We give a new framework for solving the fundamental problem of low-rank matrix completion, i.e., approximating a rank-$r$ matrix $\mathbf{M} \in \mathbb{R}^{m \times n}$ (where $m \ge n$) from random observations. First, we provide an algorithm which completes $\mathbf{M}$ on $99\%$ of rows and columns under no further assumptions on $\mathbf{M}$ from $\approx mr$ samples and using $\approx mr^2$ time. Then, assuming the row and column spans of $\mathbf{M}$ satisfy additional regularity properties, we show how to boost this partial completion guarantee to a full matrix completion algorithm by aggregating solutions to regression problems involving the observations. In the well-studied setting where $\mathbf{M}$ has incoherent row and column spans, our algorithms complete $\mathbf{M}$ to high precision from $mr^{2+o(1)}$ observations in $mr^{3 + o(1)}$ time (omitting logarithmic factors in problem parameters), improving upon the prior state-of-the-art [JN15] which used $\approx mr^5$ samples and $\approx mr^7$ time. Under an assumption on the row and column spans of $\mathbf{M}$ we introduce (which is satisfied by random subspaces with high probability), our sample complexity improves to an almost information-theoretically optimal $mr^{1 + o(1)}$, and our runtime improves to $mr^{2 + o(1)}$. Our runtimes have the appealing property of matching the best known runtime to verify that a rank-$r$ decomposition $\mathbf{U}\mathbf{V}^\top$ agrees with the sampled observations. We also provide robust variants of our algorithms that, given random observations from $\mathbf{M} + \mathbf{N}$ with $\|\mathbf{N}\|_{F} \le \Delta$, complete $\mathbf{M}$ to Frobenius norm distance $\approx r^{1.5}\Delta$ in the same runtimes as the noiseless setting. Prior noisy matrix completion algorithms [CP10] only guaranteed a distance of $\approx \sqrt{n}\Delta$.
To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the "t-matrix" model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This "t-matrix" model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and algorithms on simulated data and publicly available images and compare their performance. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts.
Non-parametric representations of dynamical systems based on the image of a Hankel matrix of data are extensively used for data-driven control. However, if samples of data are missing, obtaining such representations becomes a difficult task. By exploiting the kernel structure of Hankel matrices of irregularly measured data generated by a linear time-invariant system, we provide computational methods for which any complete finite-length behavior of the system can be obtained. For the special case of periodically missing outputs, we provide conditions on the input such that the former result is guaranteed. In the presence of noise in the data, our method returns an approximate finite-length behavior of the system. We illustrate our result with several examples, including its use for approximate data completion in real-world applications and compare it to alternative methods.
In this paper we discuss pre- and post-processing methods to induce desired consistency and/or invariance properties in blackbox systems, e.g., AI-based. We demonstrate our approach in the context of blackbox SVD-based matrix-completion methods commonly used in recommender system (RS) applications. We provide empirical results showing that enforcement of unit-consistency and shift-consistency, which have provable RS-relevant properties relating to robustness and fairness, also lead to improved performance according to generic RMSE and MAE performance metrics, irrespective of the initial chosen hyperparameter.
Obtaining rigorous statistical guarantees for generalization under distribution shift remains an open and active research area. We study a setting we call combinatorial distribution shift, where (a) under the test- and training-distributions, the labels $z$ are determined by pairs of features $(x,y)$, (b) the training distribution has coverage of certain marginal distributions over $x$ and $y$ separately, but (c) the test distribution involves examples from a product distribution over $(x,y)$ that is {not} covered by the training distribution. Focusing on the special case where the labels are given by bilinear embeddings into a Hilbert space $H$: $\mathbb{E}[z \mid x,y ]=\langle f_{\star}(x),g_{\star}(y)\rangle_{{H}}$, we aim to extrapolate to a test distribution domain that is $not$ covered in training, i.e., achieving bilinear combinatorial extrapolation. Our setting generalizes a special case of matrix completion from missing-not-at-random data, for which all existing results require the ground-truth matrices to be either exactly low-rank, or to exhibit very sharp spectral cutoffs. In this work, we develop a series of theoretical results that enable bilinear combinatorial extrapolation under gradual spectral decay as observed in typical high-dimensional data, including novel algorithms, generalization guarantees, and linear-algebraic results. A key tool is a novel perturbation bound for the rank-$k$ singular value decomposition approximations between two matrices that depends on the relative spectral gap rather than the absolute spectral gap, a result that may be of broader independent interest.
