not contradict to recent inapproximabilty results, as in [46].
A number of recent works have studied algorithms for entrywise $ell_p$-low rank approximation, namely, algorithms which given an $n times d$ matrix $A$ (with $n geq d$), output a rank-$k$ matrix $B$ minimizing $|A-B|_p^p=sum_{i,j}|A_{i,j}-B_{i,j}|^p$
Dictionary learning is a classic representation learning method that has been widely applied in signal processing and data analytics. In this paper, we investigate a family of $ell_p$-norm ($p>2,p in mathbb{N}$) maximization approaches for the comple
Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense tensor BR1Approx, and is a higher-order extension of the sparse matrix BR1Approx, is one of the most important problems in sparse tensor decompositio
We consider $ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $mtimes n$ matrix ${bf A}$ and a positive integer $r$, one seeks a binary matrix ${bf B}$ of rank at most $r$, minimizing the column-sum norm $||{bf A} -{bf B}||_1$. We show th
Simulating quantum algorithms on classical computers is challenging when the system size, i.e., the number of qubits used in the quantum algorithm, is moderately large. However, some quantum algorithms and the corresponding quantum circuits can be si