Low-rank matrix factorization is a problem of broad importance, owing to the ubiquity of low-rank models in machine learning contexts. In spite of its non-convexity, this problem has a well-behaved geometric landscape, permitting local search algorithms such as gradient descent to converge to global minimizers. In this paper, we study low-rank matrix factorization in the distributed setting, where local variables at each node encode parts of the overall matrix factors, and consensus is encouraged among certain such variables. We identify conditions under which this new problem also has a well-behaved geometric landscape, and we propose an extension of distributed gradient descent (DGD) to solve this problem. The favorable landscape allows us to prove convergence to global optimality with exact consensus, a stronger result than what is provided by off-the-shelf DGD theory.

Low-rank matrix factorization is a problem of broad importance, owing to the ubiquity of low-rank models in machine learning contexts. In spite of its non- convexity, this problem has a well-behaved geometric landscape, permitting local search algorithms such as gradient descent to converge to global minimizers. In this paper, we study low-rank matrix factorization in the distributed setting, where local variables at each node encode parts of the overall matrix factors, and consensus is encouraged among certain such variables. We identify conditions under which this new problem also has a well-behaved geometric landscape, and we propose an extension of distributed gradient descent (DGD) to solve this problem. The favorable landscape allows us to prove convergence to global optimality with exact consensus, a stronger result than what is provided by off-the-shelf DGD theory.

Approximate nearest neighbor (ANN) search is a key component in many modern machine learning pipelines; recent use cases include retrieval-augmented generation (RAG) and vector databases. Clustering-based ANN algorithms, that use score computation methods based on product quantization (PQ), are often used in industrial-scale applications due to their scalability and suitability for distributed and disk-based implementations. However, they have slower query times than the leading graph-based ANN algorithms. In this work, we propose a new supervised score computation method based on the observation that inner product approximation is a multivariate (multi-output) regression problem that can be solved efficiently by reduced-rank regression. Our experiments show that on modern high-dimensional data sets, the proposed reduced-rank regression (RRR) method is superior to PQ in both query latency and memory usage. We also introduce LoRANN, a clustering-based ANN library that leverages the proposed score computation method.

I would appreciate if you could incorporate the reposes from the rebuttal into the paper (in case of acceptance). Especially, -better highlighting of the contributions (moving some of the cited auxiliary results to the appendix might also be worth considering) -discussion of stochastic updates -for the promised discussion on alterative approaches: please check again if all primal-dual methods really require a'star-topology', and not just a connected graph; and include references to the literature; similar with the comment on gossip averaging: the literature discusses already many variants (respective'orders' of averaging/update steps); so please check again there as well. However, I would encourage to highlight contributions w.r.t. to previous works more clearly (e.g. Quality: The theorems all look sound; however, only asymptotic results are provided. Further, comparisons to alternative baselines (instead of DSG) would strengthen the paper. Originality & Significance: As outlined above, the significance could potentially be estimated more favorably if discussion/comparisons to strong baselines could be added.

The paper studies distributed matrix factorization problems. It takes the view that distributed (sub)gradient descent relates to a regularized version of the original optimization problem, and then shows that stationary points of the distributed matrix completion problem satisfy the consensus constraint. While the significance of the paper caused some discussions, the reviewers remained mostly positive in the final assessment, resulting in a narrow accept decision. We hope the suggested changes and comments help improve the paper for the camera ready version, in particular, this concerns the clarity of contributions, related work on primal-dual and gossip methods as mentioned e.g. by Reviewer 3 and other comments by the reviewers.

Low-rank methods have shown success in accelerating simulations of a collisionless plasma described by the Vlasov equation, but still rely on computationally costly linear algebra every time step. We propose a data-driven factorization method using artificial neural networks, specifically with convolutional layer architecture, that trains on existing simulation data. At inference time, the model outputs a low-rank decomposition of the distribution field of the charged particles, and we demonstrate that this step is faster than the standard linear algebra technique. Numerical experiments show that the method effectively interpolates time-series data, generalizing to unseen test data in a manner beyond just memorizing training data; patterns in factorization also inherently followed the same numerical trend as those within algebraic methods (e.g., truncated singular-value decomposition). However, when training on the first 70% of a time-series data and testing on the remaining 30%, the method fails to meaningfully extrapolate. Despite this limiting result, the technique may have benefits for simulations in a statistical steady-state or otherwise showing temporal stability.

Low-rank matrix factorization is a problem of broad importance, owing to the ubiquity of low-rank models in machine learning contexts. In spite of its non- convexity, this problem has a well-behaved geometric landscape, permitting local search algorithms such as gradient descent to converge to global minimizers. In this paper, we study low-rank matrix factorization in the distributed setting, where local variables at each node encode parts of the overall matrix factors, and consensus is encouraged among certain such variables. We identify conditions under which this new problem also has a well-behaved geometric landscape, and we propose an extension of distributed gradient descent (DGD) to solve this problem. The favorable landscape allows us to prove convergence to global optimality with exact consensus, a stronger result than what is provided by off-the-shelf DGD theory.

Low-rank matrix factorization is a problem of broad importance, owing to the ubiquity of low-rank models in machine learning contexts. In spite of its non- convexity, this problem has a well-behaved geometric landscape, permitting local search algorithms such as gradient descent to converge to global minimizers. In this paper, we study low-rank matrix factorization in the distributed setting, where local variables at each node encode parts of the overall matrix factors, and consensus is encouraged among certain such variables. We identify conditions under which this new problem also has a well-behaved geometric landscape, and we propose an extension of distributed gradient descent (DGD) to solve this problem. The favorable landscape allows us to prove convergence to global optimality with exact consensus, a stronger result than what is provided by off-the-shelf DGD theory. Papers published at the Neural Information Processing Systems Conference.