Large language models like GPT-4 exhibit emergent capabilities across general-purpose tasks, such as basic arithmetic, when trained on extensive text data, even though these tasks are not explicitly encoded by the unsupervised, next-token prediction objective. This study investigates how small transformers, trained from random initialization, can efficiently learn arithmetic operations such as addition, multiplication, and elementary functions like square root, using the next-token prediction objective. We first demonstrate that conventional training data is not the most effective for arithmetic learning, and simple formatting changes can significantly improve accuracy. This leads to sharp phase transitions as a function of training data scale, which, in some cases, can be explained through connections to low-rank matrix completion. Building on prior work, we then train on chain-of-thought style data that includes intermediate step results. Even in the complete absence of pretraining, this approach significantly and simultaneously improves accuracy, sample complexity, and convergence speed. We also study the interplay between arithmetic and text data during training and examine the effects of few-shot prompting, pretraining, and model scale. Additionally, we discuss length generalization challenges. Our work highlights the importance of high-quality, instructive data that considers the particular characteristics of the next-word prediction objective for rapidly eliciting arithmetic capabilities.
Despite the recent development in machine learning, most learning systems are still under the concept of "black box", where the performance cannot be understood and derived. With the rise of safety and privacy concerns in public, designing an explainable learning system has become a new trend in machine learning. In general, many machine learning problems are formulated as minimizing (or maximizing) some loss function. Since real data are most likely generated from non-linear models, the loss function is non-convex in general. Unlike the convex optimization problem, gradient descent algorithms will be trapped in spurious local minima in solving non-convex optimization. Therefore, it is challenging to provide explainable algorithms when studying non-convex optimization problems. In this thesis, two popular non-convex problems are studied: (1) low-rank matrix completion and (2) neural network learning.
Low-rank matrix completion is the task of recovering unknown entries of a matrix by assuming that the true matrix admits a good low-rank approximation. Sometimes additional information about the variables is known, and incorporating this information into a matrix completion model can lead to a better completion quality. We consider the situation where information between the column/row entities of the matrix is available as a weighted graph. In this framework, we address the problem of completing missing entries in air temperature data recorded by weather stations. We construct test sets by holding back data at locations that mimic real-life gaps in weather data. On such test sets, we show that adequate spatial and temporal graphs can significantly improve the accuracy of the completion obtained by graph-regularized low-rank matrix completion methods.
We propose Gibbs-Duhem-informed neural networks for the prediction of binary activity coefficients at varying compositions. That is, we include the Gibbs-Duhem equation explicitly in the loss function for training neural networks, which is straightforward in standard machine learning (ML) frameworks enabling automatic differentiation. In contrast to recent hybrid ML approaches, our approach does not rely on embedding a specific thermodynamic model inside the neural network and corresponding prediction limitations. Rather, Gibbs-Duhem consistency serves as regularization, with the flexibility of ML models being preserved. Our results show increased thermodynamic consistency and generalization capabilities for activity coefficient predictions by Gibbs-Duhem-informed graph neural networks and matrix completion methods. We also find that the model architecture, particularly the activation function, can have a strong influence on the prediction quality. The approach can be easily extended to account for other thermodynamic consistency conditions.
We consider a variant of matrix completion where entries are revealed in a biased manner. We wish to understand the extent to which such bias can be exploited in improving predictions. Towards that, we propose a natural model where the observation pattern and outcome of interest are driven by the same set of underlying latent (or unobserved) factors. We devise Mask Nearest Neighbor (MNN), a novel two-stage matrix completion algorithm: first, it recovers (distances between) the latent factors by utilizing matrix estimation for the fully observed noisy binary matrix, corresponding to the observation pattern; second, it utilizes the recovered latent factors as features and sparsely observed noisy outcomes as labels to perform non-parametric supervised learning. Our analysis reveals that MNN enjoys entry-wise finite-sample error rates that are competitive with corresponding supervised learning parametric rates. Despite not having access to the latent factors and dealing with biased observations, MNN exhibits such competitive performance via only exploiting the shared information between the bias and outcomes. Finally, through empirical evaluation using a real-world dataset, we find that with MNN, the estimates have 28x smaller mean squared error compared to traditional matrix completion methods, suggesting the utility of the model and method proposed in this work.
Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a non-negative weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $X,Y\in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - X Y^\top) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. It naturally generalizes the well-studied low rank matrix completion problem. Such a problem is known to be NP-hard and even hard to approximate assuming the Exponential Time Hypothesis [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for weighted low rank approximation. In particular, [LLR16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization that allows the alternating updates to be computed approximately. For weighted low rank approximation, this improves the runtime of [LLR16] from $\|W\|_0k^2$ to $\|W\|_0 k$ where $\|W\|_0$ denotes the number of nonzero entries of the weight matrix. At the heart of our framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.
Given only a few observed entries from a low-rank matrix $X$, matrix completion is the problem of imputing the missing entries, and it formalizes a wide range of real-world settings that involve estimating missing data. However, when there are too few observed entries to complete the matrix, what other aspects of the underlying matrix can be reliably recovered? We study one such problem setting, that of "one-sided" matrix completion, where our goal is to recover the right singular vectors of $X$, even in the regime where recovering the left singular vectors is impossible, which arises when there are more rows than columns and very few observations. We propose a natural algorithm that involves imputing the missing values of the matrix $X^TX$ and show that even with only two observations per row in $X$, we can provably recover $X^TX$ as long as we have at least $\Omega(r^2 d \log d)$ rows, where $r$ is the rank and $d$ is the number of columns. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing. In these settings, our algorithm substantially outperforms standard matrix completion and a variety of direct factorization methods.
Most of the existing works on provable guarantees for low-rank matrix completion algorithms rely on some unrealistic assumptions such that matrix entries are sampled randomly or the sampling pattern has a specific structure. In this work, we establish theoretical guarantee for the exact and approximate low-rank matrix completion problems which can be applied to any deterministic sampling schemes. For this, we introduce a graph having observed entries as its edge set, and investigate its graph properties involving the performance of the standard constrained nuclear norm minimization algorithm. We theoretically and experimentally show that the algorithm can be successful as the observation graph is well-connected and has similar node degrees. Our result can be viewed as an extension of the works by Bhojanapalli and Jain [2014] and Burnwal and Vidyasagar [2020], in which the node degrees of the observation graph were assumed to be the same. In particular, our theory significantly improves their results when the underlying matrix is symmetric.
Works on implicit regularization have studied gradient trajectories during the optimization process to explain why deep networks favor certain kinds of solutions over others. In deep linear networks, it has been shown that gradient descent implicitly regularizes toward low-rank solutions on matrix completion/factorization tasks. Adding depth not only improves performance on these tasks but also acts as an accelerative pre-conditioning that further enhances this bias towards low-rankedness. Inspired by this, we propose an explicit penalty to mirror this implicit bias which only takes effect with certain adaptive gradient optimizers (e.g. Adam). This combination can enable a degenerate single-layer network to achieve low-rank approximations with generalization error comparable to deep linear networks, making depth no longer necessary for learning. The single-layer network also performs competitively or out-performs various approaches for matrix completion over a range of parameter and data regimes despite its simplicity. Together with an optimizer's inductive bias, our findings suggest that explicit regularization can play a role in designing different, desirable forms of regularization and that a more nuanced understanding of this interplay may be necessary.
Self-supervised methods received tremendous attention thanks to their seemingly heuristic approach to learning representations that respect the semantics of the data without any apparent supervision in the form of labels. A growing body of literature is already being published in an attempt to build a coherent and theoretically grounded understanding of the workings of a zoo of losses used in modern self-supervised representation learning methods. In this paper, we attempt to provide an understanding from the perspective of a Laplace operator and connect the inductive bias stemming from the augmentation process to a low-rank matrix completion problem. To this end, we leverage the results from low-rank matrix completion to provide theoretical analysis on the convergence of modern SSL methods and a key property that affects their downstream performance.
