Recovery Guarantee of Non-negative Matrix Factorization via Alternating Updates

By Yuanzhi Li, Yingyu Liang, Andrej Risteski. 2016

30 citations recorded by the source corpus.

Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and shows that assuming a generative model of the data, it provably recovers the ground-truth under fairly mild conditions. In particular, its only essential requirement on features is linear independence. Furthermore, the algorithm uses ReLU to exploit the non-negativity for decoding the weights, and thus can tolerate adversarial noise that can potentially be as large as the signal, and can tolerate unbiased noise much larger than the signal. The analysis relies on a carefully designed coupling between two potential functions, which we believe is of independent interest.

Subfields: Optimization & Training, Probabilistic & Bayesian Methods

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Optimization & Training

1. Zeyuan Allen-Zhu
2. Yuanzhi Li
3. Zehao Dou
4. Yingyu Liang
5. Quanquan Gu
6. Peter Richtárik
7. Tengyu Ma
8. Greg Yang

In Probabilistic & Bayesian Methods

1. Zeyuan Allen-Zhu
2. Yuanzhi Li
3. Yingyu Liang

Citations in the corpus

• Feature Purification: How Adversarial Training Performs Robust Deep Learning (2020, methodology): “[10] provides a neurally-plausible algorithm for learning sparse coding along with other works using alternative minimization [1, 2, 37, 56, 59].”
• Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers (2018, background): “ distribution (such as being random Gaussian), and/or make strong assumptions about the network (such as using linear activations). Other works focusing on structured data such as sparse coding model [5, 9, 37, 40] and Li and Liang [38] shows that two-layer ReLU networks can learn classication tasks”
• Making Method of Moments Great Again? -- How can GANs learn the target distribution (2020, background): “sparse coding/non-negative matrix factorization [18, 19, 49, 47, 7]”
• Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning (2020, background): “Most of the existing works proving that neural networks can learn features only focus on the case when the input is Gaussian or Gaussian-like [5, 11, 13, 17, 38, 39, 46, 53, 54, 56, 56–58, 60, 69, 78– 80, 84, 85, 87, 89].”
• Backward Feature Correction: How Deep Learning Performs Deep Learning (2020, background): “It is thus not surprising that most existing theoretical works on the efficient learning regime of neural networks either focus on (1) two-layer networks [4, 11, 15, 17, 27, 28, 43, 47, 48, 50, 50–52, 64, 69, 70, 73, 75, 77, 79, 82, 83] which do not have the hierarchical structure, or (2) a multi-la”
• Provable Alternating Gradient Descent for Non-negative Matrix Factorization with Strong Correlations (2017, unknown): “Since A and A ∗ are in the same subspace, inspired by (Li et al., 2016) we can write A ∗ as A ( Σ + E ) for a diagonal matrix Σ and an off-diagonal matrix E , and thus the decoding becomes z Let us focus on one coordinate of z ( t ) , that is, z t ) .”