Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by Forrow et al., 2018, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.
Although Sinkhorn divergences are now routinely used in data sciences to compare probability distributions, the computational effort required to compute them remains expensive, growing in general quadratically in the size $n$ of the support of these distributions. Indeed, solving optimal transport (OT) with an entropic regularization requires computing a $n\times n$ kernel matrix (the neg-exponential of a $n\times n$ pairwise ground cost matrix) that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-\log\dotp{φ(x)}{φ(y)}$ where $φ$ is a map from the…
Data Science Institute, Columbia University（哥伦比亚大学数据科学研究所）；New York University（纽约大学）；Sciences Po（巴黎政治学院）；Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Department of Management Science & Engineering, Stanford University（斯坦福大学管理科学与工程系）
本文证明基于 Luce 选择公理的一类选择模型的最大似然估计等价于矩阵平衡问题，从而统一现有算法为 Sinkhorn 算法实例，并首次给出一般非负矩阵下 Sinkhorn 算法的全局线性收敛性及精确渐近速率。
It is of increasing importance to develop learning methods for ranking. In contrast to many learning objectives, however, the ranking problem presents difficulties due to the fact that the space of permutations is not smooth. In this paper, we examine the class of rank-linear objective functions, which includes popular metrics such as precision and discounted cumulative gain. In particular, we observe that expectations of these gains are completely characterized by the marginals of the corresponding distribution over permutation matrices. Thus, the expectations of rank-linear objectives can al…
Optimal transport (OT) naturally arises in a wide range of machine learning applications but may often become the computational bottleneck. Recently, one line of works propose to solve OT approximately by searching the \emph{transport plan} in a low-rank subspace. However, the optimal transport plan is often not low-rank, which tends to yield large approximation errors. For example, when Monge's \emph{transport map} exists, the transport plan is full rank. This paper concerns the computation of the OT distance with adequate accuracy and efficiency. A novel approximation for OT is proposed, in …
Many problems at the intersection of combinatorics and computer science require solving for a permutation that optimally matches, ranks, or sorts some data. These problems usually have a task-specific, often non-differentiable objective function that data-driven algorithms can use as a learning signal. In this paper, we propose the Sinkhorn Policy Gradient (SPG) algorithm for learning policies on permutation matrices. The actor-critic neural network architecture we introduce for SPG uniquely decouples representation learning of the state space from the highly-structured action space of permuta…
In this paper we consider the problem of computing the likelihood of the profile of a discrete distribution, i.e., the probability of observing the multiset of element frequencies, and computing a profile maximum likelihood (PML) distribution, i.e., a distribution with the maximum profile likelihood. For each problem we provide polynomial time algorithms that given $n$ i.i.d.\ samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (…
Recently, Factorization Machines (FM) has become more and more popular for recommendation systems, due to its effectiveness in finding informative interactions between features. Usually, the weights for the interactions is learnt as a low rank weight matrix, which is formulated as an inner product of two low rank matrices. This low rank can help improve the generalization ability of Factorization Machines. However, to choose the rank properly, it usually needs to run the algorithm for many times using different ranks, which clearly is inefficient for some large-scale datasets. To alleviate thi…
University of California（加利福尼亚大学）；University of Washington（华盛顿大学）
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a non-linear link function used within factorization. In particular, by applying the round function on a factorization to obtain ordinal-valued matrices, we introduce generalized round-rank (GRR). We show that not only are there many full-rank matrices that are low GRR, but further, that these matrices cannot be approximated well by low-rank linear factorization. We pr…
Tsinghua University（清华大学）；Tencent AI Lab（腾讯人工智能实验室）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Factorization machines (FM) are a popular model class to learn pairwise interactions by a low-rank approximation. Different from existing FM-based approaches which use a fixed rank for all features, this paper proposes a Rank-Aware FM (RaFM) model which adopts pairwise interactions from embeddings with different ranks. The proposed model achieves a better performance on real-world datasets where different features have significantly varying frequencies of occurrences. Moreover, we prove that the RaFM model can be stored, evaluated, and trained as efficiently as one single FM, and under some re…