We consider offline Reinforcement Learning (RL), where the agent does not interact with the environment and must rely on offline data collected using a behavior policy. Previous works provide policy evaluation guarantees when the target policy to be evaluated is covered by the behavior policy, that is, state-action pairs visited by the target policy must also be visited by the behavior policy. We show that when the MDP has a latent low-rank structure, this coverage condition can be relaxed. Building on the connection to weighted matrix completion with non-uniform observations, we propose an offline policy evaluation algorithm that leverages the low-rank structure to estimate the values of uncovered state-action pairs. Our algorithm does not require a known feature representation, and our finite-sample error bound involves a novel discrepancy measure quantifying the discrepancy between the behavior and target policies in the spectral space. We provide concrete examples where our algorithm achieves accurate estimation while existing coverage conditions are not satisfied. Building on the above evaluation algorithm, we further design an offline policy optimization algorithm and provide non-asymptotic performance guarantees.
The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local minimum with the smallest rank that still fits the training data. While rank-underestimating minima can be avoided since they do not fit the data, GD might get stuck at rank-overestimating minima. We show that with SGD, there is always a probability to jump from a higher rank minimum to a lower rank one, but the probability of jumping back is zero. More precisely, we define a sequence of sets $B_{1}\subset B_{2}\subset\cdots\subset B_{R}$ so that $B_{r}$ contains all minima of rank $r$ or less (and not more) that are absorbing for small enough ridge parameters $\lambda$ and learning rates $\eta$: SGD has prob. 0 of leaving $B_{r}$, and from any starting point there is a non-zero prob. for SGD to go in $B_{r}$.
Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not possess any optimality guarantees. We reexamine matrix completion with an optimality-oriented eye. We reformulate low-rank matrix completion problems as convex problems over the non-convex set of projection matrices and implement a disjunctive branch-and-bound scheme that solves them to certifiable optimality. Further, we derive a novel and often near-exact class of convex relaxations by decomposing a low-rank matrix as a sum of rank-one matrices and incentivizing that two-by-two minors in each rank-one matrix have determinant zero. In numerical experiments, our new convex relaxations decrease the optimality gap by two orders of magnitude compared to existing attempts, and our disjunctive branch-and-bound scheme solves $n \times m$ rank-$r$ matrix completion problems to certifiable optimality or near optimality in hours for $\max \{m, n\} \leq 2500$ and $r \leq 5$. Moreover, this improvement in the training error translates into an average $2\%$--$50\%$ improvement in the test set error.
Tropical semiring has proven successful in several research areas, including optimal control, bioinformatics, discrete event systems, or solving a decision problem. In previous studies, a matrix two-factorization algorithm based on the tropical semiring has been applied to investigate bipartite and tripartite networks. Tri-factorization algorithms based on standard linear algebra are used for solving tasks such as data fusion, co-clustering, matrix completion, community detection, and more. However, there is currently no tropical matrix tri-factorization approach, which would allow for the analysis of multipartite networks with a high number of parts. To address this, we propose the triFastSTMF algorithm, which performs tri-factorization over the tropical semiring. We apply it to analyze a four-partition network structure and recover the edge lengths of the network. We show that triFastSTMF performs similarly to Fast-NMTF in terms of approximation and prediction performance when fitted on the whole network. When trained on a specific subnetwork and used to predict the whole network, triFastSTMF outperforms Fast-NMTF by several orders of magnitude smaller error. The robustness of triFastSTMF is due to tropical operations, which are less prone to predict large values compared to standard operations.
In this paper we address the rotation synchronization problem, where the objective is to recover absolute rotations starting from pairwise ones, where the unknowns and the measures are represented as nodes and edges of a graph, respectively. This problem is an essential task for structure from motion and simultaneous localization and mapping. We focus on the formulation of synchronization via neural networks, which has only recently begun to be explored in the literature. Inspired by deep matrix completion, we express rotation synchronization in terms of matrix factorization with a deep neural network. Our formulation exhibits implicit regularization properties and, more importantly, is unsupervised, whereas previous deep approaches are supervised. Our experiments show that we achieve comparable accuracy to the closest competitors in most scenes, while working under weaker assumptions.