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 groundtruth 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.

Because of the ubiquitous applications of NMF, many efficient algorithms have been proposedforsolving(1).

Across fields such as machine learning, social science, geography, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extend similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five popular models, latent budget analysis (LBA), latent class analysis (LCA), end-member analysis (EMA), probabilistic latent semantic analysis (PLSA), and nonnegative matrix factorization (NMF). We focus on an essential issue-identifiability-of these models and prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. We also provide a brief review for algorithms of these models. We illustrate the models with a time budget dataset from social science, and end the paper with a discussion of closely related models such as archetypal analysis.

This paper tackles post-hoc interpretability for audio processing networks. Our goal is to interpret decisions of a trained network in terms of high-level audio objects that are also listenable for the end-user. To this end, we propose a novel interpreter design that incorporates non-negative matrix factorization (NMF). In particular, a regularized interpreter module is trained to take hidden layer representations of the targeted network as input and produce time activations of pre-learnt NMF components as intermediate outputs. Our methodology allows us to generate intuitive audio-based interpretations that explicitly enhance parts of the input signal most relevant for a network's decision. We demonstrate our method's applicability on popular benchmarks, including a real-world multi-label classification task.

Meshes are important representations of physical 3D entities in the virtual world. Applications like rendering, simulations and 3D printing require meshes to be manifold so that they can interact with the world like the real objects they represent. Prior methods generate meshes with great geometric accuracy but poor manifoldness. In this work, we propose NeuralMeshFlow (NMF) to generate two-manifold meshes for genus-0 shapes. Specifically, NMF is a shape auto-encoder consisting of several Neural Ordinary Differential Equation (NODE)(1) blocks that learn accurate mesh geometry by progressively deforming a spherical mesh. Training NMF is simpler compared to state-of-the-art methods since it does not require any explicit mesh-based regularization. Our experiments demonstrate that NMF facilitates several applications such as single-view mesh reconstruction, global shape parameterization, texture mapping, shape deformation and correspondence. Importantly, we demonstrate that manifold meshes generated using NMF are better-suited for physically-based rendering and simulation compared to prior works.