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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 $|A-B|_p^p=sum_{i,j}|A_{i,j}-B_{i,j}|^p$ when $p > 0$; and $|A-B|_0=sum_{i,j}[A_{i,j} eq B_{i,j}]$ for $p=0$. On the algorithmic side, for $p in (0,2)$, we give the first $(1+epsilon)$-approximation algorithm running in time $n^{text{poly}(k/epsilon)}$. Further, for $p = 0$, we give the first almost-linear time approximation scheme for what we call the Generalized Binary $ell_0$-Rank-$k$ problem. Our algorithm computes $(1+epsilon)$-approximation in time $(1/epsilon)^{2^{O(k)}/epsilon^{2}} cdot nd^{1+o(1)}$. On the hardness of approximation side, for $p in (1,2)$, assuming the Small Set Expansion Hypothesis and the Exponential Time Hypothesis (ETH), we show that there exists $delta := delta(alpha) > 0$ such that the entrywise $ell_p$-Rank-$k$ problem has no $alpha$-approximation algorithm running in time $2^{k^{delta}}$.
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 t
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, t
We provide a number of algorithmic results for the following family of problems: For a given binary mtimes n matrix A and integer k, decide whether there is a simple binary matrix B which differs from A in at most k entries. For an integer r, the sim
In applications such as natural language processing or computer vision, one is given a large $n times d$ matrix $A = (a_{i,j})$ and would like to compute a matrix decomposition, e.g., a low rank approximation, of a function $f(A) = (f(a_{i,j}))$ appl
We show how to approximate a data matrix $mathbf{A}$ with a much smaller sketch $mathbf{tilde A}$ that can be used to solve a general class of constrained k-rank approximation problems to within $(1+epsilon)$ error. Importantly, this class of problem