Summary of Symmetric Matrix Completion with Relu Sampling, by Huikang Liu et al.
Symmetric Matrix Completion with ReLU Sampling
by Huikang Liu, Peng Wang, Longxiu Huang, Qing Qu, Laura Balzano
First submitted to arxiv on: 9 Jun 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Machine Learning (stat.ML)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary This study explores symmetric positive semi-definite low-rank matrix completion (MC) under deterministic entry-dependent sampling, focusing on rectified linear unit (ReLU) sampling and a generalization to threshold-based sampling. The authors demonstrate that the MC problem’s landscape is not globally benign, with gradient descent (GD) converging to non-globally optimal stationary points. However, they prove that when the matrix factor has a small rank and satisfies mild assumptions, the objective function is geodesically strongly convex on the quotient manifold near a planted low-rank matrix. The authors also show that their assumptions are met by a matrix factor with i.i.d. Gaussian entries. To solve the formulation, they design an initialization for GD, which empirically achieves convergence to global minima. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This study looks at how to fill in missing parts of a matrix when only some of the numbers are known. The researchers tested different ways of doing this and found that some methods don’t always find the best solution. However, they were able to prove that if the matrix is structured in a certain way, then their method works well. They also showed that their method works well with matrices made up of random numbers. The study uses a technique called gradient descent to solve the problem and finds that it can be very effective. |
Keywords
» Artificial intelligence » Generalization » Gradient descent » Objective function » Relu
