We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs.

To efficiently express tensor data using the Tucker format, a critical task is to minimize the multilinear rank such that the model would not be over-flexible and lead to overfitting. Due to the lack of rank minimization tools in tensor, existing works connect Tucker multilinear rank minimization to trace norm minimization of matrices unfolded from the tensor data. While these formulations try to exploit the common aim of identifying the low-dimensional structure of the tensor and matrix, this paper reveals that existing trace norm-based formulations in Tucker completion are inefficient in multilinear rank minimization. We further propose a new interpretation of Tucker format such that trace norm minimization is applied to the factor matrices of the equivalent representation, rather than some matrices unfolded from tensor data. Based on the newly established problem formulation, a fixed point iteration algorithm is proposed, and its convergence is proved. Numerical results are presented to show that the proposed algorithm exhibits significant improved performance in terms of multilinear rank learning and consequently tensor signal recovery accuracy, compared to existing trace norm based Tucker completion methods.

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs.

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization under affine constraints ARMP and show that SVP recovers the minimum rank solution for affine constraints that satisfy a Restricted Isometry Property} (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newton-step for our SVP framework to speed-up the convergence with substantial empirical gains. Next, we address a practically important application of ARMP - the problem of low-rank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold.

A new study from researchers at MIT and Brown University characterizes several properties that emerge during the training of deep classifiers, a type of artificial neural network commonly used for classification tasks such as image classification, speech recognition, and natural language processing. The paper, "Dynamics in Deep Classifiers trained with the Square Loss: Normalization, Low Rank, Neural Collapse and Generalization Bounds," published today in the journal Research, is the first of its kind to theoretically explore the dynamics of training deep classifiers with the square loss and how properties such as rank minimization, neural collapse, and dualities between the activation of neurons and the weights of the layers are intertwined. In the study, the authors focused on two types of deep classifiers: fully connected deep networks and convolutional neural networks (CNNs). A previous study examined the structural properties that develop in large neural networks at the final stages of training. That study focused on the last layer of the network and found that deep networks trained to fit a training dataset will eventually reach a state known as "neural collapse." When neural collapse occurs, the network maps multiple examples of a particular class (such as images of cats) to a single template of that class.

A central puzzle in the theory of deep learning is how neural networks generalize even when trained without any explicit regularization, and when there are far more learnable parameters than training examples. In such an underdetermined optimization problem, there are many global minima with zero training loss, and gradient descent seems to prefer solutions that generalize well (see Zhang et al. (2017)). Hence, it is believed that gradient descent induces an implicit regularization (or implicit bias) (Neyshabur et al., 2015, 2017), and characterizing this regularization/bias has been a subject of extensive research. Several works in recent years studied the relationship between the implicit regularization in linear neural networks and rank minimization. A main focus is on the matrix factorization problem, which corresponds to training a depth-2 linear neural network with multiple outputs w.r.t. the square loss, and is considered a well-studied test-bed for studying implicit regularization in deep learning.

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization under affine constraints ARMP and show that SVP recovers the minimum rank solution for affine constraints that satisfy a Restricted Isometry Property} (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newton-step for our SVP framework to speed-up the convergence with substantial empirical gains. Next, we address a practically important application of ARMP - the problem of low-rank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold.

This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and wish to recover the missing entries under this assumption. We show that we can identify the complete matrix of minimum intrinsic dimension by minimizing the rank of the matrix in a high dimensional feature space. We develop a new formulation of the resulting problem using the kernel trick together with a new relaxation of the rank objective, and propose an efficient optimization method. We also show how to use our methods to complete data drawn from multiple nonlinear manifolds. Comparative studies on synthetic data, subspace clustering with missing data, motion capture data recovery, and transductive learning verify the superiority of our methods over the state-of-the-art.

There has been an increased interest in multimodal language processing including multimodal dialog, question answering, sentiment analysis, and speech recognition. However, naturally occurring multimodal data is often imperfect as a result of imperfect modalities, missing entries or noise corruption. To address these concerns, we present a regularization method based on tensor rank minimization. Our method is based on the observation that high-dimensional multimodal time series data often exhibit correlations across time and modalities which leads to low-rank tensor representations. However, the presence of noise or incomplete values breaks these correlations and results in tensor representations of higher rank. We design a model to learn such tensor representations and effectively regularize their rank. Experiments on multimodal language data show that our model achieves good results across various levels of imperfection.

The Schatten quasi-norm was introduced to bridge the gap between the trace norm and rank function. However, existing algorithms are too slow or even impractical for large-scale problems. Motivated by the equivalence relation between the trace norm and its bilinear spectral penalty, we define two tractable Schatten norms, i.e.\ the bi-trace and tri-trace norms, and prove that they are in essence the Schatten-$1/2$ and $1/3$ quasi-norms, respectively. By applying the two defined Schatten quasi-norms to various rank minimization problems such as MC and RPCA, we only need to solve much smaller factor matrices. We design two efficient linearized alternating minimization algorithms to solve our problems and establish that each bounded sequence generated by our algorithms converges to a critical point. We also provide the restricted strong convexity (RSC) based and MC error bounds for our algorithms. Our experimental results verified both the efficiency and effectiveness of our algorithms compared with the state-of-the-art methods.