Determinantal point processes (DPPs) have garnered attention as an elegant probabilistic model of set diversity. They are useful for a number of subset selection tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. In this work we present a new method for learning the DPP kernel from observed data using a low-rank factorization of this kernel. We show that this low-rank factorization enables a learning algorithm that is nearly an order of magnitude faster than previous approaches, while also providing for a method for computing product recom…
The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low-tubal-rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this paper, we propose a new objective funct…
Matrix factorization is a simple and natural test-bed to investigate the implicit regularization of gradient descent. Gunasekar et al. (2017) conjectured that Gradient Flow with infinitesimal initialization converges to the solution that minimizes the nuclear norm, but a series of recent papers argued that the language of norm minimization is not sufficient to give a full characterization for the implicit regularization. In this work, we provide theoretical and empirical evidence that for depth-2 matrix factorization, gradient flow with infinitesimal initialization is mathematically equivalent…
We develop an efficient alternating framework for learning a generalized version of Factorization Machine (gFM) on steaming data with provable guarantees. When the instances are sampled from $d$ dimensional random Gaussian vectors and the target second order coefficient matrix in gFM is of rank $k$, our algorithm converges linearly, achieves $O(ε)$ recovery error after retrieving $O(k^{3}d\log(1/ε))$ training instances, consumes $O(kd)$ memory in one-pass of dataset and only requires matrix-vector product operations in each iteration. The key ingredient of our framework is a construction of an…
Istituto Italiano di Tecnologia（意大利技术研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Regularization for matrix factorization (MF) and approximation problems has been carried out in many different ways. Due to its popularity in deep learning, dropout has been applied also for this class of problems. Despite its solid empirical performance, the theoretical properties of dropout as a regularizer remain quite elusive for this class of problems. In this paper, we present a theoretical analysis of dropout for MF, where Bernoulli random variables are used to drop columns of the factors. We demonstrate the equivalence between dropout and a fully deterministic model for MF in which the…
What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections. We examine the approach of first performing factorization in the compressed domain, and then reconstructing the original high-dimensional factors from the recovered (compressed) factors. In both the matrix and tensor settings, we establish conditions under which this natural approach will provably recover the original factors. While it is well-known that ra…
This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function …
University College London（伦敦大学学院）；Computational Statistics and Machine Learning - Istituto Italiano di Tecnologia（意大利技术研究院计算统计与机器学习）
Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster than standard convex solvers such as proximal gradient methods. Nevertheless, it is not guaranteed to converge to a global optimum, and the optimization can be trapped at poor stationary points. In this paper we show that it is possible to characterize all critical points of the non…
In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it suffices to identify the latent factors if the rows of one factor are \emph{sufficiently scattered} over the nonnegative orthant, while no structural assumption is imposed on the other factor except being full-rank. This is by far the mildest condition under which the latent factors are provably identifiable from the NMF model.
Motivated by an application in computational biology, we consider low-rank matrix factorization with $\{0,1\}$-constraints on one of the factors and optionally convex constraints on the second one. In addition to the non-convexity shared with other matrix factorization schemes, our problem is further complicated by a combinatorial constraint set of size $2^{m \cdot r}$, where $m$ is the dimension of the data points and $r$ the rank of the factorization. Despite apparent intractability, we provide - in the line of recent work on non-negative matrix factorization by Arora et al. (2012) - an algo…
Université de Mons（蒙斯大学）；Univ. Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；I3S（索菲亚安提波利斯信息学、信号与系统研究所）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Grenoble INP（格勒诺布尔国立理工学院）；GIPSA-lab（格勒诺布尔图像、语音信号与自动化实验室）
We propose an extension of the canonical polyadic (CP) tensor model where one of the latent factors is allowed to vary through data slices in a constrained way. The components of the latent factors, which we want to retrieve from data, can vary from one slice to another up to a diffeomorphism. We suppose that the diffeomorphisms are also unknown, thus merging curve registration and tensor decomposition in one model, which we call registered CP. We present an algorithm to retrieve both the latent factors and the diffeomorphism, which is assumed to be in a parametrized form. At the end of the pa…
