A PTAS for ℓ_p-Low Rank Approximation
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A number of recent works have studied algorithms for entrywise ℓ_p-low rank approximation, namely algorithms which given an n × d matrix A (with n ≥ d), output a rank-k matrix B minimizing A-B_p^p=∑_i,j |A_i,j - B_i,j|^p. We show the following: On the algorithmic side, for p ∈ (0,2), we give the first n^poly(k/ϵ) time (1+ϵ)-approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting for which A∈{0,1}^n× d and B = U · V, where U∈{0,1}^n × k and V∈{0,1}^k × d. There are also various notions of multiplication U · V, such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first PTAS for what we call the Generalized Binary ℓ_0-Rank-k Approximation problem, for which these variants are special cases. Our algorithm runs in time (1/ϵ)^2^O(k)/ϵ^2· nd ·^2^k d. For the specific case of finite fields of constant size, we obtain an alternate algorithm with time n · d^poly(k/ϵ). On the hardness front, for p ∈ (1,2), we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2^k^δ for a constant δ > 0, showing an exponential dependence on k is necessary. For p = 0, we observe that there is no approximation algorithm for the Generalized Binary ℓ_0-Rank-k Approximation problem running in time 2^2^δ k for a constant δ > 0. We also show for finite fields of constant size, under the ETH, that any fixed constant factor approximation algorithm requires 2^k^δ time for a constant δ > 0.
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