We revisit the landscape of the simple matrix factorization problem. For low-rank matrix factorization, prior work has shown that there exist infinitely many critical points all of which are either global minima or strict saddles. At a strict saddle the minimum eigenvalue of the Hessian is negative. Of interest is whether this minimum eigenvalue is uniformly bounded below zero over all strict saddles. To answer this we consider orbits of critical points under the general linear group. For each orbit we identify a representative point, called a canonical point. If a canonical point is a strict saddle, so is every point on its orbit. We derive an expression for the minimum eigenvalue of the Hessian at each canonical strict saddle and use this to show that the minimum eigenvalue of the Hessian over the set of strict saddles is not uniformly bounded below zero. We also show that a known invariance property of gradient flow ensures the solution of gradient flow only encounters critical points on an invariant manifold $\mathcal{M}_C$ determined by the initial condition. We show that, in contrast to the general situation, the minimum eigenvalue of strict saddles in $\mathcal{M}_{0}$ is uniformly bounded below zero. We obtain an expression for this bound in terms of the singular values of the matrix being factorized. This bound depends on the size of the nonzero singular values and on the separation between distinct nonzero singular values of the matrix.
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, w…
Embedding based models have been the state of the art in collaborative filtering for over a decade. Traditionally, the dot product or higher order equivalents have been used to combine two or more embeddings, e.g., most notably in matrix factorization. In recent years, it was suggested to replace the dot product with a learned similarity e.g. using a multilayer perceptron (MLP). This approach is often referred to as neural collaborative filtering (NCF). In this work, we revisit the experiments of the NCF paper that popularized learned similarities using MLPs. First, we show that with a proper …
A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization
School of Data Science, Fudan University, Shanghai, China（复旦大学数据科学学院，上海，中国）；Department of Computer and Information Science, University of Macau, Macau SAR, China（澳门大学计算机与信息科学系，澳门特别行政区，中国）
Generating gradients in the energy landscape using rectified linear type cost functions for efficiently solving 0/1 matrix factorization in Simulated Annealing
Fujitsu Limited（富士通株式会社）；DXR Laboratory Inc.（DXR实验室股份有限公司）；The University of Tokyo（东京大学）
We study the non-convex matrix factorization approach to matrix completion via Riemannian geometry. Based on an optimization formulation over a Grassmannian manifold, we characterize the landscape based on the notion of principal angles between subspaces. For the fully observed case, our results show that there is a region in which the cost is geodesically convex, and outside of which all critical points are strictly saddle. We empirically study the partially observed case based on our findings.
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…
The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. In this work, we focus on the situation where the corresponding population risk is a degenerate non-convex loss function, namely, the Hessian of the population risk can have zero eigenvalues. Instead of analyzing the non-convex empirical risk directly, we first study the landscape of the corresponding population risk, which is usually easier to characterize, and then build a connection between the land…
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…
Motivated by the emerging role of interpolating machines in signal processing and machine learning, this work considers the computational aspects of over-parametrized matrix factorization. In this context, the optimization landscape may contain spurious stationary points (SSPs), which are proved to be full-rank matrices. The presence of these SSPs means that it is impossible to hope for any global guarantees in over-parametrized matrix factorization. For example, when initialized at an SSP, the gradient flow will be trapped there forever. Nevertheless, despite these SSPs, we establish in this …
We provide the first positive result on the nonsmooth optimization landscape of robust principal component analysis, to the best of our knowledge. It is the object of several conjectures and remains mostly uncharted territory. We identify a necessary and sufficient condition for the absence of spurious local minima in the rank-one case. Our proof exploits the subdifferential regularity of the objective function in order to eliminate the existence quantifier from the first-order optimality condition known as Fermat's rule.
Graph-based Clustering Revisited: A Relaxation of Kernel $k$-Means Perspective
School of Computer Science and Engineering（计算机科学与工程学院）；Division of Computer Science and Engineering（计算机科学与工程系）；Yam Pak Charitable Foundation School of Computing and Information Sciences（Yam Pak慈善基金会计算与信息科学学院）；Department of Computer Science（计算机科学系）
本文提出 LoRD 和 B-LoRD 两种低秩双随机聚类模型，仅松弛正交约束并引入块对角正则化，通过将非凸约束转化为线性凸约束，实现全局收敛的投影梯度下降优化，显著提升聚类性能。
It is well-known that the Burer-Monteiro (B-M) factorization approach can efficiently solve low-rank matrix optimization problems under the RIP condition. It is natural to ask whether B-M factorization-based methods can succeed on any low-rank matrix optimization problems with a low information-theoretic complexity, i.e., polynomial-time solvable problems that have a unique solution. In this work, we provide a negative answer to the above question. We investigate the landscape of B-M factorized polynomial-time solvable matrix completion (MC) problems, which are the most popular subclass of low…
Gradient-related first-order methods have become the workhorse of large-scale numerical optimization problems. Many of these problems involve nonconvex objective functions with multiple saddle points, which necessitates an understanding of the behavior of discrete trajectories of first-order methods within the geometrical landscape of these functions. This paper concerns convergence of first-order discrete methods to a local minimum of nonconvex optimization problems that comprise strict-saddle points within the geometrical landscape. To this end, it focuses on analysis of discrete gradient tr…