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank. In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian trea…
There has recently been considerable interest in completing a low-rank matrix or tensor given only a small fraction (or few linear combinations) of its entries. Related approaches have found considerable success in the area of recommender systems, under machine learning. From a statistical estimation point of view, the gold standard is to have access to the joint probability distribution of all pertinent random variables, from which any desired optimal estimator can be readily derived. In practice high-dimensional joint distributions are very hard to estimate, and only estimates of low-dimensi…
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization variables appear only locally at a single node in the network. We term the resulting algorithm DGD+LOCAL. Using algorithmic connections to gradient descent and geometric connections to the well-behaved landscape of the centralized low-rank matrix approximation problem, we identify sufficient conditions where DGD+LOCAL is guaranteed to converge with exact consensus…
Stability of Low-Rank Implicit Regularization in Perturbed Deep Matrix Factorization
Department of Informatics and Networked Systems, University of Pittsburgh（信息学与网络系统系，匹兹堡大学）；Department of Mathematics, University of Pittsburgh（数学系，匹兹堡大学）
Department of Mathematics, UC Santa Barbara（加州大学圣巴巴拉分校数学系）；Department of ECE, UC Santa Barbara（加州大学圣巴巴拉分校电气工程与计算机科学系）
Streaming tensor factorization is a powerful tool for processing high-volume and multi-way temporal data in Internet networks, recommender systems and image/video data analysis. Existing streaming tensor factorization algorithms rely on least-squares data fitting and they do not possess a mechanism for tensor rank determination. This leaves them susceptible to outliers and vulnerable to over-fitting. This paper presents a Bayesian robust streaming tensor factorization model to identify sparse outliers, automatically determine the underlying tensor rank and accurately fit low-rank structure. We…
Ferdowsi University of Mashhad（马什哈德菲尔多西大学）；University of Tehran（德黑兰大学）
In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the …
The Chinese University of Hong Kong（香港中文大学）；Arizona State University（亚利桑那州立大学）
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint/mixed relaxations (i.e., l_{1}-norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable structured low-rank matrix factorization meth…
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）；Paul G. Allen School of Computer Science and Engineering（保罗·G·艾伦计算机科学与工程学院）；University of Washington（华盛顿大学）
We study the asymmetric low-rank factorization problem: \[\min_{\mathbf{U} \in \mathbb{R}^{m \times d}, \mathbf{V} \in \mathbb{R}^{n \times d}} \frac{1}{2}\|\mathbf{U}\mathbf{V}^\top -\mathbfΣ\|_F^2\] where $\mathbfΣ$ is a given matrix of size $m \times n$ and rank $d$. This is a canonical problem that admits two difficulties in optimization: 1) non-convexity and 2) non-smoothness (due to unbalancedness of $\mathbf{U}$ and $\mathbf{V}$). This is also a prototype for more complex problems such as asymmetric matrix sensing and matrix completion. Despite being non-convex and non-smooth, it has be…
This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimiz…
We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex quadratic loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is small, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample complexity and…
Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correction
Graduate School of Science and Engineering（研究生院）；PAF-Karachi Institute of Economics and Technology（卡拉奇经济和技术学院）；Department of Mathematics and Operational Research（数学与运筹学系）
Graphical model for factorization and completion of relatively high rank tensors by sparse sampling
D3 Center, Osaka University（大阪大学D3中心）；Department of Physics, Graduate School of Science, Osaka University（大阪大学理学研究生院物理学科）；Department of Systems Science, Graduate School of Informatics, Kyoto University（京都大学信息学研究生院系统科学学科）；RIKEN center for AIP（RIKEN AIP研究中心）
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimiz…
Robust principal component analysis (RPCA) has been widely used for recovering low-rank matrices in many data mining and machine learning problems. It separates a data matrix into a low-rank part and a sparse part. The convex approach has been well studied in the literature. However, state-of-the-art algorithms for the convex approach usually have relatively high complexity due to the need of solving (partial) singular value decompositions of large matrices. A non-convex approach, AltProj, has also been proposed with lighter complexity and better scalability. Given the true rank $r$ of the und…
