We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Cascaded regression is prevailing in face alignment thanks to its accurate and robust localization of facial landmarks, but typically demands numerous annotated training examples of low discrepancy between shape-indexed features and shape updates. In this paper, we propose a self-reinforced strategy that iteratively expands the quantity and improves the quality of training examples, thus upgrading the performance of cascaded regression itself. The reinforced term evaluates the example quality upon the consistence on both local appearance and global geometry of human faces, and constitutes the example evolution by the philosophy of "survival of the fittest." We train a set of discriminative classifiers, each associated with one landmark label, to prune those examples with inconsistent local appearance, and further validate the geometric relationship among groups of labeled landmarks against the common global geometry derived from a projective invariant. We embed this generic strategy into two typical cascaded regressions, and the alignment results on several benchmark data sets demonstrate the effectiveness of training regressions with automatic example prediction and evolution starting from a small subset.

Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is to visualize them. We present the Embedding Projector, a tool for interactive visualization and interpretation of embeddings.

A popular method for exploring high-dimensional data is something called t-SNE, introduced by van der Maaten and Hinton in 2008. The technique has become widespread in the field of machine learning, since it has an almost magical ability to create compelling two-dimensonal "maps" from data with hundreds or even thousands of dimensions. Although impressive, these images can be tempting to misread. The purpose of this note is to prevent some common misreadings. We'll walk through a series of simple examples to illustrate what t-SNE diagrams can and cannot show.