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We study the optimization landscape of deep linear neural networks with the square loss. It is known that, under weak assumptions, there are no spurious local minima and no local maxima. However, the existence and diversity of non-strict…

Deep neural networks have become a foundational tool for addressing imaging inverse problems. They are typically trained for a specific task, with a supervised loss to learn a mapping from the observations to the image to recover. However,…

Combinatorial optimization (CO) is the fundamental problem at the intersection of computer science, applied mathematics, etc. The inherent hardness in CO problems brings up challenge for solving CO exactly, making deep-neural-network-based…

The growing use of deep neural networks in safety-critical applications makes it necessary to carry out adequate testing to detect and correct any incorrect behavior for corner case inputs before they can be actually used. Deep neural…

Inverse optimization, determining parameters of an optimization problem that render a given solution optimal, has received increasing attention in recent years. While significant inverse optimization literature exists for convex…

In recent years, deep learning has been connected with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary…

We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel…

Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent…

In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of the loss function and show that critical points detected by…

Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems…

Recent progress in deep learning has been driven by increasingly larger models. However, their computational and energy demands have grown proportionally, creating significant barriers to their deployment and to a wider adoption of deep…

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…

We address a core problem of computer vision: Detection and description of 2D feature points for image matching. For a long time, hand-crafted designs, like the seminal SIFT algorithm, were unsurpassed in accuracy and efficiency. Recently,…

Point cloud filtering is a fundamental problem in geometry modeling and processing. Despite of significant advancement in recent years, the existing methods still suffer from two issues: 1) they are either designed without preserving sharp…

Deep neural networks are powerful tools for solving nonlinear problems in science and engineering, but training highly accurate models becomes challenging as problem complexity increases. Non-convex optimization and sensitivity to…

We present a comprehensive survey and benchmark of both traditional and learning-based methods for surface reconstruction from point clouds. This task is particularly challenging for real-world acquisitions due to factors such as noise,…

Deep learning algorithms are growing in popularity in the field of exoplanetary science due to their ability to model highly non-linear relations and solve interesting problems in a data-driven manner. Several works have attempted to…

Recent deep networks that directly handle points in a point set, e.g., PointNet, have been state-of-the-art for supervised learning tasks on point clouds such as classification and segmentation. In this work, a novel end-to-end deep…

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…

This paper sheds light on the risk landscape of unsupervised least squares in the context of deep auto-encoding neural nets. We formally establish an equivalence between unsupervised least squares and principal manifolds. This link provides…