Sharper dimension-free bounds on the Frobenius distance between sample covariance and its expectation

08/28/2023
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by   Nikita Puchkin, et al.
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We study properties of a sample covariance estimate Ξ£= (𝐗_1 𝐗_1^⊀ + … + 𝐗_n 𝐗_n^⊀) / n, where 𝐗_1, …, 𝐗_n are i.i.d. random elements in ℝ^d with 𝔼𝐗_1 = 0, 𝔼𝐗_1 𝐗_1^⊀ = Ξ£. We derive dimension-free bounds on the squared Frobenius norm of (Ξ£- Ξ£) under reasonable assumptions. For instance, we show that | Ξ£- Ξ£_ F^2 - 𝔼Σ- Ξ£_ F^2| = π’ͺ(Tr(Ξ£^2) / n) with overwhelming probability, which is a significant improvement over the existing results. This leads to a bound the ratio Ξ£- Ξ£_ F^2 / 𝔼Σ- Ξ£_ F^2 with a sharp leading constant when the effective rank πš›(Ξ£) = Tr(Ξ£) / Ξ£ and n / πš›(Ξ£)^6 tend to infinity: Ξ£- Ξ£_ F^2 / 𝔼Σ- Ξ£_ F^2 = 1 + π’ͺ(1 / πš›(Ξ£)).

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