Theoretical Analysis of Sparse Subspace Clustering with Missing Entries

Sparse Subspace Clustering (SSC) is a popular unsupervised machine learning method for clustering data lying close to an unknown union of low-dimensional linear subspaces; a problem with numerous applications in pattern recognition and computer vision. Even though the behavior of SSC for complete data is by now well-understood, little is known about its theoretical properties when applied to data with missing entries. In this paper we give theoretical guarantees for SSC with incomplete data, and analytically establish that projecting the zero-filled data onto the observation pattern of the point being expressed leads to a substantial improvement in performance. The main insight that stems from our analysis is that even though the projection induces additional missing entries, this is counterbalanced by the fact that the projected and zero-filled data are in effect incomplete points associated with the union of the corresponding projected subspaces, with respect to which the point being expressed is complete. The significance of this phenomenon potentially extends to the entire class of self-expressive methods.

Mathematics of Deep Learning

Recently there has been a dramatic increase in the performance of recognition systems due to the introduction of deep architectures for representation learning and classification. However, the mathematical reasons for this success remain elusive. This tutorial will review recent work that aims to provide a mathematical justification for several properties of deep networks, such as global optimality, geometric stability, and invariance of the learned representations.

Stretching Domain Adaptation: How far is too far?

While deep learning has led to significant advances in visual recognition over the past few years, such advances often require a lot of annotated data. While unsupervised domain adaptation has emerged as an alternative approach that doesn't require as much annotated data, prior evaluations of domain adaptation have been limited to relatively simple datasets. This work pushes the state of the art in unsupervised domain adaptation through an in depth evaluation of AlexNet, DenseNet and Residual Transfer Networks (RTN) on multimodal benchmark datasets that shows and identifies which layers more effectively transfer features across different domains. We also modify the existing RTN architecture and propose a novel domain adaptation architecture called "Deep MagNet" that combines Deep Convolutional Blocks with multiple Maximum Mean Discrepancy losses. Our experiments show quantitative and qualitative improvements in performance of our method on benchmarking datasets for complex data domains.

Dropout as a Low-Rank Regularizer for Matrix Factorization

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 factors are regularized by the sum of the product of squared Euclidean norms of the columns. Additionally, we inspect the case of a variable sized factorization and we prove that dropout achieves the global minimum of a convex approximation problem with (squared) nuclear norm regularization. As a result, we conclude that dropout can be used as a low-rank regularizer with data dependent singular-value thresholding.

Structured Low-Rank Matrix Factorization: Global Optimality, Algorithms, and Applications

Recently, convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it challenging to apply them to large scale datasets. Moreover, in many applications the data can display structures beyond simply being low-rank, e.g., images and videos present complex spatio-temporal structures that are largely ignored by standard low-rank methods. In this paper we study a matrix factorization technique that is suitable for large datasets and captures additional structure in the factors by using a particular form of regularization that includes well-known regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is non-convex, we show that if the size of the factors is large enough, under certain conditions, any local minimizer for the factors yields a global minimizer. A few practical algorithms are also provided to solve the matrix factorization problem, and bounds on the distance from a given approximate solution of the optimization problem to the global optimum are derived. Examples in neural calcium imaging video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional datasets.

3D Pose Regression using Convolutional Neural Networks

3D pose estimation is a key component of many important computer vision tasks such as autonomous navigation and 3D scene understanding. Most state-of-the-art approaches to 3D pose estimation solve this problem as a pose-classification problem in which the pose space is discretized into bins and a CNN classifier is used to predict a pose bin. We argue that the 3D pose space is continuous and propose to solve the pose estimation problem in a CNN regression framework with a suitable representation, data augmentation and loss function that captures the geometry of the pose space. Experiments on PASCAL3D+ show that the proposed 3D pose regression approach achieves competitive performance compared to the state-of-the-art.

Curriculum Dropout

Dropout is a very effective way of regularizing neural networks. Stochastically "dropping out" units with a certain probability discourages over-specific co-adaptations of feature detectors, preventing overfitting and improving network generalization. Besides, Dropout can be interpreted as an approximate model aggregation technique, where an exponential number of smaller networks are averaged in order to get a more powerful ensemble. In this paper, we show that using a fixed dropout probability during training is a suboptimal choice. We thus propose a time scheduling for the probability of retaining neurons in the network. This induces an adaptive regularization scheme that smoothly increases the difficulty of the optimization problem. This idea of "starting easy" and adaptively increasing the difficulty of the learning problem has its roots in curriculum learning and allows one to train better models. Indeed, we prove that our optimization strategy implements a very general curriculum scheme, by gradually adding noise to both the input and intermediate feature representations within the network architecture. Experiments on seven image classification datasets and different network architectures show that our method, named Curriculum Dropout, frequently yields to better generalization and, at worst, performs just as well as the standard Dropout method.

Hyperplane Clustering Via Dual Principal Component Pursuit

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the $\ell_1$ non-convex problem associated with DPCP, we develop a geometric analysis of a closely related continuous optimization problem. Then transferring this analysis to the discrete problem, our results state that as long as the hyperplanes are sufficiently separated, the dominant hyperplane is sufficiently dominant and the points are uniformly distributed inside the associated hyperplanes, then the non-convex DPCP problem has a unique global solution, equal to the normal vector of the dominant hyperplane. This suggests the correctness of a sequential hyperplane learning algorithm based on DPCP. A thorough experimental evaluation reveals that hyperplane learning schemes based on DPCP dramatically improve over the state-of-the-art methods for the case of synthetic data, while are competitive to the state-of-the-art in the case of 3D plane clustering for Kinect data.

Filtrated Algebraic Subspace Clustering

Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position inside the algebraic variety of a union of subspaces, and the objective is to decompose the variety into its constituent subspaces. Prior algebraic-geometric approaches to this problem require the subspaces to be of equal dimension, or the number of subspaces to be known. Subspaces of arbitrary dimensions can still be recovered in closed form, in terms of all homogeneous polynomials of degree $m$ that vanish on their union, when an upper bound m on the number of the subspaces is given. In this paper, we propose an alternative, provably correct, algorithm for addressing a union of at most $m$ arbitrary-dimensional subspaces, based on the idea of descending filtrations of subspace arrangements. Our algorithm uses the gradient of a vanishing polynomial at a point in the variety to find a hyperplane containing the subspace S passing through that point. By intersecting the variety with this hyperplane, we obtain a subvariety that contains S, and recursively applying the procedure until no non-trivial vanishing polynomial exists, our algorithm eventually identifies S. By repeating this procedure for other points, our algorithm eventually identifies all the subspaces by returning a basis for their orthogonal complement. Finally, we develop a variant of the abstract algorithm, suitable for computations with noisy data. We show by experiments on synthetic and real data that the proposed algorithm outperforms state-of-the-art methods on several occasions, thus demonstrating the merit of the idea of filtrations.

Algebraic Clustering of Affine Subspaces

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick, which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subpaces.