SQ Lower Bounds for Learning Bounded Covariance GMMs
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We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on ℝ^d of the form P= ∑_i=1^k w_i 𝒩(μ_i,Σ_i), where Σ_i = Σ≼𝐈 and min_i ≠ jμ_i - μ_j_2 ≥ k^ϵ for some ϵ>0. Known learning algorithms for this family of GMMs have complexity (dk)^O(1/ϵ). In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least d^Ω(1/ϵ). In the special case where the separation is on the order of k^1/2, we additionally obtain fine-grained SQ lower bounds with the correct exponent. Our SQ lower bounds imply similar lower bounds for low-degree polynomial tests. Conceptually, our results provide evidence that known algorithms for this problem are nearly best possible.
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