Fast Exact Matrix Completion with Finite Samples - Semantic Scholar

Fast Exact Matrix Completion with Finite Samples Prateek Jain and Praneeth Netrapalli Mircrosoft Research

Problem: Low-rank Matrix Completion

• Task: Complete ratings matrix • Applications: recommendation systems, PCA with missing entries

Result: Fast and Exact Recovery Algorithm For an n × 𝑛, rank 𝑟, incoherent matrix, • Sample complexity: O(nr 5 log 3 𝑛) 7

• Time complexity: O(nr log

3

1 𝑛 log ) 𝜖

Prior work:

• Convex relaxation: Time complexity

1 3 O(𝑛 log ) 𝜖

Slow

• Alternating minimization: Sample complexity depends on condition number and 𝜖

Algorithm: Projected Gradient Descent min 𝑋

𝑀𝑖𝑗 − 𝑋𝑖𝑗 𝑖,𝑗 ∈Ω

𝑠. 𝑡. 𝑟𝑎𝑛𝑘 𝑋 = 𝑟 (Basic) Algorithm:

𝑋𝑡+1 ← 𝑋𝑡 − 𝜂𝛻𝑓 𝑋𝑡+1 ← 𝑃𝑟 (𝑋𝑡+1 )

2

Techniques • Bound ℓ∞ norm of errors : Given rank 𝑟, incoherent matrix 𝑀 and error matrix 𝐸, obtain tight bounds for 𝑀 − 𝑃𝑟 (𝑀 + 𝐸) ∞ • Standard results bound 𝑀 − 𝑃𝑟 (𝑀 + 𝐸)

2

or 𝑀 − 𝑃𝑟 (𝑀 + 𝐸)

• Another ingredient: Extension to Davis-Kahan theorem on perturbation in low rank approximation

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