Many state-of-the-art subspace clustering methods follow a two-step process by first constructing an affinity matrix between data points and then applying spectral clustering to this affinity.

Ranked #1 on Image Clustering on coil-40

To extend this approach to data supported on a union of non-linear manifolds, numerous studies have proposed learning an embedding of the original data using a neural network which is regularized by a self-expressive loss function on the data in the embedded space to encourage a union of linear subspaces prior on the data in the embedded space.

Research in adversarial learning follows a cat and mouse game between attackers and defenders where attacks are proposed, they are mitigated by new defenses, and subsequently new attacks are proposed that break earlier defenses, and so on.

More specifically, we show that a generalization of symplectic integrators to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.

The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning.

Minimizing a non-smooth function over the Grassmannian appears in many applications in machine learning.

We also show that the global minimizer for DropBlock can be computed in closed form, and that DropConnect is equivalent to Dropout.

In this paper we study the proximal operator of the mixed $\ell_{1,\infty}$ matrix norm and show that it can be computed in closed form by applying the well-known soft-thresholding operator to each column of the matrix.

We show that similar discretization schemes applied to Newton's equation with an additional dissipative force, which we refer to as accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms -- the majority of which are new although some recover known cases in the literature.

To capture this rich visual and semantic context, we propose using two graphs: (1) an attributed spatio-temporal visual graph whose nodes correspond to actors and objects and whose edges encode different types of interactions, and (2) a symbolic graph that models semantic relationships.

Ranked #4 on Action Detection on Charades (using extra training data)

Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction.

In this paper, we develop a novel geometric analysis for a variant of SSC, named affine SSC (ASSC), for the problem of clustering data from a union of affine subspaces.

The acceleration technique introduced by Nesterov for gradient descent is widely used in optimization but its principles are not yet fully understood.

We consider the task of estimating the 3D orientation of an object of known category given an image of the object and a bounding box around it.

In the classical setting, signals are represented as vectors and the dictionary learning problem is posed as a matrix factorization problem where the data matrix is approximately factorized into a dictionary matrix and a sparse matrix of coefficients.

We introduce an end-to-end algorithm for jointly learning the weights of the CRF model, which include action classification and action transition costs, as well as an overcomplete dictionary of mid-level action primitives.

While the resulting regularizer is closely related to a variational form of the nuclear norm, suggesting that dropout may limit the size of the factorization, we show that it is possible to trivially lower the objective value by doubling the size of the factorization.

Advanced diffusion magnetic resonance imaging (dMRI) techniques, like diffusion spectrum imaging (DSI) and high angular resolution diffusion imaging (HARDI), remain underutilized compared to diffusion tensor imaging because the scan times needed to produce accurate estimations of fiber orientation are significantly longer.

While outlier detection methods based on robust statistics have existed for decades, only recently have methods based on sparse and low-rank representation been developed along with guarantees of correct outlier detection when the inliers lie in one or more low-dimensional subspaces.

High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires longer patient scan times.

In this paper, we propose a joint optimization framework --- Structured Sparse Subspace Clustering (S$^3$C) --- for learning both the affinity and the segmentation.

Image segmentation and 3D pose estimation are two key cogs in any algorithm for scene understanding.

While automatic feature learning methods such as supervised sparse dictionary learning or neural networks can be applied to learn feature representation and action classifiers jointly, the resulting features are usually uninterpretable.

Given pairwise dissimilarities between data points, we consider the problem of finding a subset of data points called representatives or exemplars that can efficiently describe the data collection.

We propose an algorithm called Sparse Manifold Clustering and Embedding (SMCE) for simultaneous clustering and dimensionality reduction of data lying in multiple nonlinear manifolds.

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